9.2 Phase equilibria and phase diagrams

Phase Equilibria and Phase Diagrams

Phase equilibria describes the conditions under which multiple phases of a substance coexist in balance, with no net mass transfer between them. This requires three simultaneous conditions: equal temperature (thermal equilibrium), equal pressure (mechanical equilibrium), and equal chemical potential for each component across all phases. Mastering phase diagrams gives you the ability to predict what phases exist under any set of conditions and how they change when you heat, cool, or pressurize a system.

Concept of Phase Equilibria

For two or more phases to coexist in dynamic balance, every component $i$ must have the same chemical potential $\mu_i$ in every phase present. "Dynamic balance" means molecules are still crossing phase boundaries in both directions, but the rates are equal, so there's no net transfer of mass.

The three equilibrium requirements are:

• Thermal equilibrium: Temperature $T$ is the same in all phases
• Mechanical equilibrium: Pressure $P$ is the same in all phases
• Chemical equilibrium: Chemical potential $\mu_i$ for each component $i$ is the same in all phases

Familiar examples: ice and liquid water coexisting at 0°C and 1 atm, or liquid water and steam coexisting at 100°C and 1 atm. At those specific conditions, the chemical potentials of water in each phase are exactly equal.

Construction of Phase Diagrams

Phase diagrams map out which phase (or phases) are stable as a function of thermodynamic variables. The type of diagram depends on how many components are in the system.

Single-component phase diagrams

The most common is the pressure–temperature ($P$–$T$) diagram. Each region on the diagram corresponds to a single stable phase (solid, liquid, or gas), and the lines separating regions are phase boundaries where two phases coexist.

• The triple point is the unique $P$–$T$ combination where all three phases coexist simultaneously. For water, this occurs at 273.16 K and 611.73 Pa.
• The critical point is the terminus of the liquid–gas boundary. Beyond it, the distinction between liquid and gas disappears, and you get a supercritical fluid.

Pressure–volume ($P$–$V$) diagrams are also used for single-component systems. Isotherms (constant-$T$ curves) show single-phase regions and two-phase regions. Within a two-phase region, horizontal tie lines connect the coexisting phases at the same pressure.

Binary phase diagrams

For two-component systems, the most common diagrams are:

• Temperature–composition ($T$–$x$) diagrams at constant pressure. These show how the phases of a mixture change with temperature and composition.
  • The liquidus line marks the temperature below which solid begins to form from the liquid.
  • The solidus line marks the temperature below which no liquid remains.
  • The eutectic point is the composition and temperature at which the mixture has its lowest melting point. At the eutectic, liquid transforms directly into two solid phases simultaneously.
• Pressure–composition ($P$–$x$) diagrams at constant temperature. These display vapor–liquid equilibrium (VLE) or liquid–liquid equilibrium (LLE) behavior under varying pressure.

Types of Phase Transitions

First-order transitions involve a discontinuous jump in properties like density and enthalpy. They always have a latent heat, meaning the system absorbs or releases energy without changing temperature during the transition.

• Examples: melting (solid → liquid), vaporization (liquid → gas), sublimation (solid → gas)

Second-order transitions are continuous. Properties like entropy and volume don't jump; instead, their derivatives (such as heat capacity or compressibility) show discontinuities. There is no latent heat.

• Examples: onset of superconductivity, ferromagnetic-to-paramagnetic transition at the Curie temperature

Solid–solid transitions occur between different crystal structures of the same substance. These are often first-order transitions with associated latent heats.

• Example: the allotropic transformation of carbon between graphite and diamond

Application of Gibbs Phase Rule

The Gibbs phase rule tells you how many intensive variables you can independently change while maintaining the same number of phases in equilibrium:

$F = C - P + 2$

where $F$ is the number of degrees of freedom (variance), $C$ is the number of components, and $P$ is the number of phases.

