We will show that the internal energy of an ideal gas is a function of temperature only. This makes physical sense because there is an assumption in ideal gas behavior that there is no interaction between the molecules when we write
Start with a reversible process for an ideal gas:
Consider two processes: one occurring at constant volume, the other occurring at constant pressure.
For process 1: ; This can be integrated because is the only thing that is changing on the righthandside ( is assumed to be independent of and ).
For process 2: ; is constant (i.e., not a function of or ) so it can be integrated directly. Using the ideal gas law:
So for process 2,
Since we can make up any quasi-static curve with segments of processes and processes
Evidently, the sum of any such processes is a function only of . Therefore, for an ideal gas
Comparing the two equations: