1.16.2: Ionic Mobilities: Aqueous Solutions

A classic subject in physical chemistry concerns the electric conductivities of salt solutions, most interest being centred on the aqueous salt solutions. Although the electric conductivities of these systems, transport properties, do not come under the heading ‘thermodynamic properties’, these conductivities have played a major part in the task of understanding the thermodynamic properties of salt solutions.

Generally however interest in the electric conductivities of salt solutions has waned as spectroscopic properties in all its form have moved to a dominant position in physical chemistry. Nevertheless the contributions made by research into the electric properties of salt solutions have been and remain enormously important.

Conductivities

At this point there is merit in commenting on the technique, mass spectrometry. In this important experimental technique, ions are produced in an ion source and then subjected to an electric field gradient, where (usually) cations are accelerated. The ions pass through a magnetic field, the path of a given ion depending on the charge and mass of the ion.

Descriptions of the electrical conductivities of salt solutions start out from a quite different basis. To understand the point we consider a reasonably concentrated aqueous solution of sodium chloride; i.e. \(0.1 \mathrm{~mol dm}^{-3} \equiv 0.1 \mathrm{~mol}\) salt in water, mass \(1 \mathrm{~kg} \equiv 0.1 \mathrm{~mol}\) salt in \((1.0/0.018) \mathrm{~mol} \text{ water } \equiv 0.1 \mathrm{~mol Na}^{+} \text { ions } + 0.1 \mathrm{~mol Cl}^{-} \text { ions } + 55.6 \mathrm{~mol} \text { water}(\ell)\). In other words, for every sodium ion there are 556 molecules of water in this aqueous solution. The contrast with the mass spectrometer experiment could not be more dramatic. Further in conventional experiments studying the electric conductivities of salt solutions, the effect of a modest electric potential gradient is simply to bias the otherwise Brownian motion of the ion in a direction depending on the sign of the charge on a given ion. As each ion makes its way through the solution it is jostled and impeded by the large number of solvent molecules. Nevertheless in theoretical treatments of the electric conductivities of salt solutions the theory envisages a slow direct progress through the solution, in the case of, for example, a cation down the electric potential gradient. The key experimental fact is that the electric properties of salt solutions at low electric currents and low electric potential gradients obey the phenomenological law, Ohm’s Law. Deviations from this law are observed for example at high electrical field gradients; e.g. Wien Effects.

Molar Conductivities

The key term in the context of the electric conductivities of a salt solution, concentration of salt \(\mathrm{c}_{j}\) is the molar conductivity \(\Lambda\) defined by equation (a) where \(\kappa\) is the electrolytic conductivity [1,2].

\[\Lambda=\kappa / \mathrm{c}_{j}\]

For a salt solution prepared using a 1:1 salt , the molar conductivity can be expressed as the sum of ionic conductivities , \(\lambda_{+}\) and \(\lambda_{-}\). Thus

\[\Lambda=\lambda_{+}+\lambda_{-}\]

Using equation (a), the electrolytic conductivity \(\kappa\) is related to the ionic conductivities using equation (c)

\[\kappa=\mathrm{c}_{\mathrm{j}} \,\left(\lambda_{+}+\lambda_{-}\right)\]

The electric mobility of a given ion, \(\mathrm{u}_{j}\) is related to the mobility \(v_{j}\) using equation (d) [3].

\[\mathrm{u}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} / \mathrm{E}\]

Footnotes

[1]

\[\begin{aligned}
\kappa &=(\text { electric current density) } / \text { (electric field strength }) \\
&=[\mathrm{j}] /[\mathrm{E}] \\
&=\left[\mathrm{A} \mathrm{m}^{-2}\right] /\left[\mathrm{V} \mathrm{m}^{-1}\right]=\left[\mathrm{S} \mathrm{m}^{-1}\right]
\end{aligned}\]

[2] \(\Lambda=\left[\mathrm{S} \mathrm{m}^{-1}\right] /\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{S} \mathrm{} \mathrm{m}^{2} \mathrm{~mol}^{-1}\right]\)

[3] \(\mathrm{u}_{\mathrm{j}}=\left[\mathrm{m} \mathrm{s}^{-1}\right] /\left[\mathrm{V} \mathrm{m}^{-1}\right]=\left[\mathrm{m}^{2} \mathrm{~s}^{-1} \mathrm{~V}^{-1}\right]\)