Inverse Legendre transform

Now the inverse Legendre transform given in equation 4.10-1 is needed. The differential of the Gibbs energy is given by... 

To do this all the reactants have to be in the database, except for one or two, The following three programs, which make this calculation in one step, are based on the concept of the inverse Legendre transform. The program to be used depends on the number of species in the reactant. The programs produce entries for the database on species. 

R. A. Alberty, Inverse Legendre transform in biochemical thermodynamics Applied to the last five reactions of glycolysis, J. Phys. Chem. 106, 6594-6599 (2002). 

The calculations in Chapters 3 to 5 have been based on the use of Legendre transforms to introduce pH and pMg as independent intensive variables. But now we need to discuss the reverse process - that is the transformation of Af G ° values calculated from measured apparent equilibrium constants in the literature to Af G° values of species and the transformation of Af° values calculated from calorimetric measurements in the literature to AfH° of species. This is accomplished by use of the inverse Legendre transform defined by (7) ... 

This shows that the fundamental equation for G can be obtained from the fundmental equation for G by use of the inverse Legendre transform. 

The program calcHlsp carries out this calculation at 298.15 K. The programs given in this section are based on the inverse Legendre transform, and they have been used to calculate 32 new species matrices providing Af C ° and 8 new species matrices providing Af H° values (7). 

We have seen that calculating species properties from experimental values of K and A // ° is more complicated than calculating K and Ar ° from species values. Thermodynamic calculations can be made by alternate paths, and so there is more than one way to calculate species properties from experimental properties. This chapter emphasizes the concept of the inverse Legendre transform discussed by Callen (8). Biochemical reaction systems are described by transformed thermodynamic properties, and the inverse transform given in equation 6.2-1 provides the transformation from experimental reactant properties to calculated species properties. In this ehapter we first considered calculations of species properties at 298.15 K from measurements of K and Ar ° at 298.15 K. Then we considered the more difficult problem of calculating Af G°(298.15 K) and Af //°(298.15 K) from Ar G "(313.15 K) and Ar H (313.15 K). The programs developed here make it possible to go from Ar G and Ar H (F.pH,/) to Af G (298.15 K,/=0) and Af H (298.15 K,/=0) in one step. 

This result, above of its inverse Legendre transformation form, stands as an implicit equation for the particle s position further identification as... 

The inverse of H determines the geometric compliance matrix (Nalewajski, 1993, 1995, 1997, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008) describing the open system in the Qi,F)-representation. The relevant thermodynamic potential is defined by the total Legendre transform of the system BO potential, which replaces the state-parameters (N, Q) with their energy conjugates (/a, F), respectively ... 

The inverse of the generalized hardnesses are the generalized global softnesses. These can most elegantly be introduced in a representation using the quantities [jU v.Ms,v,vxl as the basic variables to describe the system. Indeed, note that the variables N and Ns now have been replaced by the chemical potentials and /is, this results from the following Legendre transformation ... 

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