the thermodynamics of dry clean
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The kinetic theory of gases
The kinetic theory of gases describes a gas as a large number of small particles (atoms or
molecules), all of which are in constant, random motion. The rapidly moving particles
constantly collide with each other and with the walls of the container. Kinetic theory explains
macroscopic properties of gases, such as pressure, temperature, or volume, by considering
their molecular composition and motion. These particles have the same mass. The number of
molecules is so large that statistical treatment can be applied. These molecules are in
constant, random and rapid motion.
The rapidly moving particles constantly collide among themselves and with the walls of the
container. All these collisions are perfectly elastic. This means, the molecules are considered
to be perfectly spherical in shape, and elastic in nature.
xcept during collisions, the interactions among molecules are negligible. (That is, they exert
no forces on one another.)
The theory for ideal gases makes the following assumptions!
• The gas consists of very small particles. This smallness of their si"e is such that the
total volume of the individual gas molecules added up is negligible compared to the
volume of the container. This is e#uivalent to stating that the average distance
separating the gas particles is large compared to their si"e.
• These particles have the same mass.
• The number of molecules is so large that statistical treatment can be applied.
• These molecules are in constant, random and rapid motion.
• The rapidly moving particles constantly collide among themselves and with the walls
of the container. All these collisions are perfectly elastic. This means, the moleculesare considered to be perfectly spherical in shape, and elastic in nature.
• xcept during collisions, the interactions among molecules are negligible. (That is,
they exert no forces on one another.)
This implies!
$. %elativistic effects are negligible.
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&. 'uantummechanical effects are negligible. This means that the interparticle
distance is much larger than the thermal de roglie wavelength and the molecules
are treated as classical ob*ects.
+. ecause of the above two, their dynamics can be treated classically. This
means, the e#uations of motion of the molecules are timereversible.
• The average kinetic energy of the gas particles depends only on the temperature of the
system.
• The time during collision of molecule with the containers wall is negligible as
compared to the time between successive collisions.
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The thermodynamics of dry clean Air
&.- The rigin of the Atmosphere
The origin of the earths atmosphere is still a matter of speculation. /owever, most scientists
believe, based on their knowledge of the gases in the universe, the earths first atmosphere
was composed of helium, hydrogen, ammonia and methane. thers believe that the first
atmosphere was probably released gases from volcanoes venting the earths hot inner core,
including 0&, 1&, /&, 1, and /&-. The earths second atmosphere can be traced to the
planets heating and differentiation. 2t probably consisted of the same gases that are released
from volcanoes today! carbon dioxide, nitrogen, water vapour, and hydrogen and other trace
gasses. 3lanetary differentiation caused the lighter elements to rise to the outer layers of the
earth and initiated the escape of the lighter gases from the planets interior. The lighter gases
eventually formed the atmosphere and the oceans. 4or more information consult 5allace and
/obbs ($677).
2.1 Chemical Composition of the Atmosphere
The atmosphere is a mixture of solids, li#uids and gases. The gases in the atmosphere are
classified as either permanent (concentration remains constant) or variable (concentration
varies with time). The permanent gases include oxygen, nitrogen, neon, argon, helium and
hydrogen. The most abundant of these permanent gases are nitrogen (789) and oxygen
(&$9). The remainder of the permanent gasses and the variable gases exist in small
concentrations in the atmosphere. They are referred to as trace gasses. The atmosphere also
includes sulphur, chlorofluorocarbons, and dust and ice particles.
0ote that the air is defined as a mixture of gases and that most weather is contained within
the troposphere. 2n this chapter we will deal with so called dry air, i.e. a gas not containing
water vapour. /owever, meteorologists often consider unsaturated air to be :dry;.
2.2 The equation of state of a perfect gas
<efinition
5hen a gas or vapour is so rarefied that the proportion of space occupied by the molecules
and the attractive forces between the latter are negligible, the gas is referred to as a perfect
gas. 2n real practice no gas can be exactly perfect, but under natural conditions, the mixture of
gases (dry air) is sufficient close to perfect for most meteorological purposes. The e#uation of
state for a perfect gas, involves the three variables p, T where
• p denotes pressure in 3ascal (3a)
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• T denotes temperature in Kelvin
= denotes specific volume (volume>mass) or inverse of the density (mass>volume). 0ote that
the temperature T will always be assumed to be in Kelvin (K ? temperature in 1 @ &7+),
unless otherwise specified, and that 2 units are used in all calculations. The derivation of the
e#uation of state is by the combination of two experimental laws. The e#uation of state is
therefore not a law, but a relationship based on experiment.
