What is the criterion for equilibrium at constant temperature and volume?

The criterion for equilibrium at constant temperature and volume is expressed by the condition that the change in the Helmholtz free energy, denoted as ( dA_{T,V} ), is equal to zero. This reflects the fundamental thermodynamic principle that at equilibrium, the system's free energy is at a minimum.

In more detail, the Helmholtz free energy ( A ) is a thermodynamic potential that is particularly useful in systems that are held at constant temperature and volume. When a system reaches equilibrium, no spontaneous changes occur, and thus the differential change in Helmholtz free energy must be zero. This means that any reversible process occurring in the system cannot lead to a further decrease in ( A ), and the system is essentially in a state of balance.

This concept aligns well with the broader principles of thermodynamics, where equilibrium states are characterized by stationary values of thermodynamic potentials. It's also worth noting that the other choices reflect different aspects of thermodynamic concepts: the change in Gibbs free energy or specific relationships in thermodynamic cycles, but they do not specifically denote the criterion for equilibrium under the specified conditions of constant temperature and volume.