Entropy as a function of temperature and volume

Since Eq. (9.26) expresses the change in entropy in terms of changes in T and V, it must be identical with Eq. (9.24), which does the same thing. In view of this identity, we may write 

Since CJT is always positive (Section 9.2), Eq. (9.27) expresses the important fact that at constant volume the entropy increases with increase in temperature. Note that the dependence of entropy on temperature is simple, the differential coefficient being the appropriate heat capacity divided by the temperature. For a finite change in temperature at constant volume 

Note that if 2 mol were used, C would be doubled and so the entropy change would be doubled. 

In contrast to the simplicity of the temperature dependence, the volume dependence at constant temperature given by Eq. (9.28) is quite complicated. Remember that the volume dependence at constant energy, Eq. (9.23), was very simple. We can obtain a simpler expression for the isothermal volume dependence of the entropy by the following device. We differentiate Eq. (9.27) with respect to volume, keeping temperature constant this yields 

In the right-hand side we have replaced C by (dU/dT)v. Similarly, we differentiate Eq. (9.28) with respect to temperature keeping volume constant, to obtain 

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It is often useful as w cll to have the entropy as a function of temperature and volume. We can find this by integrating the equations... 

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