10.3 Chemical Potential and Phase Stability Criteria

Chemical potential describes how a substance's Gibbs free energy changes as material moves between phases or is added to a mixture. It's the central quantity you need for determining whether a phase is stable, metastable, or unstable, and it connects directly to fugacity-based equilibrium calculations used in engineering practice.

Chemical potential of a component

Definition and interpretation

Chemical potential ($\mu$) is defined as the partial molar Gibbs free energy of a component at constant temperature, pressure, and composition of all other components:

$\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T, P, n_{j \neq i}}$

This tells you how much the total Gibbs energy of the system changes when you add an infinitesimal amount of component $i$. It governs the direction of every spontaneous process: mass transfer between phases, chemical reactions, and mixing all proceed in the direction that lowers the total chemical potential.

For a pure substance, the chemical potential simplifies to the molar Gibbs free energy at that temperature and pressure: $\mu = g(T, P)$.

Factors influencing chemical potential

Chemical potential depends on temperature, pressure, and composition, as well as intermolecular interactions within the mixture.

For an ideal solution, the chemical potential of component $i$ is:

$\mu_i = \mu_i^{\circ}(T, P) + RT \ln(x_i)$

where $\mu_i^{\circ}$ is the chemical potential of pure $i$ at the same $T$ and $P$, $R$ is the gas constant, $T$ is absolute temperature, and $x_i$ is the mole fraction.

Because $x_i \leq 1$, the $\ln(x_i)$ term is always negative or zero. This means mixing always lowers the chemical potential of each component relative to its pure-component value in an ideal solution.

Example: In a binary ideal solution of ethanol and water, increasing the mole fraction of ethanol raises its chemical potential (the $\ln(x_{\text{ethanol}})$ term becomes less negative) while simultaneously lowering the mole fraction of water and thus lowering water's chemical potential.

Conditions for phase stability

Phase stability and metastability

• A phase is stable when its chemical potential is the lowest of all possible phases (or phase combinations) at the given $T$, $P$, and overall composition.
• A phase is metastable when another phase has a lower chemical potential, but an activation energy barrier prevents the transformation from occurring on any practical timescale.
• A phase is unstable when any infinitesimal fluctuation in composition or density will cause it to spontaneously separate or transform.

Example: Diamond at ambient conditions has a higher chemical potential than graphite, making it thermodynamically metastable. The carbon-carbon bond rearrangement required to reach the graphite structure has such a large activation barrier that the transformation is imperceptibly slow.

Derivation and application of stability conditions

Consider transferring a small amount $dn$ of a component from phase $\alpha$ to phase $\beta$ at constant $T$ and $P$. The resulting change in total Gibbs energy is:

$dG = (\mu_{\beta} - \mu_{\alpha})\, dn$

Three cases follow:

1. $\mu_{\alpha} < \mu_{\beta}$: Transferring material from $\alpha$ to $\beta$ raises $G$, so phase $\alpha$ is stable against this transfer.
2. $\mu_{\alpha} = \mu_{\beta}$: No driving force for transfer in either direction. This is the equilibrium condition.
3. $\mu_{\alpha} > \mu_{\beta}$: Material spontaneously moves from $\alpha$ to $\beta$ because doing so lowers $G$. Phase $\alpha$ is either metastable (if kinetically hindered) or unstable.

Example: Liquid water and water vapor coexist at 100 °C and 1 atm because $\mu_{\text{liquid}} = \mu_{\text{vapor}}$ at those conditions. Raise the temperature slightly and $\mu_{\text{liquid}} > \mu_{\text{vapor}}$, so vaporization becomes spontaneous.

Phase stability and Gibbs free energy

Gibbs free energy and phase stability

The Gibbs free energy determines phase stability at constant $T$ and $P$:

$G = H - TS$

At any given conditions, the system adopts whichever phase (or combination of phases) minimizes the total $G$. At low temperatures the enthalpy term $H$ dominates, favoring ordered phases (solids). At high temperatures the $-TS$ term dominates, favoring disordered phases (liquids, gases) with high entropy.

Derivatives of Gibbs free energy and stability analysis

The connection between $G$ and stability becomes precise through its derivatives with respect to composition.

• First derivative: $\left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}} = \mu_i$. This gives the chemical potential, which determines the direction of mass transfer.
• Second derivative: $\left(\frac{\partial^2 G}{\partial x_i^2}\right)_{T,P}$ determines the curvature of the Gibbs energy surface with respect to composition:
  • Positive curvature → the phase is stable against compositional fluctuations
  • Negative curvature → the phase is unstable (inside the spinodal region) and will spontaneously demix
  • Zero curvature → marks the spinodal boundary; at a critical point, both the second and third derivatives are zero

The Gibbs phase rule connects the degrees of freedom $F$ to the number of components $C$ and phases $P$:

$F = C - P + 2$

This tells you how many intensive variables ($T$, $P$, composition) you can independently vary while maintaining the same number of coexisting phases. For pure water ($C = 1$) at a two-phase boundary ($P = 2$), $F = 1$: specifying temperature fixes the equilibrium pressure.

Chemical potential vs. fugacity in equilibrium

Fugacity and its relationship to chemical potential

Fugacity ($f$) is an "effective pressure" that corrects for non-ideal behavior. It replaces pressure in the ideal-gas chemical potential expression so that the same functional form works for real gases, liquids, and mixtures.

The general relation is:

$\mu_i = \mu_i^{\circ}(T) + RT \ln\!\left(\frac{f_i}{f_i^{\circ}}\right)$

where $f_i^{\circ}$ is the fugacity in the chosen standard state (typically 1 bar for gases).

For an ideal gas, fugacity equals partial pressure ($f_i = P_i = y_i P$), and the expression reduces to:

$\mu_i = \mu_i^{\circ}(T) + RT \ln\!\left(\frac{P_i}{P^{\circ}}\right)$

The practical advantage of fugacity is that it converts the abstract equality of chemical potentials into a quantity you can calculate from equations of state or experimental data.

Phase equilibrium and fugacity equality

Because equal chemical potentials imply equal fugacities, the equilibrium condition becomes:

$f_i^{\alpha} = f_i^{\beta} = \cdots \quad \text{for each component } i$

The fugacity coefficient quantifies deviation from ideal-gas behavior:

$\phi_i = \frac{f_i}{y_i P}$

For an ideal gas, $\phi_i = 1$. For real gases at moderate to high pressures, $\phi_i$ deviates from unity and must be calculated from an equation of state (e.g., Peng-Robinson, SRK).

For vapor-liquid equilibrium (VLE), the fugacity equality for component $i$ gives:

$y_i \, \phi_i \, P = x_i \, \gamma_i \, P_i^{\text{sat}}$

where:

• $y_i$, $x_i$ = vapor and liquid mole fractions
• $\phi_i$ = vapor-phase fugacity coefficient
• $\gamma_i$ = liquid-phase activity coefficient (accounts for liquid non-ideality)
• $P_i^{\text{sat}}$ = saturation (vapor) pressure of pure component $i$

Note: this form already incorporates the Poynting correction being negligible at low to moderate pressures. At high pressures, an additional exponential correction factor is included.

Example: For a hydrocarbon mixture in a flash drum, you'd calculate $\phi_i$ from a cubic equation of state for the vapor phase and $\gamma_i$ from an activity coefficient model (e.g., NRTL, UNIQUAC) for the liquid phase. Setting the fugacities equal in both phases lets you solve for the equilibrium compositions $x_i$ and $y_i$ at a given $T$ and $P$.