Phase Equilibria in One- and Multi-Component Systems

A phase diagram compresses an enormous amount of thermodynamics into a single picture: every line on it is a statement that two (or three) phases have equal chemical potential, and every region is a statement about which single phase has the lowest chemical potential at that temperature and pressure. This chapter builds the machinery that draws those lines rigorously — the Gibbs phase rule, which counts how many variables you are even free to choose, and the Clausius–Clapeyron equation, which gives the exact slope of every phase boundary — then extends the picture from one-component systems (water, carbon dioxide) to two-component liquid mixtures, where Raoult's law from the previous chapter reappears as the tool that builds boiling-point diagrams and predicts azeotropes.

You should have to hand the chemical potential and its equilibrium condition, and Raoult's law, from Chapter 3, along with the critical point and critical constants from Chapter 1. This chapter develops the phase rule, the Clapeyron and Clausius–Clapeyron equations, one-component $p$–$T$ phase diagrams including the triple and critical points, two-component liquid–vapour diagrams and the lever rule, azeotropes, and liquid–liquid partial miscibility.

1. The Equilibrium Condition and the Gibbs Phase Rule

Phase equilibrium, from Chapter 3, requires the chemical potential of every species to be equal in every phase it occupies: $\mu_i(\alpha) = \mu_i(\beta) = \cdots$ for each component $i$, across every pair of phases $\alpha,\beta$ in contact. The Gibbs phase rule counts how many intensive variables remain free to choose once this condition is imposed.

For a system of $C$ independent components distributed among $P$ phases, specifying each phase's composition requires $C-1$ mole fractions (the last is fixed by $\sum x_i=1$), giving $P(C-1)$ composition variables, plus the two shared variables $T$ and $p$ (common to every phase in equilibrium): $P(C-1)+2$ variables in total. Phase equilibrium imposes $\mu_i(\alpha)=\mu_i(\beta)=\cdots=\mu_i(P\text{-th phase})$ for each of the $C$ components — that is $P-1$ independent equality constraints per component, or $C(P-1)$ constraints overall. The number of remaining free variables, the degrees of freedom $F$, is variables minus constraints:

Worked example. For a pure substance ($C=1$) at its triple point, three phases coexist simultaneously ($P=3$): $F=1-3+2=0$. Zero degrees of freedom means the triple point is a single, fixed $(T,p)$ point — nothing is left to choose, consistent with water's triple point sitting at the fixed, reproducible condition $273.16~\text{K}$, $611.7~\text{Pa}$. (Historically this exact value of $273.16~\text{K}$ defined the kelvin; since the 2019 SI redefinition, the kelvin is instead fixed via the Boltzmann constant, with the water triple point now an extremely precisely measured — rather than defining — temperature.) Along a two-phase coexistence line ($P=2$), $F=1-2+2=1$: one degree of freedom, meaning you may choose $T$ freely, but $p$ is then fixed by the coexistence curve (or vice versa) — exactly why phase boundaries are lines, not regions, on a one-component $p$–$T$ diagram.

2. The Clapeyron Equation

Along any two-phase coexistence curve, $\mu_\alpha(T,p)=\mu_\beta(T,p)$ holds at every point. Moving an infinitesimal distance along the curve, both sides must change equally: $d\mu_\alpha = d\mu_\beta$. Using $d\mu = -S_m\,dT + V_m\,dp$ (the molar form of the fundamental equation from Chapter 3, applied to a pure substance where $\mu=G_m$) for each phase,

Collecting terms gives $(S_{m,\beta}-S_{m,\alpha})\,dT = (V_{m,\beta}-V_{m,\alpha})\,dp$, or the Clapeyron equation:

(the second form uses $\Delta_{\text{trs}}S = \Delta_{\text{trs}}H/T$, valid exactly at the equilibrium transition temperature). This single equation is exact for any phase transition of a pure substance — solid–liquid, liquid–vapour, or solid–solid — and gives the precise slope of every line on a one-component phase diagram directly from the transition's enthalpy and volume change.

2.1 The solid–liquid boundary and water's anomaly

For fusion, $\Delta_{\text{fus}}H>0$ always (melting absorbs heat), while $\Delta_{\text{fus}}V$ is small in magnitude (liquids and solids are both nearly incompressible) but can carry either sign. For most substances the solid is denser than the liquid ($\Delta_{\text{fus}}V>0$), giving a steep, positive-slope fusion line. Water is the famous exception: ice is less dense than liquid water ($\Delta_{\text{fus}}V<0$), giving a negative Clapeyron slope — the solid–liquid line tilts backward, and increasing pressure lowers the melting point.

