Chapter 3
The Second Law of Thermodynamics 94
to state 3. From Eq. (3.23), S is zero for a reversible adiabatic process. Hence S3 S2. (As always, state functions refer to the system unless otherwise specified. Thus S3 and S2are the system’s entropies in states 3 and 2.) We next either add or withdraw enough heat q3→4isothermally and reversibly at temperature Thrto make the entropy of the system equal to S1. This brings the system to state 4 with S4 S1. (q3→4is pos-itive if heat flows into the system from the reservoir during the process 3 → 4 and neg-ative if heat flows out of the system into the reservoir during 3 → 4.) We have
Since states 4 and 1 have the same entropy, they lie on a line of constant S, an isen-trop. What is an isentrop? For an isentrop, dS 0 dqrev/T, so dqrev 0; an isentrop is a reversible adiabat. Hence to go from 4 to 1, we carry out a reversible adiabatic process (with the system doing work on the surroundings). Since S is a state function, we have for the cycle 1 → 2 → 3 → 4 → 1
The sign of S2 S1is thus the same as the sign of q3→4. We have for the cycle
The work done on the system in the cycle is thus w q3→4. The work done by the system on the surroundings is w q3→4. Suppose q3→4were positive. Then the work
w done on the surroundings would be positive, and we would have a cycle (1 → 2
→ 3 → 4 → 1) whose sole effect is extraction of heat q3→4 from a reservoir and its complete conversion to work w q3→4 0. Such a cycle is impossible, since it violates the second law. Hence q3→4cannot be positive: q3→4 0. Therefore
(3.36) We now strengthen this result by showing that S2 S1 0 can be ruled out. To do this, consider the nature of reversible and irreversible processes. In a reversible process, we can make things go the other way by an infinitesimal change in circumstances.
When the process is reversed, both system and surroundings are restored to their orig-inal states; that is, the universe is restored to its origorig-inal state. In an irreversible process, the universe cannot be restored to its original state. Now suppose that S2 S1 0. Then q3→4, which equals Thr(S2 S1), would be zero. Also, w, which equals q3→4, would be zero. (Points 3 and 4 would coincide.) After the irreversible process 1 → 2, the path 2 → 3 → 4 → 1 restores the system to state 1. Moreover, since q 0 w for the cycle 1 → 2 → 3 → 4 → 1, this cycle would have no net effect on the surroundings, and at the end of the cycle, the surroundings would be restored to their original state. Thus we would be able to restore the universe (system surroundings) to its original state. But by hypothesis, the process 1 → 2 is irreversible, and so the universe cannot be restored to its original state after this process has occurred. Therefore S2 S1cannot be zero.
Equation (3.36) now tells us that S2 S1must be positive.
S2 S1 q3S4>Thr 0
冯
dU 0冯
1dq dw2 q3S4 w S2 S1 q3S4>Thr冯
dSsyst1S2 S12 0 q3S4>Thr 0 00
冯
dSsyst 1S2 S12 1S3 S22 1S4 S32 1S1 S42S4 S3
冮
34 dqTrev 1Thr
冮
34dqrevqT3S4hr lev38627_ch03.qxd 2/29/08 3:12 PM Page 94We have proved that the entropy of a closed system must increase in an irre-versible adiabatic process:
(3.37) A special case of this result is important. An isolated system is necessarily closed, and any process in an isolated system must be adiabatic (since no heat can flow be-tween the isolated system and its surroundings). Therefore (3.37) applies, and the entropy of an isolated system must increase in any irreversible process:
(3.38) Now consider Suniv Ssyst Ssurrfor an irreversible process. Since we want to examine the effect on Sunivof only the interaction between the system and its surround-ings, we must consider that during the irreversible process the surroundings interact only with the system and not with any other part of the world. Hence, for the duration of the irreversible process, we can regard the system plus its surroundings (syst surr) as forming an isolated system. Equation (3.38) then gives Ssystsurr⬅ Suniv 0 for an irreversible process. We have shown that Sunivincreases in an irreversible process:
(3.39) where Sunivis the sum of the entropy changes for the system and surroundings.
We previously showed Suniv 0 for a reversible process. Therefore
(3.40)*
depending on whether the process is reversible or irreversible. Energy cannot be cre-ated or destroyed. Entropy can be crecre-ated but not destroyed.
The statement that
dqrev/T is the differential of a state function S that has the property Suniv 0 for every process
can be taken as a third formulation of the second law of thermodynamics, equivalent to the Kelvin–Planck and the Clausius statements. (See Prob. 3.23.)
We have shown (as a deduction from the Kelvin–Planck statement of the second law) that Suniv increases for an irreversible process and remains the same for a re-versible process. A rere-versible process is an idealization that generally cannot be pre-cisely attained in real processes. Virtually all real processes are irreversible because of phenomena such as friction, lack of precise thermal equilibrium, small amounts of tur-bulence, and irreversible mixing; see Zemansky and Dittman, chap. 7, for a full dis-cussion. Since virtually all real processes are irreversible, we can say as a deduction from the second law that Suniv is continually increasing with time. See Sec. 3.8 for comment on this statement.
Entropy and Equilibrium
Equation (3.38) shows that, for any irreversible process that occurs in an isolated sys-tem, S is positive. Since all real processes are irreversible, when processes are oc-curring in an isolated system, its entropy is increasing. Irreversible processes (mixing, chemical reaction, flow of heat from hot to cold bodies, etc.) accompanied by an in-crease in S will continue to occur in the isolated system until S has reached its maxi-mum possible value subject to the constraints imposed on the system. For example, Prob. 3.19 shows that heat flow from a hot body to a cold body is accompanied by an increase in entropy. Hence, if two parts of an isolated system are at different temper-atures, heat will flow from the hot part to the cold part until the temperatures of the parts are equalized, and this equalization of temperatures maximizes the system’s
¢Suniv 0
¢Suniv 7 0 irrev. proc.
¢Ssyst 7 0 irrev. proc., isolated syst.
¢Ssyst 7 0 irrev. ad. proc., closed syst.
Section 3.5 Entropy, Reversibility, and Irreversibility 95 lev38627_ch03.qxd 2/29/08 3:12 PM Page 95
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The Second Law of Thermodynamics 96
entropy. When the entropy of the isolated system is maximized, things cease happen-ing on a macroscopic scale, because any further processes can only decrease S, which would violate the second law. By definition, the isolated system has reached equilib-rium when processes cease occurring. Therefore (Fig. 3.11):
Thermodynamic equilibrium in an isolated system is reached when the system’s entropy is maximized.
Thermodynamic equilibrium in nonisolated systems is discussed in Chapter 4.
Thermodynamics says nothing about the rate at which equilibrium is attained. An isolated mixture of H2 and O2 at room temperature will remain unchanged in the absence of a catalyst. However, the system is not in a state of true thermodynamic equilibrium. When a catalyst is introduced, the gases react to produce H2O, with an increase in entropy. Likewise, diamond is thermodynamically unstable with respect to conversion to graphite at room temperature, but the rate of conversion is zero, so no one need worry about loss of her engagement ring. (“Diamonds are forever.”) It can even be said that pure hydrogen is in a sense thermodynamically unstable at room tem-perature, since fusion of the hydrogen nuclei to helium nuclei is accompanied by an increase in Suniv. Of course, the rate of nuclear fusion is zero at room temperature, and we can completely ignore the possibility of this process.