Equilibrium Conditions for a System of Fixed Volume in Contact with a Heat Reservoir: Helmholtz Free Energy

Consider a closed thermodynamic system \(A\) in thermal contact with a large heat reservoir \(A'\) maintained at a constant temperature \(T_0\).

The system is enclosed within rigid walls, so its volume remains constant. Only heat is exchanged between the system and the reservoir, and no external work is performed.

Let the system absorb an amount of heat \(Q\) from the reservoir.

Since the reservoir loses the same amount of heat, its entropy change is

\[ \boxed{ \Delta S'=-\frac{Q}{T_0} } \tag{1} \]

From the first law of thermodynamics,

\[ \Delta U=Q-W. \]

At constant volume, assuming only pressure–volume work,

\[ W=\int P_{\mathrm{ext}}\,dV=0. \]

Therefore,

\[ \boxed{Q=\Delta U} \tag{2} \]

Substituting equation (2) into equation (1),

\[ \boxed{ \Delta S'=-\frac{\Delta U}{T_0} } \tag{3} \]

The combined system consisting of \(A\) and the heat reservoir \(A'\) is thermally isolated from the surroundings.

Hence, the total entropy change is

\[ \Delta S_{\mathrm{total}} = \Delta S+\Delta S'. \]

Using equation (3),

\[ \Delta S_{\mathrm{total}} = \Delta S-\frac{\Delta U}{T_0}. \]

Taking \(T_0\) as constant,

\[ \Delta S_{\mathrm{total}} = -\frac{\Delta U-T_0\Delta S}{T_0}. \]

Therefore,

\[ \boxed{ \Delta S_{\mathrm{total}} = -\frac{\Delta(U-T_0S)}{T_0} } \tag{4} \]

Definition of Helmholtz Free Energy

A thermodynamic state function called the Helmholtz free energy is defined as

\[ \boxed{F=U-TS} \tag{5} \]

where \(U\) is the internal energy, \(T\) is the absolute temperature, and \(S\) is the entropy.

For initial and final equilibrium states at the reservoir temperature \(T_0\), equation (4) becomes

\[ \boxed{ \Delta S_{\mathrm{total}} = -\frac{\Delta F}{T_0} } \tag{6} \]

Condition for Thermodynamic Equilibrium

According to the second law of thermodynamics, the total entropy of an isolated system cannot decrease during a spontaneous process:

\[ \Delta S_{\mathrm{total}}\geq0. \]

Using equation (6),

\[ -\frac{\Delta F}{T_0}\geq0. \]

Since \(T_0>0\),

\[ \boxed{\Delta F\leq0} \tag{7} \]

Thus, for a closed system maintained at constant temperature and volume, the Helmholtz free energy decreases during a spontaneous process.

At stable thermodynamic equilibrium, the Helmholtz free energy attains a minimum with respect to the allowed variations:

\[ \boxed{ F=\text{Minimum at constant }T,V,N } \]

Here \(N\) denotes the fixed amount of substance. For a reversible equilibrium change, \(\Delta F=0\) under the stated constraints.

Thermodynamic Differential of Helmholtz Free Energy

From the definition,

\[ F=U-TS. \]

Differentiating,

\[ dF=dU-T\,dS-S\,dT. \]

For a simple compressible closed system of fixed composition undergoing a reversible change,

\[ dU=T\,dS-P\,dV. \]

Substituting,

\[ dF=T\,dS-P\,dV-T\,dS-S\,dT. \]

Therefore,

\[ \boxed{ dF=-S\,dT-P\,dV } \tag{8} \]

Consequently,

\[ \boxed{ S=-\left(\frac{\partial F}{\partial T}\right)_V } \]

and

\[ \boxed{ P=-\left(\frac{\partial F}{\partial V}\right)_T } \]

Relation Between Helmholtz and Gibbs Free Energies

The Gibbs free energy is defined as

\[ G=U+PV-TS. \]

Since

\[ F=U-TS, \]

we obtain

\[ \boxed{G=F+PV} \tag{9} \]

Helmholtz Free Energy and Maximum Work

For a reversible isothermal change of a closed system with fixed composition,

\[ dF=-P\,dV. \]

The reversible pressure–volume work done by the system is

\[ \delta W_{\mathrm{rev}}=P\,dV. \]

Hence,

\[ \delta W_{\mathrm{rev}}=-dF. \]

Integrating between equilibrium states 1 and 2,

\[ \boxed{ W_{\mathrm{rev}}=F_1-F_2 } \tag{10} \]

For an ideal gas undergoing reversible isothermal expansion,

\[ P=\frac{nRT}{V}. \]

Therefore,

\[ W_{\mathrm{rev}} = \int_{V_1}^{V_2}P\,dV = nRT\int_{V_1}^{V_2}\frac{dV}{V}. \]

Integrating,

\[ W_{\mathrm{rev}} = nRT\left[\ln V\right]_{V_1}^{V_2}. \]

Thus,

\[ \boxed{ W_{\mathrm{rev}} = nRT\ln\left(\frac{V_2}{V_1}\right) } \]

Consequently,

\[ \boxed{ F_1-F_2 = nRT\ln\left(\frac{V_2}{V_1}\right) } \tag{11} \]

The decrease in Helmholtz free energy represents the maximum total work obtainable from a system in an isothermal process under the appropriate constraints.