1.14.44: Maxwell Equations
Important relationships[1] in thermodynamics are based on Maxwell Equations [2-4]. Consider the state variable G for a given closed system characterized by the two independent variables, \(\mathrm{T}\) and \(\mathrm{p}\). Hence,
\[\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T} \nonumber \]
or,
\[\left(\frac{\partial[\partial \mathrm{G} / \partial \mathrm{T}]_{\mathrm{p}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
But at both fixed composition \(\xi\) and at equilibrium, \(\mathrm{A} = 0\), \(\mathrm{V}=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\) and \(\mathrm{S}=-[\partial \mathrm{G} / \partial \mathrm{T}]_{\mathrm{p}}\)
Then
\[\mathrm{E}_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \nonumber \]
For the most part we use this relationship in the context of an equilibrium displacement; i.e. at \(\mathrm{A} = 0\). Equation (c) shows that at equilibrium the isothermal dependence of entropy on pressure equals, with opposite signs, the isobaric dependence of volume on temperature. \(\mathrm{E}_{\mathrm{p}}\) is the isobaric expansion.
This equation has practical importance. Suppose we require for either practical or theoretical reasons the dependence of the molar entropy of water(\(\ell\)), \(\mathrm{S}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\) on pressure at a given temperature. This has all the signs of being a difficult project. However the Maxwell Equation (c) shows that the information is obtained by measuring the dependence of molar volume \(\mathrm{V}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\) on temperature at constant pressure, a simpler approach to the problem. Equation (c) finds several important applications. One application concerns the isothermal dependence of enthalpy on pressure. We start with the equation, \(\mathrm{H}=\mathrm{G}-\mathrm{T} \, \mathrm{S}\). We are interested in the dependence of the properties of a given system on pressure at, for example, equilibrium, \(\mathrm{A} = 0\) and constant temperature. Then,
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
But at \(\mathrm{A} = 0, \mathrm{V}=(\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}}\). Using equation (c),
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{V}-\mathrm{T} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
We see that the isothermal dependence of enthalpy on pressure is readily obtained knowing the volume of a system and its isobaric dependence on temperature. This is another interesting way in which Maxwell equations often simplify tasks facing chemists when probing the properties of systems. In fact equation (e) is fascinating bearing in mind that we can never know the enthalpy \(\mathrm{H}\) of a system but we can calculate in a straightforward manner using volumetric properties the isothermal dependence of enthalpy on pressure. In fact the integrated form of equation (e) is also useful. For a system at constant temperature [and at either constant composition \(\xi\) or at equilibrium, \(\mathrm{A} = 0\)],
\[\mathrm{H}\left(\mathrm{T}, \mathrm{p}_{2}\right)-\mathrm{H}\left(\mathrm{T}, \mathrm{p}_{1}\right)=\int_{\mathrm{p}_{1}}^{\mathrm{p}_{2}}\left[\mathrm{~V}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\right] \, \mathrm{dp} \nonumber \]
Another important Maxwell Equation is based on the Helmholtz energy, \(\mathrm{F}\), of a closed system.
\[\mathrm{F}=\mathrm{F}[\mathrm{V}, \mathrm{T}, \xi] \nonumber \]
For a closed system at fixed composition \(\xi\) (or at equilibrium when \(\mathrm{A} = 0\))
\[\left(\frac{\partial[\partial \mathrm{F} / \partial \mathrm{T}]_{\mathrm{V}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial[\partial \mathrm{F} / \partial \mathrm{V}]_{\mathrm{T}}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \nonumber \]
But, \(\mathrm{S}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) and \(\mathrm{p}=-\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\). Hence,
\[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \nonumber \]
The right-hand-side of equation (i) involves the three practical properties, \(\mathrm{p}\), \(\mathrm{V}\) and \(\mathrm{T}\). In summary, the isochoric dependence of pressure on temperature equals the isothermal dependence of entropy on volume.
Two interesting Maxwell Equations develop from the Gibbs energy \(\mathrm{G}\). For a system at fixed pressure,
\[\frac{\partial}{\partial T}\left(\frac{\partial G}{\partial \xi}\right)_{T}=\frac{\partial}{\partial \xi}\left(\frac{\partial G}{\partial T}\right)_{\xi} \nonumber \]
But \(\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \dot{\xi}}\right)_{\mathrm{T}, \mathrm{p}}\), and \(S=-\left(\frac{\partial G}{\partial T}\right)_{p, \xi}\), Then,
\[\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=\left(\frac{\partial \mathrm{S}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
This interesting equation concerns the temperature dependence of the affinity for spontaneous reaction at fixed pressure and composition. In fact this dependence equals the isothermal-isobaric entropy of reaction, \((\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\).
Also with respect to the Gibbs energy we explore the properties of a closed system at fixed temperature. Thus,
\[\frac{\partial}{\partial p}\left(\frac{\partial G}{\partial \xi}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial G}{\partial p}\right) \nonumber \]
But, \(\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=-\mathrm{A}\), and \(\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\mathrm{V}\). Then,
\[-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
In other words at constant composition the isothermal dependence of the affinity for spontaneous change on pressure equals (minus) the volume of reaction, \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\).
