14.09.04 · genchem-pchem / solutions-phase

Phase diagrams: single-component, binary, and the lever rule

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Gibbs — Trans. Conn. Acad. 3, 108 (1876)

Intuition Beginner

A phase diagram is a map. It tells you which phase -- solid, liquid, or gas -- is stable at any combination of temperature and pressure. For a pure substance, the diagram has three regions separated by curves. Each curve is a phase boundary: the set of conditions where two phases coexist in equilibrium. Where all three curves meet, you get the triple point, the single combination of temperature and pressure at which solid, liquid, and gas all coexist.

At the other extreme, the liquid-vapor curve ends at the critical point. Above this temperature and pressure, the liquid and gas become indistinguishable -- you have a supercritical fluid, a single dense phase with properties between those of a liquid and a gas. For water, the critical point is at 374 degrees C and 218 atm.

The slopes of the phase boundaries carry physical information. The liquid-vapor and solid-vapor boundaries slope upward (higher temperature requires higher pressure to maintain equilibrium). The solid-liquid boundary slopes upward for most substances because the liquid is less dense than the solid. Water is the exception: ice floats, so the solid-liquid boundary slopes slightly downward -- pressure favors the denser liquid phase, which is why ice melts under pressure.

For mixtures of two components, the phase diagram adds a composition axis. A binary T-x diagram plots temperature against mole fraction at fixed pressure. These diagrams show liquidus curves (above which everything is liquid), solidus curves (below which everything is solid), and two-phase regions in between. The lever rule lets you determine how much of each phase is present at any point in a two-phase region.

The eutectic point is the composition with the lowest possible melting temperature in the system. At the eutectic, the liquid freezes to a mixture of two solid phases at a single temperature, behaving like a pure substance. Solder (Pb-Sn) and many metal alloys exploit eutectic compositions for their low and sharp melting behavior.

Visual Beginner

  Single-component P-T phase diagram:

  Pressure
  |
  |               * critical point
  |              / \
  |             /   \   supercritical fluid
  |            / liq  \
  |           /  uid    \
  |          /            \
  |   solid/               \
  |        /  * triple       * gas (vapor)
  |       /    point
  |      /
  |     /
  |____/__________________________________
       Temperature

  The sublimation curve (solid-gas) goes from
  the origin to the triple point.
  The melting curve (solid-liquid) goes from
  the triple point upward (left for water).
  The vapor-pressure curve (liquid-gas) goes
  from the triple point to the critical point.

  Binary eutectic T-x diagram (constant P):

  T
  |  A . . . . . . . . . . . . . . . . . B
  |  |       liquid (L)
  |  |      .  . .
  |  |    /       . .
  |  |   /  L +    .  .
  |  |  /  solid A      .
  |  | .                  .
  |  |                     .
  |  |  * eutectic           .
  |  |/                       .
  |  | solid A                 .
  |  |    + solid B   L + solid B
  |  |_______________________________
  |       composition (A -> B)

  Horizontal tie line through the eutectic
  connects liquid (eutectic composition) to
  both pure solid A and pure solid B.

Worked example Beginner

Problem: The vapor pressure of benzene is 100 torr at 26.1 degrees C and 400 torr at 60.6 degrees C. Use the Clausius-Clapeyron equation to estimate the enthalpy of vaporization of benzene and predict the normal boiling point.

Solution:

The two-point form of the Clausius-Clapeyron equation relates vapor pressure to temperature:

ln \frac{P _{2}}{P _{1}} = \frac{- Δ H _{vap}}{R} (\frac{1}{T _{2}} - \frac{1}{T _{1}})

Substituting $P_{1} = 100$ torr, $T_{1} = 299.3$ K, $P_{2} = 400$ torr, $T_{2} = 333.8$ K, and $R = 8.314$ J/(mol K):

ln \frac{400}{100} = \frac{- Δ H _{vap}}{8.314} (\frac{1}{333.8} - \frac{1}{299.3})

ln 4 = 1.386 = \frac{- Δ H _{vap}}{8.314} (0.002996 - 0.003341)

1.386 = \frac{- Δ H _{vap}}{8.314} (- 0.000345)

Δ H_{vap} = \frac{1.386 \times 8.314}{0.000345} = 33, 400 J/mol = 33.4 kJ/mol

The accepted value is 30.8 kJ/mol. The discrepancy comes from assuming $Δ H_{vap}$ is constant over this temperature range.

