VisualChem The interactive chemistry textbook
中文

Lesson 6 · 30 min

Binary phase diagrams

Why does distillation never give 100 % alcohol, and why does solder melt more easily than tin or lead? Put composition across: Raoult’s law draws the liquid–vapour lens, the lever rule gives the amounts of two phases, and azeotropes and eutectics explain the limits of distillation and the arrests of cooling curves.

In this lesson

  • Build the temperature–composition diagram of an ideal binary solution from Raoult’s law, and read bubble points, dew points and the compositions of coexisting phases
  • Use the lever rule to find how much of each phase a two-phase mixture contains
  • Explain why fractional distillation separates benzene from toluene but takes ethanol only to 95.6 %
  • Read eutectic and solid-solution diagrams, and explain the breaks and arrests of cooling curves with the phase rule

A one-component phase diagram has two variables, temperature and pressure. Mix two substances and composition becomes a third, and familiar puzzles follow: however often you distil it, alcohol never reaches 100 %; solder melts lower than both tin and lead; salted ice melts at well below zero. Fix the pressure at 1 atm, put composition across and temperature up, and every phase equilibrium of a binary system fits on one flat diagram.

Predict first

An equimolar mixture of benzene and toluene is heated at 1 atm until it just begins to boil. What is the composition of the first vapour?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Raoult’s law and the temperature–composition diagram

Raoult’s law: in an ideal solution, each component’s partial vapour pressure is its mole fraction times the vapour pressure of the pure liquid. At 1 atm, the temperature at which the total vapour pressure reaches 1 atm is the bubble point, and the vapour there has yA=xApA∗/py_\text{A} = x_\text{A}p^*_\text{A}/p. Join the bubble points into the liquid line and the matching vapour compositions into the vapour line, and the two enclose a lens: liquid below, vapour above, liquid and vapour together inside. Draw a horizontal line at any temperature in the lens; where it meets the two lines are the compositions of the two phases. It is called the tie line.

p=xA pA∗(T)+xB pB∗(T),nLnV=y−zz−xp = x_\text{A}\,p^*_\text{A}(T) + x_\text{B}\,p^*_\text{B}(T), \qquad \frac{n_\text{L}}{n_\text{V}} = \frac{y - z}{z - x}
Left: Raoult’s law gives the bubble point. Right: the lever rule — the overall composition z splits the tie line in two, and each phase’s amount is proportional to the length of the segment opposite it.
x,yx, y
mole fraction of A in the liquid and the vapour
zz
mole fraction of A in the whole system
p∗(T)p^*(T)
vapour pressure of the pure component
nL,nVn_\text{L}, n_\text{V}
amounts of liquid and vapour
Raoult’s law · azeotrope · eutectic · solid solution · all computed

Binary phase diagrams: a tie line and a lever

liquid vapour 00.20.40.60.81mole fraction of benzene80859095100105110temperature / °C
  • liquid line (bubble points)
  • vapour line (dew points)
  • tie line
Click anywhere on the diagram to place the state point

liquid + vapour. Two phases: liquid of composition 0.402 and vapour of composition 0.625; by the lever rule, 56.1 % is liquid.

Phase rule (constant pressure): F = C − P + 1 = 2 − 2 + 1 = 1

Lever rule: amounts of the two phases
Benzene–toluene follows Raoult’s law (activity coefficients of 1) with Antoine constants from the NIST Chemistry WebBook; ethanol–water uses a three-suffix Margules model calibrated to the 1 atm azeotrope (x = 0.895, 78.15 °C). The bismuth–cadmium liquidus is the ideal-solution Schröder–van Laar equation with both solids treated as pure; its eutectic lies about 10 K below the measured one (≈ 145 °C at x(Cd) ≈ 0.55). The cooling curve is a heat balance of Newtonian cooling against the latent heat of freezing. Copper–nickel assumes ideal liquid and solid solutions and is only a sketch; real Cu–Ni solid solutions also have a miscibility gap at low temperature.
Drag the temperature and watch the state point leave the liquid, cross the lens and enter the vapour: inside the two-phase region the tie line’s ends give the two compositions and the bar below gives their amounts by the lever rule. Then switch to ethanol–water and find where the two lines touch: the azeotrope.

Worked example

A single flash

A benzene–toluene mixture with 50 mol % benzene is heated to 95 °C at 1 atm. The liquid then contains 0.402 and the vapour 0.625 mole fraction of benzene. How much is liquid and how much vapour?

Fractional distillation and the azeotrope

Condense the first vapour and boil it again, and the benzene is enriched another step; a fractionating column stacks many such vaporise–condense cycles, each one a theoretical plate. For benzene–toluene the liquid and vapour lines stay apart across the whole diagram, so with enough plates nearly pure benzene comes off the top.

Ethanol–water is different. It deviates positively from Raoult’s law: both activity coefficients exceed 1, the vapour pressure lies above Raoult’s prediction, and the bubble-point curve dips to a minimum at x = 0.895: the azeotrope, at 78.15 °C. There the liquid and the vapour have the same composition, and no number of plates can cross it — which is why commercial ethanol is 95.6 % by mass. Absolute ethanol needs another trick, such as adding a third component or drying with molecular sieves.

