Temperature Dependence of Errors in Parameters Derived from Van't Hoff Studies Steven F. Dec and Stanley J. Gill University of Colorado, Boulder, CO 80309
Measurements of the equilibrium constant for chemical reactions as a function of temperature provide a powerful means to determine the changes in the thermodynamic parameters of the reaction. Employment of the well-accepted methods of least-squares data analysis ( I ) facilitates the evaluation of the standard-state reaction properties from such measurements. However, many investigators have not made full use of available data analysis methods to estimate the statistical errors in these derived Darameters. such as heats of reactlonor heat capaclty rhanges. In t h ~ paper s we wish to draw attention to the use of a orocedure devrloued bv Clarke and Glew ( 2 ) for the evaluation of various thermodynamic parameters and their error estimates as a function of temperature (see ref. (3)for a condensed version). King (4) has also discussed the temperature dependence of the standard deviations in the thermodynamic parameters based on error equations (5) derived for a particular fitting equation (6). Timimi has also shown by extensive computer calculations how the statistical errors affect the evaluation of thermodynamic parameters a t different temperatures (7). We shall apply the method of Clarke and Glew ( 2 )to high-precision gas solubility data as an example which serves to illustrate how one can determine various thermodynamic parameters and their associated errors a t various temperatures. The evaluation of standard errors for various parameters derived from van't Hoff equations is especially important when one has direct calorimetric measurements of the same parameters. The full power of thermodynamic logic can then be brought to bear upon the assessment of how accurate a d articular auantitv has been determined since the two procedures should agree within assigned errors. We have recentlv been faced with makinr a comparison between van? ~ o f f e n t h a l p ychanges for dissolving gases into water (8-1 i ) and calorimetric measurements (12.1:jJand have found that error assignments have usually not been specifically given to the thermodynamic parameters obtained by the van't Hoff approach (8-11). The method of Clarke and Glew ( 2 ) serves t o solve this problem in a versatile and straightforward manner. Because it is broadly applicable to other studies of the temperature dependence of equilibrium constant measurements, we wish to describe the method in more detail than was originally given (2) and t o illustrate its use in comparing calorimetric results and temperature-dependent gas soluhility studies. The Method of Clarke and Glew Clarke and Glew (2)have shown that the standard enthalpy change for any reaction a t temperature T and constant pressure may he expanded in a Taylor series about a reference temnerature 8. i.e..
AHo(T) is arbitrarily truncated after taking the third temperature derivative of the heat capacity change and that the error in measurements of T are assumed to be negligibly small. Similarly, AC;(T) is given by ( 2 )
The standard entropy change, A S o ( T ) ,may he obtained by evaluating
The equilibrium constant a t constant pressure, K, is related to the standard free energy change, AGo(T), through the equation
where R is the gas constant. Substitution of eqn. (2) into eqn. (3) and integrating yields ASo(T)as a function of T , 8, AC;(H), and various temperature derivatives of AC; evaluated a t 8. Substitution of the inteerated ex~ressionfor ASo(T) . . and eon. ( 1 j in(