Worked example. For water at $T=273.15~\text{K}$, $\Delta_{\text{fus}}H=6.01~\text{kJ}\,\text{mol}^{-1}$, with molar volumes $V_m(\text{ice})=19.65~\text{cm}^3\,\text{mol}^{-1}$ and $V_m(\text{water})=18.02~\text{cm}^3\,\text{mol}^{-1}$ (densities $0.917$ and $0.9998~\text{g}\,\text{cm}^{-3}$), so $\Delta_{\text{fus}}V = 18.02-19.65=-1.63~\text{cm}^3\,\text{mol}^{-1}=-1.63\times10^{-6}~\text{m}^3\,\text{mol}^{-1}$:

so roughly $130$–$140~\text{atm}$ of extra pressure is needed to lower ice's melting point by just $1~\text{K}$ — a large pressure, far more than an ice skate blade actually exerts (see the pitfalls below).

3. The Clausius–Clapeyron Equation (Liquid–Vapour and Solid–Vapour)

For transitions into a vapour phase, two simplifications make the Clapeyron equation far more useful. First, the molar volume of the vapour vastly exceeds that of the condensed phase, so $\Delta V \approx V_m(\text{gas})$. Second, treating the vapour as a perfect gas, $V_m(\text{gas})=RT/p$. Substituting into the Clapeyron equation,

the Clausius–Clapeyron equation. Treating $\Delta_{\text{vap}}H$ as approximately constant over the temperature range of interest and integrating between two states $(T_1,p_1)$ and $(T_2,p_2)$,

the same exponential relation that governs any Boltzmann-weighted activated process, here linking vapour pressure directly to the enthalpy needed to escape the liquid.

Worked example. Water boils at $T_1=373.15~\text{K}$ at $p_1=1.000~\text{atm}$, with $\Delta_{\text{vap}}H=40.7~\text{kJ}\,\text{mol}^{-1}$. At what temperature does water boil at $p_2=0.700~\text{atm}$ (roughly the atmospheric pressure at $3000~\text{m}$ altitude)?

water boils almost $10^\circ\text{C}$ cooler at this altitude — the everyday reason high-altitude cooking requires longer times.

4. One-Component Phase Diagrams: Triple Point and Critical Point

A one-component $p$–$T$ phase diagram consists of three coexistence lines (solid–liquid, liquid–vapour, solid–vapour), each with the Clapeyron slope of Section 2, meeting at the triple point, where $F=0$ (Section 1) fixes a unique $T$ and $p$ at which all three phases coexist simultaneously. The liquid–vapour line does not extend indefinitely: it terminates at the critical point $(T_c,p_c)$ from Chapter 1, beyond which no liquid–vapour distinction exists and the two phases merge into a single supercritical fluid — exactly the critical behaviour the van der Waals equation was built to predict.

Key distinction. The triple point is where three different phases coexist at a single, invariant condition; the critical point is where the distinction between two phases (liquid and vapour) disappears. They are conceptually opposite kinds of special point, and do not generally occur at similar temperatures or pressures.

5. Two-Component Liquid–Vapour Phase Diagrams

For an ideal binary liquid mixture obeying Raoult's law (Chapter 3), the total vapour pressure at fixed $T$ as a function of liquid composition is linear in $x_A$:

Plotting boiling temperature instead (at fixed total pressure, typically $1~\text{atm}$) gives the more commonly used boiling-point diagram: a lens-shaped region bounded above by the vapour-composition curve and below by the liquid-composition curve, both running between the two pure-component boiling points. At any temperature within the two-phase region, a horizontal tie-line connects the liquid composition $x_B$ (where the tie-line meets the lower curve) to the vapour composition $y_B$ (where it meets the upper curve) in equilibrium with it.

5.1 The Lever Rule

For an overall mixture composition $z_B$ lying between $x_B$ and $y_B$ on a tie-line, the relative amounts (not compositions) of liquid and vapour present are given by the lever rule:

exactly analogous to a mechanical lever balanced about the point $z_B$, with the "weights" of liquid and vapour inversely proportional to their distances from $z_B$ along the tie-line. This is the quantitative basis of fractional distillation: at each stage, the vapour (richer in the more volatile component) is drawn off and re-condensed, progressively enriching it further with each theoretical plate.

Worked example. A tie-line at some temperature has liquid composition $x_B=0.20$ and vapour composition $y_B=0.65$; the overall mixture is $z_B=0.35$. The ratio of liquid to vapour present is

twice as much liquid (by moles) as vapour — consistent with $z_B=0.35$ lying closer to the liquid composition ($0.20$) than to the vapour composition ($0.65$).

6. Non-Ideal Two-Component Systems: Azeotropes

Real mixtures with activity coefficients $\gamma_i\ne1$ (Chapter 3) distort the simple lens-shaped diagram. Strong positive deviations from Raoult's law ($\gamma_i>1$, weaker-than-average A–B attraction) can push the total vapour pressure above both pure-component values at some intermediate composition, producing a minimum-boiling azeotrope: a composition at which liquid and vapour have identical composition, so the mixture boils at constant composition exactly like a pure substance (ethanol–water, famously, forms one near $96\%$ ethanol by mass). Strong negative deviations ($\gamma_i<1$) analogously produce a maximum-boiling azeotrope (e.g. HCl–water). Away from the azeotrope, the two-component, two-phase system has $F=C-P+2=2-2+2=2$: fixing $p$ and the liquid composition $x_B$ independently determines $T$ and the vapour composition $y_B$. At the azeotropic composition, however, the extra constraint $x_B=y_B$ (liquid and vapour compositions coincide) removes one further degree of freedom, so $F=1$ there — fixing $p$ alone now fixes both $T$ and the (shared) composition, which is exactly why the azeotrope boils at a constant temperature for a given pressure, just like a pure substance. The tie-line shrinks to zero length, and no amount of further simple distillation can separate the two components past that point, since removing vapour and re-condensing it reproduces the same composition it started from.