To predict the normal boiling point ( $P = 760$ torr), use $P_{1} = 100$ torr, $T_{1} = 299.3$ K:

ln \frac{760}{100} = \frac{- 33 , 400}{8.314} (\frac{1}{T _{b}} - \frac{1}{299.3})

2.028 = - 4017 (\frac{1}{T _{b}} - 0.003341)

\frac{1}{T _{b}} = 0.003341 - 0.000505 = 0.002836

T_{b} = 353 K = 80 degrees C

The accepted boiling point of benzene is 80.1 degrees C -- the estimate is remarkably close.

Worked example (lever rule) Beginner

Problem: A binary mixture of A and B at 60 mol% B is cooled into the two-phase region. The liquidus composition is 40 mol% B and the solidus composition is 80 mol% B. Use the lever rule to find the fraction of liquid and solid.

Solution:

Let $x_{0} = 0.60$ (overall composition), $x_{liq} = 0.40$ (liquid composition), $x_{sol} = 0.80$ (solid composition).

The lever rule states:

f_{liq} = \frac{x _{sol} - x _{0}}{x _{sol} - x _{liq}} = \frac{0.80 - 0.60}{0.80 - 0.40} = \frac{0.20}{0.40} = 0.50

f_{sol} = \frac{x _{0} - x _{liq}}{x _{sol} - x _{liq}} = \frac{0.60 - 0.40}{0.80 - 0.40} = \frac{0.20}{0.40} = 0.50

The system is 50% liquid and 50% solid by moles. The fulcrum of the lever is at the overall composition $x_{0}$ .

Check your understanding Beginner

Formal definition Intermediate+

Single-component phase diagrams. For a pure substance, the equilibrium among phases is governed by the equality of chemical potentials. The coexistence of two phases along a boundary requires $μ^{α} (T, P) = μ^{β} (T, P)$ . This constrains $P$ as a function of $T$ (or vice versa) along each phase boundary, defining three curves:

Sublimation curve (solid-gas): extends from absolute zero to the triple point.
Vapor-pressure curve (liquid-gas): extends from the triple point to the critical point.
Melting curve (solid-liquid): extends from the triple point upward in pressure (for most substances) or downward (for water).

Gibbs phase rule. The number of intensive degrees of freedom available to a system at equilibrium is:

F = C - P + 2

where $C$ is the number of components and $P$ is the number of coexisting phases. For a single-component system ( $C = 1$ ): at the triple point ( $P = 3$ ), $F = 0$ (invariant); along a two-phase boundary ( $P = 2$ ), $F = 1$ (univariant); in a single-phase region ( $P = 1$ ), $F = 2$ (bivariant). The critical point is the terminus of the liquid-vapor curve: beyond it, no phase boundary exists.

Clausius-Clapeyron equation. The slope of any phase boundary in a P-T diagram is given by the Clapeyron equation:

\frac{d P}{d T} = \frac{Δ S _{trans}}{Δ V _{trans}} = \frac{Δ H _{trans}}{T Δ V _{trans}}

For liquid-vapor and solid-vapor equilibria where the vapor is treated as an ideal gas and $V_{vapor} ≫ V_{condensed}$ , this simplifies to the Clausius-Clapeyron equation:

\frac{d ln P}{d T} = \frac{Δ H _{vap}}{R T ^{2}}

Integration (assuming temperature-independent $Δ H_{vap}$ ) gives the two-point form:

ln \frac{P _{2}}{P _{1}} = \frac{- Δ H _{vap}}{R} (\frac{1}{T _{2}} - \frac{1}{T _{1}})

Near the critical point, $Δ H_{vap}$ is not constant and the integrated form fails. The full Clapeyron equation must be used with an equation of state that captures the vanishing enthalpy and volume changes of vaporization at $T_{c}$ .

Binary T-x phase diagrams. For a two-component system at fixed pressure, the phase diagram is plotted as temperature versus composition (mole fraction or weight fraction). The key features depend on the type of system:

Liquid-vapor (T-x-y) diagrams. The bubble-point curve (liquidus) gives the temperature at which the first bubble of vapor forms on heating a liquid of given composition. The dew-point curve gives the temperature at which the first drop of liquid forms on cooling a vapor. Between these curves lies the two-phase liquid-vapor region. Tie lines are horizontal (isothermal) and connect coexisting liquid and vapor compositions. For ideal solutions (Raoult's law), the vapor is always enriched in the more volatile component. Fractional distillation exploits this enrichment: each vaporization-condensation cycle shifts the composition toward the more volatile component. The number of theoretical plates in a distillation column determines how many such cycles occur.

