Gibbs free energy

Definition and physical meaning

For a thermodynamic system with internal energy $U$, pressure $p$, volume $V$, temperature $T$, and entropy $S$, the Gibbs free energy is the state function

Equivalently, it is $G = H - TS$ in terms of enthalpy $H$. It is the natural potential when a system can exchange heat and volume with its environment so that $T$ and $p$ are effectively controlled (common in laboratory and chemical settings).

Differential form and natural variables

For a simple compressible single-component system,

where $\mu$ is the chemical potential and $N$ the particle number .

Thus $G$ is naturally a function of $(T,p,N)$, and it generates the key derivatives:

• $S = -\left(\partial G/\partial T\right)_{p,N}$,
• $V = \left(\partial G/\partial p\right)_{T,N}$,
• $\mu = \left(\partial G/\partial N\right)_{T,p}$.

For multicomponent systems, the last relation generalizes to $\mu_i = (\partial G/\partial N_i)_{T,p,N_{j\neq i}}$.

Spontaneity and maximum non-expansion work

At fixed $(T,p,N)$, the second law implies that a spontaneous change satisfies

with equality at thermodynamic equilibrium .

With the standard sign convention for work , the maximum work obtainable from the system other than $p\,dV$ expansion work in a process at constant $T$ and $p$ is

again achieved in a reversible process .

Extensivity link: $G$ and the chemical potential

If the system satisfies the extensivity postulate so that the fundamental relation is a homogeneous function of degree one , the Euler relation gives (single component)

Substituting into the definition of $G$ yields

highlighting that $G$ is the extensive quantity conjugate to the intensive $\mu$.

Consistency among intensive variables is then encoded by the Gibbs–Duhem relation , which constrains how $T$, $p$, and $\mu$ can vary for a single-component system.