7. Liquid–Liquid Phase Diagrams: Partial Miscibility

Some pairs of liquids are only partially miscible: below some composition-dependent boundary, mixing produces not one homogeneous phase but two liquid layers of different composition, each saturated with the other component. Plotting the boundary temperature against composition typically gives a dome: below the upper consolute temperature (or upper critical solution temperature), two liquid phases coexist over a range of overall compositions (again governed by a lever rule identical in form to Section 5.1); above it, the two liquids become fully miscible at every composition, because the increased thermal energy overcomes the unfavourable A–B interactions responsible for immiscibility. The classic laboratory example is phenol–water, with an upper consolute temperature near $66^\circ\text{C}$. A smaller number of systems (e.g. triethylamine–water) instead show a lower consolute temperature, below which the liquids are fully miscible and above which they separate — a less common but thermodynamically equally valid pattern, arising when increasing temperature specifically weakens a favourable (often hydrogen-bonding) interaction holding the mixture together.

8. Common Exam Pitfalls

• Treating $\Delta_{\text{trs}}H$ and $\Delta_{\text{trs}}V$ as always positive. $\Delta_{\text{fus}}H$ and $\Delta_{\text{vap}}H$ are positive for essentially all ordinary transitions, but $\Delta_{\text{fus}}V$ can be negative (water, and a few other substances like bismuth and gallium), giving a negative-slope fusion line — check the sign before assuming a positive Clapeyron slope.
• Overstating the "pressure melting" explanation of ice skating. The pressure a skate blade exerts, even concentrated on a thin blade, falls far short of the $\sim\!130~\text{atm}$ needed (Section 2) to produce a meaningful melting-point depression at typical skating temperatures; frictional heating is now understood to be the dominant mechanism for the thin liquid layer that enables skating.
• Applying the integrated Clausius–Clapeyron equation over a very wide temperature range where $\Delta_{\text{vap}}H$ can no longer be treated as constant — the approximation is reliable only over a modest range around the reference temperature.
• Confusing the triple point with the critical point, or assuming every substance's triple point lies near room conditions — CO$_2$'s triple point, for instance, is at $5.1~\text{atm}$, well above atmospheric pressure, which is exactly why solid CO$_2$ ("dry ice") sublimes directly to gas at $1~\text{atm}$ rather than melting.
• Misreading the lever rule's proportionality. The amount of a phase is proportional to the opposite arm of the tie-line (the distance from the overall composition to the other phase's composition), not the near arm — a frequent sign/direction error.
• Believing an azeotrope can always be separated by adding more distillation plates. At the exact azeotropic composition the liquid and vapour compositions are identical by definition, so no amount of further simple (single-pressure) distillation can cross that point; separation requires a different technique (e.g. pressure-swing distillation, which shifts the azeotropic composition, or addition of an entrainer).

Summary

• Phase equilibrium requires $\mu_i$ equal in every phase; the Gibbs phase rule, $F=C-P+2$, counts the resulting degrees of freedom (e.g. $F=0$ at a one-component triple point, $F=1$ along a one-component two-phase line).
• The Clapeyron equation, $dp/dT=\Delta_{\text{trs}}H/(T\Delta_{\text{trs}}V)$, is exact for any pure-substance phase transition and sets the slope of every line on a $p$–$T$ phase diagram.
• Water's fusion line has an anomalous negative slope because ice is less dense than liquid water ($\Delta_{\text{fus}}V<0$); most substances have a steep positive slope.
• The Clausius–Clapeyron equation, $\ln(p_2/p_1) = -(\Delta_{\text{vap}}H/R)(1/T_2-1/T_1)$, follows from approximating the vapour as a perfect gas and $\Delta V\approx V_m(\text{gas})$, and explains why boiling point falls with decreasing (e.g. high-altitude) pressure.
• The triple point ($F=0$, three phases, one-component) is a fixed point; the critical point (Chapter 1) is where the liquid–vapour distinction disappears — conceptually opposite special points.
• Two-component liquid–vapour diagrams follow from Raoult's law; the lever rule, $n_{\text{liquid}}/n_{\text{vapour}}=(y_B-z_B)/(z_B-x_B)$, gives the relative amounts of each phase at a given overall composition, underlying fractional distillation.
• Strong positive or negative deviations from Raoult's law produce minimum- or maximum-boiling azeotropes, compositions where liquid and vapour coincide and simple distillation cannot separate the components further.
• Partially miscible liquids show an upper (or occasionally lower) consolute temperature, above (or below) which the two liquids become fully miscible at every composition.