1.1.6: Activity of Solvents- Classic Analysis
Thermodynamics underpins a classic topic in physical chemistry concerning the depression of freezing point, \(\Delta \mathrm{T}_{\mathrm{f}}\) of a liquid by added solute [1,2]. We note the superscript ‘id’ in equation (a) relating the activity of a solvent in an ideal solution, molality \(\mathrm{m}_{j}\).
\[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \nonumber \]
If the properties of a given aqueous solution are determined to a significant extent by solute-solute interactions, a determined molar mass for a given solute will be in error. Otherwise an observed depression is not a function of solute-solute interactions. Glasstone [2] comments that the ratio \(\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}\) decreases with increasing concentration of solute, emphasising that a simple analysis is only valid for dilute solutions. Nevertheless the general idea is that the depression for a given mj is not a function of the hydration of a solute. Barrow [3] noted that the ratio \(\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}\) for mannitol(aq) in very dilute solutions [4] is effectively constant. A similar opinion is advanced by Adam [5] who nevertheless comments on the importance of the condition ’dilute solution’ in a determination of the molar mass of a given solute.
In summary, classic physical chemistry emphasises the importance of the superscript ‘id’ in equation (a). For very dilute solutions in a given solvent \(\ln \left(a_{1}\right)\) is linear function of \(\mathrm{m}_{j}\), leading to description of such properties as ‘depression of freezing point ‘ and ‘elevation of boiling point ’ under the heading ‘colligative properties. Only the molality of solute mj is important; solute-solute interactions and hydration characteristics of solutes apparently play no part in determining these colligative properties.
The key to these statements is provided by the Gibbs-Duhem equation. For a solution prepared using 1 kg of water(\(\lambda\)) and \(\mathrm{m}_{j}\) moles of a simple solute \(j\), the Gibbs energy is given by equation (b).
\[\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
Then, \(\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\)
\[+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \text { (c) } \nonumber \]
According to the Gibbs - Duhem Equation, the chemical potentials of solvent and solute are linked. At fixed \(\mathrm{T}\) and \(p\),
\[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0 \nonumber \]
Or, \(\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\)
\[+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \nonumber \]
\[\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0 \nonumber \]
\[\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0 \nonumber \]
\[\text { Hence, } \quad(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}}+\mathrm{d} \phi=\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \nonumber \]
The latter equation forms the basis of the oft-quoted statement that if the thermodynamic properties of a solute are ideal then so are the properties of the solvent. Similarly if the thermodynamic properties of the solute are ideal then so are the properties of the solvent.
\[\text { From equation }(h), \phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right) \nonumber \]
The importance of equation (i) emerges from the idea that \(\gamma_{j}\) describes the impact of solute-solute interactions on the properties of a given solution. If we can formulate an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the properties of the solution, we obtain \(\phi\) from equation (i). If the properties of a real solution containing a neutral solute are not ideal, both \(\gamma_{j}\) and \(\phi\) are linked functions of the solute molality. Pitzer [6] suggests equations (j) and (k) for \(\ln \left(\gamma_{j}\right)\) and \(\phi\) in terms of solute molalities using two parameters, \(\lambda\) and \(\mu\).
\[\ln \left(\gamma_{\mathrm{j}}\right)=2 \, \lambda \, \mathrm{m}_{\mathrm{j}}+3 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \nonumber \]
\[\phi-1=\lambda \, \mathrm{m}_{\mathrm{j}}+2 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \nonumber \]
For example in the case of mannitol(aq) and butan-1-ol(aq), Pitzer [6] suggests the following equations for \(\ln \left(\gamma_{j}\right)\).
\[\text { For mannitol(aq) } \quad \ln \left(\gamma_{\mathrm{j}}\right)=-0.040 \, \mathrm{m}_{\mathrm{j}} \nonumber \]
\[\text { For butan-1-ol(aq) } \ln \left(\gamma_{\mathrm{j}}\right)=-0.38 \, \mathrm{m}_{\mathrm{j}}+0.51 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \nonumber \]
Guggenheim[7] using the mole fraction scale suggests equation (n) where \(\mathrm{A}\) and \(\mathrm{B}\) are characteristic of the solute.
\[1-\phi=A \, x_{j}+B \,\left(x_{j}\right)^{2} \nonumber \]
Prigogine and Defay [1] comment that the non-ideal properties of solutions can be understood in terms of the different molecular sizes of solute and solvent. A similar comment is made by Robinson and Stokes [8] who use a parameter describing the ratio of molar volumes of solute and solvent. The extent to which the properties of a solution differ from ideal can often be traced to a variety of causes including solvation, molecular size and shape.
Footnotes
[1] I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green , London 1953.
[2] S. Glasstone, Physical Chemistry, McMillan, London, 2nd. edn., 1948,page 645.
[3] G. M. Barrow, Physical Chemistry, McGraw-Hill, New York, 4th edn.,1979, page 298.
[4] L. H. Adams, J. Am. Chem. Soc.,1915,37,481.
[5] Neil Kensington Adam, Physical Chemistry, Oxford,1956, page 284.
[6] K. S. Pitzer, Thermodynamics , McGraw-Hill, New York, 3rd. edn., 1995.
[7] E. A. Guggenheim, Thermodynamics, North-Holland Publishing Co., Amsterdam, 1950, pages 252-3
[8] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965.