1.18.1: Liquid Mixtures: Regular Mixtures
A given binary liquid mixture is prepared using liquid-1 and liquid –2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The chemical potentials, \(\mu_{1}\left(\operatorname{mix} ; x_{1}\right)\) and \(\mu_{2}\left(\operatorname{mix} ; x_{2}\right)\) are related to the mole fraction composition, \(x_{1}\) and \(x_{2} (= 1 - x_{1})\) using equations (a) and (b) where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\);
\[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
\[\mu_{2}\left(\operatorname{mix} ; x_{2}\right)=\mu_{2}^{*}(\ell)+R \, T \, \ln \left(x_{2} \, f_{2}\right) \nonumber \]
Here for both \(\mathrm{i} = 1,2\) at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(x_{i} \rightarrow 1\right) f_{i}=1 \nonumber \]
The term “regular mixture” describes a liquid mixture for which the rational activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are given by equations (d) and (e) where the property \(\mathrm{w}\) is independent of temperature and liquid mixture composition [1-5].
\[\ln \left(f_{1}\right)=(w / R \, T) \, x_{2}^{2} \nonumber \]
\[\ln \left(f_{2}\right)=(w / R \, T) \,\left(1-x_{2}\right)^{2} \nonumber \]
Then, for example, at all \(\mathrm{T}\) and \(\mathrm{p}\) [6],
\[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \ln \left(f_{1}\right)=0 ; f_{1}=1 \nonumber \]
Similarly,
\[\operatorname{limit}\left(x_{2} \rightarrow 1\right) \ln \left(f_{2}\right)=0 ; f_{2}=1 \nonumber \]
Interest in regular liquid mixtures stems from the observation that the properties of such real (as opposed to ideal) mixtures are simply described. Of course the term “real” only means that the dependences of rational activity coefficients on mole fraction composition are defined by equations (d) and (e) and that the thermodynamic properties of the liquid mixture are not ideal.
For example, according to equation (d),
\[d \ln \left(f_{1}\right) / d T=-\left(w / R \, T^{2}\right) \, x_{2}^{2} \nonumber \]
With reference to the dependence of the properties of binary liquid mixtures on temperature (at fixed pressure), equation (a) yields equation (i).
\[\frac{\mathrm{d}\left[\mu_{1}(\mathrm{mix}) / \mathrm{T}\right]}{\mathrm{dT}}=\frac{\mathrm{d}\left[\mu_{1}^{*}(\ell) / \mathrm{T}\right]}{\mathrm{dT}}+\mathrm{R} \,\left[\frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{\mathrm{dT}}\right] \nonumber \]
From the Gibbs - Helmholtz equation,
\[-\frac{\mathrm{H}_{1}(\mathrm{mix})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{1}^{*}(\ell)}{\mathrm{T}^{2}}-\mathrm{R} \,\left(\frac{\mathrm{w}}{\mathrm{R} \, \mathrm{T}^{2}}\right) \, \mathrm{x}_{2}^{2} \nonumber \]
Hence
\[\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)+\mathrm{w} \, \mathrm{x}_{2}^{2} \nonumber \]
Here
\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{mix})=\mathrm{H}_{1}^{*}(\ell) \nonumber \]
According to equation (k), \(\mathrm{H}_{\mathrm{l}}(\mathrm{mix})\) is a quadratic function of the mole fraction composition.
Further [7],
\[\mathrm{S}_{1}(\operatorname{mix})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
Hence entropic properties of regular mixtures do not deviate from the properties of an ideal liquid mixture. In terms of excess properties for regular mixtures, \(S_{m}^{E}=0\) and therefore \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\). Equations (d) and (e) can be written as explicit equations for \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) respectively.
\[f_{1}=\exp \left[\left(\frac{w}{R \, T}\right) \, x_{2}^{2}\right] \nonumber \]
\[f_{2}=\exp \left[\left(\frac{w}{R \, T}\right) \,\left(1-x_{2}\right)^{2}\right] \nonumber \]
The partial pressures of the two chemical substances are given by Raoult’s Law.
