Natural variables transformed Gibbs energy

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) ... 

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. 

The successive Legendre transformations of E yield a state function, G, for which the natural variables p and T, are both intensive properties (independent of the size of the system). Furthermore, for dp = 0 and dT = 0 (isobaric, isothermal system), the state of the system is characterized by dG. This is clearly convenient for chemical applications under atmospheric pressure, constant-temperature conditions (or at any other isobaric, isothermal conditions). Then, in place of equation (21) for internal energy variation, we state the conditions for irreversible or reversible processes in terms of the Gibbs energy as... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

---