Atmospheric Thermodynamics

Thermodynamic Systems

This section examines the atmosphere as a system, and several atmospheric phenomena that have

become critical in the understanding and guardianship of our environment. The current state of

the atmosphere is the result of a multitude of facts. The energy from the sun produces the

movements or currents in the atmosphere. This energy, the Earth's movement relative to the sun

and the components of the atmosphere and of the Earth’s surface maintains the long-term

climate, the short-term weather, and the temperature conditions. These provide conditions fit for

the forms of life found on Earth. The condition of the physical world affects and is affected by

the life present. The entire system is therefore called the biogeochemical system. In the last

century especially, this system, which evolved over billions of years, has been subject to rapid

changes due to industrial activities increasing at unprecedented rates.

This section discusses some of the basic science and details of the interactions involving the

atmosphere. We begin by examining the nature of the sun's energy, and the actions and reactions

it produces in the atmosphere. We then discuss how industrial activity has perturbed atmospheric

conditions, and what policy actions are being taken to reduce our impact. Due the complexity of

the atmospheric system, there are still large amounts of scientific uncertainty in predicting

changes precisely, but we do know enough to describe and project qualitative features of the

system that lead to an understanding of the impacts of large scale human activity.

A thermodynamic system is a quantity of matter or any region of space to which one directs

attention for the purposes of analysis. The quantity of matter or region of space must be within

prescribed boundary which may be deformable or rigid. It may even be imaginary.

Surrounding

everything outside a system boundary is referred to as the surrounding. Usually, term

surrounding is restricted to those things outside the system that in some way influence the

system.

Types of Systems

The First Law of Therodynamics

the first law of thermodynamics is often applied to process as the system changes from state to

another. Heat is a form of energy that cannot be destroyed. Because of this, there is an amount of

energy associated with every thermodynamics system called its internal energy, U which can be

changed by adding or subtracting energy of any form, and that the algebraic sum of the added or

subtracted amount is equal to the change in of internal energy, dU of the system. Assume that a

system transform from state (1) to another state (2), then there would be a change in internal

energy considering the difference in energy, Q supplied to the system and the work, W done by

the system. This can be expressed as:

dQ−dW=dU

dQ=dU+dW - - - - - - - - - - - - - - (1)

Equation (1) can every thermodynamic process such as (i) isobaric, (ii) isochoric, (iii) isothermal

and (iv) adiabatic processes.

Enthalpy

In trying to offer solution to the problems involving systems, certain products or sum of

properties occur with regularity. One of such a combination can be demonstrated by going

through a simple experiment below. Just add heat to a constant-pressure cylinder containing a

gas trapped a piston.

Heat is added slowly to the system (the gas in the cylinder) maintained at a constant pressure

under a condition where there is no friction between the cylinder and the piston. Also if the

kinetic and potential changes are neglected, and all other work mode are absent, then the first law

of thermodynamics requires that:

Q−W=U2−U1

- - - - - - - - - - - - - ()

where Q is the energy put into the system, W the work done in raising the piston up and U2 –U1 is

the change in internal energy.

But the work done on the piston is given as:

dW=pdV

W=p [V 2−V 1 ] - - - - - - - - - - - - - - ()

Putting equation () in equation () gives:

Q−p [V 2−V 1 ]=U2−U1

Q=p [V 2−V 1 ]+U 2−U 1

The quantity in the parenthesis is a combination of thermodynamic properties (pressure, volume

and internal energy) of the system. It is called the enthalpy, H of the system.

H=U+ pV

Equation () is energy equation of the system can be expressed as:

Q=H 2−H 1

Assignment

A frictionless piston is used to provide a constant pressure of 400 kPa in a cylinder containing

steam originally on 200ºC with a volume of 2m3. If 3500 kJ of heat is added, determine (i) the

initial enthalpy, (ii) the final enthalpy, (iii) the final volume and (iv) the final temperature of the

system.

A Cyclic Process

let us consider a cyclic process. It is a transformation which brings the system back to its initial

state. It can be represented a closed curve. The total work done in the cyclic process is given

geometrical by the area enclosed by curve which represents the curve. There is amount energy,

QH absorbed in the cycle while another amount of energy, QL also rejected. The net work done

by the system is given as:

QH=QL=W NET

-- - -- - - - - - - - - - - - ()

The change in internal energy over the cycle is zero.

