Environment Takes Up Volume Energy

We construct the environment as an engine that can take up volume energy, Uenv = UenviVsys)- In particular, the environment knows about the volume of the system, thus 

We now make a closer assumption on the nature of f(Vsys) given in Eq. 2.65. We assume that 

We adjust C2 — 0 and find, introducing a new (positive) constant C that gathers all the former constants, 

If = 0, then the entropy of the system will not change provided that the volume of the system changes. This is a reversible change of the state. This does not mean that the temperature will be constant in this case, because the temperature is also a function of the volume. The temperature will decrease with expanding volume according to energy equation of the ideal gas. 

then the entropy of the system will increase provided that the volume of the system increases. This is an irreversible change of the state. 

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