1.14.6: Extent of Reaction - General

The variable \(\xi\) describes in quite general terms the molecular composition/organisation. For a closed system at fixed \(\mathrm{T}\) and \(\mathrm{p}\), there is a composition/organisation \(\xi^{\mathrm{eq}}\) corresponding to a minimum in Gibbs energy where the affinity for spontaneous change is zero. In general terms there is an extent of reaction \(\xi\) corresponding to a given affinity \(\mathrm{A}\) at defined \(\mathrm{T}\) and \(\mathrm{p}\). In fact we can express \(\xi\) as a dependent variable defined by the independent variables \(\mathrm{T}\), \(\mathrm{p}\), and \(\mathrm{A}\). Thus

\[\xi=\xi[\mathrm{T}, \mathrm{p}, \mathrm{A}]\]

The general differential takes the following form.

\[\mathrm{d} \xi=\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}\]

The change in chemical composition occurs spontaneously. The change in composition is described in terms of the extent of chemical reaction, \(\xi\). In a given aqueous solution, the chemical reaction is:

\[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})\]

At each stage the extent of chemical reaction is represented by the symbol \(\xi\) [1].

\[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})\]

At \(t = 0\) (i.e. as prepared)

\[n(\text { ester })^{0} \quad n\left(\mathrm{OH}^{-}\right)^{0} \quad 0 \quad 0\]

After extent of reaction \(\xi\) (at some time later)

\[n(\text { ester })^{0} - \xi \quad n\left(\mathrm{OH}^{-}\right)^{0} -\xi \quad \xi \quad \xi\]

As the reaction proceeds so \(\xi\) increases. [NB The zero superscript signals ‘at time zero’.] At a given stage of the reaction and time \(t\), (accepting \(\mathrm{dt}\) is positive), Rate of Reaction = \(\mathrm{d}\xi / \mathrm{dt}\)

We now ask ‘why did chemical reaction proceed in this direction?’. The answer is ---- the chemical reaction was driven by the affinity for spontaneous change, symbol \(\mathrm{A}\). By identifying these two ideas, affinity for spontaneous change and the rate of reaction \(\mathrm{d}\xi / \mathrm{dt}\), we arrive at two important criteria for chemical equilibrium.
Affinity for spontaneous change \(\mathrm{A} = 0\)
Rate of change \(\mathrm{d}\xi / \mathrm{dt} = 0\)

However we need to stand back a little and examine how we might advance generalizations concerning the direction of Spontaneous Chemical Reaction. What macroscopic property can be identified which accounts for the fact that alkaline hydrolysis of ethyl ethanoate is spontaneous? To make further progress we introduce two laws of thermodynamics. Actually these are not laws in the sense of being laid down by government or by religious doctrine. Rather these laws are AXIOMS. We explore these axioms in the context for which ξ refers to a change in composition resulting from chemical reaction [2].

Footnotes

[1] For a discussion of the significance of extent of reaction \(\xi\), see:

1. K. J. Laidler and N. Kallay, Kem. Ind. (Sofia) 1988, 37, 182.
2. F. R. Cruikshank, A. J. Hyde and D. Pugh, J.Chem.Educ., 1977, 54, 88.
3. P. G. Wright, Educ. Chem., 1986,23, 111.
4. M. J. Blandamer, Educ. in Chem.,1999,36,78.

[2] The usefulness of the concept of extent of chemical reaction \(\xi\) is further illustrated by the following examples.

1. A closed system (at fixed temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)) is prepared using \(\mathrm{n}_{\mathrm{x}}^{0}\) moles of chemical substance \(\mathrm{X}\) in \(\mathrm{w}_{1} \mathrm{~kg}\) of solvent, water. Spontaneous chemical reaction forms chemical substances \(\mathrm{Y}\) and \(\mathrm{Z}.

   Thus,	\(2\mathrm{X}\)	\(\rightarrow 3\mathrm{Y}\)	\(+ \mathrm{~Z}\)
   At \(t = 0\)	\(\mathbf{n}_{X}^{0}\)	\(0\)	\(0 \mathrm{~mol}\)
   After extent of reaction, \(\xi\) with \(\mathbf{n}_{X}^{0}\)	\(\mathrm{n}_{\mathrm{x}}^{0}-2 . \xi\)	\(3 . \xi\)	\(2 . \xi \mathrm{~mol}\)

2. If the chemical reaction in (A) proceeds to completion,

   \(2 \mathrm{X}\)	\(\rightarrow 3 \mathrm{Y}\)	\(+ \mathrm{~Z}\)
   \(0\)	\(3 \, n_{x}^{0} / 2\)	\(\mathrm{n}_{\mathrm{x}}^{0} / 2 \mathrm{~mol}\)

3. If the chemical reaction in examp1e (A) proceeds to chemical equilibrium, then with \(\xi = \xi^{\mathrm{eq}\),

   	\(2\mathrm{X }\rightleftarrows\)	\(3\mathrm{Y } +\)	\(\mathrm{Z}\)
   Amounts	\(\mathrm{n}_{\mathrm{x}}^{0}-2 \, \xi^{\mathrm{eq}}\)	\(\mathrm{n}_{\mathrm{x}}^{0}-2 \, \xi^{\mathrm{eq}}\)	\(3 \, \xi^{\mathrm{eq}} \mathrm{~mol}\)
   Molalities	\(\left(n_{x}^{0}-2 \, \xi^{e q}\right) / w_{1}\)	\(3 \, \xi^{\mathrm{eq}} / \mathrm{w}_{1}\)	\(\xi^{e q} / w_{1} \mathrm{~mol kg}^{-1}\)