Thermodynamics species/reactants, transformed properties

When the pH is specified, we enter into a whole new world of thermodynamics because there is a complete set of new thermodynamic properties, called transformed properties, new fundamental equations, new Maxwell equations, new Gibbs-Helmholtz equations, and a new Gibbs-Duhem equation. These new equations are similar to those in chemical thermodynamics, which were discussed in the preceding chapter, but they deal with properties of reactants (sums of species) rather than species. The fundamental equations for transformed thermodynamic potentials include additional terms for hydrogen ions, and perhaps metal ions. The transformed thermodynamic properties of reactants in biochemical reactions are connected with the thermodynamic properties of species in chemical reactions by equations given here. 

Calculation of Standard Thermodynamic Properties of Species of a One-species Reactant from the Apparent Equilibrium Constant at 298.15 K and the Standard Transformed Enthalpy of Reaction at 313.15 K... 

These tables apply to single sets of values of pH and ionic strength. A more general approach is to use the functions of ionic strength and pH for each reactant that give the values of standard transformed thermodynamic properties at 298.15 K. For reactants for which A,//0 is known for all species, functions of temperature, pH, and ionic strength can be used to calculate standard transformed thermodynamic properties at temperatures in the approximate range 273.15 to 313.15 K, as discussed in Section 4.9. 

These tables have been given to 0.01 kJ mol-1. In general this overemphasizes the accuracy with which these formation properties are known. However for some reactants for which species are in classical tables, this accuracy is warranted. An error of 0.01 kJ mol-1 in the standard transformed Gibbs energy of a reaction at 298 K corresponds with an error of about 1 % in the value of the apparent equilibrium constant. It is important to understand that the large number of digits in these tables is required because the thermodynamic information is in differences between entries. 

The standard formation properties of species are set by convention at zero for the elements in their reference forms at each temperature. The standard formation properties of in aqueous solution at zero ionic strength are also set at zero at each temperature. For other species the properties are determined by measuring equilibrium constants and heats of reaction. Standard transformed Gibbs energies of formation can be calculated from measurements of K, and so it is really these Maxwell relations that make it possible to calculate five transformed thermodynamic properties of a reactant. 

Now we are in a position to calculate the standard transformed thermodynamic properties of reactants from the standard properties of the species that make them up. In this chapter the transformed thermodynamic properties are calculated only at 298.15 K. Caleulations at other temperatures are presented in the next chapter. The first step is to adjust the properties at zero ionic strength to the desired ionic strength in the range 0-0.35 M. Equations 1.3-5 and 1.3-6 Chapter 1 show how these calculations can be made using the extended Debye-HUckel equation. Substituting equation 1.3-5 in equation 3.5-3 in two places yields... 

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... 

Chapters 3-5 have described the calculation of various transformed thermodynamic properties of biochemical reactants and reactions from standard thermodynamic properties of species, but they have not discussed how these species properties were determined. Of course, some species properties came directly out of the National Bureau of Standard Tables (1) and CODATA Tables (2). One way to calculate standard thermodynamic properties of species not in the tables of chemical thermodynamic properties is to express the apparent equilibrium constant K in terms of the equilibrium constant K of a reference chemical reaction, that is a reference reaction written in terms of species, and binding polynomials of reactants, as described in Chapter 2. In order to do this the piiTs of the reactants in the pH range of interest must be known, and if metal ions are bound, the dissociation constants of the metal ion complexes must also be known. For the hydrolysis of adenosine triphosphate to adenosine diphosphate, the apparent equilibrium constant is given by... 

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 chapter has been about calculating species properties from apparent equilibrium constants and transformed enthalpies of reaction, but there is a prior question. Where is the experimental data Fortunately, Goldberg, Tewari, and coworkers have searched the literature for these data, have evaluated it, and have published a series of review articles (10-15). These review articles provide thermodynamic data on about 500 enzyme-catalyzed reactions involving about KXX) reactants. In principle all these reactants can be put into thermodynamic tables. Goldberg, Tewari, and Bhat (16) have produced a web site to assist in the acquisition of data from the review articles. 

R. A. Alberty, Calculation of thermodynamic properties of species of biochemical reactants using the inverse Legendre transform, J. Phys. Chem. 109 B, 9132-9139 (2005). 

Chemical reaction systems are discussed in terms of species, and many chemical thermodynamic properties can be calculated from the species properties given in the next section, for example pKs. However, in making calculations on enzyme-catalyzed reactions it is useful to take the pH as an independent variable. When this is done the principal thermodynamic properties of a reactant are the standard transformed Gibbs energy of formation Af G the standard transformed enthalpy of formation A( H the standard transformed entropy of formation Af 5 and the average number of hydrogen atoms in the reactant 77h. These properties are related by the following equations ... 

The calculation of Af G° and Af H° of species from experimental data on apparent equilibrium constants and transformed enthalpies of reaction is described in R. A. Alberty, Thermodynamics of Biochemical Reactions, Wiley, Hoboken, NJ (2003) and a number of places in the literature. That is not discussed here because this package is oriented toward the derivation of mathematical functions to calculate thermodynamic properties at specified T, pH, and ionic strength. There are two types of biochemical reactants in the database ... 

This package provides data on the species of 131 reactants at 298.15 K and programs for calculating various transformed thermodynamic properties. Programs are given for the calculation of apparent equilibrium constants and other transformed thermodynamic properties of enzyme-catalyzed reactions by simply typing in the reaction. 

BasicBiochemData3 contains functions of pH and ionic strength that give standard transformed thermodynamic properties of 199 reactants for which values of Af G are known for all species that have significant concentrations in the pH 5 to pH 9 range. Using ATP as an example, these two functions are as follows ... 

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