Examples of cycle process are Carnot, Otto and others. Their principles form the basis of

engines. The efficiency of the engines can be determined thus:

Efficiency of an engine, η=Net work done, WNET

the total energyput into the system, QH

η=W NET

QH

By substitution, the efficiency becomes:

η=QH -Q L

QH=1−

QL

QH

- - - - - - - - - - - - - - - - - ()

Entropy

It can also be proved that efficiency can be defined in terms of temperatures TH and TL at which

energy QH are QL are absorbed and rejected respectively.

η=1−T L

TH=1−

QL

QH

- - - - - - - - - - - - - - - - - ()

Hence,

T L

TH=QL

QH

- - - - - - - - - - - - - - - - -()

Rearranging equation () gives:

QH

T H=QL

T L

- - - - - - - - - - - - - - - - -()

the quantity

QT

is known as entropy with a symbol, S. It is preferably expressed in differential

form as:

dS=dQT

- - - - - - - - - - - - - - - - -()

The energy Q, can be express as:

dQ=TdS - - - - - - - - - - - - - - - - -()

Helmholtz Functions

Helmholtz Functions also known as Helmholtz free energy is defined as: F=U−TS

Gibbs Function

Gibbs Functions also known as Gibbs free energy is defined as: G=H−TS

Atmospheric Systems and Maxwell Relation

Atmospheric systems are given masses and composition of the atmosphere under study. Once a

system has been identified the rest of the atmosphere is the surrounding of the identified system.

The following can be the system of the atmosphere: (i) the composition of air with a humidity

that ranges from 0% to saturation and clouds of saturated air and either water drops or ice

crystals. Unsaturated air is a mixture of gases in constant composition with molecular weight

being 28.964 and water vapour whose content can be expressed by the vapour pressure and by its

molar fraction.

Molar Fraction

molar fraction can be defined as the ratio of the number of moles of water vapour to the total

number of moles. Since the number of moles of water vapour is always much smaller than that of

dry air, molar fraction can be given another definition. It is by approximation. It is given as the

ratio of the moles of water vapour to the number of moles of dry air.

The atmospheric systems can be regarded as ideal gases. These systems; dry air, water vapour

and moist air; all obey ideal gas equation given as

pV=mRTM

=nRT

- - - - - - - - - - - - - - - - - ()

where p is the pressure, V the volume, m mass R the universal gas constant (8.3 Jmol–1K–1) M the

average molecular weight (28.9) and n the number of m. the presence of any of the following (i)

water vapour, (ii) cloud and (iii) water droplets changes the the atmospheric parameters.

First Law of Thermodynamics

From our previous studies, the first law can be stated thus:

dU=δQ+δA - - - - - - - - - - - - - - - - - - - - (1)

Where U is the internal energy of the system; Q, the heat absorbed by the system and A the

amount of work done. The internal energy is state function while work done is a function of the

path. In the atmosphere, the work done is causing the system to expand. This is given as

dA=−pdV. By substituting this in equation gives:

dU=δQ−pdA - - - - - - - - - - - - - - - - - - - - (2)

Consider the definition of enthalpy, given asH=U+ pV

, and differentiating equation gives:

dH=dU+ pdV +Vdp - - - - - - - - - - - - (3)

Combining equations (2) and (3) gives dH=δQ−pdV +pdV +Vdp

and hence a change in

enthalpy is:

dH=δQ+Vdp - - - - - - - - - - - - (4)

Molar Heat Capacity of Air at Constant Pressure

Molar Heat Capacity of Air at Constant Pressure can be defined as the heat necessary to increase

the temperature of one mole of air by one degree at constant pressure, given as

CP=∂Q|PdT

and

consider equation (4) for a process at constant pressure where the change in pressure is zero,

dp=0, then the enthalpy at constant pressure is given as:

dH|P=CP dT+0 - - - - - - - - - - - - - - - - (5)

Substituting equation (5) into equation (4), and a very important equation in the study

atmospheric processes is:

CPdT=∂Q+Vdp - - - - - - - - - - - - - - - - (6)

Saturated Air

Saturated air contains water droplets or water vapour molecules. At constant pressure, the system

absorbs energy in two parts; (i) energy which increases the temperature and (ii) another energy

which absorbs energy as latent heat turning water into vapour.

