- #1

A Dhingra

- 211

- 1

This is a section from Callen, Herbert B - Thermodynamics and an Introduction to Thermostatistics

"Any equilibrium state can be characterized either as a state of maximum entropy for given energy or as a state of minimum energy for given entropy. But these two criteria nevertheless suggest two different ways of attaining equilibrium. Let us consider a piston originally fixed at some point in a closed cylinder. We are interested in bringing the system to equilibrium without the constraint on the position of the piston. We can simply remove the constraint and allow the equilibrium to establish itself spontaneously. Here the entropy increases and the energy is maintained constant by the closure condition. This process is suggested by the maximum entropy principle. Alternatively, we can permit the piston to move very slowly, reversibly doing work on an external agent until it has moved to the position that equalises the pressure on the two sides. During this process energy is withdrawn from the system and entropy remains a constant (the process is reversible and no heat flows).This is the process suggested by the minimum energy principle.

Independent of whether the equilibrium is brought about by either of these two processes, or by any other process, the final equilibrium state in each case satisfies both extrema conditions."

According to the two processes described in the text,

the first process refers to this till the equilibrium state is reached: (∂S/∂V)

_{T}

and the second process refers to (∂P/∂T)

_{V}

And according to the Maxwell's relations (the Helmholtz free energy minimized ) these two are equal. But according to the last statement : both of these satisfy the extremal condition, but need not be necessarily equal.

So please tell me what is to be added in the theory so that the equivalence seems visible for a process. (i am not asking for a mathematical proof, the one based on the definition of exact differentials)

Or someone please help me visualize that these two processes are actually same (also in what respect as per the Maxwell's relations?)

Thank you for any help

(I was confused in this section hence had to put it here, but i had no intention of copyright violation in any way)