1.14.45: Moderation
For a closed system, the dependence of chemical composition \(\xi\) on temperature \(\mathrm{T}\) at affinity \(\mathrm{A}\) and constant pressure is given by equation (a).
\[\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left[\frac{\mathrm{A}+(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{\mathrm{T} \,(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}\right] \nonumber \]
Similarly for a closed system, the dependence of chemical composition \(\xi\) on pressure at fixed temperature is given by equation (b).
\[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=\left[\frac{(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}\right] \nonumber \]
These two equations form the basis of ‘Laws of Moderation’ for closed systems at chemical equilibrium. These equations yield the sign for the two quantities \(\left(\frac{\partial \xi}{\partial T}\right)_{p, A=0}\) and \(\left(\frac{\partial \zeta}{\partial p}\right)_{T, A=0}\) which describe the change in composition when a system at equilibrium is perturbed to a neighboring equilibrium state.
We recall that by definition \(\xi\) is positive for displacement in composition from reactants to products; \(\left(\frac{\partial V}{\partial \xi}\right)_{T, A=0}\) is the volume of reaction. If \(\left(\frac{\partial V}{\partial \xi}\right)_{T, A=0}\) is positive, \(\left(\frac{\partial \xi}{\partial p}\right)_{T, A=0}\) is negative because \(\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}<0\). According to equation (b), an increase in pressure favors a swing in the equilibrium position towards more reactants [1].
Similarly it follows from equation (a) that an increase in temperature favors a swing in the equilibrium position towards more reactants for an exothermic reaction [2].
Moderation is a striking example of the Second Law of Thermodynamics in action with reference to the direction of spontaneous changes in a closed system following changes in either \(\mathrm{T}\) or \(\mathrm{p}\). Here the stress on the word ‘closed’ reminds us that these laws of moderation do not apply to open system although the point is not always stressed. Therefore controversy often surrounds what is often called Le Chatelier’s Principle.
Consider a closed system in which the following chemical equilibrium is established at defined \(\mathrm{T}\) and \(\mathrm{p}\).
\[x X+y Y \Leftrightarrow z Z \nonumber \]
As often argued, if \(\delta \mathrm{n}_{\mathrm{Y}}\) moles of chemical substance \(\mathrm{Y}\) are added to the system, then the equilibrium amount of chemical substance \(\mathrm{Z}\) increases. In fact such moderation of composition only occurs if \(\sum_{j=1}^{j=i} v_{j}\) is zero for a chemical equilibrium involving i chemical substances. An interesting case concerns the Haber Synthesis.
\[\mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) \Leftrightarrow 2 \mathrm{NH}_{3}(\mathrm{g}) \nonumber \]
If in the equilibrium system mole fraction \(\mathrm{x}\left(\mathrm{N}_{2}\right) < 0.5\), addition of a small amount of \(\mathrm{N}_{2}(\mathrm{g})\) leads to an increase in the amount of ammonia. However if \(\mathrm{x}\left(\mathrm{N}_{2}\right) > 0.5\) addition of a small amount of \(\mathrm{N}_{2}(\mathrm{g})\) leads to dissociation of ammonia to form more \(\mathrm{N}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2}(\mathrm{g})\) [3].
Footnotes
[1] This conclusion is called a Theorem of Moderation. Co-author MJB was taught that the outcome was “Nature’s Law of Cussedness” (\(\equiv\) Obstinacy). An exothermic reaction operates to generate heat so the system responds when the temperature is raised in the direction for which the process is endothermic. This line of argument is not good thermodynamics but does make the point.
[2] Another example of Nature’s Obstinacy; see [1]. Note the switch in sign on the r.h.s of equations (a) and (b).
[3] I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans - Green, London, 1953, page 268.