1.10.26: Gibbs Energies- Binary Liquid Mixtures
Interest in the thermodynamic properties of binary liquid mixtures extended over most of the 19th Century. Interest was concerned with the vapour pressures of the components of a given binary mixture. Experimental work led Raoult to propose a law of partial vapour pressures which characterises a liquid mixture whose thermodynamic properties are ideal.
If liquid 1 is a component of a binary liquid mixture, components liquid 1 and liquid 2, the mixture is defined as having ideal thermodynamic properties if the partial vapour pressure at equilibrium \(\mathrm{p}_{1}(\text{mix})\) is related to the mole fraction composition using equation (a).
\[) p (mix) x p ( * 1 1 1 = ⋅ l (a)
Here \(\mathrm{p}_{1}^{*}(\ell)\) is the vapour pressure of pure liquid 1 at the same temperature. A similar equation describes the vapour pressure of liquid 2, \(\mathrm{p}_{2}(\text{mix})\) in the mixture. Raoult’s Law as given in equation (a) forms the basis for examining the properties of real binary liquid mixtures.
Binary liquid mixtures [1-13] are interesting in their own right. Nevertheless chemists also use binary liquid mixtures as solvents for chemical equilibria and for media in which to carry out chemical reactions between solutes. In fact an enormous amount of our understanding of the mechanisms of inorganic and organic reactions is based on kinetic data describing the rates of chemical reactions in 80/20 ethanol and water mixtures. Because so much experimental information in the literature concerns the properties of solutes in binary aqueous mixtures we concentrate our attention on these systems. Indeed the properties of such solvent systems cannot be ignored when considering the properties of solutes in these mixtures. In examining the properties of binary aqueous mixtures we adopt a convention in which liquid water is chemical substance 1 and the non-aqueous liquid component is chemical substance 2. For the most part we describe the composition of a given liquid mixture (at defined temperature and pressure) using the mole fraction scale. Then if \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) are the amounts of chemical substances 1 and 2, the mole fractions are defined by equation (b) [12,13].
\[\mathrm{x}_{1}=\mathrm{n}_{1} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \quad \mathrm{x}_{2}=\mathrm{n}_{2} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
The chemical potentials of each component in a given liquid mixture, mole fraction composition \(\mathrm{x}_{2}\left(=1-\mathrm{x}_{1}\right)\) are compared with the chemical potential of the pure liquid chemical potential at the same temperature and pressure, \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\).
The starting point is a description of the equilibrium at temperature \(\mathrm{T}\) between pure liquid and its saturated vapour in a closed system; i.e. a two phase system. [In terms of the Gibbs phase rule the number of components = 1; the number of phases = 2. Hence the number of degrees of freedom = 1. Then at a specified temperature the vapour pressure is defined.] The equilibrium is described in terms of the equality of chemical potentials of substance 1 in the two phases; equation (c).
\[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right) \nonumber \]
Thus \(\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)\) is the molar Gibbs energy of the pure liquid 1, otherwise the chemical potential.
However our interest concerns the properties of binary liquid mixtures. Using the Gibbs phase rule, number of phases = 2; number of components = 2; hence number of degrees of freedom = 2. Then for a defined temperature and mole fraction composition the vapour pressure \(\mathrm{p}(\text{mix})\) is fixed. Equation (d) describes the equilibrium between liquid and vapour phases with reference to liquid substance 1.
\[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1} ; \mathrm{T}\right)=\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right) \nonumber \]
Equations (c) and (d) offer a basis for comparing the chemical potentials of chemical substance 1 in the two phases, liquid and vapour.
\[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)-\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)-\mu_{1}\left(\mathrm{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right) \nonumber \]
Analysis of vapour pressure data is not straightforward because account has to be taken of the fact that the properties of real gases are not those of an ideal gas. However here we assume that the properties of chemical substance in the gas phase are ideal. Consequently we use the following equation to provide an equation for the r.h.s. of equation (e) in terms of two vapour pressures.
\[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right) \nonumber \]
Equation (a) is combined with equation (f) to yield the following equation for chemical substance 1 as component of the binary liquid mixture.
\[\mu_{1}\left(\operatorname{mix} ; \mathrm{id} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
Similarly for chemical substance 2 in the liquid mixture,
\[\mu_{2}\left(\operatorname{mix} ; \mathrm{id} ; \mathrm{x}_{2}\right)=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right) \nonumber \]
Hence for an ideal binary liquid mixture, formed by mixing \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles respectively of chemical substance 1 and 2, the Gibbs energy of the mixture is given by equation (i).
