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If we divide through by the total mass of the gas (m

) and note that n

¼ m

where M

is the molecular weight of the species i, and total mass is

m ¼

ð13Þ

The statement of the ideal gas law (equation of state) for a mixture of gases is then

p ¼ r



T ¼ r



RRT ð14Þ

where



MM is the mean molecular weight of the mixture, which is given by:



MM 

ð15Þ

where N

is the molar fraction of a particular ga s given by N

¼ p

=p ¼ V

=V.

Composition of Air The current atmosphere of Earth is composed of ‘‘air,’’

which we ﬁnd to contain:

1. Dry air, which is a mixture of gases described below

2. Water, which can be in any of the three states of liquid, solid, or vapor

3. Aerosols, which are solid or liquid particles of small sizes

The chemical composition of dry air is given in Table 2.

Water vapor and the liquid and solid forms of water vary in its volume fraction

from 0% in the upper atmosphere to as high as 3 to 4% near the surface under humid

conditions. Because of this variation, dry air is treated separately from vapor in

thermodynamic theory.

Among the trace constituents are carbon dioxide, ozone, chloroﬂuorocarbons

(CFCs), and methane, which, despite their small amounts, have a very large

impact on the atmosphere because of their interaction with terrestrial radiation

passing through the atmosphere, or in the case of CFCs because of their impact

on the ozone formation process.

Note from Table 2 that the molecular weighted average gas constant for dry air is



RR ¼ 287:05 J=kg=K, which we also call the dry air gas constant.

Work by Expansion

If a system is not in mechanical equilibrium with its surroundings, it will expand or

contract. Assume that l is the surface to the system that expands inﬁnitesimally to l

186 PHYSICAL ATMOSPHERIC SCIENCE

in the direction ds (see Fig. 2). Then the surface element ds has performed work

against the external pressure P. The work perfor med is

ðdW Þ

¼ Pds dl cos f ¼ P

surr

dV ð16Þ

where dV is the change in volume.

For a ﬁnite expansion, then

W ¼

pdV ð17Þ

TABLE 2 Composition of Dry Air near the Earth’s Surface

Gas Symbol

Molecular

Weight

Molar

(or Volume)

Fraction

Mass

Fraction

Speciﬁc

Gas

Constant

(J=kg K)

=m ð



RRÞ

(J=kg K)

Nitrogen N

28.013 0.7809 0.7552 296.80 224.15

Oxygen O

31.999 0.2095 0.2315 259.83 60.15

Argon Ar 39.948 0.0093 0.0128 208.13 2.66

Carbon Dioxide CO

44.010 0.0003 0.0005 188.92 0.09

Helium He 0.0005

Methane CH

0.00017

Hydrogen H 0.00006

Nitrous Oxide N

O 0.00003

Carbon Monoxide CO 0.00002

Neon Ne 0.000018

Xenon X 0.000009

Ozone O

0.000004

Krypton Kr 0.000001

Sulfur Dioxide SO

0.000001

Nitrogen Dioxide NO

0.000001

Chloroﬂuorocarbons CFC 0.00000001

Total 1.0000 1.0000



RR ¼ 287.05

Figure 2 Work of expansion (from Iribarne and Godson, 1973).

2 ATMOSPHERIC THERMODYNAMICS 187

where i, f stand for the initial and ﬁnal states. Since the integrand is not a total

derivative, total work over a cyclic process can be nonzero. Therefore

W ¼

pdV ¼

PdV





PdV



ð18Þ

which is deﬁned to be positive if integrated in the clockwise sense. This is the area

enclosed by the trajectory in the graph given in Figure 3.

The work by expansion is the only kind of work that we shall consider in our

atmospheric systems. Assuming the pressure of the system is homogeneous and in

equilibrium with the surroundings, we can adopt the notation

dW ¼ PdV ð19Þ

where W is the work perfor med on the surroundings by the system.

First Law (or Principle) of Thermodynamics

The ﬁrst law of thermodynamics can be simply stated: The energy of the universe is

constant. Let us consider the general concept of just what is meant by energy. We

know that if we apply force to an object over some distance and as a result we

accelerate the object, we will have performed work on the object, measured in the

energy units of force  distance ¼ joules. That work will precisely equal the

increase in kinetic energy of the object, the total of which is measured by half its

mass multiplied by the square of its new velocity. Moreover, we understand that our

effort expends energy from the working substance. Depending on the method used

to accelerate the object, the energy was previously stor ed in some other form;

perhaps the ‘‘potentia l energy’’ of a compressed spring, ‘‘chemical energy’’ stored

in the muscle cells of one’s arm, or perhaps the ‘‘kinetic energy’’ of another object

that strikes the object that was accelerated. There are obviously many possibilities

for working substances featuring different types of stored energy. Energy, in a

general sense, quantiﬁes a potential to apply a force and so perform work.

