7.2 Clapeyron Equation and Phase Change Analysis

Phase Equilibrium in Thermodynamics

Concept and Importance

Phase equilibrium occurs when two or more phases of a substance coexist at the same temperature, pressure, and chemical potential. Under these conditions, there's no net transfer of mass or energy between the phases. The formal condition comes from requiring that the chemical potential of each component be equal across all phases present.

Understanding phase equilibrium matters because it determines what state a substance occupies under given conditions and governs how properties like density, enthalpy, and entropy behave near phase boundaries.

Gibbs Phase Rule and Phase Transitions

The Gibbs Phase Rule relates the degrees of freedom in a system at equilibrium:

$F = C - P + 2$

where $F$ is the number of independent intensive variables you can change without disturbing equilibrium, $C$ is the number of components, and $P$ is the number of phases. For a pure substance ($C = 1$) with two phases coexisting ($P = 2$), you get $F = 1$, meaning specifying temperature alone fixes the equilibrium pressure (and vice versa). That's exactly why the phase boundaries on a P-T diagram are curves, not regions.

Phase transitions occur when the system crosses from one equilibrium state to another due to changes in temperature, pressure, or composition:

• Melting: solid to liquid (ice → water)
• Vaporization: liquid to vapor (water → steam)
• Sublimation: solid to vapor (dry ice → $CO_2$ gas)

The latent heat of a phase transition is the energy required to change phase without changing temperature. This quantity is central to energy balance calculations and appears directly in the Clapeyron equation.

Clapeyron Equation Derivation

Starting from the Fundamental Relations

The derivation begins by considering a reversible phase transition between two phases at equilibrium. At equilibrium, both phases share the same temperature and pressure, and the specific Gibbs function is equal in both phases:

$g_f = g_g$

where $g$ denotes specific Gibbs free energy. If you move along the phase boundary (changing $T$ and $P$ together so equilibrium is maintained), the changes in Gibbs function for each phase must remain equal:

$dg_f = dg_g$

Arriving at the Clapeyron Equation

Recall that for a simple compressible substance, the differential of specific Gibbs function is:

$dg = v\,dP - s\,dT$

Applying this to both phases and setting them equal:

$v_f\,dP - s_f\,dT = v_g\,dP - s_g\,dT$

Rearranging:

$(s_g - s_f)\,dT = (v_g - v_f)\,dP$

$\frac{dP}{dT} = \frac{s_g - s_f}{v_g - v_f} = \frac{\Delta s}{\Delta v}$

Since the phase change occurs at constant temperature and pressure, the entropy change relates to the latent heat by $\Delta s = \Delta h / T$. Substituting gives the Clapeyron equation:

$\frac{dP}{dT} = \frac{\Delta h}{T\,\Delta v}$

where $\Delta h$ is the specific enthalpy of the phase transition (latent heat) and $\Delta v$ is the specific volume change. This equation is exact and applies to any first-order phase transition with no approximations.

Phase Transitions Analysis

Applying the Clapeyron Equation

The Clapeyron equation gives the slope of the phase equilibrium curve on a P-T diagram at any point. How you use it depends on the type of transition:

• Solid-liquid transitions: $\Delta v$ is typically small, so $dP/dT$ is large, meaning the melting curve is steep on a P-T diagram. For most substances $\Delta v > 0$ (the liquid is less dense), so the slope is positive. Water is the classic exception: ice is less dense than liquid water, so $\Delta v < 0$ and the solid-liquid curve has a negative slope. This is why increasing pressure on ice lowers its melting point.
• Liquid-vapor and solid-vapor transitions: $\Delta v$ is large (the vapor phase occupies much more volume), so the slope $dP/dT$ is comparatively gentle.

The Clausius-Clapeyron Equation

For vaporization and sublimation, two simplifying assumptions let you integrate the Clapeyron equation analytically:

1. The vapor phase behaves as an ideal gas: $v_g \approx RT/P$
2. The specific volume of the condensed phase is negligible compared to the vapor: $\Delta v \approx v_g$

Substituting into the Clapeyron equation:

$\frac{dP}{dT} = \frac{\Delta h_{vap} \cdot P}{RT^2}$

Separating variables and integrating between states 1 and 2 (assuming $\Delta h_{vap}$ is roughly constant over the temperature range):

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

This is the Clausius-Clapeyron equation. Note that $R$ here is the specific gas constant for the substance (or the universal gas constant if you use molar enthalpy).

Estimating Phase Change Properties

Practical applications of these equations include:

• Boiling point at altitude: Atmospheric pressure drops at higher elevations. Using the Clausius-Clapeyron equation with the known boiling point at 1 atm (100°C for water), you can estimate the boiling point at a reduced pressure. For example, at about 2000 m elevation (~80 kPa), water boils near 93°C.
• Sublimation temperature of $CO_2$: Dry ice sublimes at -78.5°C at 1 atm. The Clausius-Clapeyron equation lets you find the sublimation temperature at other pressures.
• Phase equilibrium curves: Integrating the Clapeyron equation traces out the full phase boundary on a P-T diagram, mapping the stability regions of each phase.

Phase Diagrams Interpretation

Single-Component Phase Diagrams

A single-component P-T phase diagram has three main curves, each representing two-phase equilibrium as described by the Clapeyron equation:

• Solid-liquid curve (fusion curve): Steep slope; positive for most substances, negative for water.
• Liquid-vapor curve (vaporization curve): Extends from the triple point up to the critical point.
• Solid-vapor curve (sublimation curve): Extends from the triple point down to lower temperatures.

Key features to know:

• The triple point is where all three curves meet. It's the unique temperature and pressure at which solid, liquid, and vapor coexist simultaneously. For water, this is 273.16 K and 611.7 Pa. By the Gibbs Phase Rule, $F = 1 - 3 + 2 = 0$, so the triple point is invariant.
• The critical point terminates the liquid-vapor curve. Beyond the critical temperature and pressure, there's no distinct phase boundary between liquid and vapor; the substance exists as a supercritical fluid. For water, the critical point is approximately 647 K and 22.06 MPa.

The slope of each curve at any point is given directly by the Clapeyron equation, connecting the diagram's geometry to measurable thermodynamic quantities ($\Delta h$ and $\Delta v$).

Multi-Component Phase Diagrams

For binary and multi-component systems, phase diagrams become more complex because composition is an additional variable. The Gibbs Phase Rule still governs the degrees of freedom, but now $C > 1$:

• The iron-carbon phase diagram is used extensively in metallurgy to predict the phases present in steel (austenite, ferrite, cementite) at various temperatures and carbon concentrations.
• The ethanol-water phase diagram is essential for understanding distillation, showing how the vapor and liquid compositions differ at equilibrium.

These diagrams incorporate the effects of composition on phase equilibrium and are constructed using extensions of the same thermodynamic principles that underlie the Clapeyron equation.