Van't Hoff analysis of K° (T): How good…or bad?

Binding constant data K° (T) are commonly subjected to van't Hoff analysis to extract estimates of ΔH°, ΔS°, and ΔCP° for the process in question. When such analyses employ unweighted least-squares fitting of ln K° to an appropriate function of the temperature T, they are tacitly assuming constant relative error in K°. When this assumption is correct, the statistical errors in ΔG°, ΔH°, ΔS°, ΔCP°, and the T-derivative of ΔCP° (if determined) are all independent of the actual values of K° and can be computed from knowledge of just the T values at which K° is known and the percent error in K°. All of these statistical errors except that for the highest-order constant are functions of T, so they must normally be calculated using a form of the error propagation equation that is not widely known. However, this computation can be bypassed by defining ΔH° as a polynomial in (T-T0), the coefficients of which thus become ΔH°, ΔCP°, and 1 / 2 dΔCP° / dT at T = T0. The errors in the key quantities can then be computed by just repeating the fit for different T0. Procedures for doing this are described for a representative data analysis program. Results of such calculations show that expanding the T range from 10-40 to 5-45 °C gives significant improvement in the precision of all quantities. ΔG° is typically determined with standard error a factor of ∼30 smaller than that for ΔH°. Accordingly, the error in TΔS° is nearly identical to that in ΔH°. For 4% error in K°, the T-derivative in ΔCP° cannot be determined unless it is ∼10 cal mol− 1 K− 2 or greater; and ΔCP° must be ∼50 cal mol− 1 K− 1. Since all errors scale with the data error and inversely with the square root of the number of data points, the present results for 4% error cover any other relative error and number of points, for the same approximate T structure of the data.