\[\mathrm{p}_{1}=\mathrm{p}_{1}^{*} \, \mathrm{x}_{1} \, \exp \left[\left(\frac{\mathrm{W}}{\mathrm{R} \, \mathrm{T}}\right) \, \mathrm{x}_{2}^{2}\right] \nonumber \]
\[\mathrm{p}_{2}=\mathrm{p}_{2}^{*} \, \mathrm{x}_{2} \, \exp \left[\left(\frac{\mathrm{w}}{\mathrm{R} \, \mathrm{T}}\right) \,\left(1-\mathrm{x}_{2}\right)^{2}\right] \nonumber \]
For both liquid components, deviations from ideal thermodynamic properties increase with increase in the magnitude of \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\). If \(\mathrm{w} >0\), the deviations are called positive whereas if \(\mathrm{w} <0\) the deviations are called negative. In the event that \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\) equals 2, the plots of \(\mathrm{p}_{1}\) and \(\mathrm{p}_{2}\) against mole fraction composition are horizontal when \(x_{1} = x_{2} = 0.5\). But in the event that \((\mathrm{w} / \mathrm{R} \, \mathrm{T})\) equals 3, a range of binary liquid mixtures exist having intermediate mole fraction compositions and are unstable. These mixtures separate into two liquid mixtures, one rich in component 1 and the other rich in component 2.
Footnotes
[1] J. Hildebrand, J Am. Chem. Soc.,1929, 51 ,69. Accounts of this class of mixtures are given in references [2]-[5].
[2] E. A. Guggenheim, Thermodynamics, North Holland Publishing Company, Amsterdam, 1950, chapter 5; note that Guggenheim uses the symbol \(x\) to represent the mole fraction composition of a binary liquid mixture \(x_{2}\); see page 173.
[3] E. A. Guggenheim, Mixtures, Clarendon Press, Oxford, 1952, chapter IV.
[4] M.L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, chapter 16.
[5] G. N. Lewis and M. L. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 1961, chapter 21.
[6]
\[\mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \nonumber \]
Then, \(\mathrm{w}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\), a molar energy.
[7] From, \(\mu_{1}(\operatorname{mix})=\mathrm{H}_{1}(\operatorname{mix})-\mathrm{T} \, \mathrm{S}_{1}(\operatorname{mix})\)
\[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{w} \, \mathrm{x}_{2}^{2}=\mathrm{H}_{1}^{*}(\ell)+\mathrm{w} \, \mathrm{x}_{2}^{2}-\mathrm{T} \, \mathrm{S}_{1}(\mathrm{mix}) \nonumber \]
Or,
\[\mu_{1}^{*}(\ell)-\mathrm{H}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=-\mathrm{T} \, \mathrm{S}_{1}(\operatorname{mix}) \nonumber \]
Or,
\[-\mathrm{T} \, \mathrm{S}_{1}^{\prime \prime}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=-\mathrm{T} \, \mathrm{S}_{1}(\mathrm{mix}) \nonumber \]
[8] From equation (p) with \((w / R \, T)=2\),
\[\begin{aligned}
&\mathrm{p}_{1}=\mathrm{p}_{1}^{*} \, \mathrm{x}_{1} \, \exp \left(2 \, \mathrm{x}_{2}^{2}\right)\\
&p_{1}=p_{1}^{*} \, x_{1} \, \exp \left[2 \,\left(1-x_{1}\right)^{2}\right]\\
&\frac{\mathrm{d}\left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right)}{\mathrm{dx}_{1}}=\exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right]-\mathrm{x}_{1} \, 4 \,\left(1-\mathrm{x}_{1}\right) \, \exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right]\\
&=\exp \left[2 \,\left(1-x_{1}\right)^{2}\right] \,\left[1-4 \, x_{1} \,\left(1-x_{1}\right)\right]\\
&=\exp \left[2 \,\left(1-x_{1}\right)^{2}\right] \,\left[1-4 \, x_{1}+4 \, x_{1}^{2}\right]\\
&\frac{\mathrm{d}\left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right)}{\mathrm{dx_{1 }}}=\exp \left[2 \,\left(1-\mathrm{x}_{1}\right)^{2}\right] \,\left(1-2 \, \mathrm{x}_{1}\right)^{2} \\
&\frac{\mathrm{d}\left(\mathrm{p}_{2} / \mathrm{p}_{2}^{*}\right)}{\mathrm{dx}_{2}}=\exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right]+\mathrm{x}_{2} \,(-4) \,\left(1-\mathrm{x}_{2}\right) \, \exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right] \\
&\frac{\mathrm{d}\left(\mathrm{p}_{2} / \mathrm{p}_{2}^{*}\right)}{\mathrm{dx_{2 }}}=\exp \left[2 \,\left(1-\mathrm{x}_{2}\right)^{2}\right] \,\left(1-2 \, \mathrm{x}_{2}\right)^{2}
\end{aligned} \nonumber \]