The relation arises from integrating the general partial-derivative expansion
or replacing the partial derivatives with the corresponding material properties
with constant-pressure heat capacity , temperature T, bulk modulus K, thermal expansion coefficient , and volume V , for the specific cases of an ideal gas, for which , or constant pressure (), thus giving and then
To calculate for instance, you'd express dS in the variables you wish to use, e.g.,
Then you'd figure out what material properties those partial derivatives refer to, simplify, and integrate over the temperature range of interest. Here, , so at constant pressure and then
The same general strategy would be applied to calculate
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