\[\begin{aligned}
&\mathrm{G}(\operatorname{mix} ; \mathrm{id})= \\
&\quad \mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right]
\end{aligned} \nonumber \]
Bearing in mind that Gibbs energies cannot be measured for either a pure liquid or a liquid mixture, it is useful to rephrase equation (i) in terms of the change in Gibbs energy that accompanies mixing to form an ideal binary liquid mixture. We envisage a situation where before mixing the molar Gibbs energy of the system defined as \(\mathrm{G}(\text{no}-\text{mix})\) is given by equation (j).
\[\mathrm{G}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mu_{2}^{*}(\ell) \nonumber \]
The change in Gibbs energy on forming the ideal binary liquid mixture \(\Delta_{\text{mix}} \mathrm{G}\) is given by the equation (k).
\[\Delta_{\text {mix }} \mathrm{G}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{n}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{n}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
We re-express \(\Delta_{\text{mix}} \mathrm{G}\) in terms of the Gibbs energy of mixing forming one mole of the ideal binary liquid mixture.
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}=\Delta_{\text {mix }} \mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
Hence,
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right] \nonumber \]
As required,
\[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 0\right) \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=0 \nonumber \]
\[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \Delta_{\text {mix }} G_{m}(\text { id })=0 \nonumber \]
The dependence of \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}\) on mole fraction composition is defined by equation (m). For a mixture where \(x_{1}= x_{2}=0.5\),
\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}=2.0 \times 0.5 \times \ln (0.5)=-0.693 \nonumber \]
In fact the molar Gibbs energy of mixing for an ideal binary liquid mixture is negative across the whole composition range.
Equation (m) is the starting point of most equations used in the analysis of the properties of binary liquid mixtures. In fact most of the chemical literature concerned with liquid mixtures describes the properties of aqueous mixtures, at ambient pressure and \(298.15 \mathrm{~K}\). Nevertheless an extremely important subject concerns the properties of liquid mixtures at high pressures [14,15].
Footnotes
[1] K. N. Marsh, Pure Appl. Chem., 1983, 55 , 467.
[2] K. N. Marsh, Annu. Rep. Prog. Chem., Sect. C, Phys. Chem., 1994, 91 ,209.
[3] J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, 3rd. edn., London, 1982.
[4] G. Scatchard, Chem.Rev., 1931, 8 ,321;1949, 44 ,7.
[5] E. A. Guggenheim, Liquid Mixtures, Clarendon Press, Oxford, 1952.
[6] J. H. Hildebrand and R. L. Scott, Solubility of Non-Electrolytes, Reinhold, New York,3rd. edn.,1950.
[7] J. H. Hildebrand, J. M. Prausnitz and R. L. Scott, Regular and Related Solutions, van Nostrand Reinhold, New York,1970.
[8] A. G. Williamson, An Introduction to Non-Electrolyte Solutions, Oliver and Boyd, Edinburgh, 1967.
[9] Y. Koga, J.Phys.Chem.,1996, 100 ,5172.
[10] L. S. Darken, Trans. Metallurg. Soc., A.M.E., 1967, 239 ,80.
[11] A. D. Pelton and C. W Bale, Metallurg. Trans.,!986, 17A ,211.
[12] According to Phase Rule, P = 2 (for liquid and vapour), C = 2 then F = 2 + 2 - 2 = 2. Having defined temperature and pressure there remain no degrees of freedom - the system is completely specified.
[13] Other methods of defining the composition include the following.
Mass % [or w%] Mass of component \(1=\mathrm{n}_{1} \, \mathrm{M}_{1}\) Mass of component \(2=\mathrm{n}_{2} \, \mathrm{M}_{2}\)
Then
\[\mathrm{w}_{1} \%=\frac{\mathrm{n}_{1} \, \mathrm{M}_{1} \, 100}{\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]} \quad \mathrm{w}_{2} \%=100-\mathrm{w}_{1} \% \nonumber \]
Volume % This definition often starts out by defining the volumes of the two liquid components used to prepare a given mixture at defined temperature and pressure. The definition does not normally refer to the volume of the actual mixture. The volume after mixing is often less than the sum of the component volumes before mixing.
\[\begin{aligned}
&\mathrm{V}_{1}^{*}(\ell)=\mathrm{n}_{1} \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell) \quad \mathrm{V}_{2}^{*}(\ell)=\mathrm{n}_{2} \, \mathrm{M}_{2} / \rho_{2}^{*}(\ell) \\
&\mathrm{V}_{2} \%=\frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{V}_{2}^{*}(\ell)}
\end{aligned} \nonumber \]
[14] G. Schneider, Pure Appl. Chem.,1983, 55 ,479; 1976, 47 ,277.
[15] G. Schneider, Ber. Bunsenges, Phys.Chem.,1972, 76 ,325.