|Resources for the Thermodynamics of Biochemical Reactions|
6. Maxwell Relations
6. Maxwell Relations
As we have seen above, Legendre transforms are used to define new thermodynamic potentials. For each thermodynamic potential like U, S, H, G, G’ , and G’’ there is a fundamental equation that gives the expression for the total differential of the thermodynamic potential in terms of the differentials of its independent variables. If G’ or G’’ can be expressed as a function of its independent variables, the coefficients of the terms in the fundamental equation can be obtained by taking partial derivatives of G’ or G’’. For example, when a single reactant is involved, the fundamental equation for G ’ can be written as
dG’ = –S’dT + VdP + ∆fG’dn’ + RTln(10)nc(H)dpH
Thus –S’ = (∂G’/∂T), V = (∂G’/∂P), ∆fG’ = (∂G’/∂n’), and (∂G’/∂pH) = RTln(10)nc(H), where the subscripts have been left out. These are often called equations of state because they deal with state properties. However, the problem in using these equations is that there is no direct way to experimentally determine G’.
If a reaction occurs in a system at a specified pH, the fundamental equation for G’ becomes
dG ’ = –S’dT + VdP + ∆rG’dx’ + RTln(10)nc(H)dpH
where x’ is the extent of the biochemical reaction. Now the equations of state are
–S’ = (∂G’/∂T), V = (∂G’/∂P), ∆rG’ = (∂G’/∂x’), and (∂G’/∂pH) = RTln(10)nc(H).
We still have the problem that there is no direct way to experimentally determine G’, but we need to look at the Maxwell relations (second cross partial derivatives). If we ignore the VdP term, this fundamental equation has three Maxwell relations:
–(∂S’/∂x’) = (∂∆rG’/∂T), that can be written as ∆rS’ = -(∂∆rG’/∂T) or ∆rS’º = -(∂∆rG’º/∂T )
(∂∆rG’/∂pH) = RTln(10)(∂nc(H)/∂x’), that can be written as (∂∆rG’º/∂pH) = RTln(10)∆rNH, where ∆rNH is the change in the binding of hydrogen ions in the reaction. Note that ∆rNH can be determined experimentally using a pHstat.
(∂S’/∂pH) = Rln(10)(∂(Tnc(H))/∂T)
If both sides of this equation are divided by dx’, this Maxwell relation can be written as
(∂∆rS’º/∂pH) = Rln(10)(∂(T∆rNH)/∂T)
By substituting S’ = H’/T – G’/T in the fundamental equation, we can obtain two more Maxwell relations
∆rH’º = –T2(∂(∆rG’º/T)/∂T) (the Gibbs-Helmholtz equation)
(∂∆rH’º/∂pH) = RT2ln(10)(∂∆rNH/∂T)
Thus there are five Maxwell relations for a reaction system at specified pH.
The standard transformed Gibbs energy of reaction can be calculated from measured apparent equilibrium constants: ∆rG’º = –RTlnK’. When ∆rG’º can be expressed as a function of temperature and pH, it is the Maxwell relations that make it possible to calculate ∆rS’º, (∂∆rS’º/∂pH), ∆rH’º, (∂∆rH’º/∂pH), and ∆rNH.
If there is another independent variable like pMg, the number of Maxwell relations increases to nine. Thus as more independent variables are introduced by use of Legendre transforms, more thermodynamic properties can be measured (in this case ∆rNMg) and there are more relations between measurable thermodynamic properties.
Another example is provided by the binding of oxygen by a protein at a site coupled with acid groups. Here again there are nine Maxwell relations if [O2] is considered to be an independent variable. As mentioned above, the maximum number of Legendre transforms that can be used with a given system is one less than the number of components. There is an advantage in making as many Legendre transforms as possible. When this is feasible, the advantage is that the thermodynamics of the system can be represented by a single function of the independent variables. This function contains all the thermodynamic information on the system in the sense that all the properties and relations between them can be calculated by taking partial derivatives. Mathematica can be used to construct the equation for G’ or G’’ because of its symbolic capabilities, and it can be used to calculate the partial derivatives no matter how complicated the function is. Mathematica utilizes a systematic procedure based on the chain rule that effectively allows any derivative to be worked out.
Department of Chemistry
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