The analysis of equilibria in solution by the partition function method has shown how the total chemical amounts [T[M], [TA], and [TH] of the components M, A, and H respectively can be expressed and determined as functions of different powers of site affinity constants kj and cooperativity functions γ(ij,bj), whereas the cumulative formation constants βPQR cannot be used as statistically independent parameters to be refined in least-squares processes. The same drawback holds for molar enthalpies, ΔHPQR. A special algorithm has been developed by which the heat evolved can be expressed as a function of specific site enthalpies Δj and specific cooperativity enthalpies Δhγj for each class j of sites. The algorithm represents the mathematical analog of the interconnections between components in the types of complex macrospecies MPAQHR, MPAQ, AQHR, MPHR, MP(AQHR), etc., of the chemical model assumed and their deconvolution into microspecies MPAQHR. Each class j of sites with site constant kj and cooperativity coefficient bj is described by a polynomial {A figure is presented} where Yj is any ligand M, A, or H and it is the number of sites in the class j. The concentrations of the microspecies are calculated as single terms of the polynomials Jj or by products of more terms, each of which belongs to a different Jj. Each term of the polynomial is labeled by its own index (pj, or qj, or rj, or in general ij), which is the exponent of the term and contains the statistical factor mij. calculated as the coefficient of the term of the polynomial. The product of more terms is labeled by the indices of the component terms. The relations are therefore represented by combinations of indices {A figure is presented}. In order to perform the calculation of concentrations of the microspecies and macrospecies by a procedure suitable for computer programming, each polynomial Jj is associated with a vector Jj whose elements [ij] are the terms of Jj. The cooperativity factors are set in a diagonal matrix Γj, whose elements are{A figure is presented} and then introduced into the non-cooperative polynomials Jj by vector products {A figure is presented}. The product of terms giving for each microspecies the contribution to the total concentrations [TM], [TA], and [tH] is calculated as the element of a matrix {A figure is presented} obtained as tensor product: (1,2= j1J2 or {A figure is presented} etc. Depending on the chemical model, there are additional different matrices Ll. The combination of indices of each element of Ll is {A figure is presented}. The indices are said to define an index space {i.s.}, parallel to the affinity cooperativity space. The elements of the matrices Ll are also used to set a matrix ΔC, whose elements Δcpqr are the changes of concentration of the microspecies during the thermochemical reaction. The i.s. is parallel to the concentration space also. The enthalpy change at the nth experimental point for each microspecies Mp,AqHr is calculated from the quantities {A figure is presented}{A figure is presented} which are then summed for all the indices p,q,r. This relation can also be put in a matrix form: {A figure is presented}. All these matrices define spaces which are parallel to {i.s.}. The observed heat at the nth experimental point is the scalar product {A figure is presented} where the elements of the vector Xj, with 1 ≤ j' ≤2jmax are the whole set of j couples of variables Δhj, Δhyj, and the elements of the vector aj' are weighted experimental thermochemical values. By repeating the calculations and summing successively the values for all the n experimental points, the system of normal equations Axj' = ΔQ is set up, where the elements of ΔQ are sums of n weighted experimental heats. By solution of the system, the values of Δhj and Δhyj for all the j classes are obtained. © 1990.
