Salby M.L. Fundamentals of Atmospheric Physics

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80 3

The Second Law and Its Implications

Figure 3.1 A gas at the system pressure Ps that is acted on by a piston with pressure pp and

is maintained at constant temperature through contact with a heat reservoir.

isothermal compression at the same temperature (Fig. 3.2). If the cycle is ex-

ecuted very slowly and without friction, the system pressure p~ is uniform

throughout the gas and equals that exerted by the piston

p p

at each stage of

the process. Therefore, the work performed by the system during expansion is

given by

W12 __

ppdv

f2 RT12 dv

,11

- RTI2 In (~11) 9

Likewise, the work performed on the system during compression is

--W21 -- --

f21

ppdv

=RT121n(V2) '01

(3.1.1)

3.1

Natural and Reversible Processes

q41 = 0

T12 > T34

T12 (q 12 > O)

(~ Isothermal Cycle

Carnot Cycle

T34 (q34 < O)

q23 = 0

Figure

3.2 Isothermal cycle (dashed) and Carnot cycle (solid) in the

p-v

plane.

which equals the work performed by the system during expansion. Hence, the

net work performed during the cycle vanishes. Since

f du

= 0, (2.13) implies

f, q-fpdv-o

and the net heat absorbed during the cycle also vanishes. Consequently, both

the system and the environment are restored to their original states, so the

compression from state 1 to state 2 is entirely reversible. The same follows for

the expansion and for the cycle as a whole.

If the cycle is executed rapidly, some of the gas is accelerated. Pressure is

then no longer uniform across the system and

does not equal

pp.

During

rapid compression, the piston must exert a pressure which exceeds that exerted

when the compression is executed slowly (e.g., to offset the reaction of the gas

being accelerated). Therefore, the work performed on the system exceeds that

performed under reversible conditions. ~ Since Au vanishes, the first law implies

that heat transferred out of the system during rapid compression likewise

exceeds that when the compression is executed slowly. During rapid expansion,

the piston must exert a pressure which is smaller than that exerted when the

expansion is executed slowly. The work performed by the system is then less

than that performed under reversible conditions and, since Au ---_ 0, so is the

1Kinetic energy generated by the excess work is eventually dissipated and lost through heat

transfer to the environment.

The Second Law and Its Implications

heat transferred into the system. Thus, executing the isothermal cycle rapidly

results in

f pdv = f 6q < O.

Net work is performed on the system, which must be compensated by net

rejection of heat to the environment.

This example illustrates that completing a transition between two states

reversibly minimizes the work that must be performed on a system and maxi-

mizes the work that is performed by the system. These results can be extended

to a system that performs work through a conversion of heat transferred into

it, namely, to a heat engine. With (2.13), it follows that the cyclic work per-

formed by such a system is maximized when the cycle is executed reversibly.

Irreversibility reduces the net work performed by the system over a cycle,

which must be associated with some heat rejection to the environment.

3.1.1 The Carnot Cycle

Consider a cyclic process comprised of two isothermal legs and two adiabatic

legs (Fig. 3.2). During isothermal expansion and compression, the system is

maintained at constant temperature through contact with a thermal reservoir,

which serves as a heat source or heat sink. During adiabatic compression and

expansion, the system is thermally isolated. If executed reversibly, this process

describes a

Carnot cycle.

The leg from state 1 to state 2 is isothermal expansion

at temperature T12, so

AL/12 -- 0.

Then the heat absorbed during that leg is given by

q12 -- w12

- f12pdv

=RTa21n(V2) "vl

The leg from state 2 to state 3 is adiabatic, so

(3.2.1)

q23 -- 0

and

--W23 -- AU23

= co(T34 -

Z12 ).

(3.2.2)

3.1

Natural and Reversible Processes

Similarly, the leg between states 3 and 4 is isothermal compression, so the heat

transfer and work are given by

q34-w34-RT341n(V-~33), (3.2.3)

whereas the leg between states 4 and 1 is adiabatic, so it involves work

W41 --

cv(Zl2 -

T34 ). (3.2.4)

If the cycle is executed in reverse, the work and heat transfer during each leg

are exactly cancelled by those during the corresponding leg of the forward

cycle. Therefore, the Carnot cycle is reversible.

During the two adiabatic legs of (3.2.2) and (3.2.4), the work performed

cancels, while the heat transfer vanishes identically. Hence, the net work and

heat transfer over the cycle follow from the isothermal legs alone. The volumes

in (3.2.1) and (3.2.3) are related to the changes of temperature along the

adiabatic legs, which follow from Poisson's identity (2.30.1)

y-1

T12T34 --(V~-~)')'-I--(~I 4)

Thus,

1) 2 1) 3

1) 1 1) 4

Collecting the heat transferred during the isothermal legs yields

1)4

f6q--RT1eln(~11 )

-i- RT34 In (~33)

= R(T12 - T34)In (v~-12) ,

which is positive for T12

> T34

if v2 > va. Then

(3.3)

(3.4)

f pdv-

> O. (3.5)

The system performs net work, which is converted from heat absorbed during

the cycle.

