Gibbs Free Energy

Gibbs free energy $G$ is the thermodynamic potential that determines whether a process is spontaneous at constant temperature and pressure:

$G = H - TS$ $\Delta G = \Delta H - T\Delta S$

Spontaneity Criterion

Notice that a process can be spontaneous even if $\Delta H > 0$ (endothermic), provided the entropy gain $T\Delta S$ is large enough.

Relation to the Equilibrium Constant

At standard conditions, the standard Gibbs energy change $\Delta G^\circ$ is related to the equilibrium constant $K$ by:

$\Delta G^\circ = -RT\ln K \implies K = e^{-\Delta G^\circ/(RT)}$

A large negative $\Delta G^\circ$ gives $K \gg 1$ (products strongly favoured).

Helmholtz Free Energy

At constant volume (rather than pressure), the relevant potential is the Helmholtz free energy:

$\Delta A = \Delta U - T\Delta S$

Your Task

Implement the three functions below.