Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 3.30 – 3.31.
In a quasistatic process, the relation between entropy, temperature and the heat flow is
where is the (infinitesimal) amount of heat flowing into or out of the system at temperature . For a process at constant pressure but changing temperature, can be written in terms of the heat capacity at constant pressure, since the amount of heat required to change the temperature by is . In that case, the entropy change between temperatures and is
Example 1 From Schroeder’s Figure 1.14, we can estimate a linear relation for for a mole of diamond between and . Reading off the graph we get
Between these temperatures, a formula for is therefore a straight line:
If we assume this is valid over the range of temperature from 298 K up to 500 K, we can get the entropy change over that range for a mole of diamond (incidentally, if you want to try this experiment, you’ll need a very big diamond. A mole of diamond (carbon) is around 12 grams, and there are 5 carats per gram, so you’re looking for a 60 carat diamond). The entropy change is
The entropy of a mole of diamond at is given in Schroeder’s appendix as so the total entropy at is
Example 2 An empirical formula obtained by fitting to measured data for for one mole of graphite is
where the constants are
The entropy change of a mole of graphite over the range of temperature from 298 K up to 500 K is therefore
Adding on the tabulated value for we get
The entropy of graphite is larger than that of diamond which we’d expect since diamond’s crystal structure is more ordered than that of graphite.