This result seems to contradict the equation d F = − S d T − P d V, as keeping T and V constant seems to imply d F = 0, and hence F = constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation d F = − S d T − P d V is restricted, no process can occur at constant T and V, since there is a unique P (T, V) relation, and thus T, V, and P are all fixed. To allow for spontaneous processes at constant T and V, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers N j of particles of each type j. The differential of the free energy then generalizes to