1.14.29: Functions of State
One Chemical Substance
For a system containing one chemical substance we define the volume using equation (a).
\[\mathrm{V}=\mathrm{V}[\mathrm{T}, \mathrm{p}, \mathrm{n},] \nonumber \]
The variables in the square brackets are the Independent Variables . The term `independent' means that, within limits [1], we can change \(\mathrm{T}\) independently of the pressure and \(\mathrm{n}_{j}\); change \(\mathrm{p}\) independently of \(\mathrm{T}\) and \(\mathrm{n}_{\mathrm{j}}\); and \(\mathrm{n}_{\mathrm{i}}\) independently of \(\mathrm{T}\) and \(\mathrm{p}\). There are some restrictions in our choice of independent variables. At least one variable must define the amount of all substances in the system and one variable must define the `hotness' of the system.
Actually there is merit in writing equation (a) in terms of three intensive variables which in turn define, for example the, the molar volume of liquid chemical substance 1 at specified temperature and pressure, \(\mathrm{V}_{1}^{*}(\ell)\).
\[\mathrm{V}_{1}^{*}(\ell)=\mathrm{V}(\ell)\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}=1\right] \nonumber \]
Two Chemical Substances
If the chemical composition of a given closed system is specified in terms of two chemical substances 1 and 2, four independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) define the dependent variable \(\mathrm{V}\) [2]. Thus
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right] \nonumber \]
i - Chemical Substances
For a system containing i-chemical substances where the amounts can be independently varied, the dependent variable \(\mathrm{V}\) is defined by the following equation.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
In a general analysis, we start out with a closed system having Gibbs energy at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), molecular composition (organisation) \(\xi\) and affinity for spontaneous change \(\mathrm{A}\). We define the Gibbs energy as follows.
\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \boldsymbol{\xi}] \nonumber \]
In the state defined by equation (e), there is an affinity for spontaneous change (chemical reaction) \(\mathrm{A}\). We imagine that starting with the system in the state defined by equation (e), it is possible to change the pressure and perturb the system to a series of neighbouring states for which the affinity for spontaneous change remains constant. The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant affinity is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\).
Returning to the original state characterised by \(\mathrm{T}\), \(\mathrm{p}\) and \(\xi\), we imagine it is possible to perturb the system by a change in pressure in such a way that the system remains at fixed extent of reaction \(\xi\). The differential dependence of \(\mathrm{G}\) on pressure for the original state along the path at constant \(\xi\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\). We explore these dependences of \(\mathrm{G}\) on pressure at fixed temperature and at (i) fixed composition \(\xi\) and (ii) fixed affinity for spontaneous change, \(\mathrm{A}\). \((\partial G / \partial p)_{\mathrm{T}, \mathrm{A}}\) and \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\) are related using a standard calculus operation [1]. Thus at fixed temperature,
\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}} \nonumber \]
This interesting equation shows that the differential dependence of Gibbs energy (at constant temperature) on pressure at constant affinity for change does NOT equal the corresponding dependence at constant extent of chemical reaction. This is inequality is not surprising. But our interest is drawn to the case where the system under discussion is, at fixed temperature and pressure, at thermodynamic equilibrium where \(\mathrm{A}\) is zero, \(\mathrm{d} \xi / \mathrm{dt}\) is zero, the Gibbs energy is a minimum AND significantly \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. We conclude that
\[\mathrm{V}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \mathrm{A}=0}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi(\mathrm{eq})} \nonumber \]
This rather long winded argument confirms that volume \(\mathrm{V}\) is a state variable, the dependence of \(\mathrm{G}\) on pressure for differential displacement at constant '\(\mathrm{A} =0\)' and \(\sim^{e} q\) being identical. These comments may seem trivial but the point is made if we go on to consider the volume of a system as a function of temperature at constant pressure. We again use a calculus operation [1] to derive the relationship in equation (h).
\[\left[\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{p}} \nonumber \]
We are not surprised to discover that in general terms the differential dependence of \(\mathrm{V}\) on temperature at constant affinity does not equal the differential dependence of \(\mathrm{V}\) on temperature at constant composition/organisation. Indeed unlike the simplification we used in connection with equation (e), we cannot assume that the volume of reaction \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p} p}\) is zero at equilibrium. In other words for a closed system at thermodynamic equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) {where \(\mathrm{A}=0, \xi=\xi^{\mathrm{eq}} \text { and } \mathrm{d} \xi / \mathrm{dt}=0\)}, there are two thermal expansions [2].
We consider a closed system in equilibrium state I defined by the set of variables, \(\left\{\mathrm{T}[\mathrm{I}], \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{I}]\right\}\). The equilibrium composition is represented by \(\xi^{\mathrm{eq}}[\mathrm{I}]\) at zero affinity for spontaneous change. This system is perturbed to nearby state at constant pressure .
