1.5.6: Chemical Potentials- Liquid Mixtures- Raoult's Law
A given closed system contains two volatile miscible liquids. The closed system is connected to a pressure-measuring device which records that at temperature T the pressure is ptot. The composition of the liquid mixture is known; i.e. mole fractions \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) (where \(\mathrm{x}_{2} = 1 - \mathrm{x}_{1}\)). The system contains two components so that in terms of the Phase Rule, \(\mathrm{C} = 2\) There are two phases, vapour and liquid so that \(\mathrm{P} = 2\). From the rule, \(\mathrm{P} + \mathrm{F} = \mathrm{C} + 2\), we have fixed the composition and temperature using up the two degrees of freedom. Hence the pressure ptot is fixed.
We imagine that the mixture under examination is a binary aqueous mixture; water is chemical substance 1. If we measure the partial pressure of, say, liquid 1, \(\mathrm{p}_{1}\) we find that \(\mathrm{p}_{1}\) is close to a linear function of mole fraction \(\mathrm{x}_{1}\).
\[\text { At equilibrium and temperature } \mathrm{T}, \quad \mathrm{p}_{1}^{\mathrm{eq}} \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1} \nonumber \]
As the mole fraction \(\mathrm{x}_{1}\) approaches unity (i.e. the composition of the mixture approaches pure water) the equilibrium vapour pressure of water \(\mathrm{p}_{1}^{\mathrm{eq}}\) approaches that of pure liquid water at the same temperature, \(\mathrm{p}_{1}^{*}(\ell)\) We have linked the equilibrium vapour pressure of water to the composition of the liquid mixture.
In fact it turns out that as the composition of the mixture approaches pure liquid 1, the latter relationship becomes an equation. We assert that if the thermodynamic properties of the mixture were ideal then \(\mathrm{p}_{1}\) would be related to mole fraction \(\mathrm{x}_{1}\) using the following equation.
\[\mathrm{p}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{p}_{1}^{*}(\ell) \nonumber \]
Returning to experiment, we invariably find that as a real solution becomes more dilute (i.e. as \(\mathrm{x}_{1}\) approaches unity) \(\mathrm{p}_{1}^{\mathrm{eq}}\) for real solutions approaches \(p_{1}^{e q}(a q ; i d)\). Therefore we rewrite equation (b) as an equation for a real solution by introducing a new property called the (rational) activity coefficient \(\mathrm{f}_{1}\).
\[\mathrm{p}_{1}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{f}_{1} \, \mathrm{p}_{1}^{*}(\ell) \nonumber \]
Here \(\mathrm{f}_{1}\) is the (rational) activity coefficient for liquid component 1 defined as follows.
\[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \nonumber \]
\[\text { Similarly for volatile liquid } 2 ; p_{2}=x_{2} \, f_{2} \, p_{2}^{*} \nonumber \]
\[\operatorname{limit}\left(x_{2} \rightarrow 1\right) f_{2}=1 \nonumber \]
Although equations (d) and (f) have simple forms, rational activity coefficients carry a heavy load in terms of information. For a given aqueous system, \(\mathrm{f}_{1}\) describes the extent to which interactions involving water molecules in a real system differ from those in the corresponding ideal system. The challenge of expressing this information in molecular terms is formidable.
We carry over these ideas to the task of formulating an equation for the chemical potential of water in the liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). We make the link between partial pressure and the tendency for liquid 1 to escape to the vapour phase, down a gradient of chemical potential
By definition (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)),
\[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
\[\text { where, at all } \mathrm{T} \text { and } \mathrm{p}, \quad \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1.0\right) \mathrm{f}_{1}=1.0 \nonumber \]
\(\mu_{1}^{*}(\ell)\) l is the chemical potential of pure liquid water (at the same \(\mathrm{T}\) and \(\mathrm{p}\)). In other words, pure liquid water is the reference state against which we compare the properties of water in an aqueous mixture. For the pure liquid at temperature \(\mathrm{T}\), \(V_{1}^{*}(\ell)=\mathrm{d} \mu_{1}^{*}(\ell) / \mathrm{dp}\). If \(\mathrm{p}^{0}\) is the standard pressure [1] \(\left(10^{5} \mathrm{~N} \mathrm{~m}^{-2}\right)\),
\[\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{d} \mu_{1}^{*}(\ell)=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
\[\text { Then }[2], \quad \mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mu_{1}^{0}(\ell ; \mathrm{T})=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
\(\mu_{1}^{0}(\ell ; \mathrm{T})\) is the standard chemical potential of water(\(\ell\)) at temperature \(\mathrm{T}\).
\[\text { Therefore, } \quad \mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
This is an important equation although, at this stage, we can go no further. Without information concerning the dependence on pressure of \(\mathrm{V}_{1}^{*}(\ell)\) {or density} we cannot evaluate the integral in equation (k). However, we can comment on possible patterns in these chemical potentials. If the thermodynamic properties of the liquid mixture are ideal then \(\mathrm{f}_{1}\) equals \(1.0\). Hence equation (k) takes the following simple form (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)).
\[\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
In a solution the mole fraction x1 is less than unity and so \(\ln \left(x_{1}\right)<0\) [3]. Hence \(\mu_{1}(\operatorname{mix} ; \mathrm{id})<\mu_{1}^{*}(\ell)\) at the same \(\mathrm{T}\) and \(\mathrm{p}\) [4]
Footnotes
[1] J.D. Cox, Pure Appl. Chem., 1982, 54 , 1239; R.D. Freeman, Bull. Chem. Thermodyn., 1982, 25 , 523.
[2] \(\mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\)
[3] With increase in the amount of component 2 so \(\mathrm{x}_{1}\) tends to zero. In this limit \(\mu_{1}(\mathrm{aq})\) is minus infinity.
[4] Note that \(\mathrm{p}_{1}^{\mathrm{eq}}(\operatorname{mix} ; \mathrm{id})-\mathrm{p}_{1}^{*}<0\). Adding a solute lowers the vapour pressure of the water. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of the second component; G. Bertrand and C. Treiner, J. Solution Chem.,1984, 13 ,43.