The entropy that arises because mixing increases the number of ways atoms can be arranged. In an ideal solution it is −R[x ln x + (1 − x) ln(1 − x)], with its maximum at x = 0.5.
Japanese name: 混合エントロピー
Entropy of mixing appears in 2 practice questions across 1 topic, including Thermodynamics of materials.
Field: Thermodynamics of materials
The number of ways to place N_A atoms of A and N_B atoms of B on the lattice sites of a crystal is W = N!/(N_A!N_B!), while a pure substance can be arranged in only one way. Applying Stirling's approximation to Boltzmann's formula S = k ln W gives the entropy of mixing per mole as ΔS_mix = −R[x ln x + (1 − x) ln(1 − x)]. This is configurational entropy and does not include entropy from atomic vibrations or magnetism. It takes its maximum value R ln 2 ≈ 5.76 J/(mol·K) at x = 0.5. The same form of expression applies to mixing ideal gases.
The entropy of mixing is not negligible even for small amounts of solute. For example, at x = 0.01, ΔS_mix ≈ 0.47 J/(mol·K), which at 1000 K lowers the Gibbs energy by −TΔS_mix ≈ −470 J/mol. As x approaches 0, the slope of the curve becomes infinitely steep, so even sparingly soluble elements always dissolve to a tiny extent, and removing impurities becomes harder the closer purity is pushed to its limit. Because the entropy of mixing acts on G in the form −TΔS, the drive toward mixing is stronger at higher temperature, which is why in many alloys the solubility limit widens at higher temperatures.
Mixing equal amounts of n elements gives an entropy of mixing of R ln n; for five elements it is R ln 5 ≈ 13.4 J/(mol·K) (about 1.61R). Since around 2004, alloys containing many principal elements in nearly equal proportions have been studied as high-entropy alloys. The name comes from the idea that a large entropy of mixing suppresses the formation of intermetallic compounds and stabilizes simple solid solutions, and one definition calls alloys with an entropy of mixing of 1.5R or more high-entropy alloys. In practice, however, many cases have been found in which compounds or multiple phases form because of the enthalpy of mixing and differences in atomic size.
When long-chain polymers are mixed, the atoms in each chain move together while connected, so the number of arrangements is far smaller than for small molecules of the same mass. In Flory–Huggins theory, the entropy of mixing becomes smaller in inverse proportion to the degree of polymerization, which is why polymer blends phase-separate readily even with a slightly positive heat of mixing. Also, when an alloy orders, the positions of atoms become fixed, so its configurational entropy becomes smaller and reaches 0 for complete order. Order–disorder transformations can be understood as a competition between enthalpy, which favors order at low temperature, and the entropy of mixing, which wins at high temperature.
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.
For ideal mixing of two components it is largest at x = 0.5, where its value is R ln 2 ≈ 5.76 J/(mol·K). For n components mixed in equal amounts it is R ln n, which grows as the number of components increases, but only slowly because the growth is logarithmic.
The configurational entropy of an ideal solution depends only on composition, not on temperature. Its contribution to the Gibbs energy, however, is −TΔS_mix, so its effect grows with temperature. In real solutions the degree of short-range order changes with temperature, so the entropy of mixing also changes somewhat with temperature.
Not always. Whether they mix is decided by the Gibbs energy of mixing ΔH_mix − TΔS_mix, and if the enthalpy of mixing is large and positive, separation is more stable at low temperature. If ΔH_mix = 0, as in an ideal solution, they mix at every composition.
Above Tc, the contribution of the entropy of mixing wins, and a single-phase solid solution forms at all compositions.
In the regular solution model, the enthalpy of mixing is ta…In an ideal solution the enthalpy of mixing is 0, and ΔG_mix = −TΔS_mix arises only from the entropy of mixing.
Which statement about the Gibbs energy of mixing of an idea…Regular solution Gibbs energy Activity Chemical potential Ellingham diagram Equilibrium oxygen partial pressure Spinodal decomposition