Changes in Chemical Potential and Pressure

In the train of thoughts above we have mingled the concept of energy as a function of the variables as such and the constraint of fixed volume. The constraint is not a general constraint in that way that it is an inherent property of the substance itself. Thus, we could imagine that the substance can be placed in a container in that the volume can be varied to some extent. In that situation the partial derivative to the volume is associated with an admissible real physical process. 

In the case of fixing the volume we face the situation of a virtual process. Even when the volume is fixed, basically we could change the volume. However, we must have some previous knowledge on how the function p(5, V, n) looks like. There is one way to get the fundamental equation U(S, V, n) purely via theoretical considerations. But in order to measure C/(5, y, n), variations of the respective extensive variables are necessary. 

the mathematics of thermodynamics suggests that an intensive variable such as the pressure is only experimentally accessible via a physical variation of its associated extensive variable, here the volume V. This means that we cannot construct a pressure sensor that is completely stiff in its volume. 

We can express this situation also from the view of the system itself. Any measurement of a property of the system would change the state of the system itself, due to the variation of a certain extensive variable. Of course, in practice the change of the state of the system due to measurement may be negligible. 

The chemical potential of a one-component system depends on temperature and pressure. If two vessels (0 and ( ) are connected via a capillary, a flow would occur, if the chemical potentials are not equal, p, p . We start from a condition where the chemical potentials are equal. For this reason, also the pressures and temperatures are equal  

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