A more useful thermodynamic potential

Now we have two parameters, S y and Ug y, that will tell us which way processes will go, but they refer to processes which virtually never occur, except perhaps in classroom exercises - that is, processes which occur at constant values of U and V, or of S and V, our two constraints. We need a parameter which will refer to processes at constant T and P, our most common case.  

It is usual to speak of processes occurring at constant U and V, or constant T and P. It would be more accurate to speak of processes having the same values of U and V, or of T and P, before and after the process. It doesn t really matter what the system does between the two states that is, the system need not be at constant T and P during the process. 

Thus our definition of the second law has led to a function, G, which will always decrease to a minimum in spontaneous processes in systems having specified values of T and P. It is an extremely useful thermodynamic potential. All we have to do is to find a way to get measurable values of this function for all pure compounds and solutes, and to find how they change with T, P, and concentration, and we will then be able to predict the equilibrium configuration of any system by minimizing G. 

To see how G changes with T and P is fairly simple. To see how it changes with composition is a little more difficult (see 7.6 8.2). Combining (4.9) and (4.36), we find another fundamental equation,  

Because (4.40) could also be written as a total differential. 

---