1.22.11: Volumes: Liquid Mixtures: Binary: Method of Tangents
- Page ID
- 397795
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The ‘Method of Tangents’ is an important technique which is readily illustrated using the volumetric properties of binary liquid mixtures. The starting point is (as always?) the Gibbs - Duhem Equation which leads to equation (a) for systems at fixed temperature and pressure.
\[\mathrm{n}_{1} \, \mathrm{dV}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{dV}_{2}(\operatorname{mix})=0\]
Dividing by \(\left(\mathrm{n}_{1} + \mathrm{~n}_{2}\right)\),
\[\mathrm{x}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{dV}_{2}(\operatorname{mix})=0\]
The molar volume is given by equation (c).
\[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]
Hence (at equilibrium, fixed temperature and pressure) the differential dependence of \(\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\) on mole fraction \(\mathrm{x}_{1}\) is given by equation (d).
\[\begin{aligned}
\frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\mathrm{mix}) &+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \\
&+\mathrm{V}_{2}(\mathrm{mix}) \,\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx}}\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]
\end{aligned}\]
From the Gibbs-Duhem equation,
\[\mathrm{x}_{1} \, \frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}=0\]
[Note the common denominator.] Also \(\mathrm{x}_{1} + \mathrm{~x}_{2} = 1\). And so,
\[\mathrm{dx}_{1}=-\mathrm{dx}_{2}\]
Therefore,
\[\frac{\mathrm{dV} \mathrm{m}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{2}(\operatorname{mix})\]
Combination of equations (c) and (g) yields the following equation.
\[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]\]
Further,
\[\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{1}(\mathrm{mix})\]
Hence,
\[\mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})+\left(1-\mathrm{x}_{1}\right) \, \frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}\]
At a given mole fraction, we determine the molar volume of the mixture \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and its dependence on mole fraction. \(\left[\mathrm{dV} \mathrm{m}_{\mathrm{m}}(\operatorname{mix}) / \mathrm{dx}_{1}\right]\) is the gradient of the tangent at mole fraction \(\mathrm{x}_{1}\) to the curve recording the dependence of \(\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\) on \(\mathrm{x}_{1}\); hence the name of this method of data analysis. This analysis is relevant because, as commented above, we can determine the variables \(\mathrm{V}_{1}^{*}(\ell), \mathrm{V}_{2}^{*}(\ell) \text { and } \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\).
Another approach is based on excess molar volumes \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) and their dependence on mole fraction at fixed temperature and pressure. Since
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]
And
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})-\mathrm{V}_{\mathrm{m}}(\mathrm{id})\]
From equations (c), and (k),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]
We define excess partial molar volumes;
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\]
and
\[\mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\]
Hence the excess molar volume of the mixture is related to two excess partial molar volumes.
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix})\]
We use equation (m) to obtain the differential dependence of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) on mole fraction \(\mathrm{x}_{1}\).
\[\begin{aligned}
\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx} \mathrm{x}_{1}}\right] } \\
&+\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx} \mathrm{x}_{1}}\right] \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]
\end{aligned}\]
We write the Gibbs - Duhem equation in the form shown in equation (e) together with equation (p). Hence,
\[\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{V}_{2}^{\mathrm{E}}\]
or,
\[\mathrm{V}_{2}^{\mathrm{E}}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]
Hence using equation (o),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}+\mathrm{x}_{2} \, \mathrm{V}_{1}^{\mathrm{E}}-\mathrm{x}_{2} \, \mathrm{dV} \mathrm{V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]
Thus,
\[\mathrm{V}_{1}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, d \mathrm{~V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]
Equation (t) is the excess form of equation (j). A plot of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\) shows a curve passing through '\(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} = 0\)' at \(\mathrm{x}_{1} = 0\) and \(\mathrm{x}_{1} = 1\). Other than these two reference points, thermodynamics does not define the shape of the plot of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\). Thermodynamics does not define the shape of the plot of \(\mathrm{V}_{1}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\) other than to require that at \(\mathrm{x}_{1} = 1\), \(\mathrm{V}_{1}^{\mathrm{E}}\) is zero. An interesting feature is the sign and magnitude of \(\mathrm{V}_{1}^{\mathrm{E}}\) in the limit that \(\mathrm{x}_{1} = 0\); i.e. at \(\mathrm{x}_{2} = 1\).
The volumetric properties of a binary liquid (homogeneous) mixture is summarized by a plot of excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against, for example, mole fraction \(\mathrm{x}_{1}\). In fact this type of plot is used for many excess molar properties including \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) and \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\). Here we consider a general excess molar property \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\). The corresponding excess partial molar property of chemical substance 1 is \(\mathrm{X}_{\mathrm{1}}^{\mathrm{E}}\) which is related to \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) and the dependence of \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) on \(\mathrm{x}_{1}\) at mole fraction \(\mathrm{x}_{1}\),
\[\mathrm{X}_{1}^{\mathrm{E}}=\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, \mathrm{dX} \mathrm{X}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]
Calculation of \(\mathrm{X}_{\mathrm{1}}^{\mathrm{E}}\) requires the gradient \(\mathrm{dX} \mathrm{m} / \mathrm{dx}_{1}\) as a function of mole fraction composition. The way forward involves fitting the dependence of \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) on \(\mathrm{x}_{1}\) to a general equation and then calculating \(\mathrm{dX} \mathrm{m} / \mathrm{dx}_{1}\) using the derived parameters [1-4].
Footnotes
[1] C. W. Bale and A. D. Pelton, Metallurg. Trans.,1974,5,2323.
[2] C. Jambon and R. Philippe, J.Chem.Thermodyn.,1975,7,479.
[3] M. J. Blandamer, N. J. Blundell, J. Burgess, H. J. Cowles and I. M. Horn, J. Chem. Soc. Faraday Trans.,1990,86,277.
[4] A description of a useful procedure for non-linear least squares analysis is given by W. E. Wentworth, J.Chem.Educ.,1965,42,96.