Molar Latent Heat of Vaporization, LV

Molar latent heat of vaporization Lv can be defined as the heat necessary to evaporate one mole

of water. This can be expressed as:

LV=dQE

dN - - - - - - - - - (7)

Hence the energy taken in to evaporate the water is:

dQE=LV dN - - - - - - - - - (8)

The total energy absorbed during the process is

dQT=dQA+dQE

- - - - - - - - - (9)

Hence by putting equations (5) and (8) together gives the change in enthalpy during the process

at constant pressure as:

[dH ] p=[dQ ]P=CP dT+LV dN - - - - - - (10)

Hence simply the change in enthalpy is:

dH=CP dT+LV dN - - - - - - - - - - - - - - - (11)

Comparing equations (4) and (11) and equating gives:

CP dT+LV dN=∂Q+Vdp - - - - - - - - - - - - - - - (11)

CLAUSIUS-CLAPEYRON EQUATION

Exact Differential Equation

Consider the variable z as s function of x and y, that is z = f(x,y), then:

dz=( ∂ z∂ x )

ydx+( ∂ z

∂ y )x

dy- - - - - - - - - - - - (1)

Putting M=(∂ z∂ x )

y andN=( ∂ z

∂ y )x , then equation (1) can be express as:

dz=Mdx+Ndy - - - - - - - - - - - - - - - - - - - (2)

Differentiating M and N with respect to y and x respectively gives:

(∂M∂ y )= ∂∂ y ( ∂ z

∂ x )= ∂2 z∂ x ∂ y - - - - - - - - - - - - - - - - - (3)

(∂N∂ y )= ∂∂ x ( ∂ z

∂ y )= ∂2 z∂ x ∂ y - - - - - - - - - - - - - - - - - (4)

Comparing equations (3) and (4), it is evident that:

(∂M∂ y )x=(∂N∂ x )

y - - - - - - - - - - - - - - - - - - - (5)

Entropy of an Ideal Gas

If a system absorbs an infinitesimal amount of heat dQR during a reversible process, then the

entropy change of the system can be defined as:

dS=dQR

T - - - - - - - - - - - - - - - - -- (6)

Hence the quantity of heat absorbed is given as:

dQR=TdS - -- - - - - - - - - - - - - - - - (7)

Enthalpy of a System

Let heat be added slowly to a system which is maintained at constant pressure. It is assumed that

the contact point between the seal of the piston and the cylinder is frictionless. Also the kinetic

and potential energy changes are negligible. In this particular situation, the first law of

thermodynamics can be stated thus:

dQ=dU+dW - - - - - - - - - - - - - - - (8)

As the cylinder changes volume from V1 to V2, the internal energy of the system also changes

from U1 to U2. This is because, the temperature also changes. The work done in raising the piston

up for this constant-pressure process is given as:

dW=pdV=p (V 2−V 1 ) - - - - - - - - - - - - - - - - (9)

Putting equation (9) in equation (8), the first law of thermodynamics becomes:

dQ=U2−U1+ p (V 2−V 1)

dQ=(U2+pV2 )−(U1+pV1) - - - - - - - - - - - (10)

The quantity in the parenthesis is a combination of properties which is a property of itself and it

is known as enthalpy.

H= (U+pV ) - - - - - - - - - - - - - - - - - - - - - - - (11)

Maxwell Relations

For infinitesimal changes in the internal and enthalpy of a compressible system, the first law of

thermodynamics can be rewritten as:

dU=TdS−pdV - - - - - - - - - - - - - - - - - - - - (12)

dH=TdS+Vdp - - - - - - - - - - - - - - - - - - - - (13)

In addition to these functions, there are two other functions that are very important when

discussing saturations.

Helmholtz Function, F

Helmholtz function is also known Helmholtz free energy and can be defined as:

F=U−TS - - - - - - - - - - - - - - - - - - - - - - - (14)

Differentiating equation (14), gives:

dF=dU−TdS- SdT - - - - - - - - - - - - - - - - - (15)

Putting equation (12) in equation (15) gives:

dF=−SdT- pdV - - - - - - - - - -- - - - - - - - - - (16)

Gibbs Function, G

Gibbs functions also known as Gibbs free energy can be stated thus:

G=H−TS - - - - - - - - - - - - - -- - - - - - - - - (17)

By differentiating and making substitution into the differential gives:

dG=−SdT+Vdp - - - - - - - - - - - - - - - - - - -(18)

When isobaric and isothermal processes take place at the same time, that is when there are no

changes in temperature and pressure, then the change in Gibbs function is zero, meaningdG=0

and Gibbs function is constant, that is G = constant. This condition is very important in processes

involving change of phases. Sublimation, fusion and vaporization take place at isothermal and

isobaric processes. During these processes, Gibbs functions are constant. If molar Gibbs

functions are denoted by symbols g', g'' and g''' of saturated solid, saturated liquid and saturated

vapour respectively, then at (i) fusion curve, g' = g'', (ii) vaporization, g'' = g''', (iii) sublimation g'

= g''' and (iv) triple point, any two of the equation holds and therefore, g' = g'' = g'''.

