Excess Properties

The deviation of a property from its ideal-solution value is called the excess property. For a generic property M, the excess property is given the symbol M^E, and M^E is the value of the property for the mixture relative to the property for an ideal mixture, M^E = M – M^is. Ideal solutions were discussed in Section 10.10. The molar volume of an ideal solution is just the weighted sum of the molar volumes of the components, . The excess volume is then,

 Excess volume.

Although the excess volumes of liquids are typically a very small percentage of the volume, the concepts of excess properties are easily grasped by first studying the excess volume and then exploring the more abstract quantities of excess enthalpy, entropy, or Gibbs energy.

The excess volume of the system 3-pentanone (1) + 1-chlorooctane (2) at 298.15 K has been measured by Lorenzana, et al.,⁸ and is shown in Fig. 11.9. The molar volumes of the pure components are V₁ = 106.44 cm³/mol and V₂ = 171.15 cm³/mol. At the equimolar concentration, the excess volume is 0.204 cm³/mol. Therefore, the molar volume is V = V^E + V^is = 0.204 + 0.5 · 106.44 + 0.5 · 171.15 = 139.00 cm³/mol. The excess volume is only 0.15% of the total volume. The partial molar excess volume is calculated in a manner analogous to the partial molar volume, . If an algebraic expression is available for the excess volume, it may be differentiated by this relation to yield formulas for the excess volumes. Graphically, the partial molar volumes at any point may be found by drawing the tangent line to the excess volume curve and reading the intercepts. At the composition shown at the tangent point in Fig. 11.9, the intercepts give  and . The partial molar volumes depend on composition.

Figure 11.9. Excess volume for the 3-pentanone (1) + 1-chlorooctane (2) system at 298.15 K.

The excess enthalpy is very similar to the excess volume,

A solution with H^E > 0 has an endothermic heat of mixing, and when H^E < 0, the heat of mixing is exothermic. In an adiabatic mixing process an endothermic mixing process will cool and an exothermic mixing process will heat.

In directly analogous fashion, the excess Gibbs energy can be defined as the difference between the Gibbs energy of the mixture and the Gibbs energy of an ideal solution, G^E = G – G^is. Then, instead of speaking of partial molar volumes, we speak of partial molar Gibbs energies. But you should recognize the partial molar Gibbs energy as the chemical potential as introduced in Section 10.8, and the significance of the chemical potential to phase equilibrium calculations should resonate strongly after reading Chapter 10. What remains is to rearrange the mathematics into the final relations to show how this property is related to the activity coefficients.