Enthalpy, Helmholtz Function, Gibbs Function
Definition
The enthalpy $H$ is defined as follows.
$$ H := U + PV $$
The helmholtz function $F$ is defined as follows.
$$ F := U - TS $$
The gibbs function $G$ is defined as follows.
$$ G := H - TS $$
Explanation
Enthalpy is a function whose name is nearly as well known as entropy, but the field that treats it as important is mainly chemistry. For physics, rather than struggling desperately to grasp its physical meaning, it is perfectly fine to regard these three as a set and view them as little more than algebraic manipulation. If you also consider the expression for internal energy derived from the first law of thermodynamics, you can feel a formulaic comfort.
- Internal energy: $dU = T dS - p dV$
- Enthalpy: $dH = T dS + V dp$
- Helmholtz function $dF = -S dT - p dV$
- Gibbs function: $dG = -S dT + V dp$
They boast a beauty reminiscent of the complex representations of trigonometric and hyperbolic functions. As for the derivation, once you derive the enthalpy the way you derive the internal energy, the rest follows naturally, so it is not difficult at all.