Properties of a pure substance can conveniently be represented in the functions:

(i) Internal energy, U = Q – pV.

(ii) Enthalpy, H = U + pV.

(iii) Helmholtz function, F = U – TS.

(iv) Gibbs Functions, G = H – TS

All these are functions of any two of the following; pressure p, volume V and temperature, T.

Maxwell Relations

The functions above can be expressed in exact differential equations forms as below:

(i) Internal energy; dU = TdS – pdV

(ii) Enthalpy, dH = TdS + Vdp

(iii) Helmholtz functions, dF = - SdT – pdV

(iv) Gibbs functions, dG = -SdT + Vdp

Applying the results of exact differential ofdz=M ydx+N xdy , Maxwell relations can be

obtained. Bearing in mind that(∂M y

∂ y )x=(∂N x

∂ x )y

Internal Energy

From change in Internal Energy, dU = TdS – pdV can be expressed as U=f (S,V ) and it exact

differential equation can be stated as:

dU=(∂U∂ S )V

dS+(∂U∂V )S

dV - - - - - - - - - - - - - - - - - - - (20)

Comparing equation (20) with internal energy equation:

T=(∂U∂ S )V - - - - - - - - - -- - - - - - - - - - - (21)

-p=(∂U∂V )S - - - -- - - - - - - - - -- - - - - - (22)

Differentiating equation () with respect to V keeping S constant gives:

( ∂T∂V )

S= ∂

∂V (∂U∂ S )V= ∂2U

∂V ∂ S - - - - - - - (23)

Also differentiating equation () with respect to S keeping V constant gives:

(−∂ p∂S )

V= ∂

∂ S (∂U∂V )S= ∂2U

∂V ∂ S - - - - - - - (24)

Comparing equations (23) and (24) gives:

( ∂T∂V )

S=−( ∂ p

∂ S )V - - - - - - - - - - - - - - - - - (20)

From Enthalpy

From change in enthalpy, dH = TdS + Vdp can be expressed as H= f (S,p ) and it exact

differential equation can be stated as:

(∂T∂ p )S=( ∂V∂ S )

p . - - - - - - - -- - -- - - - - - - - (21)

From Helmholtz Function, dF = - SdT – pdV

(∂ p∂T )

V=( ∂ S

∂V )T - - - - - - - - - - - - - - - - - - (22)

From Gibbs Function, dG = - SdT + Vdp

(∂ S∂ p )

T=−(∂V∂T )

p - - - - - - - - - - - - - - - - - - (23)

These equations can be used in variety of ways. It makes it possible for certain immeasurable

properties or quantities like entropy S and enthalpy H to be expressed in terms of measurable

quantities like pressure p, volume V and temperature T during phase changes. Let us consider the

quantity enthalpy of vaporization. The temperature remains constant during the phase change.

( ∂ S∂V )

T=K=

Sg−SfV g−V f - - - - - - - - - - - - - - - - - - - - (24)

To express this expression in specific values, both the numerator and the denominator must be

divided by m, the mass of the system to give:

( ∂ S∂V )

T=K=

sg−sfvg−v f - - - - - - - - - - - - - - - - - - - - (25)

From Maxwell relations, (∂ p∂T )

V=( ∂ S

∂V )T and hence:

(∂ p∂T )

V=K=

sfg

v fg -- - - - - - - - - - - - - -- - - - - - - - - - (26)

where sfg=sg−sf and vfg=vg−v f

Also from specific point of view, dh = Tds + vdp and integrating both sides gives:

∫ dh=∫T K ds+∫ vdp - - - - - - - - - - - - - - - - - - - - - - (27)

But at phase changes, the temperature T and pressure remain constant where and it follows that

dp = 0:

∫ dh=∫T K ds , Hence hfg=T K sfg

sfg=

hfg

T=K - - - - - - - - - - - - - - - - - - - (28)

Putting equation (28) in equation (26) gives Clapeyron equation;

(∂ p∂T )

V=K=

hfg

T T=K vfg - - - - - - - -- - - - - - (29a)

hfg=T=K vfg(∂ p∂T )

v=K - - - - - - - - - - - - - (29b)

But (∂ p∂T )

V=K≃

p2−p1

T 2−T 1 - - - - - - - - - - - -- - (30)

where, T2 and T1 are selected at equal interval above and below the constant temperature, TK. At

a relative low pressure, Vg Vf equation (29a) can be modified. Take saturated vapour as an

ideal gas, then vfg=vg−v f≃

RTp and:

pRT

= 1vfg -- - - - - - - - - - - - - - - - - - - - - - - (31)

