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Thermodynamics of materials

Which statement about the Gibbs energy of mixing of an ideal solution (ideal solid solution), ΔG_mix = RT[x ln x + (1 − x) ln(1 − x)], is correct? x is the mole fraction of one component.

  1. AIt is negative at all compositions and its slope tends to minus infinity as x → 0, so a tiny amount of solute dissolves in any substance
  2. BIt arises from the enthalpy of mixing, so it is independent of temperature and turns positive at some compositions, causing separation
  3. CIt reaches its largest positive value at x = 0.5, and compositions close to a pure substance mix less readily
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Answer: A. It is negative at all compositions and its slope tends to minus infinity as x → 0, so a tiny amount of solute dissolves in any substance

In an ideal solution the enthalpy of mixing is 0, and ΔG_mix = −TΔS_mix arises only from the entropy of mixing. For 0 < x < 1, ln x is negative, so ΔG_mix is negative at all compositions and takes its minimum value RT ln 0.5 at x = 0.5 (about −5.76 kJ/mol at 1000 K). As x → 0, the slope d(ΔG_mix)/dx = RT ln[x/(1 − x)] tends to minus infinity, so dissolving even a tiny amount of solute always lowers the free energy. This is the thermodynamic reason why it is difficult to make a perfectly pure substance. ΔG_mix is proportional to T and has nothing to do with enthalpy. What is largest at x = 0.5 is the entropy of mixing; ΔG_mix is at its minimum there.

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