Figure 3 Work of expansion in a cycle (from Iribarne and Godson, 1973).

188

PHYSICAL ATMOSPHERIC SCIENCE

The ﬁrst law requires that for any thermodynamic system an energy budget must

exist that requires any net energy ﬂowing into a system to be accounted for by

changes in the external energy of the system or the internal energy stored within

the system. The external energy of the system includes its kinetic energy of motion

and its energy of position, relative to forces outside the system such as gravitational,

electrical, magnetic, chemical, and many other forms of potential energy between the

system and its surroundings. The same energies can also exist within the system

between the molecules, atoms, and subatomic particles composing the system and

are termed internal energies. Kinetic energy of molecular motions relative to the

movement of the center of mass of the system is measured by the system’s tempera-

ture and represen t the thermal internal energy. The other internal energies are

potential internal energies and may include a potential energy against the inter-

molecular forces of attraction (latent heat), chemical potential energy (e.g., a gas

composed of oxygen and hydrogen can potentially react), and so on.

Because we probably are not even aware of all of the forms of internal forces that

exist in a system, we make no attempt to evaluate the total internal energy. Instead,

for the purposes of classical thermodynamics, we need only consider changes in

internal energy occurring during allowable processes. For instance, for atmospheric

studies we choose to ignore nuclear reactions and even most chemical reactions. But

we cannot neglect the changes in internal potential energy due to intermolecular

forces of liquid and ice water phases in the atmosphere. The intermolecular forces

are substantial, and the energy needed to overcome them is the latent heat of

vaporization and melting. Thermodynamic energy transfers between thermal and

these internal potential energies drive the general circulation of Earth’s atmosphere!

Our limited thermodynamic discussion will ignore the treatment of some internal

energies such as surface energy of droplets, chemical energy in photochemical

processes, electrical energy in thunderstorms and the upper atmosphere, chemical

processes involving CFCs, an d other processes known to have important secondary

impacts on the evolution of the atmospheric thermodynamic system. These

processes can be added to the system when needed, following the methods described

in this chapter.

In most texts of classical thermodynamics, closed thermodynamic systems are

assumed. A closed system allows no mass exchange between the parcel (system) and

the surroundings. This approximation greatly simpliﬁes the thermodynamic formu-

lation. Later, we can still account for molecular transfer by representing it as a

noninteracting and externally speciﬁed source of mass or energy rather than as an

explicitly interacting component of the formulation.

One such open mass ﬂux that must be accounted for with an open system is the

mass ﬂux of precipitation. We afﬁx our parcel coordinate relative to the center of

mass of the dry air, which is assumed to move identically to vapor. The liquid and ice

components of the system, however, may attain a terminal velocity that allows it to

ﬂow into and out of the parcel diabatically. As a result, we will derive thermo-

dynamic relationships for an open system, or one where external ﬂuxes of mass

into and out of the system are allowed. To retain some simplicity, however, we will

make the assumption that external mass ﬂuxes into and out of the system will be of

2 ATMOSPHERIC THERMODYNAMICS 189

constituents having the same state as those internal to the system. Henc e we will

allow rain to fall into our parcel, but it will be assum ed to have the same temperature

as the system. Although the effect of this assumption can be scaled to be small, there

are instances where it can be important. One such instance is the case of frontal fog,

where warm droplets falling into a cool air mass result in the formation of fog. These

moist processes will be described in detail in the next chapter.

As demanded by the ﬁrst law, we form an energy budget, ﬁrst for a simple one-

component system of an ideal gas:

Energy exchanged with surrounding s ¼ change in energy stored

Q  W ¼ DU 

ð20Þ

where Q repres ents the ﬂow of thermal energy into the system including radiative

energy and energy transferred by conduction, W is the work performed on the

surroundings by the system, U is the thermal internal energy, and A

is the kth

component of potential internal energy, summed over all of the many possible

sources of internal energy. The work, as described in the previous section, is the

work of expansion performed by the parcel on the surroundings. It applies only to

our gas system and does not exist in the same form for our liquid and ice systems. In

those systems, it should be reﬂected as a gravitational potential term; however, it is

typically neglected.

In general, we treat the air parcel as a closed system, whereby there are no mass

exchanges with the surroundings. This is generally a good approximation as mole-

cular transfers across the boundaries can usually be neglected or included in other

ways. One exception occurs and that is when we are considering liquid and ice

hydrometeors. If we ﬁx our parcel to the center of mass of the dry air, then there

can be a substantial movement of liquid or ice mass into and out of the parcel.

