Properties of the Helmholtz free energy

Consider the special case (a) that the only heat transferred to the system is from a heat reservoir which remains at the constant temperature T (5) that the initial and final temperatures, and Tg of the system are equal, and are equal to the temperature T of the reservoir. 

From (1 16) we have for the heat taken in by the system in terms of its entropy change  

It may be noted that this relation remains valid even if the system passes through intermediate temperatures different from jT (= 2), provided that the only heat reservoir which is affected is also at In relation (2 15) the inequality sign refers to an irreversible process and the equality sign to a reversible one. Between the assigned initial and final states the value of A2—A1 is, of course, the same whether the path is a reversible one or not, since is a function of state. Relation (2 16) may thus be interpreted as follows. The work done by a system during a process in which the initial and final temperatures and the temperature of the heat reservoir are all equal, is either less than or equal to the decrease in A. Thus, for such a process when carried out reversibly. 

The reason why A has been called a free energy may be seen as follows. Equation (2 11) may be written 

Another important characteristic of A is that it provides a criterion of equilibrium. For an infinitesimal change at constant temperature we have from (2 15) 

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