5.2 Phase Equilibria and Phase Diagrams

Phase diagrams for single-component systems

Phase equilibria and phase diagrams give you a way to predict what phase a substance will be in at any given temperature and pressure. They map out the boundaries between solid, liquid, and gas, and reveal special points where phases merge or coexist. For Physical Chemistry II, you need to not only read these diagrams but also connect them to thermodynamic quantities like Gibbs free energy and the Clapeyron equation.

Graphical representations of equilibrium relationships

A phase diagram plots pressure on the y-axis against temperature on the x-axis for a pure substance. Each region of the diagram corresponds to a single stable phase (solid, liquid, or gas), meaning that within that region, one phase has the lowest Gibbs free energy and is therefore thermodynamically favored.

The lines separating these regions are coexistence curves (phase boundaries), where two phases have equal molar Gibbs free energies: $G_m^{\alpha} = G_m^{\beta}$. At any point on a coexistence curve, both phases are in equilibrium and can exist together.

To use a phase diagram, find the point corresponding to your temperature and pressure. If it falls inside a region, the substance exists as that single phase. If it falls on a boundary line, two phases coexist.

Phase boundaries and their slopes

Each coexistence curve has a slope governed by the Clapeyron equation:

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

where $\Delta S_m$, $\Delta V_m$, and $\Delta H_m$ are the molar entropy, volume, and enthalpy changes for the phase transition. This equation is exact and applies to all phase boundaries.

The three main boundaries and their typical behavior:

• Solid-liquid (melting curve): The slope is usually positive and steep because $\Delta V_m$ for melting is small and positive for most substances. Water is a famous exception: ice is less dense than liquid water, so $\Delta V_m < 0$ and the melting curve has a negative slope.
• Solid-gas (sublimation curve): Always has a positive slope. Both $\Delta H_{sub}$ and $\Delta V_m$ are positive (the gas phase occupies far more volume), so $dP/dT > 0$.
• Liquid-gas (vaporization curve): Also has a positive slope, but it terminates at the critical point. Beyond that point, liquid and gas phases become indistinguishable.

For the liquid-gas and solid-gas boundaries, where one phase is a gas, the Clapeyron equation is often approximated by the Clausius-Clapeyron equation:

$\frac{d \ln P}{dT} = \frac{\Delta H_m}{RT^2}$

This assumes the vapor behaves ideally and that the molar volume of the condensed phase is negligible compared to the gas.

Triple points, critical points, and phase boundaries

Triple points

The triple point is the single temperature-pressure combination where solid, liquid, and gas coexist simultaneously. All three coexistence curves meet here.

At the triple point, the molar Gibbs free energies of all three phases are equal:

$G_m^{solid} = G_m^{liquid} = G_m^{gas}$

This point is invariant, meaning you cannot change temperature or pressure at all without losing one of the three phases (the Gibbs phase rule confirms $F = 0$; see below).

For water, the triple point occurs at 273.16 K (0.01°C) and 611.73 Pa (0.006 atm). This value is so precisely reproducible that it was historically used to define the kelvin.

Critical points

The critical point is the terminus of the liquid-gas coexistence curve. It represents the highest temperature and pressure at which distinct liquid and gas phases can coexist.

• For water: $T_c = 647.10 \text{ K}$ (373.95°C), $P_c = 22.06 \text{ MPa}$ (218 atm)

Above the critical temperature, no amount of pressure can liquefy the gas. Instead, the substance exists as a supercritical fluid, which has properties intermediate between liquid and gas (liquid-like density, gas-like viscosity and diffusivity). Supercritical $CO_2$, for example, is widely used as an industrial solvent because of these hybrid properties.

One key physical feature: as you approach the critical point along the vaporization curve, the densities of the liquid and gas phases converge. At the critical point itself, the distinction vanishes and the meniscus between liquid and gas disappears.

Phase boundaries

To summarize how the three boundaries relate:

• All three curves intersect at the triple point
• The vaporization curve extends from the triple point up to the critical point, where it terminates
• The sublimation curve extends from the triple point toward lower temperatures
• The melting curve extends from the triple point toward higher pressures and has no known upper terminus (no solid-liquid critical point has been observed for simple substances)

Degrees of freedom using the Gibbs phase rule

The Gibbs phase rule equation

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

$F = C - P + 2$

where:

• $F$ = degrees of freedom (number of independently variable intensive parameters)
• $C$ = number of components
• $P$ = number of phases in equilibrium

The "+2" accounts for temperature and pressure as the two external variables. For a single-component system ($C = 1$), this simplifies to:

$F = 3 - P$

Applying the phase rule to a phase diagram

Here's how the phase rule maps onto different locations on a single-component phase diagram:

The $F = 0$ result at the triple point is why it's a single, unique point on the diagram rather than a line or region. This also explains why triple points are so useful as thermometric fixed points: the conditions are exactly reproducible.

Phase behavior under varying conditions

Predicting phase behavior using phase diagrams

To predict what happens to a substance as you change conditions, trace a path on the phase diagram:

1. Locate your starting point (initial $T$ and $P$).
2. Draw the path corresponding to the change (e.g., heating at constant pressure is a horizontal line moving right; compressing at constant temperature is a vertical line moving up).
3. Each time the path crosses a coexistence curve, a phase transition occurs.
4. If the path enters a single-phase region, the substance exists entirely in that phase.

Phase transitions and their direction

When your path crosses a phase boundary, the type of transition depends on which boundary you cross and which direction you're moving:

• Crossing the melting curve from the solid side → melting; from the liquid side → freezing
• Crossing the vaporization curve from the liquid side → vaporization; from the gas side → condensation
• Crossing the sublimation curve from the solid side → sublimation; from the gas side → deposition

A useful thermodynamic rule: increasing temperature at constant pressure (or decreasing pressure at constant temperature) favors the phase with higher molar entropy. Since $S_m^{gas} > S_m^{liquid} > S_m^{solid}$, heating generally drives transitions toward the gas phase.

Conversely, increasing pressure at constant temperature generally favors the denser, lower-entropy phase. For most substances that means solid > liquid > gas. Again, water is the notable exception: because ice is less dense than liquid water, increasing pressure on ice near 0°C can actually cause it to melt.

One more subtlety worth noting: if you heat a substance above its critical temperature and then increase the pressure, you can go from gas to a supercritical fluid to a liquid-like state without ever crossing a phase boundary. This continuous path around the critical point means there's no sharp gas-to-liquid transition, which is a direct consequence of the vaporization curve terminating at the critical point.