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Thermodynamics of materials

In the regular solution model, the enthalpy of mixing is taken as ΔH_mix = Ωx(1 − x) (Ω is an interaction parameter independent of temperature), and the entropy of mixing is the same as for an ideal solution. For Ω = +20 kJ/mol, which value is closest to the critical temperature Tc = Ω/(2R) at which the region where the solid solution separates into 2 phases (the miscibility gap) disappears? Take the gas constant as 8.314 J/(mol·K).

  1. AAbout 601 K
  2. BAbout 2406 K
  3. CAbout 1203 K
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Answer: C. About 1203 K

Tc = Ω/(2R) = 20000/(2 × 8.314) ≈ 1203 K. This formula is obtained as the temperature at which the second derivative of ΔG_mix, −2Ω + RT/[x(1 − x)], becomes 0 at the symmetric composition x = 0.5. Below Tc, the middle of the ΔG_mix curve becomes convex upward and 2 minima appear, so separating into 2 phases of different composition lowers the free energy. Above Tc, the contribution of the entropy of mixing wins, and a single-phase solid solution forms at all compositions. About 601 K uses 4R in the denominator, and about 2406 K uses R. If Ω is negative (unlike atoms attract), no miscibility gap forms, and there is a tendency toward ordering or compound formation.

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