Enthalpy, Helmholtz Free Energy, Gibbs Free Energy
Definition
Enthalpy $H$ is defined as follows:
$$ H := U + PV $$
Helmholtz function $F$ is defined as follows:
$$ F := U - TS $$
Gibbs function $G$ is defined as follows:
$$ G := H - TS $$
Explanation
Enthalpy is a function almost as well-known as entropy, but its main area of importance is chemistry. For physics, it is advisable to view these three together as a set for algebraic manipulation rather than striving too hard to understand their physical significance. Considering the expression for internal energy derived from the first law of thermodynamics, one can feel a mathematical comfort.
- Internal energy: $dU = T dS - p dV$
- Enthalpy: $dH = T dS + V dp$
- Helmholtz function $dF = -S dT - p dV$
- Gibbs function: $dG = -S dT + V dp$
It boasts a beauty that reminds one of the complex representation of trigonometric and hyperbolic functions. The derivation is not at all difficult, proceeding naturally from deriving the internal energy to the enthalpy and beyond.