Definitions The statement The definitions given above are mathematically equivalent appears to me to be meaningless The three possible definitions correspond to three different ways of defining partial molar free energy There is no way in which the three definitions can be considered to be equivalent as there is no way in which Gibbs free energy G Helmholtz free energy H and internal energy U are equivalent The second and third definitions are of little practical value because they apply at constant entropy S a condition which is virtually impossible to achieve in practice I have included them for the sake of completeness because they can be found in texts on thermodynamics