Putting equation () in equation () gives:

Or hfg=T=K (RT

p )(∂ p∂T )

v=K

hfg=(RT2

p )( ∂ p∂T )

v=K - - - - - - - - - - - - - - (32a)

(∂ p∂T )

v=K=

phfg

RT2 - - - - - - - - - - - - - - - -- - (32b)

This equation is known as Clausius-Clapeyron equation and can be expanded. During the phase

change, the pressure depends only on the temperature and hence using ordinary derivative:

(dpdT )

SAT=

phfg

RT2 - - -- - - - - - - - - - - - - - - - - (33)

(dpp )

SAT=hfg

R (dTT 2 )

SAT - - -- - - - - - - - - - - - (34)

Integrating both sides of equation (34) gives:

ln ( p2

p1)=−

hfg

R [ 1T 2

− 1T 1 ]

ln ( p2

p1)=hfg

R [ 1T1

− 1T 2 ] - - - - - - - - - - - - - - - (35)

Clausius-Clapeyron equation allows us to estimate the vapour pressure at another temperature if

the vapour pressure is known at some temperature and if the enthalpy of vaporization is known.

Worked Example 1: The vapour pressure of water is 1.0 atm at 373 K, and the enthalpy of

vaporization is 40.7 kJ mol-1. Estimate the vapour pressure (i) at temperature 363K and (ii)

383K respectively.

Solution: Using the Clausius-Clapeyron equation given as:

ln ( p2

p1)=hfg

R [ 1T1

− 1T 2 ]

(i) By substitution

ln (P363

1 .0 )=40.7×103

8. 31 [ 1363

− 1373 ]

( P363

1. 0 )=exp[40 . 7×103

8 .31 ( 1363

− 1373 )]

P363=1. 0exp[40 . 7×103

8. 31 ( 1363

− 1373 )]

P363 = 0.697 atm

(ii) Similarly, P383=1. 0exp[40 . 7×103

8. 31 ( 1383

− 1373 )]

P383 = 1.409 atm

Comment:

Note that the increase in vapor pressure from 363 K to 373 K is 0.303 atm, but the increase from

373 to 383 K is 0.409 atm. The increase in vapor pressure is not a linear process.

The Clausius-Clapeyron equation applies to any phase transition. The following example shows

its application in estimating the heat of sublimation.

Assigment 2: Water has a vapor pressure of 24 mmHg at 25oC and a heat of vaporization of

40.7 kJ/mol. What is the vapor pressure of water at 67oC?

Assignment 3: An unknown liquid has a vapor pressure of 88mmHg at 45oC and 39 mmHg at

25oC. What is its heat of vaporization?

ATMOSPHERIC SYSTEM

A system is any body of a given mass in the atmosphere. The rest of the bodies in the atmosphere

the system interacts is known as its surroundings. There are two systems existing in the

atmosphere. They are (i) the composition of the air the humidity of which varies from zero to

saturation and (ii) the clouds

Unsaturated Air

(i) Dry air: this is a mixture of gases in their constant composition. It has a mean molecular

weight of 28.964 and (ii) water vapour which can be considered under the variable of molar

fraction.

Molar Fraction

molar fraction NV can be defined as the ration of the number of moles of water vapour to the total

number of moles. This can be expressed mathematically as:

NV=The Number of Moles of Water VapourThe Total Number of Moles - - - - - - - - - - - - - (1)

Approximately this can be expressed as:

NV=The Number of Moles of Water Vapour

The Total Number of Moles Dry Air - - - - - - - - - - - - - (2)

Ideal Gas Law

dry air, water vapour and the constituents can be regarded approximately as ideal gases and they

obey the ideal gas law which can be stated thus:

pV=mRTM

=nRT - - - - - - - - - - - - - - - (3)

where p is the pressure exerted on the system or by the system, V the volume of the system, R

the universal gas constant of value 8.3 Jmol–1 K–1 and T the absolute temperature. The total

number of mole is given asn= m

M .

The First Law of Thermodynamics

a process involving only small changes in thermodynamic coordinates of a system is known as

infinitesimal process. For such process, the first law can be stated thus:

∂Q=dU+dW - - - - - - - - - - - - - - - - (4)

In equation (4) dU is an exact differential representing the change in internal energy. The internal

energy U is an extensive property of the system. It represents the energy of the system at a

particular time. From equation (4), the change in internal energy when there is change in

temperature is given by the difference in input heat and the work done:

dU=∂Q−dW - - - - - - - - - - - - -- - - - - (5)

dU=∂Q−pdV - - - - - - - - - - - - - - - - - (6)

The internal energy U is a function of state. The changes are independent of path. The work done

and the energy absorbed or put into the system depend on the path.

