Contributions to Mineralogy and Petrology

, Volume 66, Issue 4, pp 389–400

The thermodynamic properties of reciprocal solid solutions

  • Bernard J. Wood
  • J. Nicholls

DOI: 10.1007/BF00403424

Cite this article as:
Wood, B.J. & Nicholls, J. Contr. Mineral. and Petrol. (1978) 66: 389. doi:10.1007/BF00403424


A multisite solid solution of the type (A, B) (X, Y) has the four possible components AX, AY, BX, BY. Taking the standard state to be the pure phase at the pressure and temperature of interest, the mixing of these components is shown not to be ideal unless the condition:
$$\Delta G^0 = (\mu _{AX}^0 + \mu _{BY}^0 - \mu _{AY}^0 - \mu _{BX}^0 = 0$$
applies. Even for the case in which mixing on each of the individual sublattices is ideal, ΔG0 contributes terms of the following form to the activity coefficients of the constituent components:
$$RT\ln \gamma _{AX} = - X_{B_1 } X_{Y_2 } \Delta G^0$$
(XJi refers to the atomic fraction of J on sublattice i). The above equation, which assumes complete disorder on (A, B) sites and on (X, Y) sites is extended to the general n-component case. Methods of combining the “cross-site” or reciprocal terms with non-ideal terms for each of the individual sites are also described. The reciprocal terms appear to be significant in many geologically important solid solutions, and clinopyroxene, garnet and spinel solid solutions are all used as examples.
Finally, it is shown that the assumption of complete disorder only applies under the condition:
$$\Delta G^0 \ll zn_1 RT$$
where z is the number of nearest-neighbour (X, Y) sites around A and n1 is the number of (A, B) sites in the formula unit. If ΔG0 is relatively large, then substantial short range oder must occur and the activity coefficient is given by (ignoring individual site terms):
$$\gamma _{AX} = \left( {\frac{{1 - X'_{Y_2 } }}{{1 - X_{Y_2 } }}} \right)^{zn_1 }$$
where XY2 is the equilibrium atomic fraction of Y atoms surrounding A atoms in the structure. The ordered model may be developed for multicomponent solutions and individual site interactions added, but numerical methods are needed to solve the simultaneous equations involved.

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Bernard J. Wood
    • 1
  • J. Nicholls
    • 2
  1. 1.Department of GeologyUniversity of ManchesterManchesterGreat Britain
  2. 2.Department of GeologyUniversity of CalgaryAlbertaCanada