Affinity and characteristic functions

The variation of internal energy is, according to the first law of thermod5rnamics  

If all the exchanges are reversible, apart from the transformation under study (we then say that the system is at physical equilibrium), then the woik 

The sum that appears in the above expression is extended to all couples of conjugate variables, with the exception of the temperature-entropy couple (which is why the index k begins at the value of 2). By substituting this back into expression [1.13] and taking account of relation [1.12], we obtain  

This gives us a new expression of the affinity, which is therefore the opposite of the differential of the function U in relation to the fractional extent with constant entropy and extensive variables  

We can generalize this expression for any characteristic function F, defined by  

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