Thermodynamic Similarity of Transformed Functions

We can use this fact to check whether the variable set and the functional dependence of the variable set obey the demand that the complete Legendre transformation is zero. Only then, we have a function of the energy that obeys thermodynamic similarity. 

There is a potential pitfall in the use of the Legendre transformation of Eulerian homogeneous equations. We start from the representation of the energy [/ of a single phase in natural variables as a total differential. We use purposely a sloppy notation, as sometimes common, 

Observe that the variables T, —p, p-sne not directly proportional to the mol number n. Therefore, if all the variables of the system are scaled by a common factor, the energy is scaled by the same factor. 

We inspect now the Legendre transformation of the energy with respect to volume, the free energy, or Helmholtz energy F  

Obviously, this function behaves on scaling of entropy, volume, and mol number in the same way like the energy. If these variables of the system are scaled by a 

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