1.14.50: Osmotic Pressure
A semi-permeable membrane [1] separates an aqueous solution (where the mole fraction of water equals \(\mathrm{x}_{1}\)) and pure solvent at temperature \(\mathrm{T}\) and ambient pressure. Solvent water flows spontaneously across the membrane thereby diluting the solution. This flow is a consequence of the chemical potential of the solvent in the solution being lower than the chemical potential of pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). If a pressure (\(\mathrm{p} + \pi\)) is applied to the solution, the spontaneous process stops because the solution at pressure (\(\mathrm{p} + \pi\)) and the solvent at pressure \(\mathrm{p}\) are in thermodynamic equilibrium; \(\pi\) is the osmotic pressure. Thus at equilibrium,
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}+\pi)=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \nonumber \]
Under this equilibrium condition solvent flows in both directions across the semi-permeable membrane but the net flow is zero. In the analysis presented here we take account of the fact, writing \(\mathrm{p}^{\prime}\) for (\(\mathrm{p} + \pi\)), the chemical potential of water in the aqueous solution is given by equation (b).
\[\begin{aligned}
&\mu_{1}^{\mathrm{eq}}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{\prime}\right)= \\
&\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0)+\mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \mathrm{K}_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{p}^{\prime}\right] \\
& \\
&+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)
\end{aligned} \nonumber \]
Here \(\mathrm{f}_{1}\) is the activity coefficient expressing the extent to which the thermodynamic properties of water in the aqueous solution are not ideal. For the pure solvent water at pressure \(\mathrm{p}\) (i.e. on the other side the of the semi-permeable membrane),
\[\begin{aligned}
&\mu_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\
&\quad \mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0)+\mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{p}\right]
\end{aligned} \nonumber \]
But osmosis experiments explore an equilibrium characterized by equation (d).
\[\mu_{1}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{\prime}\right)=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \nonumber \]
Therefore using equations (b) and (c),
\[\begin{array}{r}
\mathrm{p}^{\prime} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right]+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)= \\
\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]
\end{array} \nonumber \]
Or,
\[\begin{aligned}
&\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right] \\
&-\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)
\end{aligned} \nonumber \]
The terms \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}^{\prime}\right]\) and \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]\) describe the molar volumes of water at pressures \(\mathrm{p}\) and \(\mathrm{p}^{\prime}\); i.e. at pressure \(\mathrm{p}\) and (\(\mathrm{p}+\pi\)). We assume that both terms can be replaced by the molar volumes at average pressure \([(2 \, \mathrm{p}+\pi) / 2]\); namely \(\mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ;(2 \, \mathrm{p}+\pi) / 2]\). Therefore
\[\pi \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ;(2 \, \mathrm{p}+\pi) / 2]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
In the event that \(\pi<<2 \, p\),
\[\pi \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right) \nonumber \]
If the thermodynamic properties of the solutions are ideal, \(\mathrm{f}_{1}\) equals unity. Then
\[\pi^{\mathrm{id}} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
In the latter two equations \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\) is treated as a constant, independent of the thermodynamic properties of the solution. A further interesting development of equation (i) is possible for a solution prepared using \(\mathrm{n}_{1}\) moles of solvent water and \(\mathrm{n}_{j}\) moles of solute. Thus
\[-\ln \left(\mathrm{x}_{1}\right)=\ln \left(\frac{1}{\mathrm{x}_{1}}\right)=\ln \left(\frac{\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)=\ln \left(1+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right) \nonumber \]
But . \(\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}\right)<<1\). We expand the last term in equation (j).
\[\ln \left(1+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}-\frac{1}{2} \,\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)^{2}+\frac{1}{3} \,\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}}\right)^{3}-\ldots . . \nonumber \]
If we retain only the first term;
\[\pi^{\mathrm{id}} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]=\mathrm{R} \, \mathrm{T} \,\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}\right) \nonumber \]
But for a dilute solution, the volume of the solution \(\mathrm{V}\) is given by \(\mathbf{n}_{1} \, \mathrm{V}_{1}^{*}[\ell ; \mathrm{T} ; \mathrm{p}]\).
Or [2],
\[\pi^{\mathrm{id}} \, \mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \nonumber \]
But concentration
\[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V} \nonumber \]
Then [3]
\[\pi^{\mathrm{id}}=\mathrm{c}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \nonumber \]
Agreement between \(\pi(\mathrm{obs})\) and \(\pi^{\mathrm{id}}\) for aqueous solutions containing neutral solutes (e.g. sucrose) confirms the validity of the thermodynamic analysis.
Footnotes
[1] The term ‘semi-permeable’ in the present context means that the membrane is only permeable to the solvent. Perhaps the optimum semipermeable membrane is the vapor phase.
[2] Historically, equation(o) owes much to the equation of state for an ideal gas; i.e. \(\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T}\). From an experimentally found proportionality between \(\pi\) and \(\mathrm{c}_{j}\), van’t Hoff showed that the proportionality constant can be approximated by \(\mathrm{R} \, \mathrm{T}\).
[3]
\[\pi=\left[\mathrm{mol} \mathrm{m}^{-3}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{m}^{-3}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right] \nonumber \]