1.22.11: Volumes: Liquid Mixtures: Binary: Method of Tangents
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.
\[\mathrm{n}_{1} \, \mathrm{dV}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{dV}_{2}(\operatorname{mix})=0 \nonumber \]
Dividing by \(\left(\mathrm{n}_{1} + \mathrm{~n}_{2}\right)\),
\[\mathrm{x}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{dV}_{2}(\operatorname{mix})=0 \nonumber \]
The molar volume is given by equation (c).
\[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix}) \nonumber \]
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).
\[\begin{aligned}
\frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\mathrm{mix}) &+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \\
&+\mathrm{V}_{2}(\mathrm{mix}) \,\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx}}\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]
\end{aligned} \nonumber \]
From the Gibbs-Duhem equation,
\[\mathrm{x}_{1} \, \frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}=0 \nonumber \]
[Note the common denominator.] Also \(\mathrm{x}_{1} + \mathrm{~x}_{2} = 1\). And so,
\[\mathrm{dx}_{1}=-\mathrm{dx}_{2} \nonumber \]
Therefore,
\[\frac{\mathrm{dV} \mathrm{m}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{2}(\operatorname{mix}) \nonumber \]
Combination of equations (c) and (g) yields the following equation.
\[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \nonumber \]
Further,
\[\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{1}(\mathrm{mix}) \nonumber \]
Hence,
\[\mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})+\left(1-\mathrm{x}_{1}\right) \, \frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}} \nonumber \]
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
\[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
And
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})-\mathrm{V}_{\mathrm{m}}(\mathrm{id}) \nonumber \]
From equations (c), and (k),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
We define excess partial molar volumes;
\[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell) \nonumber \]
and
\[\mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell) \nonumber \]
Hence the excess molar volume of the mixture is related to two excess partial molar volumes.
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix}) \nonumber \]
We use equation (m) to obtain the differential dependence of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) on mole fraction \(\mathrm{x}_{1}\).
\[\begin{aligned}
\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx} \mathrm{x}_{1}}\right] } \\
&+\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx} \mathrm{x}_{1}}\right] \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]
\end{aligned} \nonumber \]
We write the Gibbs - Duhem equation in the form shown in equation (e) together with equation (p). Hence,
\[\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{V}_{2}^{\mathrm{E}} \nonumber \]
or,
\[\mathrm{V}_{2}^{\mathrm{E}}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1} \nonumber \]
Hence using equation (o),
\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}+\mathrm{x}_{2} \, \mathrm{V}_{1}^{\mathrm{E}}-\mathrm{x}_{2} \, \mathrm{dV} \mathrm{V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1} \nonumber \]
Thus,
\[\mathrm{V}_{1}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, d \mathrm{~V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1} \nonumber \]
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 \(\mathrm{x}_{1}\),
\[\mathrm{X}_{1}^{\mathrm{E}}=\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, \mathrm{dX} \mathrm{X}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1} \nonumber \]
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.