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Fig. 1.7 Reversible path R and irreversible path I connecting states
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Proof Let R and I denote respectively any reversible and any irreversible path joining A to B, as shown in Fig. 1.7. For path R the assertion holds by definition of S. Now consider the cyclic transformation made up of I plus the reverse of R. From Clausius' theorem we have
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dQ f -T == S(B)- SeA)
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(b) The entropy of a thermally isolated system never decreases.
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Proof A thermally isolated system cannot exchange heat with the external world. Therefore dQ = 0 for any transformation. By the previous property we immediately have
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S(B) - SeA) ~ 0
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The equality holds if the transformation is reversible.
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An immediate consequence of this is that for a thermally isolated system the state of equilibrium is the state of maximum entropy consistent with external constraints. For a physical interpretation of the entropy, we consider the following example. One mole of ideal gas expands isothermally from volume VI to V2 by two routes: Reversible isothermal expansion and irreversible free expansion. Let us calculate the change of entropy of the gas and of the external surroundings.
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Reversible Isothermal Expansion. The arrangement is illustrated in Fig. 1.8. In the P- V diagram the states of the gas (and not its surroundings) are represented. Since the gas is ideal, U = U(T). Henge!:J.U = O. The amount of heat absorbed is equal to the work done, which is the shaded area in the P- V diagram:
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V2 !:J.Q = RTlogVI
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Reservoir T
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T Fig. 1.8 Reversible isothermal expansion of an ideal gas.
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The reservoir supplies the amount of heat -I1Q. Hence,
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I1Q V2 (I1S)reservoir= - - = -RlogT VI
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The change in entropy of the whole arrangement is zero. An amount of work
V2 W = I1Q = RT log VI
is stored in the spring connected to the piston. This can be used to compress the gas, reversing the transformation.
Free Expansion. This process is illustrated in Fig. 1.3. The initial and final states are the same as in the reversible isothermal expansion. Therefore (AS)gas is the
same as in that case because S is a state function. Thus (I1S)gas = R log V2 VI
Since no heat is supplied by the reservoir we have (AS)reservoir = 0 which leads to an increase of entropy of the entire system of gas plus reservoir:
(I1S ) total = R log -
In comparison with the previous case, an amount of useful energy
W = T(I1S)total
is "wasted," for it could have been extracted by expanding the gas reversibly. This example illustrates the fact that irreversibility is generally "wasteful," and is marked by an increase of entropy of the total system under consideration. For this reason the entropy of a state may be viewed as a measure of the unavailability of useful energy in that state. It may be noted in passing that heat conduction is an irreversible process and thus increases the total entropy. Suppose a metal bar conducts heat from reservoir T2 to reservoir T1 at the rate of Q per second. The net increase in entropy per second of the entire system under consideration is
The only reversible way to transfer heat is to operate a Carnot engine between the two reservoirs. We might indulge in the following thought. The entropy of the entire universe, which is as isolated a system as exists, can never decrease. Furthermore, we have ample evidence, by just looking around us, that the universe is not unchanging, and that most changes are irreversible. It follows that the entropy of the universe constantly increases, and will lead relentlessly to a "heat death" of the universe-a state of maximum entropy. Is this the fate of the universe In a universe in which the second law of thermodynmnics is rigorously correct, the affirmative answer is inescapable. In fact, however, ours is not such a universe, although this conclusion cannot be arrived at within thermodynamics. Our universe is governed by molecular laws, whose invariance under time reversal denies the existence of any natural phenomenon that absolutely distinguishes between the past and the future. The proper answer to the question we posed is no. The reason is that the second law of thermodynaInics cannot be a rigorous law of nature. This leads to the new question, "In what sense, and to what extent, is the second law of thermodynamics correct " We exaInine this question in our discussion of kinetic theory (see Section 4.4) where we see that the second law of thermodynamics is correct "on the average," and that in macroscopic phenomena deviations from this law are so rare that for all practical purposes they never occur.