The Carnot cycle is a paradigm of a heat engine. For net work to be per-

formed by the system, heat must be absorbed at high temperature T12 (during

isothermal expansion) and rejected at low temperature T34 (during isothermal

compression). According to (3.2), more heat is absorbed at high temperature

than is rejected at low temperature. The first law (2.13) then implies that net

heat absorbed during the cycle is balanced by net work performed by the sys-

tem. If the preceding cycle is executed in reverse, the system constitutes a

3 The Second Law and Its Implications

refrigerator. More heat is rejected at high temperature than is absorbed at

low temperature. Then (2.13) asserts that net heat rejected during the cycle

must be compensated by work performed on the system.

Net work and heat transfer over the cycle are proportional to the tempera-

ture difference T12-

T34

between the heat source and heat sink. Not all of the

heat absorbed by the system during expansion is converted into work. Some

of that heat is rejected during compression. Therefore, the efficiency of a heat

engine is limited--even for a reversiblecycle.

3.2 Entropy and the Second Law

In the development of the first law, we observed that work is independent of

path under adiabatic conditions, which allowed the internal energy be intro-

duced as a state variable. The second law of thermodynamics is inspired by

the observation that the quantity

q/T

is independent of path under reversible

conditions. As for the first law, this allows the introduction of a state vari-

able,

entropy,

which is defined in terms of a quantity that is, in general, path

dependent.

Consider the Carnot cycle. According to (3.4),

f 6q=R

[In (V~a) (v~)]

-T +In =0,

which is equivalent to the identity

q1____22 -I- q4____l_l _ 0.

(3.6)

r12 T41

It turns out this relationship holds under fairly general circumstances.

Carnot's Theorem

The identity (3.6) holds for any reversible cycle between two heat reser-

voirs at temperatures

T12

and T41, irrespective of details of the cycle.

It can be shown that any reversible cycle can be represented as a succession of

infinitesimal Carnot cycles (e.g., Keenan, 1970). Therefore, Carnot's theorem

implies that the identity

f (T)rev--0 (3.7)

holds irrespective of path.

By the exact differential theorem (Sec. 2.1.4), it follows that the quantity

(x y)

(6q/T)rev

represents a point function. That is,

f~x'.,

x(6q/T)rev

depends only

on the thermodynamic state (x, y) and not on the ~ l~at'~ along which the system

3.2 Entropy and the Second

Law 85

evolved to that state. In the spirit that internal energy was introduced (Sec.

2.2), we define the state variable entropy

rev

which constitutes a property of the system.

Carnot's theorem and the identity (3.7) apply to a reversible process. Under

more general circumstances, the following inequality holds.

The Clausius Inequality

For a cyclic process,

f~q

-T < 0, (3.9)

where equality applies if the cycle is executed reversibly.

One of several statements of the second law, the Clausius inequality has the

following consequences that pertain to the direction of thermodynamic pro-

cesses:

1. Heat must be rejected to the environment somewhere during a cycle.

2. Under reversible conditions, more heat is exchanged at high temper-

ature than at low temperature.

3. Irreversibility reduces the net heat absorbed during a cycle.

The first consequence precludes the possibility of a process that converts heat

from a single source entirely into work: a perpetual motion machine of the sec-

ond kind. Some of the heat absorbed by a system that performs work must be

rejected. Representing a thermal loss, that heat rejection limits the efficiency

of any heat engine, even one operated reversibly. The second consequence of

(3.9) implies that net work is performed by the system during a cycle (namely,

it behaves as a heat engine) if heat is absorbed at high temperature and

rejected at low temperature. Conversely, net work must be performed on the

system during the cycle (namely, it behaves as a refrigerator) if heat is rejected

at high temperature and absorbed at low temperature. The third consequence

of (3.9) implies that irreversibility reduces the net work performed by the sys-

tem, in the case of a heat engine, and increases the net work that must be

performed on the system, in the case of a refrigerator.

A differential form of the second law, which applies to an incremental pro-

cess, may be derived from the Clausius inequality. Consider a cycle comprised

of a reversible and an irreversible leg between two states 1 and 2 (Fig. 3.3). A

cycle comprised of two reversible legs may also be constructed between those

states. For the first cycle, the Clausius inequality yields

2 8q 1

f (_~_)revq_f2 (t~q

)

~/irrev ~

The Second Law and Its Implications

Figure 3.3 Cyclic processes between two states comprised of (i) two reversible legs and

(ii)

one reversible and one irreversible leg.

whereas for the second cycle

~12(-~)revd-~21 (-~)rev--O"

Subtracting gives

or with (3.8)

f2 a ( 6q 1

--T-) -f2 (6q

irrev -T-) rev ~ 0

ds >

irrev

Since it holds between two arbitrary states, the inequality must apply to the

integrands too. Thus,

> (3.10)

- T'

where equality holds if the process is reversible.

