1.1.3: Activity- Solutions and Liquid Mixtures
The concept of activity was introduced by Lewis[1] in descriptions of the properties of liquid mixtures and solutions. By way of illustration we consider the chemical potential of chemical substance \(j\), \(\mu_{j}(\text { system })\) present in a solution at fixed pressure \(p\) and temperature \(T\). By definition,
\[\mu_{j}(\text { system })=\mu_{j}(\text { ref })+R \, T \, \ln \left(a_{j}\right) \nonumber \]
While we can never know either \(\mu_{j}(\text { system })\) or \(\mu_{j}(\text { ref })\), the difference is related to the activity \(a_{j}\), a dimensionless function of the composition of the system. We as observers of the system are required to define the reference state where the chemical potential of chemical substance \(j\) can be clearly defined. Nevertheless the terms \(\mu_{j}(\text { system })\), \(\mu_{j}(\text { ref })\) and \(a_{j}\) in equation (a) are based on somewhat abstract concepts. The link with practical chemistry is made through the differential of equation (a) with respect to pressure at constant temperature.
\[\text { Then, } \mathrm{V}_{\mathrm{j}}(\text { system })=\mathrm{V}_{\mathrm{j}}(\text { ref })+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}} \nonumber \]
\(\mathrm{V}_{\mathrm{j}}(\text { system })\) and \(\mathrm{V}_{\mathrm{j}}(\text { ref })\) are, respectively, the partial molar volumes of chemical substance \(j\) in the system and in a convenient reference state. The term \(\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\) contrasts the role of intermolecular interactions in the two states. Four applications of the concept of activity make the point.
For the binary liquid mixture, ethanol + water at defined temperature and pressure, the activity of, for example, water (substance 1) \(a_{1}\) is given by the product, \(x_{1} \, f_{1}\) where \(f_{1}\) is the (rational) activity coefficient and \(x_{1}\) is the mole fraction of water.
\[a_{1}=x_{1} \, f_{1} \nonumber \]
The activity of urea (chemical substance \(j\)) in an aqueous solution is related to the product of activity coefficient \(\gamma_{i}\) and molality \(m_{j}\) using the reference molality \(m^{0}\), namely \(1 \mathrm{ mol kg}^{–1}\).
\[a_{j}=\left(m_{j} / m^{0}\right) \, \gamma_{j} \nonumber \]
If the concentration of urea in the solution equals \(c_{j} \mathrm{ mol dm}^{-3}\), then the activity \(a_{j}\) is given by equation (e) where \(c_{r}\) is the reference concentration \(c_{r}\), \(1 \mathrm{ mol dm}^{–3}\), and \(y_{j}\) is the solute activity coefficient.
\[a_{j}=\left(c_{j} / c_{r}\right) \, y_{j} \nonumber \]
If \(x_{j}\) is the mole fraction of urea and \(\mathrm{f}_{\mathrm{j}}^{*}\) is the asymmetric activity coefficient, the activity of urea is given by equation (d).
\[a_{j}=x_{j} \, f_{j}^{*} \nonumber \]
Equations (d) to (f) describe the same property, namely activity \(a_{j}\) of solute \(j\) in a given solution.
Footnotes
[1] G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259.