Maxwell Equations are used in the analysis of parameters describing chemical equilibria. In general terms the limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) depends on pressure and the limiting volume of reaction. \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) depends on temperature. Further the entropy of reaction at temperature \(\mathrm{T}\), \(\Delta_{\mathrm{r}} \mathrm{S}^{\#}\) depends on pressure. These complexities signal more complexities in data analysis. Fortunately two Maxwell Equations assist the analysis. [Here ∆rS# refers to the difference between partial molar entropies of reactants and products in solution reference states at a pressure significantly different from the standard pressure.]
The isothermal dependence of entropy of reaction on pressure is related to the isobaric dependence of limiting volume of reactions on temperature.
\[\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{S}^{\#}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=-\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\partial \mathrm{T}}\right]_{\mathrm{p}} \nonumber \]
Further the isothermal pressure dependence of the limiting enthalpy of reaction is related to the limiting volume of reaction and its isobaric temperature dependence. Thus,
\[\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}-\mathrm{T} \,\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\partial \mathrm{T}}\right]_{\mathrm{p}} \nonumber \]
The relationships offer a check of derived quantities and the numerical analysis when equilibrium constants are reported as functions of temperature and pressure. The beauty of thermodynamics is appreciated when one realizes that these relationships are precise. Discovery that a set of data and associated analyses do not conform to these equations does not disprove these Maxwell Equations. Rather one must conclude that analysis of the original experimental results is flawed. In fact Maxwell Equations offer an interesting exercise in units of derived and measured parameters. The isentropic expansion \(\mathrm{E}_{\mathrm{S}}\) is related to the isochoric dependence of entropy on pressure [5,6]. From \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}]\),
\[\partial^{2} U / \partial S \, \partial V=\partial^{2} U / \partial V \, \partial S \nonumber \]
Then,
\[(\partial \mathrm{T} / \partial \mathrm{V})_{\mathrm{S}}=-(\partial \mathrm{p} / \partial \mathrm{S})_{\mathrm{V}} \nonumber \]
We invert the latter equation. Hence,
\[\mathrm{E}_{\mathrm{S}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{s}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}} \nonumber \]
Two other Maxwell Equations are worthy of note. From,
\[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
At equilibrium and fixed composition,
\[\left[\partial(\partial H / \partial S)_{p} / \partial p\right]_{\mathrm{S}}=(\partial T / d p)_{\mathrm{S}} \nonumber \]
and
\[\left[\partial(\partial H / \partial \mathrm{p})_{\mathrm{s}} / \partial \mathrm{T}\right]_{\mathrm{p}}=(\partial \mathrm{V} / \mathrm{dS})_{\mathrm{p}} \nonumber \]
Then,
\[(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \nonumber \]
From \((\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\). Then, \((\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}} \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\) Hence, \((\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}}=-(\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\) Then,
\[(\partial \mathrm{p} / \partial \mathrm{T})_{\mathrm{V}}=(\partial \mathrm{S} / \partial \mathrm{V})_{\mathrm{T}} \nonumber \]
Footnotes
[1] The extent of information available from thermodynamic partial derivatives is explored by:
- R. Gilmore, J. Chem. Phys., 1981,75, 5964; 1982, 77 , 5853.
- M. Ishara, Bull. Chem. Soc. Jpn., 1986, 59 , 5853.
- E. Grunwald, J. Am. Chem. Soc., 1984, 106 , 5414.
[2] E. F. Caldin comments on 1010 possible relationships; Chemical Thermodynamics, Oxford, 1958 (page 158).
[3] H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, van Nostrand 1943.
[4] Extending the observation made by A. B. Pippard [The Elements of Classical Thermodynamics, Cambridge, 1957, p. 46], Maxwell Equations are dimensionally homogeneous in that cross-multiplication yields the following pairs of variables;
- \(\mathrm{p} - \mathrm{~V}\),
- \(\mathrm{T} - \mathrm{~S}\) and
- \(\mathrm{A} - \xi\).
The product of each pair is energy, with unit ‘Joule’.
\[\begin{array}{r}
\mathrm{T} \, \mathrm{S}=[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}^{-1}\right]=[\mathrm{J}] \\
\mathrm{p} \, \mathrm{V}=\left[\mathrm{Nm}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\
\mathrm{A} \, \xi=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \,[\mathrm{mol}]=[\mathrm{J}]
\end{array} \nonumber \]
[5] With reference to equation (o),
\[\begin{aligned}
{\left[\frac{\partial \Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\partial \mathrm{p}}\right]_{\mathrm{T}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+[\mathrm{K}] \,\left[\frac{\mathrm{m}^{3} \mathrm{~mol}^{-1}}{[\mathrm{~K}]}\right] } \\
&=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}
\end{aligned} \nonumber \]
[6] S. D. Hamann, Aust. J. Chem.,1984, 37 ,867.