Question

Statistical Mechanics and Thermodynamics of Simple Systems

We know that the total energy U and the pressure P are identically the same for an assembly of distinguishable particles as for molecules of the classical ideal gas while S is different. Please explain why this makes sense. All you have to do is write in words and explain why it makes physical sense using heuristic reasoning or your physicist's intuition, that's all.

I worked out the total energy U and the pressure P and those two thermodynamic quantities are the same for both cases. But why? I know the N! factor goes away when taking the partial derivatives while holding N fixed, but what is the physical explanation for why two seemingly different cases actually have the same total energy U and pressure P?

We are working under the assumption that the independent-particle approximation holds, and the total internal energy of N particles (distinguishable or indistinguishable) is equal to N times the average energy of a single particle. So, why would this be the case?

Answer 1

IT IS BECAUSE OF GIBBS PARADOX

According to the mathematical definition of the word function, the internal energy function of a system can be of two different types. The first one depends on a single variable: temperature; the second depends on two variables: pressure and volume. The internal energy function depends on the process performed on the system (nevertheless, both types of the function are quantitatively identical). The energy function which depends only on temperature is used in processes where the volume of the system is changed not by compression or stretching; and the energy function which depends on the volume and pressure is used in processes where the volume of the system is changed by positive or negative compression. In the GIBBS PARADOX, the former case is used and the energy function of the first type must be applied. It was shown that in the former case, the system must be described by the momentum space. In the latter case, the system must be described by the traditional phase space. This explains the Gibbs paradox in statistical mechanics. The entropy is always extensive. It also can be described by two different functions which depend on the process performed on the system. The first function depends only on temperature, and the second depends on volume as well.

PLEASE LIKE. WE NEED IT.