Azeotropes. When positive or negative deviations from Raoult's law are large enough, the bubble-point and dew-point curves touch at a point where the liquid and vapor compositions are identical. This is an azeotrope. A minimum-boiling azeotrope (e.g., ethanol-water at 95.6 wt% ethanol, 78.2 degrees C) arises from positive deviations; a maximum-boiling azeotrope (e.g., HCl-water at 20.2 wt% HCl, 108.6 degrees C) from negative deviations. Azeotropes cannot be separated by simple distillation because the vapor and liquid have the same composition.

Liquid-liquid (miscibility gap) diagrams. When the enthalpy of mixing is sufficiently positive (unfavorable A-B interactions), the mixture separates into two liquid phases below an upper critical solution temperature (UCST) or above a lower critical solution temperature (LCST). The binodal curve separates the single-phase from the two-phase region. Within the two-phase region, horizontal tie lines connect the compositions of the coexisting liquids.

Solid-liquid (eutectic) diagrams. When two components are immiscible in the solid state, the T-x diagram shows a eutectic point. The liquidus consists of two branches: one giving the freezing-point depression of A by B, the other the freezing-point depression of B by A. These curves meet at the eutectic composition and temperature, which is the lowest melting point in the entire system. At the eutectic, $F = 0$ (three phases coexist: liquid, solid A, solid B).

The freezing-point depression of component A by solute B follows:

ln x_{A} = \frac{Δ H _{fus, A}}{R} (\frac{1}{T _{f, A}^{*}} - \frac{1}{T _{f}})

where $x_{A}$ is the mole fraction of A in the liquid, $T_{f, A}^{*}$ is the freezing point of pure A, and $T_{f}$ is the freezing point of the solution. This equation generates the liquidus branches of the eutectic diagram.

Lever rule. In any two-phase region of a binary phase diagram, mass conservation requires:

n_{α} (x_{0} - x_{α}) = n_{β} (x_{β} - x_{0})

where $x_{0}$ is the overall composition, $x_{α}$ and $x_{β}$ are the compositions of the two coexisting phases (read from the ends of the tie line), and $n_{α}$ and $n_{β}$ are the amounts of each phase. The fraction of phase $α$ is:

f_{α} = \frac{x _{β} - x _{0}}{x _{β} - x _{α}}

The lever rule applies to any two-phase region: liquid-vapor, liquid-liquid, liquid-solid, or solid-solid. It is a statement of material balance, not a thermodynamic law.

Peritectic reaction. In some binary systems, a liquid reacts with a solid on cooling to form a new solid phase:

liquid + solid α \to solid β

This peritectic reaction occurs at a fixed temperature (invariant at constant pressure, $F = 0$ ). The peritectic point lies between the compositions of the reacting solid and the product solid. Peritectic systems are common when the two components have different crystal structures that cannot form a continuous solid solution. Many technically important systems display peritectics, including Fe-C (steel) and Cu-Zn (brass).

Congruent melting. A compound melts congruently when it forms a liquid of the same composition as the solid. On the phase diagram, congruent melting produces a local maximum in the liquidus at the compound's stoichiometric composition -- the liquidus curves rise to a peak at that point. The compound behaves like a pure substance at that composition: the liquid and solid have identical compositions at the melting point.

Incongruent melting occurs when a compound decomposes on heating into a liquid and a different solid. The phase diagram shows a peritectic horizontal at the decomposition temperature. The compound does not have a true melting point in the usual sense because it does not pass directly from solid to liquid of the same composition.

Key result Intermediate+

Theorem (Clapeyron equation). Along a two-phase coexistence curve for a single-component system, the slope of the phase boundary is:

\frac{d P}{d T} = \frac{Δ H _{trans}}{T Δ V _{trans}}

where $Δ H_{trans}$ and $Δ V_{trans}$ are the enthalpy and volume changes of the phase transition at temperature $T$ .

Proof sketch. Along the coexistence curve, $μ^{α} (T, P) = μ^{β} (T, P)$ . An infinitesimal displacement along the curve must preserve equality: $d μ^{α} = d μ^{β}$ . Using the fundamental relation $d μ = - S_{m} d T + V_{m} d P$ :

- S_{m}^{α} d T + V_{m}^{α} d P = - S_{m}^{β} d T + V_{m}^{β} d P

Rearranging gives $(V_{m}^{β} - V_{m}^{α}) d P = (S_{m}^{β} - S_{m}^{α}) d T$ , and substituting $Δ S_{trans} = Δ H_{trans} / T$ yields the Clapeyron equation.