Equation (3.10) is the most common form of the second law. In combination

with (3.8), it implies an upper bound to the heat that can be absorbed by the

system during a given change of state:

6q <_ Tds,

3.3 Restricted

Forms of the Second Law

namely, the heat absorbed when that process is executed reversibly. If the

process is executed irreversibly, additional heat is rejected to the environment.

This reduces

and the net heat absorbed, which in turn reduces the net work

performed by the system if the process is cyclic (2.13).

Also represented in (3.10) is the direction of thermodynamic processes.

Through the inequality, the second law asserts whether or not a system is

capable of evolving along a given path. A process for which the change of

entropy satisfies (3.10) is possible. If (3.10) is satisfied through equality, that

process is reversible, whereas if it is satisfied through inequality that process

is irreversible (e.g., a natural process). By contrast, a process that satisfies the

reverse inequality is impossible.

3.3 Restricted Forms of the Second Law

The entropy of a system can either increase or decrease, depending on the

heat transfer needed to achieve the same change of state under reversible

conditions. For certain processes, the change of entropy implied by the second

law is simplified.

For an adiabatic process, (3.10) reduces to

dSad > O,

(3.11)

so the entropy can then only increase. It follows that irreversible work increases

a system's entropy. Letting the control surface of a hypothetical system pass to

infinity eliminates heat transfer to the environment and leads to the conclusion

that the entropy of the universe can only increase. For a reversible adiabatic

process, (3.11) implies

= 0, so such a process is

isentropic.

For an isochoric process, wherein expansion work vanishes, the first law

(2.23.1) transforms (3.8) into

ds-c~

-T- rev"

Since it involves only state variables, this expression must hold whether or not

the process is reversible. Hence,

- -T (3.12)

In the absence of work, entropy can either increase or decrease, depending

on the sign of

dT.

It follows that heat transfer can either increase or decrease

a system's entropy.

According to (3.11) and (3.12), changes of entropy follow from

1. Irreversible work and

2. Heat transfer.

88 3 The Second Law and Its Implications

Irreversible work only increases s, whereas heat transfer (irreversible or oth-

erwise) can either increase or decrease s. In the absence of work, the change

of entropy equals 6q/T, irrespective of path (e.g., whether or not the process

is reversible).

3.4 The Fundamental Relations

Substituting the second law (3.10) into the two forms of the first law (2.22)

leads to the inequalities

du < Tds- pdv, (3.13.1)

dh < Tds + vdp. (3.13.2)

For a reversible process, these reduce to

du = Tds- pdv, (3.14.1)

dh = Tds + vdp. (3.14.2)

Since they involve only state variables, the equalities (3.14) cannot depend on

path. Consequently, they must hold whether or not the process is reversible.

Known as the fundamental relations, these identities describe the change in

one state variable in terms of the changes in two other state variables. Even

though generally valid, the fundamental relations cannot be evaluated easily

under irreversible conditions. The values of p and T in (3.14) refer to the

pressure and temperature "of the system," which can be specified only under

reversible conditions. Under irreversible conditions, the relationship among

these variables reverts to the inequalities (3.13), wherein p and T denote

"applied values," which can be specified.

It is convenient to introduce two new state variables, that are referred to

as auxiliary functions. The Helmholtz function is defined by

f- u- Ts. (3.15.1)

The Gibbs function is defined by

g=h- Ts

= u + pv- Ts. (3.15.2)

In terms of these variables, the fundamental relations become

df = -sdT- pdv, (3.16.1)

dg = -sdT + vdp. (3.16.2)

3.4 The Fundamental Relations

The Helmholtz and Gibbs functions are each referred to as the

free energy

of the system: f for an isothermal process and g for an isothermal-isobaric

process, because they reflect the energy available for conversion into work

under those conditions (Problem 3.10).

3.4.1 The Maxwell Relations

Variables appearing in (3.14) and (3.16) are not entirely independent. Involv-

ing only state variables, each fundamental relation has the form of an exact

differential. According to the exact differential theorem (2.8), these relations

can hold only if the cross derivatives of the coefficients on the right-hand sides

are equal. Applying that condition to (3.14) and (3.16) yields the identities

(

(3.17)

which are known as the

Maxwell relations.

3.4.2 Noncompensated Heat Transfer

The inequalities in (3.13) account for additional heat rejection to the environ-

ment that occurs through irreversibility. Those inequalities can be eliminated

in favor of equalities by introducing the

noncompensated heat transfer 3q',

defined by

6q = Tds- 6q',

(3.18)

where

6q'

> 0. The noncompensated heat transfer represents the additional

heat that must be rejected to the environment as a result of irreversibility, for

example, that associated with frictional dissipation of kinetic energy.

Substituting (3.18) into the first law (2.22.1) yields

du = Tds- pdv- 6q',

(3.19)

where p and T are understood to refer to applied values. An expression

for the noncompensated heat transfer can be derived from (3.14) and (3.19),