Changes of Energy with Matter

We will investigate what will happen, when we are allowing a gas to escape from a system. The energy of a system containing a single component changes with 

By taking the partial derivative of the energy with the mol number, we have to keep the total entropy and the total volume a constant. In this case, in fact, the entropy and the volume per mol and thus per molecule increases according to 

The situation becomes different if we want to remove some gas from the system, keeping the volume V constant, and not the total entropy S, but the molar entropy 5 = C3. Then we have to apply the chain rule for functions with multiple arguments 

The last identity in Eq. (5.22) arises from the Euler relation of homogeneous functions of first order U = TS - pV + iJ,n and the definition of the enthalpy as 

When we want to remove the gas in keeping the total entropy constant and the molar volume constant, we get 

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