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Cooling curves: the bismuth–cadmium diagram

Let bismuth–cadmium melts of several compositions cool slowly, read the liquidus and eutectic temperatures from the breaks and arrests of their curves, and draw the phase diagram yourself.

Difficulty ●●○ ≈ 35 min

Aim Map the liquidus and the eutectic of the bismuth–cadmium system by thermal analysis, and locate the eutectic composition with a Tammann triangle.
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.

Procedure

  1. Choose a melt with the composition slider (the axis is the mole fraction of cadmium). The cooling curve below is the temperature–time record as the melt cools from 340 °C in a switched-off furnace.
  2. Read the temperature at which the slope breaks (crystals begin to form: the liquidus), and the temperature and length of the arrest.
  3. Repeat for pure bismuth, x = 0.2, 0.4, 0.565, 0.8 and pure cadmium, and tabulate the break and arrest temperatures.
  4. Plot the breaks against composition to get the liquidus; the arrests give the eutectic line; plot arrest length against composition (a Tammann triangle), and the two straight sides meet at the eutectic composition.
  5. Compare with the measured eutectic (about 145 °C at a cadmium mole fraction of about 0.55) and account for the ideal-solution model’s error.

What to notice

  • Pure bismuth and pure cadmium each show a single arrest, at 271.4 °C and 321.1 °C.
  • Other compositions show a break and then an arrest; every arrest is at the same temperature (135.3 °C in the model), whatever the composition.
  • The nearer the eutectic composition, the lower the break and the longer the arrest; at exactly the eutectic composition the break disappears and the arrest is longest.

In a real lab

  • Cadmium and its vapour are toxic and carcinogenic: work in a fume hood and keep the melt below 350 °C; many teaching labs use tin–bismuth or tin–lead instead, with the same physics.
  • A layer of graphite powder on the melt slows oxidation; the thermocouple sits in a glass sheath in the middle of the melt.
  • Real curves often supercool before an arrest — the temperature dips below the plateau and recovers once crystallisation starts. Read the flat part after the recovery.

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