Corollary (Clausius-Clapeyron equation). For a liquid-vapor (or solid-vapor) equilibrium where the vapor is an ideal gas and $V_{vapor} ≫ V_{condensed}$ :

\frac{d ln P}{d T} = \frac{Δ H _{vap}}{R T ^{2}}

Theorem (Gibbs phase rule). The number of intensive degrees of freedom in a system at thermodynamic equilibrium is $F = C - P + 2$ , where $C$ is the number of independent chemical components and $P$ is the number of coexisting phases.

The proof counts the number of intensive variables (temperature, pressure, and $C - 1$ independent composition variables for each of $P$ phases) and subtracts the number of constraints ( $P - 1$ chemical-potential equalities for each of $C$ components). The result is $F = (P (C - 1) + 2) - C (P - 1) = C - P + 2$ .

Theorem (Lever rule). In a binary system at equilibrium in a two-phase region, the mole fractions of the two phases are:

f_{α} = \frac{x _{β} - x _{0}}{x _{β} - x _{α}}, f_{β} = \frac{x _{0} - x _{α}}{x _{β} - x _{α}}

where $x_{0}$ is the overall composition and $x_{α}$ , $x_{β}$ are the phase compositions at the tie-line endpoints.

The lever rule follows from the material balance $x_{0} = f_{α} x_{α} + f_{β} x_{β}$ together with $f_{α} + f_{β} = 1$ .

Common binary phase-diagram types

System type	Key feature	Example
Isomorphous (complete solid solubility)	Continuous solidus and liquidus	Cu-Ni
Eutectic (no solid solubility)	Eutectic point, three two-phase regions	Pb-Sn
Eutectic (limited solid solubility)	Terminal solid solutions + eutectic	Al-Si
Peritectic	Peritectic reaction L + $α \to β$	Fe-C (at 1495 degrees C)
Congruent melting compound	Liquidus maximum at compound composition	Mg-Zn (MgZn $_{2}$ )
Incongruent melting compound	Peritectic decomposition	Na-K (Na $_{2}$ K)
Minimum-boiling azeotrope	Bubble-point and dew-point curves touch at minimum T	Ethanol-water
Maximum-boiling azeotrope	Bubble-point and dew-point curves touch at maximum T	HCl-water
Liquid-liquid miscibility gap	UCST or LCST, binodal curve	Phenol-water

Distillation on T-x diagrams

For an ideal binary mixture, the T-x-y diagram at constant pressure shows the bubble-point curve below and the dew-point curve above. Heating a liquid of composition $x_{0}$ produces the first vapor bubble at the bubble point, with composition read from the dew-point curve along the horizontal tie line. As distillation proceeds, the liquid enriches in the less volatile component.

The Fenske equation gives the minimum number of plates at total reflux:

N_{m i n} = \frac{ln [ ( \frac{x _{D}}{1 - x _{D}} ) ( \frac{1 - x _{B}}{x _{B}} ) ]}{ln α}

where $α = P_{A}^{*} / P_{B}^{*}$ is the relative volatility, $x_{D}$ is the distillate mole fraction of the more volatile component, and $x_{B}$ is the bottoms mole fraction.

When an azeotrope is present, the bubble-point and dew-point curves merge and distillation cannot cross the azeotropic composition. Breaking the azeotrope requires pressure-swing distillation, extractive distillation, or a third-component entrainer.

Derivation of the liquidus from freezing-point depression Master

The liquidus branches of a binary eutectic phase diagram follow from the freezing-point depression equation applied to each component. For component A in equilibrium with pure solid A:

μ_{A}^{* solid} (T_{f}) = μ_{A}^{* liq} (T_{f}) + R T_{f} ln x_{A}

Expanding the pure-component chemical potentials about the melting point $T_{f, A}^{*}$ and using $Δ S_{fus, A} = Δ H_{fus, A} / T_{f, A}^{*}$ :

\frac{Δ H _{fus, A}}{R} (\frac{1}{T _{f, A}^{*}} - \frac{1}{T _{f}}) = ln x_{A}

This is the Schroeder-van Laar equation for the liquidus of component A. An analogous equation holds for component B. Plotting $T_{f}$ as a function of $x_{A}$ (or $x_{B}$ ) gives the two liquidus branches of the eutectic diagram. The eutectic point is found at the intersection of the two liquidus curves, where both equations are simultaneously satisfied.

