Entropy of an Ideal Gas
One of the things which can be determined directly from this equation is the change in entropy during an isothermal expansion where N and U are constant (implying Q=W). Expanding the entropy expression for Vf and Vi with log combination rules leads to
For determining other functions, it is useful to expand the entropy expression to separate the U and V dependence.
Then making use of the definition of temperature in terms of entropy:
This gives an expression for internal energy that is consistent with equipartition of energy.
with kT/2 of energy for each degree of freedom for each atom.
For processes with an ideal gas, the change in entropy can be calculated from the relationship
This is a useful calculation form if the temperatures and volumes are known, but if you are working on a PV diagram it is preferable to have it expressed in those terms. Using the ideal gas law
But since specific heats are related by CP = CV + R,
Since entropy is a state variable, just depending upon the beginning and end states, these expressions can be used for any two points that can be put on one of the standard graphs.