Solutions, Phase Equilibria and Electrochemistry

Regular solution model activity

A regular solution is a model in which the entropy of mixing is ideal, meaning random, while the enthalpy of mixing is finite. All non-ideal behaviour is carried by one interaction parameter \Omega, which is taken as independent of temperature in the simplest model. For a binary A-B solution, \Delta H^{mix} = \Omega x_A x_B, and since the excess entropy is zero, the excess Gibbs energy is G^{XS} = \Omega x_A x_B. The activity coefficients are RT\ln\gamma_A = \Omega x_B^2 and RT\ln\gamma_B = \Omega x_A^2, and each activity follows from a_i = \gamma_i x_i. A positive \Omega gives a positive deviation from Raoult's law, while a negative \Omega gives a negative deviation.

• Interaction parameter \Omega — the constant that measures the bond energy difference, \Omega = zN_A\left(\varepsilon_{AB} - \tfrac{1}{2}(\varepsilon_{AA}+\varepsilon_{BB})\right), where z is the coordination number.
• Dilute limit — as x_B \to 0, RT\ln\gamma_B^{\infty} = \Omega, so one dilute activity coefficient gives \Omega directly.

Example: for a regular solution with \Omega = -4000 J mol^{-1} at 1000 K and x_A = x_B = 0.5, \ln\gamma_A = \Omega x_B^2/RT = -0.120, so \gamma_A = 0.887 and a_A = 0.443. The partial molar excess enthalpy of a component equals RT\ln\gamma_i in this model, so a measured partial enthalpy at a given composition fixes the activity coefficient and the activity. The interaction parameter can be recovered from the integral enthalpy at one composition through \Omega = \Delta H^{mix}/(x_A x_B), after which the activity at another composition follows from the same equations. An excess Gibbs energy written as \Omega x_A x_B identifies a regular solution, and the activity of B then follows from RT\ln\gamma_B = \Omega x_A^2. A regular solution therefore has a finite, non-zero enthalpy of mixing together with an ideal entropy of mixing, so both quantities are finite for intermediate compositions.

Electrochemical cell free energy

The Gibbs energy change of a cell reaction is tied to its emf by \Delta G = -nFE, where n is the number of electrons transferred in the balanced reaction and F is the Faraday constant, about 96500 C mol^{-1}. A positive emf therefore means a negative \Delta G and a spontaneous reaction. The standard value is \Delta G^\circ = -nFE^\circ, and for a non-standard reaction quotient Q the general relation is \Delta G = \Delta G^\circ + RT\ln Q.

• Standard cell potential — E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode}, with both values written as reduction potentials. A table that lists oxidation potentials must first have its signs reversed.
• Chemical potential of an ion — taking the metal in its standard state as the zero of reference, a reduction half-reaction gives \mu^\circ_{M^{n+}} = nFE^\circ. For example, Cu^{2+} with E^\circ = +0.34 V gives \mu^\circ_{Cu^{2+}} = +65.6 kJ mol^{-1}, which agrees with the positive standard Gibbs energy of formation of the aqueous ion.

The standard chemical potential of a metal ion follows directly from its standard half-cell potential, so a negative E^\circ gives a negative \mu^\circ for the ion. For a cell in which a compound is formed from its elements, the standard Gibbs energy of formation is obtained from the measured emf, with the activities of the species in the alloy and electrolyte, and any gas partial pressure, entering Q. When two couples are combined into a spontaneous cell, the standard cell potential is the difference of their reduction potentials, and the corresponding standard Gibbs energy is -nFE^\circ_{cell}.

Gibbs phase rule

The Gibbs phase rule relates the degrees of freedom F to the number of components C and phases P at equilibrium, F = C - P + 2, where the 2 accounts for temperature and pressure. For a reacting system the number of components is C = S - R, where S is the number of species and R is the number of independent reactions among them. All gas species together form one phase, while each solid compound and each pure solid element counts as its own phase.

• Constant pressure — with pressure fixed, the rule becomes F = C - P + 1. In a binary alloy, a point inside a two-phase field therefore has one degree of freedom, which is temperature.
• Constant temperature and pressure — the rule reduces to F = C - P, so the largest number of phases that can coexist equals the number of components.

A reacting system with several gas and solid species is reduced to its independent components by subtracting the number of independent reactions, and this step is the most common source of error. At constant temperature and pressure, the reduced form F = C - P means that the maximum number of phases in equilibrium is the number of components. In a binary alloy at constant pressure, a point inside a two-phase field has F = 3 - 2 = 1, so only temperature can change while both phases remain in equilibrium.

