Explanation:

In order to determine the spontaneity of a process, we need to consider the changes in various thermodynamic properties such as entropy (S), enthalpy (H), Helmholtz free energy (A), and Gibbs free energy (G).

Constant Volume and Temperature:
The condition of constant volume and temperature implies that the system is not performing any work other than expansion work (if any). This means that the only type of work involved is the work done by the system due to the expansion of gases against an external pressure.

Spontaneity:
A spontaneous process is one that occurs naturally without any external intervention. For a spontaneous process, the total entropy of the system and its surroundings always increases.

Change in Entropy:
Entropy is a measure of the randomness or disorder in a system. If the change in entropy (ΔS) is positive, it indicates an increase in the randomness or disorder of the system, which is favored for spontaneous processes. Therefore, option (a) is a valid criterion for spontaneity.

Change in Enthalpy:
Enthalpy is the heat energy absorbed or released during a process. If the change in enthalpy (ΔH) is negative, it indicates that the process is exothermic and releases heat to the surroundings. While this criterion is important for determining the heat flow of a process, it does not directly indicate the spontaneity of the process. Hence, option (b) is not a valid criterion for spontaneity.

Change in Helmholtz Free Energy:
Helmholtz free energy (A) is a thermodynamic potential that combines the effects of both enthalpy and entropy. It is defined as A = U - TS, where U is the internal energy of the system, T is the temperature, and S is the entropy. For a process at constant volume and temperature, the change in Helmholtz free energy (ΔA) is related to the change in entropy (ΔS) by the equation ΔA = ΔU - TΔS. If ΔA is negative, it indicates that the process is spontaneous. Therefore, option (c) is a valid criterion for spontaneity.

Change in Gibbs Free Energy:
Gibbs free energy (G) is another thermodynamic potential that takes into account both enthalpy and entropy. It is defined as G = H - TS, where H is the enthalpy, T is the temperature, and S is the entropy. For a process at constant volume and temperature, the change in Gibbs free energy (ΔG) is related to the change in enthalpy (ΔH) and entropy (ΔS) by the equation ΔG = ΔH - TΔS. If ΔG is negative, it indicates that the process is spontaneous. Therefore, option (d) is also a valid criterion for spontaneity.

Conclusion:
Among the given options, the correct criterion for spontaneity of a process in a system under constant volume and temperature and in the absence of any work other than expansion work is the change in Helmholtz free energy (option c). However, it is important to note that the change in entropy (option a) and the change in Gibbs free energy