Gibbs free energy

The Gibbs free energy is $G=E-TS+pV$. For a simple system of fixed composition,

For a single-phase ideal gas mixture with molar amounts $N_i$, common temperature $T$, and pressure $P$, extensivity and ideal mixing giveHere $N={\sum }_i{N}_i$, $x_i$ is the mole fraction, $\mathcal{R}$ is the molar gas constant, and $P^{\circ }$ is the standard pressure. Integrating $\partial {\mu }_i/\partial P=\mathcal{R}T/P$ and including the ideal mixing entropy proves ${\mu }_i={\mu }_i^{\circ }+\mathcal{R}T\log (x_{i}P/P^{\circ })$. Reactions can change $N$, so fractions alone do not set the extensive Gibbs free energy. Its composition Hessian is $\mathcal{R}T({\delta }_{ij}/N_i-1/N)$; it is positive semidefinite by the weighted Cauchy-Schwarz inequality.

Equality of mixed derivatives of $dG=-SdT+VdP$ gives

For a reaction with stoichiometric changes ${\nu }_i$, varying the reaction extent at fixed temperature and pressure givesEquilibrium requires ${\sum }_i{\nu }_i{\mu }_i=0$.

Thermochemical equilibrium requires every chemical reaction to have zero Gibbs free-energy change. For ideal reactants, the equilibrium constant is $K=\exp [-{\Delta }_r{G}^{\circ }/(RT)]$ after standard-state activity factors are included.

At fixed temperature and pressure, a closed reacting system reaches thermochemical equilibrium by minimizing Gibbs free energy over nonnegative species amounts subject to conserved elemental abundances. For an allowed stoichiometric vector $\nu$, an interior minimum has ${\sum }_i{\nu }_i{\mu }_i=0$. More generally the chemical potential vector lies in the row span of the elemental conservation matrix at positive concentrations, with the appropriate one-sided inequalities for absent species. The first law of thermodynamics gives the thermodynamic differential, while the Second law of thermodynamics supplies the minimization direction.

For an elementary reversible reaction, detailed balance gives $k_f{\prod }_r{n}_r^{{\nu }_r}=k_r{\prod }_p{n}_p^{{\nu }_p}$ at equilibrium. Hence the reverse rate coefficient is the forward coefficient divided by the equilibrium constant, with units fixed by the reaction stoichiometry.

Two phases coexist where their molar Gibbs free energies agree. Differentiating that equality along the coexistence curve relates its slope to the entropy and volume jumps.

At fixed temperature, choose a coexistence pressure $p_{\ast }$ such that the gas and liquid have equal chemical potentials. Because $d\mu =vdp$, this gives ${\int }_{v_l}^{v_g}vdp=0$, or equivalently ${\int }_{v_l}^{v_g}[p(v,T)-p_{\ast }]dv=0$. The equal areas replace the unstable part of an isotherm by a horizontal coexistence segment.

A phase diagram partitions a space of thermodynamic variables into regions occupied by different equilibrium phases and records the coexistence curves between them.

The lever rule finds phase fractions from a horizontal tie line on a phase diagram. For solid and liquid concentrations $C_s,C_l$ and bulk concentration $\overline{C}$, conservation gives $\overline{C}=(1-\varphi )C_s+\varphi C_l$. Thus $\varphi =(\overline{C}-C_s)/(C_l-C_s)$ is the liquid fraction, provided $C_s\le \overline{C}\le C_l$. For pure solid, $\varphi =\overline{C}/C_l$.

A bicritical point is where two continuous transition lines and a first-order line meet, typically separating a disordered phase from two competing ordered phases.

The liquidus is the locus in a phase diagram above which the equilibrium state is entirely liquid. In a dilute binary solution it is often approximated locally by a linear relation such as $T_L=T_0-mC$.

Freezing-point depression is the lowering of a solvent's equilibrium freezing temperature by dissolved solute. In the linear dilute approximation, $T_L=T_0-mC$ with $m>0$.

An eutectic system has a composition and temperature at which a liquid is in equilibrium with two distinct solid phases. Cooling liquid of that composition through the eutectic temperature produces both solid phases together.

The eutectic composition is the liquid concentration at the eutectic temperature, where liquid and two solid phases coexist. It is a composition coordinate on a phase diagram, not the overall concentration of every sample in an eutectic system.

The eutectic temperature is the temperature of simultaneous liquid–two-solid phase coexistence in an eutectic system. In directional freezing of a solution away from the eutectic composition, a mushy layer can end at this temperature, where its remaining liquid solidifies.

At a first-order phase transition, the Gibbs free energy is continuous while at least one first derivative, such as entropy or volume, jumps. A nonzero entropy jump produces latent heat $L=T\Delta S$.

A smooth scalar Landau free energy can end a two-phase phase coexistence curve by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic Landau free energy; the order-parameter jump then vanishes and the magnetic susceptibility diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.

Classical nucleation theory balances the positive surface free energy of a droplet against its negative bulk free-energy gain. In three dimensions,has critical radius $R^{\ast }=2\sigma /\Delta$ and barrier $F^{\ast }=16\pi {\sigma }^3/(3{\Delta }^2)$.

A critical nucleus is the unstable droplet at the maximum of the nucleation free energy. Smaller droplets shrink, while larger droplets lower their free energy by growing.

Thermal crossing of a nucleation barrier $F^{\ast }$ has rate proportional to $e^{-F^{\ast }/(k_{B}T)}$ and mean waiting time proportional to $e^{F^{\ast }/(k_{B}T)}$, up to a kinetic prefactor.

The latent heat from phase $\alpha$ to phase $\beta$ isper mole or per particle, according to the normalization used.

The Stefan number compares sensible heat $c_p\Delta T$ with latent heat $L$ per unit mass. Some solidification literature uses its reciprocal as the Stefan parameter, so the defining formula must accompany the convention.

The latent-to-sensible heat ratio is the reciprocal of the Stefan number defined by $\mathrm{Ste}=c_p\Delta T/L$. Large $S$ means that phase change uses much more energy than changing temperature by $\Delta T$.

Along a first-order coexistence curve,At a critical point the two phases merge, their entropy jump vanishes, and the latent heat tends to zero.