Clearly this process must be considered with an open thermodynamic system, where

at least ﬂuxes of liquid and ice are allowed with respect to the center of mass of the

dry air parcel. Hence we will consider the possibility that A

may change in part

because of exchanges of mass between the parcel and the environment, particularly

present with falling precipitation. Combining Eq. (20) with (19) for an inﬁnitesimal

process, we obtain the form of the ﬁrst law:

dQ ¼ dU þ PdV

ð21Þ

where it should be noted that P is the press ure of the surroundings, as PdVdeﬁnes

work performed on the surroundings and dV is the change in volume of the

surroundings. Note that for the adiabatic case, dQ ¼ 0 by deﬁnition.

Because we require conservation of energy, it is evident that the change in both

internal thermal energy (dU) and internal potential energy (dA

) must be exact

differentials, only dependent on the initial and ﬁnal states of the process and not

the path that the process takes. Hence U and A

must also be state variables.

190 PHYSICAL ATMOSPHERIC SCIENCE

Heat Capacities Heat capacity is a property of a substance and is deﬁned to be

the rate at which the substance absorbs (loses) thermal energy (Q) compared to the

rate at which its temperature (T ) rises (falls) as a result. It is a property usually

deﬁned in units of dQ=dT (energy per temperature change) or in units of dq=dT

(energy per mass per temperature change) in which case it is called speciﬁc heat

capacity. If heat is passed into a system, it may be used to either incre ase the internal

thermal or potential energy of the system or can be used to perform work. Hence

there are an inﬁnite number of possible values for heat capacity depending on the

processes allowable. It is useful to determine the heat capacity for special cases

where the allowable processes are restricted to only one. Hence we neglect the

storage of potential internal energy and restrict the system to constant volume or

constant pressure processes. Hence,



ð22Þ

where C

and C

are the heat capacities at constant volume and pressure, respec-

tively, and where c

and c

are the speciﬁc heat capacities at constant volume and

pressure, respectively. It can be expected that C

> C

, since in the case of constant

pressure, the parcel can use a portion of the energy to expand, and hence perform

work on the surroundings, reducing the rise in temperature for a given addition of

heat.

Since the ﬁrst law requires that the change in internal energy, U, be an exact

differential, U must also be a state variable; i.e., its value is not dependent upon path.

Hence U ¼ uðP; a; T Þ. If we accept that air can be treated as an ideal gas, then the

equation of state eliminates one of the variables, since the third becomes a function

of the other two. Employing Euler’s r ule,

dU ¼



dV þ



dT ð23Þ

Substituting Eq. (23) into the ﬁrst law [Eq. (20)], we get

dQ ¼



dT þ

þ P



dV 

: ð24Þ

For a constant volume process,



ð25Þ

2 ATMOSPHERIC THERMODYNAMICS 191

Hence, retur ning to Eq. (23),

dU ¼ C

dT þ



dV : ð26Þ

It has been shown experimentally that ð@U=@V Þ

¼ 0, indicating that air behaves

much as an ideal gas. Hence the internal energy of a gas is a function of temperature

only provided that C

is a function of temperature only. If we assume that possibil ity,

=@V ¼ 0. Experiments show that for an ideal gas the variation of C

with

temperature is small, and since we hold air to be approximately ideal, it follows

that C

is a constant for air. The ﬁrst law [Eq. (24)] is therefore written as:

dq ¼ C

dT þ PdV

: ð27Þ

One can also obtain an alte rnate form of the ﬁrst law by replacing dV with the ideal

gas law and then combining q

ith

Eq. (27) to yield

dQ ¼ðC

þ R*Þ dT  VdP

: ð28Þ

For an isobaric process, ðdP ¼ 0 Þ, one ﬁnds



¼ C

þ R* ð29Þ

 C

¼ R*: ð30Þ

Similarly for dry air, the speciﬁc heat capacities are

 c

¼ R

ð31Þ

Equation (30) demonstrates that C

> C

. Statistical mechanics show that speciﬁc

relationships between speciﬁc heats at constant volume and at constant pressure can

be found for a gas depending on how many atoms form the gas molecule. Table 3

TABLE 3 Relationship between Speciﬁc Heats and

Molecular Structure of Gas

Monotonic gas C

¼ C

þ R ¼

Diatomic gas C

192

PHYSICAL ATMOSPHERIC SCIENCE

shows these relationships for diatomic and monoatomic gases. Since dry air,

composed mostly of molecular oxygen and nitrogen is nearly a diatomic gas,

¼ 1004 J=kg=K; c

¼ 717 J=kg=K: ð32Þ

Substituting Eq. (30) into Eq. (28) we obtain

dQ ¼ C

dT  VdP

ð33Þ

This form of the ﬁrst law is especially useful because changes of pressure and

temperature are most commonly measured in atmospheric science applications.

Note that C

dT is not purely a change in energy, nor is Vdppurely the work.