Lever rule in binary phase diagrams

In a two-phase field the overall alloy composition C_0 and the phase compositions C_\alpha and C_\beta are linked by the lever rule. With phase fractions f_\alpha and f_\beta, we have f_\alpha + f_\beta = 1 and C_0 = f_\alpha C_\alpha + f_\beta C_\beta, so f_\alpha = \frac{C_\beta - C_0}{C_\beta - C_\alpha}. The three compositions must be on the same basis, so wt% gives mass fractions and at% gives mole fractions.

• Tie-line — the horizontal line in a two-phase field joining the two phase boundaries; its end points give the compositions of the coexisting phases.
• Molar ratio of phases — if the molar ratio of two phases is known, their fractions follow from the ratio, and the lever rule then gives the unknown phase composition.

Example: an alloy of 30 wt% B in an \alpha + \beta field whose boundaries are 10 wt% B and 70 wt% B has f_\alpha = (70-30)/(70-10) = 2/3. Of 1000 g, 666.7 g is \alpha containing 66.7 g of B. A known molar proportion of two phases fixes their fractions, and the lever rule then gives the composition of one phase from the overall composition of the alloy. On slow cooling of a hypoeutectic alloy, the primary phase forms first with its fraction set by the lever rule just above the eutectic temperature, and the remaining liquid transforms into eutectic. The ratio of primary to eutectic \alpha is therefore found from two successive lever calculations. For example, with eutectic liquid at 40 wt% B, eutectic \alpha at 10 wt% B and eutectic \beta at 90 wt% B, an alloy of 30 wt% B gives primary \alpha of 1/3 and eutectic \alpha of 2/3 \times (90-40)/(90-10) = 0.417, so the ratio is 0.8. The mass of a solute in a phase equals the mass of that phase times the composition of the phase. A 500 g charge of 25 wt% B lying between a 5 wt% B \alpha boundary and an 85 wt% B \beta boundary has f_\alpha = (85-25)/80 = 0.75, so the \alpha mass is 375 g and it holds 18.75 g of B.

Ideal solution mixing entropy

An ideal solution is one whose components mix randomly with no heat effect and no volume change, so \Delta H^{mix} = 0 and \Delta V^{mix} = 0. The entropy of mixing is then purely configurational and positive, \Delta S^{mix} = -R\sum_i x_i \ln x_i, and the molar Gibbs energy of mixing is \Delta G^{mix} = RT\sum_i x_i \ln x_i, which is negative at every composition.

• Configurational entropy — the entropy arising from the number of ways atoms can be distributed among lattice sites; for n total moles it is \Delta S = -nR\sum_i x_i\ln x_i.
• Maximum at equiatomic composition — for a binary, the entropy of mixing is largest at x = 0.5, where it equals R\ln 2 per mole of atoms, and it vanishes for pure components.

Forming an ideal solution therefore raises the entropy of the system, because random mixing increases disorder. The molar free energy of mixing of an ideal solution depends only on the mole fractions and temperature, so the masses must first be converted to moles using the atomic weights before the logarithmic term is evaluated. The total entropy of mixing for an ideal solid solution is found by counting moles of each component, not grams, so mass is divided by atomic weight before the mole fractions are formed.

Solution mixing enthalpy

The sign of the enthalpy of mixing sets the direction of deviation from ideality. For a regular solution, a negative \Delta H^{mix} gives an activity coefficient below one, so a_B < x_B, while a positive \Delta H^{mix} gives a_B > x_B. An ideal solution has \Delta H^{mix} = 0 and therefore a_B = x_B.

• Negative deviation — \Delta H^{mix} < 0, unlike pairs are favoured, and the activity lies below the mole fraction; this is typical of compound-forming systems.
• Positive deviation — \Delta H^{mix} > 0, like pairs are favoured, and the activity lies above the mole fraction; this is typical of systems with a miscibility gap.
• Phase diagram link — when the solid is stabilized by mixing more strongly than the liquid, the melting point rises at intermediate composition and a maximum appears; when the liquid is stabilized more strongly, a minimum appears.

For a dilute solution of B, a negative enthalpy of mixing makes the activity of B smaller than its mole fraction. In a binary phase diagram, the signs and magnitudes of the mixing enthalpies of the solid and liquid decide whether the liquidus and solidus show a maximum or a minimum. A strongly negative solid mixing enthalpy with an ideal liquid produces a maximum.