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].

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