Enthalpy of a System

Another way of stating the first law of thermodynamics is to consider the enthalpy of the system.

Certain products or sum of properties occur with regularity in finding solutions to problems.

Consider a system receiving a heat at a constant pressure. Assume there is no friction between

the cylinder and the piston and kinetic and potential energies and other modes of work are

neglected. Then applying the first law of thermodynamics gives:

Q=U 2−U 1+dW - - - - - - - - - - - - - - - - (7)

But the work done when there is a change of volume is:

W=P (V 2 -V1 ) - - - - - - - - - - - - - - - - - - - - (8)

Putting equation (8) in equation (7) gives:

Q=(U2+PV2)−(U1+PV1) - - - - - - - - - - - - - - - - (9)

The quantity in the parentheses is a combination of properties and is thus a property itself. The

property is known as enthalpy. Hence enthalpy is given as:

H=U+PV - - - - - - - - - - - - - - - - (10)

This can be expressed in specific value terms by dividing through equation (10) by the mass of

the system to give:

h=u+Pv - - - - - - - - - - - - - - - - (11)

By differentiating equation (10) gives:

dH=∂U+pdV+Vdp - - - - - -- - - - - - - - - - (12)

Combining equations (6) and (12) the change in the enthalpy in conformity with the first law of

thermodynamics is:

dH=∂Q+Vdp - - - - - -- - - - - - - - - - (13)

Molar Heat Capacity of Air at Constant Pressure, CP

Molar heat capacity of air at constant pressure can be defined as the heat necessary to increase

the temperature of one mole of air by one degree (1°) at constant pressure.

Consider heat added to a system (air) at constant pressure (meaning the change in pressure being

zero) resulting in a change of temperature of dT, then from equation (13), the change in enthalpy

is:

dH|P=K=∂Q|P=K+Vdp|P=K - - - - - -- - - - - - - - - - (14)

But during the process, the pressure is constant meaning there is no change in pressure. Hence

dH|P=K=∂Q|P=K=CPdT - - - - - -- - - - - - - - - - - - (15)

But the enthalpy of a system does not depend on whether the process is operated at pressure or

not. It is a function of temperature and hence for any process:

CP dT=∂Q+Vdp - - - - - -- - - - - - - - - - - - (16)

Equation (16) is applicable to atmospheric process involving air.

Molar Latent Heat of Vaporization, LV

Consider one mole of saturated air containing water droplet. During isobaric process, heat

received is two folds; (i) the one due to increase in temperature and the other due to heat

absorbed by water when evaporation is taking place. Molar heat of evaporation is the heat

necessary to evaporate one mole of water. The heat absorbed causes an increase in molar fraction

of water vapour in the air. The heat absorbed during evaporation isLV dNV . Since there is a

change in temperature during that process (at constant pressure), the total energy absorbed is

CP dT+LV dNV . This is the heat absorbed at isobaric process and it equal to:

dH|P=K=∂Q|P=K=CPdT+LV dNV - - - - - -- - - - - - - - - - - - (17)

CP dT+LV dNV=∂Q+Vdp - - - - - -- - - - - - - - - - - - (18)

Equation (18) is needed in studying atmospheric processes involving water clouds.

The change in the molar fraction dNV can be expressed as in terms of temperature T and pressure

p as independent variables. Now defines molar as the ration of partial vapour pressure (e) to the

total pressure (p). Hence:

NV= ep - - - - - - - - - - - - - ()

Differentiating equation () gives:

dNV=−dep

−edpp2

de can be linked to the variation of temperature as considered in Clausius-Clapeyron equation.

This is because the differential will appear for the case when air is saturated with water vapour

so that dNV results from the evaporation or condensation of water.

dedT

=LV e

RT2

From equation (),

de=LV edT

RT2

Putting equation () in equation () gives:

dNV=−LV e

RT2 dT− ep2 dp

Atmospheric Processes

The three processes that take place in the atmosphere are (i) cooling or warming at constant

pressure, (ii) expansion or compression and (iii) mixing.

Cooling

Dew

Dew is water in the form of droplets that appear on thin, exposed objects in the morning and

evening. As the exposed surface cools by radiating its heat, atmospheric moisture condenses at a

rate greater than that at which it can evaporate resulting in the formation of water droplets. When

the temperatures are low enough, dew takes the form of ice. This forms frost. Because dew is

related to temperature of surfaces, in summer, it is formed most easily on surfaces which are not

warmed by conducted heat from deep grounds such as such as grass, leaves, railings, car roofs

and bridges.