Bets first use oyles law, pressure (p) specific volume ( ) ? constant$ which demands that
the temperature T be held constant, while pressure and specific volume (or density) is
allowed to change as follows!
3 ? constant, or
? constant.
The second law is 1harles law, Temperature (T) specific volume ( ) ? constant& which
demands that the pressure remains constant while temperature and specific volume (ordensity) is allowed to change as follows!
This can be expressed as!
? 1onstant, or
T ? constant.
The rest of the derivation is #uite easy if one remembers that we apply oyles law for the
change of state p$, $, T$ (the initial state) to p&, $ (the intermediate state), with
temperature held constant and then apply 1harless law for the change p&, , T$ to
p&, &, T& (the final state) with pressure held constant.
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xercise $
5rite down oyles law and 1harless law for the two indicated processes and eliminate the
intermediate specific volume . 5rite your resulting e#uation so that only initial state
variables appear on the left hand side and only final state variables appear on the right hand
side. Then, motivate that the e#uation of state can be written as follows!
p ? %T or p ? %T!
The constant % is known as the specific gas constant and is measured at a volume occupied
by a unit mass of the gas and selected pressure and temperature. The gas constant % for dryair e#uals &87 Ckg$ K $, by Dordon et al ($668). The e#uation of state is fundamental to the
understanding of thermodynamics of air (gases). This is not the only valid derivation.
3hysicists derive it using statistical physics. This e#uation expresses the three way
relationship between the state variables temperature, density and pressure of a gas. 2t may be
used to explain atmospheric behaviour if one of the variables is considered constant or it can
be used to eliminate one of the state variables from a formula.
xercise &5hat is the density of a sample of dry air at the E-- h3a level if the temperature is &-F G
5hat is the density at the $--- h3a level if the temperature is +-F G (Answers! -.H6 and $.$E
kg m+)
2.3 The universal gas constant
To derive the e#uation of state, we have used 1harless and oyles laws. Avogadros Baw
can be used for calculating the Iniversal gas constant. 2t shows that molar volume of a gas atthe same pressure and temperature will be the same for all permanent gases. Jolar volume
can be defined as a volume occupied by a mass of gas e#ual to unit mass multiplied by the
molecular weight of the gas e.g. $ gram molecule is m grams where, m is the molecular
weight. 1onsider a volume of gas with mass e#ual to m kg and m a number e#ual to the
molecular weight of the gas. The volume occupied by this m kg of gas is called the molar
volume. The molar volume changes with pressure and temperature.
1onsider the e#uation of state for dry air, with gas constant % d!
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p ? % dT so that p? % dT ? %LT
where m is the mass of the molar volume and m numerically e#uivalent to the molecular
weight. ut, noting that the units of is M N ? volume>mass, we may write m ? , the
molar volume and thus
p ? %LT
%emembering that at the same temperature and pressure the molar volume is the same for all
gases (Avogadro), we can see that %L must be a universal gas constant and is given in
5allace and /obbs ($677) as 8.+$O+ $-+ C K $ kmol$.
2n general pm ?m%T ? %LT,
3roviding the relationship m% ? %L holds.
ubstitution in the e#uation of state results in!
p ? ( )
2.4 Mixture of gases
y <altons law, a gas (e.g. air) occupying a volume , can be separated into its different
components, each occupying the same volume , each having its own pressure (partial
pressure) and every component obeying its individual e#uation of state.
That is, if the partial pressures are p$, p&... ps, then
pk ? Jk % k TP k ? $, &,.......,s
5here the cubic meters of air contains the mass Jk of each gas constituent. The sum
of the partial pressures gives the pressure of the mixture! pk ? p and summing the above gaslaws above we get!
?
2f J denotes the total mass of the mixture, and % is chosen such that
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J% ? , or %?
then
p ? J%T or p ? %T
5ith ? >J the specific volume of the mixture. #uation (&.8) is the e#uation of state
for a mixture of perfect gases and conforms to the e#uation of state of one perfect gas,
with gas constant % given by (&.7), pertaining to the mixture.
