Consider a (non-isolated) thermodynamics system at room temperature and pressure, i.e. $T=T_0,P=P_0$ which is not in equilibrium. We know
$$ \Delta S_\text{Universe}=\Delta S_\text{System}+\Delta S_\text{surrondings}\ge 0 $$
$$ \begin{aligned} \Delta S_\text{Universe}&=\Delta S-\frac{Q}{T_0}=\frac{1}{T_0}(T_0\Delta S-\Delta E-P_0\Delta V)\\&=-\frac{1}{T_0}(\Delta E-T_0\Delta S+P_0\Delta V)\ge 0 \end{aligned} $$
$$ (\Delta E-T_0 \Delta S+P_0\Delta V)=\Delta(E-T_0 S+P_0V)=\Delta A\le 0 $$
where $A$ is known as the availability of the system
During any spontaneous change of a system at constant temperature and pressure: $\Delta A\le0$
| System | $\Delta E=0, \Delta V=0$ | $\Delta T=0, \Delta V=0$ | $\Delta T=0, \Delta P=0$ |
|---|---|---|---|
| Direction of spontaneous change | $\Delta S\ge 0$ | $\Delta F \le 0$ | $\Delta G \le 0$ |
| Equilibrium achieved when | $\Delta S=0$ | $\Delta F=0$ | $\Delta G=0$ |
**Helmholtz free energy** 🔴
$$ F(T,V) $$
Gibbs free energy 🟢
$$ G(T,P) $$
Enthalpy 🔵
$$ H(P,S) $$
$$ F(T,V)=E-TS $$
$$ \begin{aligned} \text dE&=T\text dS-P\text dV \\
&=T\text dS+S\text dT-S\text dT-P\text dV \\ &=\text d(TS)-S\text dT-P\text dV \\ \Rightarrow \text d(E-TS)&=-S\text dT-P\text dV \end{aligned} $$
$$ S=-\left( \frac{\partial F}{\partial T} \right)_V \quad ; \quad P=-\left ( \frac{\partial F}{\partial V} \right)_T $$
Consider change with Constant $V$ & $T$ then:
$$ \begin{aligned} \text d(E-TS)&=\text dE-T\text dS-S\text dT \\ &=\text dE-T\text dS \text{ (cons. $T$)} \\ &=\text dE-T\text dS \le 0 \text{ (cons. $V$)} \end{aligned} $$
The Helmholtz energy $\text dF\le0$, for spontaneous processes that happen at constant temperature and volume, i.e. it is a decreasing quantity that becomes zero at equilibrium
$$ \begin{aligned} \text d(E-TS)&=\text dE-T\text dS-S\text dT \\ &=\text dE-T\text dS \text{ (cons. $T$)} \\ &=\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em}W_\text{rev} \;(T\text dS=\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em}Q_\text{rev})
\end{aligned} $$
The maximal non-volume reversible work that can be done in a constant volume and constant temperature process is given by $\text dF$