For ideal systems, the eutectic composition and temperature can be calculated analytically. Setting the two liquidus equations equal and solving gives:

x_{B, eut} = \frac{1}{1 + exp [ \frac{Δ H _{fus, B}}{R} ( \frac{1}{T _{eut}} - \frac{1}{T _{f, B}^{*}} ) ] / exp [ \frac{Δ H _{fus, A}}{R} ( \frac{1}{T _{eut}} - \frac{1}{T _{f, A}^{*}} ) ]}

The component with the lower melting point and smaller enthalpy of fusion contributes more to lowering the eutectic temperature. In practice, the Schroeder-van Laar equation overestimates the eutectic temperature because it assumes ideal mixing in the liquid and complete immiscibility in the solid -- both approximations that introduce systematic errors.

Non-ideal corrections. For real systems, $ln x_{A}$ is replaced by $ln a_{A} = ln (γ_{A} x_{A})$ . The corrected liquidus becomes $\frac{Δ H _{fus, A}}{R} (\frac{1}{T _{f, A}^{*}} - \frac{1}{T _{f}}) = ln x_{A} + ln γ_{A}$ . Positive deviations ( $γ_{A} > 1$ ) depress the liquidus further than the ideal prediction.

Compound formation and solid-state transformations. When an intermediate compound A $_{m}$ B $_{n}$ exists, the phase diagram divides into two sub-diagrams, each with its own eutectic. Congruent melting produces a liquidus maximum at the compound's stoichiometric composition; incongruent melting produces a peritectic. The Fe-C diagram includes a eutectoid at 727 degrees C where austenite transforms to ferrite + cementite (pearlite), underpinning the heat treatment of steel.

CALPHAD. Modern phase diagrams are calculated from thermodynamic models. The CALPHAD method parameterizes the Gibbs energy of each phase as a function of $T$ , $P$ , and composition, then finds equilibrium by global Gibbs-energy minimization. Software packages (Thermo-Calc, Pandat, FactSage) implement these calculations and are standard tools in materials design.

Distillation column analysis using phase diagrams Master

The design of a fractional distillation column relies on the T-x-y diagram and the McCabe-Thiele method. Each equilibrium stage corresponds to one step on the diagram: a horizontal line to the dew-point curve (vaporization) followed by a vertical line down to the bubble-point curve (condensation). The reflux ratio -- the ratio of liquid returned to distillate taken off -- controls the trade-off between number of plates and product rate. Practical operation uses 1.2--1.5 times the minimum reflux.

For azeotropic systems, the equilibrium curve crosses the $y = x$ diagonal at the azeotropic composition and distillation is trapped. Extractive distillation introduces a high-boiling entrainer that alters the relative volatility and breaks the azeotrope.

Residue curve maps extend the analysis to ternary systems. Distillation boundaries -- separatrices on the residue curve map -- divide composition space into regions that cannot be crossed by simple distillation.

Exercise (medium, symbolic).

The Pb-Sn system at 1 atm has the following data: pure Pb melts at 327 degrees C, pure Sn melts at 232 degrees C, and the eutectic is at 183 degrees C and 61.9 wt% Sn. The solid solubility of Sn in Pb at the eutectic temperature is 19.2 wt% Sn, and the solid solubility of Pb in Sn is 2.5 wt% Sn. An alloy of 40 wt% Sn is cooled slowly from 350 degrees C.

(a) At what temperature does the first solid appear, and what is its composition? (b) At 200 degrees C, what are the compositions and relative amounts of the two phases? (c) Just above the eutectic temperature, what phases are present and in what amounts? (d) Just below the eutectic temperature, describe the microstructure.

Hint

Read the liquidus and solidus compositions from the phase diagram at each temperature. Use the lever rule for the relative amounts. At the eutectic, the remaining liquid (of eutectic composition) transforms to a fine mixture of the two solid phases.

Answer

(a) The liquidus for the Pb-rich side at 40 wt% Sn intersects at approximately 260 degrees C. The first solid is Pb-rich solid solution ( $α$ ) at approximately 12 wt% Sn (read from the solidus at this temperature).

(b) At 200 degrees C, the alloy is in the liquid + $α$ region. The liquid composition is approximately 55 wt% Sn (from the liquidus), and the $α$ composition is approximately 17 wt% Sn (from the solidus). Lever rule:

f_{L} = \frac{17 - 40}{17 - 55} = \frac{- 23}{- 38} = 0.605

f_{α} = \frac{40 - 55}{17 - 55} = \frac{- 15}{- 38} = 0.395

So approximately 60.5% liquid and 39.5% solid $α$ .

(c) Just above 183 degrees C, the liquid has been enriched to nearly the eutectic composition (61.9 wt% Sn). The $α$ phase is at approximately 19.2 wt% Sn. By the lever rule:

f_{L} = \frac{19.2 - 40}{19.2 - 61.9} = \frac{- 20.8}{- 42.7} = 0.487

f_{α} = 0.513

About 48.7% liquid (eutectic composition) and 51.3% proeutectic $α$ .

