Legendre transforms specified

When the chemical potentials of several species are held constant, it may be useful to write the Legendre transform in terms of matrices. For example, when the chemical potentials of several species are specified, the Legendre transform can be written as... 

The fractional saturation of tetramer YT and the fractional saturation of dimer YD are functions only of [02] at specified T, P, pH, etc., as shown by equations 7.1-18 and 7.3-6. However, since the tetramer form is partially dissociated into dimers, the fractional saturation of heme Y is a function of both [02] and [heme]. Ackers and Halvorson (1974) derived an expression for the function Y([02], [heme]). When Legendre transforms are used, a simpler form of this function is obtained, and it can be used to derive limiting forms at high and low [heme]. These limiting forms are of interest because they show that if data can be obtained in regions where Y is linear in some function of [heme], extrapolations can be made to obtain YT and YD. These fractional saturations can be analyzed separately to obtain the Adair constants for the tetramer and the dimmer (Alberty, 1997a). 

Since coenzymes, and perhaps other reactants, are in steady states in living cells, it is of interest to use a Legendre transform to define a further transformed Gibbs energy G" that provides the criterion for spontaneous change and equilibrium at a specified pH and specified concentrations of coenzymes. This process brings in a further transformed entropy S" and a further transformed enthalpy H", but the relations between these properties have the familiar form. 

Use of a Legendre Transform to Define a Transformed Gibbs Energy at a Specified pH... 

Chapters 1 and 2 dealt with species properties. But in this chapter we have found that when the pH is specified, it is necessary to use a Legendre transform to define a transformed Gibbs energy G of a system and that this automatically brings in a transformed enthalpy H and transformed entropy S. In fact we have entered a whole new world of thermodynamics where attention is focussed on reactants, which are sums of species, rather than on species. This world of biochemical... 

When the availability of oxygen atoms from h2o is specified by p (h2o), a Legendre transform can be used to define a further transformed Gibbs energy G " by (4,5)... 

It can be calculated from the Af Sj ° of species given in the preceding section. This can be done starting with equation 3.3-7 for the Legendre transform for the entropy of a system at specified pH ... 

From this thermodynamic perspective, the first Hohenberg-Kohn theorem is merely an assertion that that the Legendre transform from the electronic ensemble specified by N and v(r) to that specified by p(r) exists [73]. In analogy to classical thermodynamics, the state function for the p-ensemble is ... 

Incomplete. Additional functions could be constructed, such as Uip, V,N) or S(U,p,N) but because these involve conjugate pairs p and V, or p andN, and are missing other variables, they do not uniquely specify the state of a system. Such functions cannot be obtained by Legendre transforms of the fundamental equations. 

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