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The state I is displaced to a nearby equilibrium state II defined by the set of variables \(\left\{\mathrm{T}[\mathrm{I}]+\delta \mathrm{T}, \mathrm{p}, \mathrm{A}=0, \xi^{\mathrm{eq}}[\mathrm{I}]\right\}\). This equilibrium displacement is characterised by a volume change.
\[\Delta \mathrm{V}(\mathrm{A}=0)=\mathrm{V}[\mathrm{II}]-\mathrm{V}[\mathrm{I}] \nonumber \]
\[\mathrm{E}(\mathrm{A}=0)=\left[\frac{\mathrm{V}(\mathrm{II})-\mathrm{V}(\mathrm{I})}{\Delta \mathrm{T}}\right] \nonumber \]
The equilibrium expansivity,
\[\alpha(\mathrm{A}=0)=\mathrm{E}) \mathrm{A}=0) / \mathrm{V} \nonumber \]
In order for the system to move from one equilibrium state, I with composition \(\xi^{\mathrm{eq}}[\mathrm{I}]\) to another equilibrium state, II with composition \(\xi^{\mathrm{eq}}[\mathrm{II}]\), the chemical composition and /or molecular organisation changes. The term `expansion' indicates the isobaric dependence of volume on temperature,
\[\mathrm{E}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1}\right] \nonumber \]
\(\mathrm{E}\) is an extensive variable. The corresponding volume intensive variable is the expansivity, \(\alpha\).
\[\alpha=\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{p} \nonumber \]
where \(\alpha=\frac{1}{\left[\mathrm{~m}^{3}\right]} \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~K}}\right]=\left[\mathrm{K}^{-1}\right]\)
Hence we define the `frozen' expansion, \(\mathrm{E}(\sim=\text { fixed })\). An alternative name is the instantaneous expansion because, practically, we would have to change the temperature at such a high rate that there is no change in molecular composition or molecular organisation in the system. Thus,
\[\mathrm{E}(\xi=\text { fixed })=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \nonumber \]
Further
\[\alpha(\xi=\text { fixed })=\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \nonumber \]
Similar comments apply to isothermal compressibilities, \(\kappa_{\mathrm{T}}\); there are two limiting properties, \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) and \(\kappa_{\mathrm{T}}(\xi)\). In order to measure \(\kappa_{\mathrm{T}}(\xi)\) we have to change the pressure in a infinitely short time.
The entropy \(\mathrm{S}\) at fixed composition is given by the partial differential \(-\left(\frac{\partial G}{\partial T}\right)_{p, 5}\) and, at constant affinity of spontaneous change by \(-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}}\). At equilibrium where \(\mathrm{A} = 0\), the equilibrium entropy, \(S(A=0)=-\left(\frac{\partial G}{\partial T}\right)_{p, A=0}\). We carry over the argument described above but now concerned with a system characterised by \(\mathrm{T}, \mathrm{p}, \xi\) which is perturbed by a change in temperature. We consider two pathways, at constant \(\mathrm{A}\) and at constant \(\xi\).
\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \nonumber \]
But at equilibrium, \(\mathrm{A}\) which equals \(-\left[\frac{\partial G}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\) is zero and so \(\mathrm{S}(\mathrm{A}=0)=\mathrm{S}\left(\xi^{\mathrm{eq}}\right)\). Then just as for volumes, the entropy of a system is not a property concerned with pathways between states; entropy is a function of state.
Another important link involving Gibbs energy and temperature is provided by the Gibbs-Helmholtz equation. We explore the relationship between changes in (\(\mathrm{G}/\mathrm{T})\) at constant affinity and fixed \(\xi\) following a change in temperature. Thus,
\[\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \nonumber \]
But at equilibrium \(\mathrm{A}\) which equals \(-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Then \(\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\).
In other words the variable, enthalpy is a function of state. This is not the case for isobaric heat capacities. Thus,
\[\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \nonumber \]
We cannot assume that the triple product term in equation (q) is zero. Hence there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) and the `frozen' isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)\). In other words, an isobaric heat capacity is not a function of state because it is concerned with a pathway between states.
Footnotes
[1] The phrase `independent variable' is important. With reference to the properties of an aqueous solution containing ethanoic acid, the number of components for such a solution is 2, water and ethanoic acid. The actual amounts of ethanoic acid, water, ethanoate and hydrogen ions are determined by an equilibrium constant which is an intrinsic property of this sytem at a given \(\mathrm{T}\) and \(\mathrm{p}\). From the point of view of the Phase Rule, the number of components equals two. For the same reason when we consider the volume of a system containing \(\mathrm{n}\) moles of water we do not take account of evidence that water partly self-dissociates into \(\mathrm{H}^{+} (\mathrm{aq})\) and \(\mathrm{OH}^{-} (\mathrm{aq})\) ions.
[2] In terms of the Phase Rule, we note that for two components (C = 2) and one phase (P = 1) , the number of degrees of freedom F equals equals two. These degrees of freedom refer to intensive variables. Hence for a solution where chemical substance 1 is the solvent and chemical substance 2 is the solute, the system is defined by specifying by the three (intensive) degrees of freedom, \(\mathrm{T}, \mathrm{p} and, for example, solute molality.