(d) Just below 183 degrees C, the eutectic liquid solidifies into a lamellar mixture of $α$ (19.2 wt% Sn) and $β$ (97.5 wt% Sn). The final microstructure consists of large proeutectic $α$ grains (formed during cooling above the eutectic) surrounded by a fine eutectic mixture of $α$ + $β$ lamellae. The proportions within the eutectic mixture are determined by the lever rule applied between 19.2 and 97.5 wt% Sn at the eutectic composition of 61.9 wt% Sn.

Exercise (hard, symbolic).

The liquidus curves of the Pb-Sn system can be approximated by the Schroeder-van Laar equation. For Pb: $T_{f}^{*} = 600.6$ K, $Δ H_{fus} = 4.77$ kJ/mol. For Sn: $T_{f}^{*} = 505.1$ K, $Δ H_{fus} = 7.07$ kJ/mol.

(a) Calculate the ideal eutectic composition and temperature by solving the two liquidus equations simultaneously. (b) Compare with the experimental values ( $T_{eut} = 456$ K, $x_{Sn,eut} = 0.73$ mol fraction). Explain the discrepancy.

Hint

Write $ln x_{Pb} = (Δ H_{fus,Pb} / R) (1/ T_{f, Pb}^{*} - 1/ T_{eut})$ and $ln x_{Sn} = (Δ H_{fus,Sn} / R) (1/ T_{f, Sn}^{*} - 1/ T_{eut})$ . Since $x_{Pb} + x_{Sn} = 1$ , these are two equations in two unknowns ( $T_{eut}$ and $x_{Sn}$ ). Solve iteratively.

Answer

(a) The two liquidus equations are:

ln (1 - x_{Sn}) = \frac{4770}{8.314} (\frac{1}{600.6} - \frac{1}{T _{eut}}) = 573.7 (\frac{1}{600.6} - \frac{1}{T _{eut}})

ln x_{Sn} = \frac{7070}{8.314} (\frac{1}{505.1} - \frac{1}{T _{eut}}) = 850.5 (\frac{1}{505.1} - \frac{1}{T _{eut}})

By iteration (testing $T$ values and comparing the $x_{Sn}$ obtained from each equation), the ideal eutectic converges to approximately $T_{eut} \approx 395$ K and $x_{Sn} \approx 0.50$ .

(b) The experimental eutectic is at 456 K and $x_{Sn} = 0.73$ . The ideal calculation predicts a lower temperature and a composition near 0.5. The discrepancy arises because the Schroeder-van Laar equation assumes ideal mixing ( $γ = 1$ ), zero solid solubility, and constant $Δ H_{fus}$ . Real Pb-Sn solutions show positive deviations and nontrivial solid solubility, both of which raise the eutectic temperature and shift the composition toward Sn.

Exercise (hard, symbolic).

A binary system A-B forms a compound A $_{2}$ B that melts congruently at 800 degrees C. Pure A melts at 600 degrees C and pure B melts at 1000 degrees C. There are eutectics at 400 degrees C between A and A $_{2}$ B, and at 700 degrees C between A $_{2}$ B and B.

(a) Sketch the complete phase diagram, labeling all regions, phases, and invariant points. (b) Describe the cooling path for a melt of composition 30 mol% B. How does it differ from the cooling path for 40 mol% B? (c) At 500 degrees C, an alloy of 50 mol% B is in a two-phase region. The solid is A $_{2}$ B and the liquid is at 65 mol% B. Use the lever rule to find the fraction of each phase.

Hint

The compound A $_{2}$ B has composition $x_{B} = 1/3$ (33.3 mol% B). The diagram has two eutectics: one between A ( $x_{B} = 0$ ) and A $_{2}$ B ( $x_{B} = 0.333$ ) at 400 degrees C, and another between A $_{2}$ B ( $x_{B} = 0.333$ ) and B ( $x_{B} = 1$ ) at 700 degrees C. The congruent melting of A $_{2}$ B creates a liquidus maximum at $x_{B} = 0.333$ .

Answer

(a) The diagram has: pure A ( $x_{B} = 0$ , mp 600 degrees C), eutectic E1 ( $x_{B} \approx 0.15$ , 400 degrees C), compound A $_{2}$ B ( $x_{B} = 0.333$ , congruent mp 800 degrees C as a liquidus maximum), eutectic E2 ( $x_{B} \approx 0.65$ , 700 degrees C), and pure B ( $x_{B} = 1.0$ , mp 1000 degrees C). The diagram splits into two sub-diagrams: A-A $_{2}$ B and A $_{2}$ B-B, each with its own eutectic.

