3.6 Enthalpy and the thermodynamic potentials

The internal energy $U$ is the natural energy when entropy and volume are the variables under control. But experiments rarely hold $S$ and $V$ fixed — far more often it is temperature and pressure. For each such choice there is a different energy function, a thermodynamic potential, built from $U$ so that its natural variables are the ones actually held fixed.

Enthalpy and flow work

The first potential beyond $U$ is the enthalpy

H \;\equiv\; U + pV.

Its differential, using $dU = T\,dS - p\,dV$ , is

dH \;=\; T\,dS + V\,dp,

so $H$ is the natural energy for processes at constant pressure: at fixed $p$ the heat added equals $\Delta H$ , which is why the constant-pressure heat capacity was defined through it in 3.3. Enthalpy also carries the energy bookkeeping for flow. A fluid pushed across a boundary does work $pV$ against the pressure there; folding that flow work into the energy gives, for a steady open stream,

h_1 + \tfrac12 u_1^2 + g z_1 \;=\; h_2 + \tfrac12 u_2^2 + g z_2,

with $h = u + p/\rho$ the specific enthalpy. This is energy conservation along a streamline with the flow work included; Bernoulli’s equation is its incompressible limit, and the energy condition across a shock front is another instance.

The family of potentials

Each potential is a Legendre transform of $U$ that swaps a “stiff” variable ( $S$ or $V$ ) for the conjugate “soft” one ( $T$ or $p$ ):

| Potential | Definition | Differential | Natural variables | Minimised at fixed | |---|---|---|---|---| | Internal energy $U$ | — | $dU = T\,dS - p\,dV$ | $S, V$ | $S, V$ | | Enthalpy $H$ | $U + pV$ | $dH = T\,dS + V\,dp$ | $S, p$ | $S, p$ | | Helmholtz free energy $F$ | $U - TS$ | $dF = -S\,dT - p\,dV$ | $T, V$ | $T, V$ | | Gibbs free energy $G$ | $H - TS$ | $dG = -S\,dT + V\,dp$ | $T, p$ | $T, p$ |

The right-hand column is the key to equilibrium. The second law says the entropy of system-plus-surroundings can only increase; carried over to the system alone at fixed external conditions, that becomes a minimisation: at fixed temperature and volume a system minimises $F$ , and at fixed temperature and pressure it minimises $G$ . Which potential to watch is fixed entirely by which variables the surroundings hold constant.

▶ A Maxwell relation from a potential Derivation

Because each potential is a state function, its mixed second partial derivatives are equal regardless of order. Apply this to $F(T,V)$ , whose first derivatives are $-S = (\partial F/\partial T)_V$ and $-p = (\partial F/\partial V)_T$ :

\frac{\partial}{\partial V}\!\left(\frac{\partial F}{\partial T}\right) \;=\; \frac{\partial}{\partial T}\!\left(\frac{\partial F}{\partial V}\right) \quad\Longrightarrow\quad \left(\frac{\partial S}{\partial V}\right)_T \;=\; \left(\frac{\partial p}{\partial T}\right)_V.

This is one of the four Maxwell relations. Each potential yields one, and together they connect quantities that are hard to measure (how entropy varies with volume) to ones that are easy (how pressure varies with temperature). ✓

The Helmholtz and Gibbs free energies are the working tools of Chapter 4, where minimising them over the state of a system selects the equilibrium phase and locates the boundaries between solid, liquid, and vapour.