Gibbs energies Legendre transforms

Equation 54 implies that U is a function of S and P, a choice of variables that is not always convenient. Alternative fundamental property relations may be formulated in which other pairs of variables appear. They are found systematically through Legendre transformations (1,2), which lead to the following definitions for the enthalpy, H, Hehnholt2 energy,, and Gibbs energy, G ... 

The temperature can be introduced as an independent variable by defining the Gibbs energy G with the Legendre transform... 

The use of this Legendre transform has introduced the intensive property T as an independent variable. It can be shown that the criterion for spontaneous change and equilibrium is given by (dG)rp 0. The Gibbs energy is so useful because T and P are convenient intensive variables to hold constant and because, as we will see shortly, if G can be determined as a function of T and P, then S, V, H, and U can all be calculated. 

In this section we will consider the Legendre transforms that define the enthalpy H, Helmholtz energy A, and Gibbs energy G. 

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

The Legendre transform that defines the further transformed Gibbs energy G", which provides the criterion for spontaneous change and equilibrium in dilute... 

When the concentrations of ATP and ADP are in a steady state, these concentrations can be made natural variables by use of a Legendre transform that defines a further transformed Gibbs energy G" as follows. 

In order to introduce the chemical potential of molecular oxygen as a natural variable, the following Legendre transform is used to define a further transformed Gibbs energy G" (Alberty, 1996b) ... 

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... 

The transformed Gibbs energy provides the criterion for spontaneous change and equilibrium in systems of enzyme-catalyzed reactions when the independent variables for the system are T, P, pH, and Wc - Notice that making this Legendre transform has introduced ) as a natural variable, but it has not changed the number of natural variables because there is now one less component that is conserved, the hydrogen atom component. 

In considering systems of enzyme-catalyzed reactions at steady state concentrations of coenzymes, a Legendre transform can be used to define a further transformed Gibbs energy G " (17). 

However, the concentration of water does not appear in the expression for the equilibrium constant, and so this treatment fails in calculating the equilibrium concentrations using equcalcc (see Chapter 7). This calculational problem can be solved by using the further transformed Gibbs energy G defined by the Legendre transform (4)... 

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