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

Binding constant data K degrees(T) are commonly subjected to van't Hoff analysis to extract estimates of Delta H degrees, Delta S degrees, and Delta C-P degrees for the process in question. When such analyses employ unweighted least-squares fitting of ln K degrees to an appropriate function of the temperature T they are tacitly assuming constant relative error in K degrees. When this assumption is correct, the statistical errors in Delta G degrees, Delta H degrees, Delta S degrees, Delta C-P degrees, and the T-derivative of Delta C degrees P (if determined) are all independent of the actual values of K degrees and can be computed from knowledge of just the T values at which K degrees is known and the percent error in K degrees. 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 Delta H degrees as a polynomial in (T-T-0), the coefficients of which thus become Delta H degrees, Delta C-P degrees, and 1/2 d Delta C-P degrees/dT at T = T-0. The errors in the key quantities can then be computed by just repeating the fit for different T-0. 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 degrees C gives significant improvement in the precision of all quantities. Delta G degrees is typically determined with standard error a factor of similar to 30 smaller than that for Delta H degrees. Accordingly, the error in T Delta S degrees is nearly identical to that in Delta H degrees. For 4% error in K degrees, the T-derivative in Delta C-P degrees cannot be determined unless it is similar to 10 cal mol(-1) K-2 or greater; and Delta C-P degrees must be similar to 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. (c) 2005 Elsevier B.V. All rights reserved.