For single-component systems ($C = 1$): $F = 3 - P$

1. At the triple point, three phases coexist ($P = 3$), so $F = 0$. You can't change temperature or pressure at all without losing a phase. The triple point is a fixed, invariant point on the diagram.
2. Along a phase boundary, two phases coexist ($P = 2$), so $F = 1$. You can vary one variable (say temperature), but pressure is then determined by the boundary curve.
3. In a single-phase region ($P = 1$), $F = 2$. You can independently vary both $T$ and $P$.

For binary systems ($C = 2$): $F = 4 - P$

• At the eutectic point, three phases coexist (liquid + two solids), so $F = 4 - 3 = 1$. Since these diagrams are typically drawn at constant pressure, that one degree of freedom is already used up, making the eutectic a fixed point on a $T$–$x$ diagram.

Problem-Solving with Phase Diagrams

Reading the diagram: Given a temperature and pressure (or composition), locate the point on the diagram and identify which region it falls in. That tells you which phase(s) are present.

Using the lever rule to find relative amounts of phases in a two-phase region:

1. Draw a horizontal tie line through your point, extending to the boundaries of the two-phase region.
2. Identify the compositions at each end of the tie line. These are the compositions of the two coexisting phases.
3. The fraction of each phase is inversely proportional to the length of the lever arm on its side. If your overall composition is $x_0$, the left boundary is at $x_\alpha$, and the right boundary is at $x_\beta$:

$\text{Fraction of } \beta = \frac{x_0 - x_\alpha}{x_\beta - x_\alpha}, \quad \text{Fraction of } \alpha = \frac{x_\beta - x_0}{x_\beta - x_\alpha}$

Tracing phase transformations: To predict what happens during heating, cooling, or pressure changes, trace a path on the diagram and note each boundary you cross. At each boundary, a phase appears or disappears. You can calculate the heat absorbed or released using the latent heat and the mass of material undergoing the transition.

Thermodynamic Principles in Phase Equilibria

Gibbs free energy ($G$) is the governing potential at constant $T$ and $P$. A system at equilibrium sits at a minimum in $G$. When two phases coexist, they have equal molar Gibbs free energy for each component. If they didn't, material would spontaneously transfer to the phase with lower $G$.

Chemical potential ($\mu_i$) is the partial molar Gibbs free energy of component $i$. It quantifies the thermodynamic "drive" for component $i$ to move between phases. At phase equilibrium:

$\mu_i^{\alpha} = \mu_i^{\beta}$

for every component $i$ and every pair of phases $\alpha$ and $\beta$.

Fugacity ($f_i$) and activity ($a_i$) provide practical ways to express chemical potential. Fugacity is an "effective pressure" and activity is an "effective concentration," both correcting for non-ideal behavior. They connect to chemical potential through:

$\mu_i = \mu_i^0 + RT \ln\left(\frac{f_i}{f_i^0}\right) = \mu_i^0 + RT \ln a_i$

At phase equilibrium, the fugacity (or activity) of each component is equal across all phases.

Clapeyron equation gives the slope of any phase boundary on a $P$–$T$ diagram:

$\frac{dP}{dT} = \frac{\Delta S_m}{\Delta V_m} = \frac{\Delta H_m}{T\,\Delta V_m}$

This is exact and applies to all phase transitions (solid–liquid, solid–solid, etc.). The sign of $\Delta V_m$ determines whether the boundary slopes positively or negatively. Water's solid–liquid boundary slopes to the left because ice is less dense than liquid water ($\Delta V_m < 0$ on melting).

Clausius–Clapeyron equation is a simplified form for vaporization, assuming the vapor behaves as an ideal gas and the volume of the condensed phase is negligible compared to the vapor:

$\ln\frac{P_2}{P_1} = -\frac{\Delta H_{\text{vap}}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

This is the go-to equation for estimating how vapor pressure changes with temperature or for calculating boiling points at non-standard pressures. Keep in mind it breaks down near the critical point, where the ideal gas assumption fails and the liquid volume is no longer negligible.