Enthalpy For convenience, we introduce a new state variable called enthalpy,

deﬁned as:

H  U þ PV : ð34Þ

Differentiating across we obtain

dH ¼ dU þ pdVþ VdP; ð35Þ

and substituting Eq. (21), to eliminate dU:

dQ ¼ dH  VdP

: ð36Þ

Using Euler’s rule with Eq. (33), we obtain



¼ C

: ð37Þ

Enthalpy differs only slightly from energy but arises as the preferred energy variable

when the work term is expressed with a pressure change rather than a volume change.

The enthalpy variable exists purely for convenience (since pressure is more easily

measured than volume) and is not demanded or deﬁned by any thermodynamic law.

Reversible Adiabatic Processes in the Atmosphere: Deﬁnition of

Potential Temperature (

) These types of processes are of great importance

to the atmospheric scientist. Ignoring radiative effects, or heating effects of

condensation, the dry adiabatic process represents the thermal tendencies that air

will experience as it rises and the pressure lowers or as it sinks and the pressure rises.

In the absence of condensation, this represents the bulk of the temperature change a

parcel will experience if its rise rate is fast enough to ignore radiative effects. We

deﬁned an adiabatic process as one for which Q ¼ 0. Then for an adiabatic process,

2 ATMOSPHERIC THERMODYNAMICS 193

combining Eq. (33), the equation of state [Eq. (7)], and then integrating we obtain

T ¼ kP

ð38Þ

where k ¼ R*=C

¼ 0:29 and k is a constant of integration that can be determined

from a solution point. Equation (38) is known to atmospheric scientists as the

Poisson equation, not to be confused with the second-order partial differential

equation also known as ‘‘Poisson’s equation.’’ The Poisson equation states that,

given a relationship between temperature and pressure at some reference state, the

value of temperature can be calculated for all other pressures the system might

obtain through reversible dry adiabatic processes.

It is often convenient to deﬁne potential temperature (y) to be the temperature

that a parcel would have at 1000 kPa press ure. It follows directly from Eq. (38) that

y  T



R=c

; ð39Þ

where p

¼ 1000 kPa is the reference pressure at which potential temperature (y)

equals temperature (T). The potential temperature proves extremely useful to meteor-

ologists. For all reversible dr y adiabatic atmospheric thermodynamic processes, y

remains constant and represents a conserved property of the ﬂow. Later we will show

y also represents the dry entropy of the ﬂow and that it is one in a hierarchy of

entropy variables that are conserved in ﬂow under various permitted dry and moist

processes.

Second Law (or Principle) of Thermodynamics

The zeroth principle of thermodynamics deﬁned temperature to be a quantity that is

determined from two bodies in thermal equilibrium. The ﬁrst law stated the principle

of energy conservation during any thermodynamic process. The second law deals

with the direction of energy transfer during a thermodynamic process occurring

when two bodies are not in thermodynamic equilibrium. It is again an empirical

statement, not derived in any way from ﬁrst principles but rather from observations

of our perceived universe. The second law can be stated in two ways that can be

shown to be equivalent:

1. Thermal energy will not spontaneously ﬂow from a colder to a warmer object.

2. A ther modynamic proces s always acts down gradient in the universe to reduce

differentiation overall and hence mix things up and reduce order overall. If we

deﬁne entropy to be a measure of the degree of disorder, then the second law

states: The entropy of the universe increases or remains the same as a result of

any process. Hence, the entropy of the universe is always increasing.

Just as the ﬁrst law demanded that we deﬁne the thermal variable temperature, the

second law requires a new thermodynamic variable entropy. To form a mathematical

194 PHYSICAL ATMOSPHERIC SCIENCE

treatment for entropy, we consider several special thermodynamic processes that

reveal the nature of entropy behavior and provide guidance to its form.

Mathematical Statement of the Second Law It is illustrative to examine the

amount of work performed by a parcel on its sur roundings under isothermal

conditions, implying that the internal energy of the system is held constant. This

will demonstrate an interesting behavior addressed by the second law.

Consider a ﬁxed mass of an ideal gas conﬁned in a cylinder ﬁtted with a movable

piston of variable weight. The weight of the piston and its cross-sectional area

determine the pressure on the gas. Let the entire assembly be placed in a constant

temperature bath to maintain isothermal conditions. Let the initial pressure of the gas

be 10 atm and the initial volume be 1 liter. Now consider three isothermal processes

(Fig. 4) (Sears, 1953):

Process I The weight on the piston is changed so as to produce an effective

pressure of 1 atm and since PV ¼ nR*T, its volume becomes

10 liters. The work of expansion is given by:

exp

¼ PDVa

ð1 atm ¼ 1013  10

PaÞð40Þ

Since P

surr

is constant at 1 atm, the work is 1ð10  1Þ¼9 Latm.

This work is performed on the surroundings. With respect to the

system, the energy transfer (dQ) is therefore 9 Latm.

Figure 4 Illustration of three isothermal processes.

2 ATMOSPHERIC THERMODYNAMICS 195