Criterion of Equilibrium and Stability: Helmholtz Function

The criterion of equilibrium and stability of a system existing at constant volume and constant temperature is given by the Helmholtz function. The Helmholtz function is a thermodynamic potential that determines the equilibrium state of a system.

1. What is the Helmholtz Function?
The Helmholtz function, denoted by A, is defined as the difference between the internal energy (U) and the product of temperature (T) and entropy (S) of a system. Mathematically, it can be expressed as:

A = U - TS

2. Constant Volume and Constant Temperature System
For a constant volume and constant temperature system, the change in internal energy (ΔU) is zero. This implies that the change in Helmholtz function (ΔA) is also zero. Therefore, the equilibrium condition for such a system is given by:

ΔA = 0

3. Criterion of Equilibrium and Stability
The criterion of equilibrium and stability is determined by the behavior of the Helmholtz function. The Helmholtz function is a convex function of extensive variables, such as volume (V) and mole numbers (N). This means that the second derivative of the Helmholtz function with respect to these variables is positive:

∂²A/∂V² > 0
∂²A/∂N² > 0

4. Equilibrium Condition
At equilibrium, the Helmholtz function is at a minimum with respect to extensive variables. This means that the first-order partial derivatives of the Helmholtz function with respect to these variables are zero:

∂A/∂V = 0
∂A/∂N = 0

5. Stability Condition
The stability of the system is determined by the behavior of the second-order partial derivatives of the Helmholtz function. If the second derivatives are positive, the system is stable. If they are negative, the system is unstable and can undergo spontaneous changes.

6. Conclusion
In a system existing at constant volume and constant temperature, the criterion of equilibrium and stability is determined by the Helmholtz function. The equilibrium condition is given by ΔA = 0, and the stability condition is determined by the positive second derivatives of the Helmholtz function. Therefore, the correct option is 'C' - Helmholtz function.