A solution model in which the entropy of mixing is the same as for an ideal solution and the enthalpy of mixing is Ωx(1 − x). When Ω is positive, a miscibility gap appears at low temperature.
Japanese name: 正則溶液
Regular solution appears in 1 practice question across 1 topic, including Thermodynamics of materials.
Field: Thermodynamics of materials
The regular solution is a model that assumes atoms mix randomly, so that the entropy of mixing is the same as for an ideal solution, and adds only the enthalpy of mixing from bonds between unlike atoms. Hildebrand named it in 1929. Considering only bonds between nearest neighbors, the interaction parameter is Ω = zN_A[ε_AB − (ε_AA + ε_BB)/2], where z is the number of nearest neighbors and ε the energy of each bond. If unlike bonds are weaker than the average of like bonds, Ω is positive and atoms of the same kind tend to cluster. If they are stronger, Ω is negative, showing a tendency toward ordering, in which unlike atoms sit next to each other.
In a regular solution, the activity coefficient of component A is given by RT ln γ_A = Ω x_B². For example, with Ω = 10 kJ/mol at 1000 K, γ_A ≈ 1.35 at x_B = 0.5 and γ_A ≈ 3.33 in the limit where A is very dilute, so the activity is larger than in an ideal solution. When Ω is positive and the temperature is low enough, a miscibility gap appears. The critical temperature at its top is Tc = Ω/(2R). At T = 0.8Tc, the compositions of the two coexisting phases are x ≈ 0.145 and 0.855, and the spinodal compositions are x ≈ 0.276 and 0.724.
The regular solution is an inconsistent approximation in that it assumes atoms mix randomly even though they interact. When the magnitude of Ω is large compared with RT, short-range order actually forms in which like or unlike atoms gather, and the entropy of mixing is smaller than for an ideal solution. In real alloys, Ω also changes with temperature and composition, and miscibility gaps are often not symmetric. More precise treatments therefore use the quasi-chemical approximation or subregular solution expressions that extend Ω to a polynomial in composition. Even so, the value of this model is that it shows the tendency toward phase separation or ordering with a single number.
In the CALPHAD method, the excess Gibbs energy of a solution phase is expressed by a Redlich–Kister polynomial of the form x_A x_B Σ L_k (x_A − x_B)^k, whose first term L₀ corresponds to Ω of the regular solution. Ω can be obtained from calorimetric measurements of the heat of mixing, from activities measured by vapor pressure or electromotive force, from the shape of miscibility gaps and so on. It is also sometimes estimated from bond energies found by empirical methods or first-principles calculations. For systems with a miscibility gap in the solid solution, such as Fe-Cr, the regular solution approach gives a rough estimate of the range of temperature and composition where phase separation occurs.
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.
An ideal solution has zero enthalpy of mixing and mixes through the entropy of mixing alone. A regular solution keeps the entropy of mixing of an ideal solution and adds an enthalpy of mixing Ωx(1 − x). With Ω = 0, a regular solution reduces to an ideal solution.
Unlike atoms attract, so the enthalpy of mixing is negative and no miscibility gap forms at any temperature. The activity coefficients are less than 1 and the activities lower than in an ideal solution. In systems where Ω is strongly negative, superlattices and intermetallic compounds actually form readily, and the assumption of random mixing no longer holds.
Because the enthalpy of mixing has the form x(1 − x), symmetric in composition, the gap is also symmetric about x = 0.5, with its top at x = 0.5 and T = Ω/(2R). In real alloys the gap is often asymmetric, and then it is described by expressions in which Ω varies with composition.
Entropy of mixing Activity Chemical potential Ellingham diagram Equilibrium oxygen partial pressure Gibbs energy Spinodal decomposition