(b) For 30 mol% B (A-A $_{2}$ B sub-diagram): primary A $_{2}$ B precipitates at ~720 degrees C, liquid shifts toward A, and at 400 degrees C the remaining liquid solidifies at E1 to A + A $_{2}$ B. For 40 mol% B (A $_{2}$ B-B sub-diagram): primary A $_{2}$ B precipitates at ~780 degrees C, liquid shifts toward B, and at 700 degrees C the liquid solidifies at E2 to A $_{2}$ B + B.

Historical notes Master

The systematic study of phase equilibria began in metallurgy. In 1887, Le Chatelier attempted thermal analysis of alloys; his student Osmond succeeded in 1890, establishing the critical points in the Fe-C system that underpin ferrous metallurgy.

The theoretical foundation was laid by J. Willard Gibbs in "On the Equilibrium of Heterogeneous Substances" (Trans. Conn. Acad. 3, 108--248 and 343--524, 1875--1878). Gibbs derived the phase rule and introduced the chemical potential as the quantity equalized across coexisting phases. His work was so abstract that decades passed before the chemistry community absorbed it. Bakhuis Roozeboom independently rediscovered the phase rule in the 1880s and spent his career classifying binary phase diagrams into types (eutectic, peritectic, compound-forming, miscibility-gap), establishing the taxonomy still used today.

The Clausius-Clapeyron equation predates Gibbs. Clapeyron derived $d P / d T = Δ S /Δ V$ in 1834 from Carnot-cycle analysis, and Clausius refined it in 1850 using the first and second laws.

Wade and Merriman initiated the systematic study of azeotropes in 1911, coining the term "azeotrope" (Greek: "boiling without change"). The CALPHAD method was pioneered by Kaufman and Bernstein in 1970, transforming phase-diagram work from pure experiment to computational thermodynamics. Norman Bowen applied phase diagrams to geology in the 1920s, establishing the phase relationships governing magma crystallization (Bowen's reaction series).

Connections Master

Colligative properties (14.09.03): The liquidus branches of binary phase diagrams are generated by the same freezing-point depression equation that underpins the colligative framework — phase diagrams are the graphical representation of colligative effects extended across the full composition range.
Thermodynamics (14.06): The Clapeyron and Clausius-Clapeyron equations governing phase-boundary slopes are direct consequences of the equality of chemical potentials across coexisting phases, a central result of the thermodynamics unit. The Gibbs phase rule is a theorem about the dimensionality of the equilibrium manifold.
Acid-base equilibria (14.10): Many metal hydroxide phase boundaries depend on pH, linking binary phase diagrams to acid-base chemistry. The solubility-pH diagrams for sparingly soluble salts are effectively phase diagrams with pH as a composition variable.

Bibliography Master

Exercise (hard, symbolic).

Use the Clapeyron equation to estimate the pressure required to lower the melting point of ice by 1.00 degree C. For water at 0 degrees C: $Δ H_{fus} = 6.01$ kJ/mol, $V_{m}^{liquid} = 18.02$ cm $^{3}$ /mol, $V_{m}^{solid} = 19.64$ cm $^{3}$ /mol. Compare the slope $d P / d T$ with the liquid-vapor Clapeyron slope at the same temperature ( $Δ H_{vap} = 45.05$ kJ/mol, treating vapor as ideal gas).

Hint

Apply $d P / d T = Δ H_{fus} / (T Δ V_{fus})$ with $Δ V_{fus} = V_{m}^{L} - V_{m}^{S} = 18.02 - 19.64 = - 1.62$ cm $^{3}$ /mol. The negative $Δ V$ gives a negative slope.

Answer

For the solid-liquid boundary:

\frac{d P}{d T} = \frac{6010}{( 273.15 ) ( - 1.62 \times 1 0 ^{- 6} )} = - 1.36 \times 1 0^{7} Pa/K = - 134 atm/K

To lower the melting point by 1.00 K requires approximately 134 atm of increased pressure. This enormous value explains why pressure melting plays a negligible role in ice skating.

For the liquid-vapor boundary at 0 degrees C ( $P_{vap} = 611$ Pa):

\frac{d P}{d T} = \frac{Δ H _{vap}}{T Δ V _{vap}} = \frac{45050}{( 273.15 ) ( 3.72 )} = 44.3 Pa/K

The liquid-vapor slope (44.3 Pa/K) is nearly six orders of magnitude smaller than the solid-liquid slope ( $- 1.36 \times 1 0^{7}$ Pa/K), reflecting the huge volume change of vaporization versus the tiny volume change of melting.

