Binary Solutions and the Lever Rule Revisited

Abstract

The partition coefficient for the normalized fraction of the low‐melting component X1 in a binary solution is simply related to the exchange coefficient KD, and for many systems of interest is a nearly linear function of the solid composition, providing a satisfactory estimate of the phase diagram from limited partitioning data. The analogous coefficient for the high‐melting component is nearly linear against the liquid composition. Both relations simplify the application of the Rayleigh distillation equation, either in an integrated form or numerically calculated with variable partition coefficient and continuously reset tangent composition, yielding accurate stepwise paths of fractional crystallization and fusion. The integrated form is useful if KD is constant and only one crystal phase is present. In the stepwise calculation, all variations in KD and modal ratios can be loaded onto the exponent of the Rayleigh equation and reset continually as the calculation proceeds. The residue produced on fractional crystallization or melting on a binary loop approximates rapidly to the pure endmember composition. Review of the geological and metallurgical literature shows that the amount of residue has been successfully calculated in many different ways, starting with N. L. Browen in 1913.