Solid–liquid diagrams: eutectics and solid solutions

Bismuth and cadmium mix completely as liquids but hardly at all as solids. Add cadmium to molten bismuth and the freezing point falls — the same reason salt lowers the freezing point of water. For an ideal solution each liquidus obeys ln⁡xi=−ΔHfusR(1T−1Tm)\ln x_i = -\dfrac{\Delta H_\text{fus}}{R}\left(\dfrac{1}{T} - \dfrac{1}{T_\text{m}}\right). The point where the two liquidus lines meet is the eutectic: the ideal model puts it at 135 °C and a cadmium mole fraction of 0.565, against about 145 °C and 0.55 measured. Liquid of that composition freezes into two kinds of crystal at once, interleaved in a fine eutectic structure.

Cool a melt with 0.30 cadmium: at 203 °C bismuth crystals begin to form, the latent heat they release slows the cooling, and the curve shows a break; at the eutectic temperature the remaining liquid freezes all together and the temperature stands still in an arrest. At constant pressure the phase rule reads F = C − P + 1: with two phases F = 1 and the temperature can still change; with liquid and two solids, F = 0 and the temperature must hold until the liquid is gone.

Copper and nickel atoms are similar in size and crystal structure and mix in any proportion even as solids. Their diagram is a lens bounded by two lines — solidus below, liquidus above — the same shape as the liquid–vapour lens of benzene–toluene.

Raoult’s law · azeotrope · eutectic · solid solution · all computed

Binary phase diagrams: a tie line and a lever

ideal eutectic 135.3 °C measured eutectic ≈ 145 °C liquid liquid + Bi(s) liquid + Cd(s) Bi(s) + Cd(s) 00.20.40.60.81mole fraction of cadmium100150200250300temperature / °C
  • liquidus
  • tie line
Click anywhere on the diagram to place the state point

liquid + Bi(s). Two phases: liquid of composition 0.314 and Bi(s) of composition 0.000; by the lever rule, 95.6 % is liquid.

Phase rule (constant pressure): F = C − P + 1 = 2 − 2 + 1 = 1

Lever rule: amounts of the two phases
Cooling curve of this melt
020040060080010001200time / s100150200250300temperature / °C

Break: 203.3 °C; arrest: 135.3 °C, lasting 376 s

Benzene–toluene follows Raoult’s law (activity coefficients of 1) with Antoine constants from the NIST Chemistry WebBook; ethanol–water uses a three-suffix Margules model calibrated to the 1 atm azeotrope (x = 0.895, 78.15 °C). The bismuth–cadmium liquidus is the ideal-solution Schröder–van Laar equation with both solids treated as pure; its eutectic lies about 10 K below the measured one (≈ 145 °C at x(Cd) ≈ 0.55). The cooling curve is a heat balance of Newtonian cooling against the latent heat of freezing. Copper–nickel assumes ideal liquid and solid solutions and is only a sketch; real Cu–Ni solid solutions also have a miscibility gap at low temperature.
Change the cadmium content and watch the cooling curve: a pure metal has a single arrest; melts away from the eutectic composition show a break and then an arrest; at exactly the eutectic composition the break disappears and the arrest is longest.
Four kinds of binary diagram
SystemShapeSpecial point
benzene–tolueneliquid–vapour lensnone
ethanol–watertwo lenses touching at the azeotropeazeotrope 78.15 °C, x = 0.895
bismuth–cadmiumtwo liquidus lines meeting at the eutecticeutectic ≈ 145 °C (ideal model 135 °C)
copper–nickelsolid–liquid lensnone

Concept check

Which mixture can a sufficiently efficient fractionating column separate into two nearly pure components?

Try the calculation

A benzene–toluene mixture of overall composition z = 0.500 splits at some temperature into liquid with x = 0.402 and vapour with y = 0.625. What percentage of the total amount is liquid?

%

Key takeaways

  • Raoult’s law gives the vapour pressure of an ideal solution; at 1 atm the bubble-point and dew-point lines enclose a lens, and the vapour is always richer in the more volatile component.
  • In a two-phase region the ends of the tie line give the two compositions and the lever rule gives the two amounts.
  • Ethanol–water forms a minimum-boiling azeotrope at 78.15 °C and x = 0.895 (95.6 % by mass), which distillation cannot cross.
  • In a eutectic system the liquidus lines meet at the eutectic; with three phases at constant pressure F = 0, and the cooling curve shows an arrest.
Progress is stored only in this browser and never uploaded.

Data and references

  1. P. J. Linstrom & W. G. Mallard (eds.), NIST Chemistry WebBook, SRD 69: Antoine parameters for benzene (Eon, Pommier et al. 1971) and toluene (Williamham, Taylor et al. 1945).
  2. P. Atkins, J. de Paula & J. Keeler, Atkins’ Physical Chemistry, 11th ed. (Oxford, 2018), Topics 5C–5D (phase diagrams of binary systems).
  3. Bi–Cd eutectic: 144–146 °C at 60 wt % Bi / 40 wt % Cd (x(Cd) ≈ 0.55), as in the cooling-curve story; pure-metal melting points and enthalpies of fusion from the CRC Handbook.
  4. CRC Handbook of Chemistry and Physics, 97th ed. (2016): melting points and enthalpies of fusion of Bi, Cd, Cu and Ni; Sn–Pb and NaCl–H₂O eutectics.