Exercise (hard, symbolic).

The Antoine equation provides an accurate empirical representation of vapor pressure: $lo g_{10} P = A - B / (C + T)$ where $P$ is in mmHg and $T$ is in degrees C. For benzene: $A = 6.90565$ , $B = 1211.033$ , $C = 220.790$ . For toluene: $A = 6.95464$ , $B = 1344.800$ , $C = 219.482$ .

(a) Calculate the bubble-point temperature and vapor composition for a liquid mixture of 40 mol% benzene at $P = 760$ mmHg, assuming Raoult's law. (b) Plot the T-x-y diagram for this system and identify the range of compositions over which fractional distillation can achieve 95 mol% benzene purity in the distillate.

Hint

At the bubble point, $P = x_{A} P_{A}^{*} (T) + x_{B} P_{B}^{*} (T) = 760$ mmHg. Solve for $T$ iteratively. Then $y_{A} = x_{A} P_{A}^{*} (T) / P$ .

Answer

(a) At $x_{B} = 0.40$ , solve $0.40 \cdot P_{benz}^{*} (T) + 0.60 \cdot P_{tol}^{*} (T) = 760$ mmHg iteratively. At $T = 95$ degrees C: $P_{benz}^{*} = 1 0^{6.90565 - 1211.033/315.79} = 1178$ mmHg, $P_{tol}^{*} = 1 0^{6.95464 - 1344.800/314.48} = 485$ mmHg. $P = 0.40 (1178) + 0.60 (485) = 762$ mmHg. Bubble point $\approx 95.0$ degrees C. $y_{benz} = 0.40 \times 1178/762 = 0.619$ .

(b) Benzene-toluene has no azeotrope, so distillation can achieve 95 mol% benzene from any $x_{0} > 0$ . From $x_{0} = 0.40$ , the Fenske equation gives $N_{m i n} = ln [(0.95/0.05) (0.60/0.40)] / ln 2.5 \approx 3.7$ , so 4 plates at total reflux. In practice, 8--12 plates suffice.

Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. Transactions of the Connecticut Academy of Arts and Sciences, 3, 108--248, 343--524 (1875--1878). The foundational work on the phase rule and chemical potential.

Clapeyron, E. Memoire sur la puissance motrice de la chaleur. Journal de l'Ecole Royale Polytechnique, 14, 153--190 (1834). Original derivation of the Clapeyron equation from Carnot-cycle analysis.

Clausius, R. Uber die bewegende Kraft der Warme. Annalen der Physik, 155, 368--397 (1850). Refinement of Clapeyron's work using the first and second laws.

Roozeboom, H. W. B. Die heterogenen Gleichgewichte vom Standpunkte der Phasenlehre. Vieweg, Braunschweig, 5 vols. (1901--1918). First systematic classification of phase-diagram types.

Osmond, F. and Roberts-Austen, W. C. On the Critical Points of Iron and Steel. Philosophical Transactions of the Royal Society A, 184, 573--618 (1893). Experimental determination of the Fe-C phase diagram.

Bowen, N. L. The Evolution of the Igneous Rocks. Princeton University Press, 1928. Phase equilibria applied to magma crystallization and the Bowen reaction series.

Wade, J. and Merriman, R. W. CXLVII. -- The Influence of Water on the Boiling Points of Acetic Acid and of Ethyl Alcohol, at Different Pressures. Journal of the Chemical Society, Transactions, 99, 997--1011 (1911). Systematic study of azeotropes.

Kaufman, L. and Bernstein, H. Computer Calculation of Phase Diagrams. Academic Press, 1970. Foundation of the CALPHAD method.

Zumdahl, S. S. and DeCoste, D. J. Chemical Principles, 8e. Cengage, 2017. Ch. 11.7--11.8 for introductory phase diagrams and the Clausius-Clapeyron equation.

Atkins, P. and de Paula, J. Physical Chemistry, 12e. Oxford University Press, 2023. Ch. 5 for the thermodynamic treatment of phase equilibria.

Engel, T. and Reid, P. Thermodynamics, Statistical Thermodynamics, and Kinetics, 4e. Pearson, 2019. Ch. 8 for binary phase diagrams and the lever rule.

Massalski, T. B. (ed.) Binary Alloy Phase Diagrams, 2e. ASM International, 1990. Comprehensive compilation of experimentally determined binary phase diagrams.

Saunders, N. and Miodownik, A. P. CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide. Pergamon, 1998. Definitive reference on computational thermodynamics of phase equilibria.