On a method of numerical differentiation and its application to the evaluation of limiting activity coefficients and their first derivatives

Abstract

A method of numerical differentiation based on the measurement of intercepts rather than slopes is described and applied to the correlation and extrapolation of excess Gibbs free energy data for binary liquid mixtures as well as to their deduction from total pressure-composition data. Activity coefficients, as well as their first derivatives with respect to composition at infinite dilution, have been estimated by our method for three binary systems, both by direct differentiation of the excess Gibbs free energy data and through the use of the first and second composition derivatives of the pressure in the infinite dilution limit. The two sets of values are in good agreement with each other, thus confirming the reasonableness of our estimates of the infinite dilution first and second derivatives of the total pressure and excess Gibbs free energy. Further confirmation of our estimates is also sought by using the above values for the infinite dilution activity coefficients and their derivatives in certain empirical expressions for the excess Gibbs free energy to extrapolate the latter over the whole concentration range; such extrapolated values show good agreement with experimentally determined values. Our method has also been used to evaluate derivatives in parts of the concentration range other than near the ends; the values so obtained compare well with those derived from very closely-fitting functions of the data set.