Water vapour will condense into droplets depending on temperature. The temperature at which

droplets can form is known as dew point. When surface temperature drops, eventually reaching

the dew point, atmospheric water vapour condenses to form small drop on the surface. Sufficient

cooling of the surface typically take place when it loses more energy by infrared radiation than it

receives as solar radiation from the sun, which is especially the case on clear night. As another

important point, poor thermal conductivity restricts the replacements of such loses from deeper

ground layers which are typically warmer at night. Preferred objects of dew formation are thus

poor conducting or well isolated from the ground, and non-metallic or coated as shiny metal

surfaces are poor infrared radiator.

Hydrometeors

In meteorology, precipitation is any product of the condensation of atmospheric water vapour

that falls under gravity. The main forms of precipitation include rain, snow, sleet, hail and

graupel. It occurs when a local portion of the atmosphere becomes saturated with water vapour

and water condenses. Two processes, possibly acting together can lead to air becoming saturated

cooling the air or adding water vapour to the air. Generally, precipitation will fall to the surface

with exception of virga which evaporates before reaching the surface.

Frost

Frost is the solid deposition of water vapour from saturated air. It is formed when solid surfaces

are cooled to below the dew point of the adjacent air as well as below the freezing point of water.

Frost crystal’s size differs depending on the time and water vapour available. Frost is also

usually translucent in appearance. Frost causes economic damage when it destroys plants or

hanging fruits.

Dew forms as result of condensation of water on solid surface on the ground. The surfaces cool

during the night. Cooling is due to radiation to a temperature below those corresponding to

saturation.

Consider an air parcel whose vapour pressure and temperature are represented by the figure

above. If the point P represents the initial state of the air parcel and begin to cool isobarically

then its temperature reduces. Cooling continues until a point Q on the ice-water equilibrium. The

air becomes saturated with ice. The temperature at the point Q is known as frost point, Tf.

Freezing may not occur because this process require a surface upon which the vapour may

sublimate.

Cooling continues until a point R on the water-vapour equilibrium curve. The air parcels become

saturated with water. The temperature at this point is known as dew point, Td. At the point P', the

air parcel has more humidity than at triple point, no frost is formed except dew when occurs

isobarically.

Fog

Suppose a mass of atmosphere cools isobarically until the temperature falls below the dew point,

condensation occurs as microscopic droplets formed on condensation nuclei. Condensation

nuclei are tiny particles which are invisible to the human eyes such as dust, dirt, pollutant, which

provide surfaces on which water molecules can condense and gather into water droplets. Fog is a

collection of water droplets or ice crystals suspended in lower part of the atmosphere near the

Earth’s surface. While fog is a type of cloud, the term “fog” is typically distinguished from the

more generic term “cloud” in that fog is low-lying, and the moisture in the fog is often generated

locally (such as from a nearby body of water like lake or ocean, or from a nearby moist ground

or marshes. and the moisture in the fog is often generated locally such as from a nearby body of

water, like a lake or the ocean, or from nearby moist ground or marshes. Fog can be

distinguished from mist only by its density as expressed in the resulting decreasing visibility.

Types of Fog

Radiation Fog: It is formed at night under clear skies with calm winds when heat absorbed by

the earth’s surface during the day is radiated into space. As the earth’s surface continue to cool,

provided a deep enough layer of moist air is present near the ground, humidity will reach 100%

and fog will form. Radiation fog varies in depth from about 1 meter to 350 meters and is always

found at level. It is usually remains stationary. It can reduce visibility to near zero at times. It

makes driving very hazardous.

Valley Fog: Valley fog is a type of radiation fog. It is very common in mountainous areas. When

air along ridgetops and the upper slopes of the mountains begins to cool after sunset, the air

becomes dense and heavy. It begins to drain down into the valley floors below. As the air in the

valley floor continues to cool due to radiational cooling, the air becomes saturated and fog is

formed. Valley fog can be very dense at times and makes driving very hazardous due to reduced

visibility. This type of fog tends to dissipate very quickly once the sun comes up and starts to

evaporate the fog layers.

Advection Fog: it often looks like radiation fog and results from condensation. However, the

condensation in this case is caused not by a reduction in surface temperature but rather by the

horizontal movement of warm moist air over a cold surface. This means that advection fog can

sometime be distinguished from radiation fog by its horizontal motion along the ground.

