On the algebraic fitting of experimental thermodynamic data with special regard to employing of orthonormal polynomials

Following the Weierstrass approximation theorem the thermodynamic excess functions are representable with arbitrary high accuracy by means of polynomials of sufficient high degrees in the mole fraction x. So, algebraic fitting of experimental thermodynamic excess data can be based upon mathematical polynomial expressions without any loss of generality. With respect to the necessary scattering of experimental results, algebraic evaluation of those data can only be solved by employing the calculus of observations. The least square method is the only principle of fitting with full justification by statistical mathematics, and which can be applied directly for algebraic fitting of experimental data by means of a computer. The general linear problem of fitting is solved explicitly (i) by means of Gauss method of elimination, and (ii) by employing the property of “orthonormality” of polynomials. In the latter case the explicit form of the “orthonormal” polynomials depends strongly on the number of experimental data which has to be fitted. A convenient procedure is presented to generate polynomials which are orthonormal with respect to an actual set of experimental data. Computer-programs in PORTRAN-language are enclosed 1) to employ Gauss method of elimination, and 2) to generate discrete orthonormal polynomials.