aerosol-cloud interaction anatoli bogdan institute of physical chemistry, university of innsbruck...
TRANSCRIPT
Aerosol-cloud interaction
Anatoli Bogdan
Institute of Physical Chemistry University of Innsbruck Austria
andDepartment of Physics University of Helsinki
Finland
Contents
- role of aerosol in cloud formation - ideal gas - vapor pressure and partial vapor pressure - Kelvin equation - hygroscopic aerosol particles - Raoultrsquos law - Kohler curves
Cloud condensation nuclei or CCNs or cloud seeds are small particles (typically 02 microm or 1100 th the size of a cloud droplet) about which cloud droplets coalesce Water requires a non-gaseous surface to make the transition from a vapor to a liquid In the atmosphere this surface presents itself as tiny solid or liquid particles called CCNs When no CCNs are present water vapor can be supercooled below 0degC before droplets spontaneously form
At T gt 0 ordmC the air would have to be supersaturated to ~400 before the droplets could form The concept of cloud condensation nuclei has led to the idea of cloud seeding that tries to encourage rainfall by seeding the air with condensation nuclei It has further been suggested that creating such nuclei could be used for marine cloud brightening a geo-engineering technique
Aerosol pollution over Northern India and Bangladesh - NASA
httpenwikipediaorgwikiCloud_condensation_nuclei
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Contents
- role of aerosol in cloud formation - ideal gas - vapor pressure and partial vapor pressure - Kelvin equation - hygroscopic aerosol particles - Raoultrsquos law - Kohler curves
Cloud condensation nuclei or CCNs or cloud seeds are small particles (typically 02 microm or 1100 th the size of a cloud droplet) about which cloud droplets coalesce Water requires a non-gaseous surface to make the transition from a vapor to a liquid In the atmosphere this surface presents itself as tiny solid or liquid particles called CCNs When no CCNs are present water vapor can be supercooled below 0degC before droplets spontaneously form
At T gt 0 ordmC the air would have to be supersaturated to ~400 before the droplets could form The concept of cloud condensation nuclei has led to the idea of cloud seeding that tries to encourage rainfall by seeding the air with condensation nuclei It has further been suggested that creating such nuclei could be used for marine cloud brightening a geo-engineering technique
Aerosol pollution over Northern India and Bangladesh - NASA
httpenwikipediaorgwikiCloud_condensation_nuclei
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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- Slide 31
-
Cloud condensation nuclei or CCNs or cloud seeds are small particles (typically 02 microm or 1100 th the size of a cloud droplet) about which cloud droplets coalesce Water requires a non-gaseous surface to make the transition from a vapor to a liquid In the atmosphere this surface presents itself as tiny solid or liquid particles called CCNs When no CCNs are present water vapor can be supercooled below 0degC before droplets spontaneously form
At T gt 0 ordmC the air would have to be supersaturated to ~400 before the droplets could form The concept of cloud condensation nuclei has led to the idea of cloud seeding that tries to encourage rainfall by seeding the air with condensation nuclei It has further been suggested that creating such nuclei could be used for marine cloud brightening a geo-engineering technique
Aerosol pollution over Northern India and Bangladesh - NASA
httpenwikipediaorgwikiCloud_condensation_nuclei
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
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At T gt 0 ordmC the air would have to be supersaturated to ~400 before the droplets could form The concept of cloud condensation nuclei has led to the idea of cloud seeding that tries to encourage rainfall by seeding the air with condensation nuclei It has further been suggested that creating such nuclei could be used for marine cloud brightening a geo-engineering technique
Aerosol pollution over Northern India and Bangladesh - NASA
httpenwikipediaorgwikiCloud_condensation_nuclei
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
- Slide 2
- Slide 3
- Slide 4
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- Slide 31
-
Aerosol pollution over Northern India and Bangladesh - NASA
httpenwikipediaorgwikiCloud_condensation_nuclei
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
-
httpearthobservatorynasagovFeaturesAerosols
Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Warm Clouds
Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
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-
Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air
The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
-
Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter
General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
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General about gases and vapors
Perfect gas
Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas
In perfect gas i) the distance between molecules is much larger than the length of free
path of molecules and ii) the interaction between molecules is restricted only to their collisions
which are considered to be similar to that of the hard balls
Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
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-
Vapor pressure of water
Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation
The atmosphere is a mixture of gases including water vapor
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
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-
When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)
After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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-
Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases
The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as
DTTCT
BA
sw ep
ln
where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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- Slide 7
- Slide 8
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- Slide 10
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-
132 Air humidity Amount of water vapor in the air can be expressed by several different ways
Specific humidity Mass of water vapor per unit mass of humid air air
OH
m
m 2
Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100
RH = 100)(
Tp
ppure
ws
w (153)
Saturation ratio S Ratio
S =)( Tp
ppure
ws
w (154)
From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0
For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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For the perfect gas a following (experimental) equation of state p = f (T V n) is true
pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb
Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as
pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that
pV = RT = KE = 3
2Nmmol (13)
where KE is kinetic energy of the molecules composing the perfect gas mmol mass of
molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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-
Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i
piV = niRT (17) and the equation of state for the mixture itself
pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)
Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)
Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Kelvin equation
ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius
ps(r) = the actual vapour pressure of droplet of radius r
ps(infin)= the saturation vapour pressure over bulk water
σw = surface tension
ρw= water density
Rv - the universal gas constant T - temperature
ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
- Slide 1
- Slide 2
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- Slide 27
- Slide 28
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- Slide 30
- Slide 31
-
What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc
Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption
Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances
Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution
Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere
All soluble salts will deliquesce if the air is sufficiently humid
141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie
AA
A Xp
p or pA = XA pA (160)
The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6
Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law
The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions
The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component
Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)
Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed
p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3
wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation
Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation
Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Koumlhler Curve = Kelvin equation + Raoultrsquos law
p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3
wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]
The critical radius rc and critical supersaturation Sc are calculated as
rc= (3ba)12 and Sc= (4 a3[27 b])12
httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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httpenwikipediaorgwikiFileKohler_curvespng
Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute
As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated
Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system
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