I'm trying to understand the relation: P=T (∂S/∂V)U,N. I understand the definition of temperature as: 1/T =(∂S/∂U)V,N. The relation for pressure has a similar derivation, but how can T be thrown in there? In my textbook, the author derives the relation for two systems in thermal equilibrium. Are we supposed to only compare the pressures of systems in thermal equilibrium? I'm clearly a little confused -- any help would be appreciated.