2.5 Molecular eight of !r" air
The gas constants for the different gases in the atmosphere can be found in Dordon etal,
$668, and from section &.E the specific gas constant for dry air may be obtained by
considering the molecular weights and specific gas constants of the constituent gases. The
molecular weight of dry air can be defined using m% ? %L, where m denotes the molecular
weight of dry air (md). 2t follows that
md ? ? &8.67
2.# The first la of thermo!"namics
The first law of thermodynamics is expressed as follows!
d' ? dI @ d5 ($)
where
d' ? 1hange of heat (energy) of the system (d'). This is the energy added or taken from
the system. Typical energy added by conduction, convection, radiation.
dI ? 1hange in internal energy of the system (dI). 2t can be related to the temperature of
the system and to the molecular motion of the substance.
d5 ? 5ork done on the system by the external forces (d5). 5hen a gas expands it does work against the external pressure forces. 2t uses energy to perform this work.
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2f no energy is added or subtracted from the system (d# ? -) we call the process
A<2AAT21.
#uation ($), expressed per unit mass, becomes
d# ? du @ dw (&)
and using the e#uation for work expressed per unit mass dw ? pd>m ? pd P the familiar
expression of the first law,
d# ? du @ pd
2.$ %pecific heats of gases
AJ (&---) defines heat capacity (also called thermal capacity) as the ratio of the energy or
enthalpy absorbed (or released) by a system to the corresponding temperature rise (or fall).
/eat capacities are defined for particular processes.
4or a constant volume process,
1v ?
5here I is the internal energy of a system and T is its temperature. 4or a constant pressure
process,
1p ?
where / is the system enthalpy,
/ ? I @ p
a thermodynamic state function with I internal energy, p pressure, and volume.
AJ (&---) defines specific heat capacity (or specific heat) as the heat capacity of a system
divided by its mass. 2t is a property solely of the substance of which the system is composed.As with heat capacities, specific heats are commonly defined for processes
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occurring at either constant volume (cv) or constant pressure (cp). 4or an ideal gas, both are
constant with temperature and related by cp ? cv @ % with % the gas constant. 4or
dry air at &7+ K,
1 p ? $--E.7 &.EC K $kg$
1 ? 7$6 &.EC$kg$
and thus
% ? 1 p 1 ? &8H.7 &.EC$ K $kg$
4or moist air, the specific heat capacities of the dry air and water vapour must be combined in
proportion to their respective mass fractions.
xercises +
$. Assuming cv ? 1v>m and 1v ? prove that
cv v
&. Assuming c p ? 1 p>m and 1 p ? , and the e#uation of state p ? %T, prove
that
1 p ? cv @ %
2n an isosteric process, no expansion takes place, no work is done and pd ? -, meaning
that all the energy is used to increase the internal energy (and the temperature) of the gas.
The first law takes the simple form d# ? du ? cvdT! This provides a relationship between the
internal energy and the temperature of a gas for isosteric processes. 2n general
then, the first law is written as
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d# ? cvdT @ pd
%emarks
$. 2t is important to remember that the first law (&) expresses ($) per unit mass.
&. The specific heat at constant volume
cv ( )v
5hich gives the amount of energy (Coules) re#uired to heat $ kg of the gas by $ Kelvin
at constant volume and a value of 7$7 CK $kg$ is given for dry air.
+. The specific heat at constant pressure
c p p
5hich gives the amount of energy (Coules) re#uired to heat $ kg of the gas by $ Kelvin
at constant pressure and a value of $--O CK $kg$ is given for dry air.
O. 0ote that c p Qcv since heating that takes place at constant pressure causes work to be
done due to accompanied expansion (d ).
xercises O$. ince the term pd= is difficult to deal with, eliminate it between the e#uation of state
and the first law of thermodynamics to give
d# ? (cv @ %)dT dp!
/int! first differentiate e#uation of state
2.& A!ia'atic process
2t is important to define the so called adiabatic process. This process simply means that
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by and specific entropy s as entropy per unit mass. They add that the entropy of an isolated
system cannot decrease in any real physical process, one statement of the second
law of thermodynamics. Dordon et al. ($668) uses the symbol for specific entropy and
writes
d ?
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