1.10.7: Gibbs Energies- Solutions- Parameters Phi and ln(gamma)
The practical osmotic coefficient can be calculated knowing the dependence of \(\gamma_{\mathrm{j}}\) on molality of solute \(j\). Of course at this stage we do not know the form of the dependence of \(\gamma_{\mathrm{j}}\) on \(\mathrm{m}_{\mathrm{j}}\). In fact \(\gamma_{\mathrm{j}}\) also depends on the solute, temperature and pressure. But for a given system (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) we might express \(\phi\) as a series expansion of the molality \(\mathrm{m}_{\mathrm{j}}\). Thus,
\[\phi=1+a_{1} \, m_{j}+a_{2} \, m_{j}^{2}+a_{3} \, m_{j}^{3}+\ldots \ldots \nonumber \]
Interestingly this assumed dependence is equivalent to a series expansion in mole fraction of solute \(\mathrm{x}_{\mathrm{j}}\) for \(1 \mathrm{nf}_{1}\), where \(\mathrm{f}_{1}\) is the (rational) activity coefficient for the solvent [1,2].
\[\operatorname{lnf}_{1}=\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{2}+\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{3}+\mathrm{b}_{3} \, \mathrm{x}_{\mathrm{j}}^{4}+\ldots \ldots \nonumber \]
Here \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3} \ldots\) depend on the solute (for given \(\mathrm{T}\) and \(\mathrm{p}\)). The link between the two equations can be expressed as follows.
\[\mathrm{b}_{1}=-\left[(1 / 2)+\mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}\right] \nonumber \]
\[\mathrm{b}_{2}=-\left[(2 / 3)+2 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+\mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}\right] \nonumber \]
\[\mathrm{b}_{3}=-\left[(3 / 4)+3 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+3 \, \mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}+\mathrm{a}_{3} \, \mathrm{M}_{1}^{-3}\right] \nonumber \]
Footnotes
[1] J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Phys., 1968, 48 , 675.
[2] By definition, for a solution j in solvent, chemical substance 1,
\[\mathrm{x}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}} /\left(\mathrm{M}_{\mathrm{l}}^{-1}+\mathrm{m}_{\mathrm{j}}\right) \nonumber \]
where \(\mathrm{M}_{1}\) is the molar mass of solvent expressed in \(\mathrm{kg mol}^{-1}\). Hence molality of solute \(j\),
\[\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{l}}^{-1} \,\left(1-\mathrm{x}_{\mathrm{j}}\right)^{-1} \nonumber \]
We expand \(\left(1-x_{j}\right)^{-1}\) based on the premise that \(0<\mathrm{x}_{\mathrm{j}}<<1.0\) for dilute solutions. Then,
\[\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \mathrm{M}_{1}^{-1} \,\left[1+\mathrm{x}_{\mathrm{j}}+\mathrm{x}_{\mathrm{j}}^{2}+\mathrm{x}_{\mathrm{j}}^{3}+\ldots \ldots\right] \nonumber \]
or, m x j jj jj M x x x = ⋅+ + ++ − 1 1 234 [ .....] (d)
Here we carry all terms up to and including the fourth power of xj. But from the two methods for relating µ1(aq) to the composition of a solution, 1nx f M m 11 1 j ( ) ⋅ =−⋅ ⋅ φ (e)
Then, 1 1 11 1 1 2 2 3 3 nx f M m m m m j jjj ( ) [ .....] ⋅ =− ⋅ ⋅ + ⋅ + ⋅ + ⋅ + a a a (f)
or, 1 11 1 11 2 2 3 3 4 nf n M m m m m j jj j j x a a a =− − − ⋅ + ⋅ + ⋅ + ⋅ + ( ) [ .....] (g)
But for dilute solutions, −− =+ + + 11 2 3 4 2 34 nx x x x j jj j j ( ) ( /) ( /) ( /) x (h)
Using equation (c) for mj as a function of xj in the context of equation (f), we obtain an equation for ln(f1). 1 234 1 2 34 nf x x x jj j j ( ) ( /) ( /) ( /) x =+ + + −− − − xxxx jj jj 234 1 4 j 1 1 1 3 j 1 1 1 2 j 1 1 − M ⋅ x ⋅ a − 2 ⋅ M ⋅ x ⋅ a − 3⋅ M ⋅ x ⋅ a − − − − ⋅ ⋅ −⋅ ⋅ ⋅ − − M xa M xa 1 j j 2 3 2 1 2 4 2 3 − ⋅⋅ − M xa 1 j 3 4 3 (i)
Hence, 1n(f1) = { [( / ) ( )] } − +⋅ ⋅ − 1 2 1 1 1 2 a M x j } { [(2 / 3) (2 a M ) (a M )] x3 j 2 2 1 1 1 1 + − + ⋅ ⋅ + ⋅ ⋅ − − +− + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − −− { [ / ) ( ) ( ) ( )] } 34 3 3 1 1 1 2 1 2 3 1 3 4 aM aM aM x j (j)