1.14.12: Legendre Transformations
Many important thermodynamic equations are closely related. These relationships are often highlighted by the mathematical technique, Legendre Transformations [1,2]. With reference to thermodynamics, Callen [3] discusses application of Legendre Transformations. The essential features of Legendre Transformations can be understood in the following terms.
A primary variable \(\mathrm{Q}\) is defined by two dependent variables \(x\) and \(y\). Thus
\[Q=Q[x, y] \nonumber \]
Then
\[\mathrm{dQ}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \, \mathrm{dx}+\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{y}}\right)_{\mathrm{x}} \, \mathrm{dy} \nonumber \]
By definition,
\[\mathrm{u}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \quad \text { and } \quad \mathrm{v}=\left(\frac{\partial \mathrm{Q}}{\partial \mathrm{y}}\right)_{\mathrm{x}} \nonumber \]
Then from equation (b),
\[\mathrm{dQ}=\mathrm{u} \, \mathrm{dx}+\mathrm{v} \, \mathrm{dy} \nonumber \]
A new variable \(\mathrm{Z}\) is defined by equation (e).
\[\mathrm{Z}=\mathrm{Z}[\mathrm{u}, \mathrm{y}] \text { where } \mathrm{Z}=\mathrm{Q}-\mathrm{u} \, \mathrm{x} \nonumber \]
Then,
\[\mathrm{dZ}=\mathrm{dQ}-\mathrm{x} \, \mathrm{du}-\mathrm{u} \, \mathrm{dx} \nonumber \]
Hence using equation (d),
\[d Z=u \, d x+v \, d y-x \, d u-u \, d x \nonumber \]
Or,
\[\mathrm{d} Z=-\mathrm{x} \, \mathrm{du}+\mathrm{v} \, \mathrm{dy} \nonumber \]
Hence,
\[x=-\left(\frac{\partial Z}{\partial u}\right)_{y} \quad \text { and } \quad v=\left(\frac{\partial Z}{\partial y}\right)_{u} \nonumber \]
Comparison of equations (a) and (e) reveals the transformation, \(\mathrm{Q}[\mathrm{x}, \mathrm{y}] \rightarrow \mathrm{Z}[\mathrm{u}, \mathrm{y}]\). We now explore thermodynamic transformations [3]. The following Master Equation relates the change in thermodynamic energy \(\mathrm{U}\) with the changes in entropy \(\mathrm{S}\) at temperature \(\mathrm{T}\), volume \(\mathrm{V}\) at pressure \(\mathrm{p}\) and composition \(\xi\) at affinity \(\mathrm{A}\); \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi]\).
\[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
By definition, enthalpy \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\);
\[\mathrm{dH}=-\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
Using equation (j),
\[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp} \nonumber \]
Then \(\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\) Or,
\[\mathrm{H}=\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi] \nonumber \]
The transformation is- \(\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi] \rightarrow \mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]\) By definition, Gibbs energy
\[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S} \nonumber \]
Or using equation (k),
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
Then
\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT} \nonumber \]
Hence from equation (l)
\[\mathrm{dG}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT} \nonumber \]
Or, \(\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\) And,
\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi] \nonumber \]
The transformation is \(\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi] \rightarrow \mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\) Similarly, \(\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi] \rightarrow \mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\) Ledgendre transformations can be examined in the broad context of chemistry and biochemistry [5]. Their importance lies in establishing the general mathematical structure of thermodynamics [6].
Footnotes
[1] A. M. Legendre; an eighteenth century mathematician.
[2] C. Paus at http://web.mit.edu /8.21/www/ notes/notes/ node7.html
[3] B. Callen, Thermodynamics, Wiley, New York,1961.
[4] E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York , 1997.
[5] R. A. Alberty, Chem. Revs.,1994, 94 ,1457.
[6] D. Kondepudi and I. Prigogine, Modern Thermodynamics, Wiley, New York,1998.