Sea Fog: Sea fogs are always advection fogs because the oceans don’t radiate heat in the same

way as land and so never cool sufficient to produce radiation fog. Fog forms at sea when warm

air associated with a current drift over a cold current and condensation take place. Sometimes

such fogs are drawn inland by low pressure.

Advection fog may also form when moist maritime or ocean air drifts over a cold inland area.

This usually happens at night when the temperature of the land drops due to radiation cooling.

Upslope Fog: It is formed when light winds push moist air up a hillside or mountainside to a

level where the air becomes saturated and condensation occurs. This type of fog usually forms a

good distance from the peak of the hill or mountain and covers a large area. Upslope fog occurs

in all mountain ranges. This usually occurs during the winter months when cold air behind a cold

front drifts westward and encounters the eastward facing slopes of Rocky Mountains. As the cold

moist air rises up the slopes of the mountain, condensation occurs and extensive areas of fog

form on the lower slope of the mountains.

Ice Fog: This type of fog is formed when the air temperature is well below freezing and is

composed entirely of tiny ice crystal that are suspended in the air. Ice fog will only be witnessed

in the cold Arctic/Polar air. Generally the temperature will be 14 F or colder in order for ice fog

to occur.

Freezing Fog: Freezing fog occurs when the water droplets that the fog is composed of are

“supercooled”. Super-cooled water droplets remain in the liquid state until they come into

contact with a surface upon which they can freeze. As a result, any object the freezing fog comes

into contact with will become coated with ice. The same thing happens with freezing rain or

drizzle.

Evaporation or Mixing Fog: This type of fog is formed when sufficient water vapour is added

to the air by evaporation and the moist air mixes with cooler, relatively drier air. The two

common types are steam fog and frontal fog. Steam fog is form when colder air moves over

warm water. When the cool air mixes with the moist air over the water, the moist air cools until

its humidity reaches 100% and fog is formed. This type of fog takes on the appearance of wisps

of smoke of rising off the surface of the water.

The other type of evaporation fog is known as frontal fog. It is formed when warm raindrops

evaporate into a cooler drier layer of air near the ground. Once enough rain has evaporated into

the layer of cool surface, the humidity of this air reaches 100% and fog is formed.

During condensation, temperature drops very slowly. This is because the heat loss is being

compensated by latent heat of condensation which is being released.

Potential Temperature

When air parcel moves vertically, there is change pressure approximately exponentially altitude.

The changes take place adiabatically. The exchange of heat with the surrounding dQ is zero.

Reason; this is because the transfer of heat between the air parcel to the surrounding is very slow.

Heat processes are energy transfer by radiation and turbulent mixing at the borders. Mixing will

not penetrate much in short time. The happening at the borders are are neglected if the air mass is

large enough. Molecular heat conduction is negligible.

Considering CP dT=δQ+Vdp since the process is adiabatic that is dQ is zero, then:

δQ=CP dT−Vdp=0 ,

Dividing the equation T gives:

CPdTT

−VdpT

=0

From gas law,

VT

=Rp

By substitution equation () becomes:

CPdTT

−Rdpp

=0

Integrating equation () givesCP dlnT−Rdlnp=0

Tp−RCP=constant

This is known as Poisson’s equation for the adiabatic expansion of an ideal gas.

As the air parcel moves upwards in the atmosphere, it temperature varies with p with respect to

equation (). Let the air parcel be taken a reference level where the pressure is p0 = 1000 mb to

acquire a temperature, then θ=T ( p0

p )RCP

Atmospheric Lapse Rate ()

The atmospheric Lapse rate () refers to the changes of atmospheric variables with a changes of

attitude, the variables being temperature unless specified otherwise, such as pressure, density, or

humidity. While usually applied to the Earth atmosphere, the concert of lapse rate can be

extended to atmospheres (if any) that exist on other planets.

Lapse rates are usually expressed as the amount of temperature change associated with a

specified amount of altitude change, such as 9.8 K per kilometre or 0.0098 K per meter. If the

atmospheric air cools with increasing altitude, the lapse rate may be expressed as a negative

number. If the air is heats with increasing altitude, the lapse rate may be expressed as a positive

number.

The lapse rate is most often denoted by the Greek capital letter Gamma (). At times L is used to

denote lapse rate with reference to US standard. A few others use the Greek lower case letter

Gamma, , which is an unfortunate choice since gamma is also used for specific ratio.

Stability

Changes of phase and latent heat

Adiabatic processes, moisture

Thermodynamic diagram (chart)

RADIATION

Fundamental physics of atmospheric electricity

Radiation Laws

Solar and Terrestrial Radiation

Application

Ozone Holes

ATMOSPHERIC SYSTEM