1.14.20: Electric Conductivities of Salt Solutions- Dependence on Composition
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the molar conductivity of given salt solution Λ depends on the concentration of salts. This subject has an extensive scientific literature. One of the challenges is to calculate \(\Lambda\) for given salt solution knowing the properties of the pure solvent and the salt at specified \(\mathrm{T}\) and \(\mathrm{p}\). A key quantity is the limiting molar conductivity \(\Lambda^{\infty}\) defined for a given salt solution by equation (a).
\[\operatorname{Lt}\left(c_{j} \rightarrow 0\right) \Lambda=\Lambda^{\infty} \nonumber \]
Moreover as Kohlrausch showed in 1876 ( > 125 years ago) a given \(\Lambda^{\infty}\) can be expressed as the sum of limiting molar ionic conductances, \(\lambda_{\mathrm{j}}^{\infty}\). Thus
\[\Lambda^{\infty}=\sum_{j=1}^{j=i} \lambda_{j}^{\infty} \nonumber \]
A difficult theoretical task is to estimate \(\lambda_{\mathrm{j}}^{\infty}\) for a given ion at defined \(\mathrm{T}\) and \(\mathrm{p}\) and specified solvent. Rather more progress has been made in predicting quantitatively the dependence of \(\left(\Lambda-\Lambda^{\infty}\right)\) on concentration of salt in a given solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\) assuming that the ions in solution are characterised by their electric charges and radii. Indeed quantitative treatments of the electrical conductivities of salt solutions have attracted enormous interest and provided a challenge to scientists with good mathematical abilities. Here we summarise briefly the essence of treatments published by Onsager [1-3] and by Fuoss [3,4]. The account given below is based on that set out by N. K. Adam [5].
A relaxation effect and an electrophoretic effect contribute to the magnitude of \(\left(\Lambda-\Lambda^{\infty}\right)\) for a real salt solution for which \(\Lambda<\Lambda^{\infty}\). In a real solution under the influence of an applied electric field, anions and cations move in opposite direction. The word ‘move’ does not reflect the complexity of the real situation. In a real solution and in the absence of an applied electric field, the ions move in random directions, Brownian motion, as a consequence of the thermal energy of the system. In some sense, ions and solvent molecules are jostling continuously. When an electric potential gradient is applied across the solution, the previously random motion of ions is now biased in a particular direction depending on the ionic charge. If the solution is ‘infinitely dilute’, the velocity of a given ion is characteristic of the ion, solvent, temperature and pressure.
In a real solution the mobility decreases with increase in concentration of salt. Two retarding effects are identified, relaxation and electrophoretic effects. The latter emerges from the fact that a given \(j\) ion moves against the flow of counter-ions together with associated solvent molecules. In a real solution and in the absence of an applied electric field, a given \(j\) ion is at the centre of an ion atmosphere which has an electric charge equal in magnitude but opposite in sign to that of the \(j\) ion. Under the impact of an applied electric field the \(j\)-ion moves away from the centre of the ion atmosphere. The latter pulls the \(j\)-ion back towards this centre. In other words the \(j\)-ion is retarded by this relaxation effect. The latter term reflects the fact that the retardation depends on the rate at which the electric charge density in the ion atmosphere grows as the \(j\) ion moves through the solution and decays in the wake of the \(j\) ion.
Ionic Mobility
A given salt solution at temperature \(\mathrm{T}\) and pressure \(p\) contains a sa }\)]. In solution, the motion of ions is quite random, a pattern usually described as Brownian motion. If however an electric field is applied across the solution the movement of ions is biased in a given direction depending on the sign of the charge on the \(j\) ion. The electrical mobility \(\mathrm{u}_{j}\) of ion \(j\) describes the velocity of ion-\(j\) in an electric field gradient measured in [\(\mathrm{V m}^{-1}\)] [6]. In the absence of ion-ion charge-charge interaction , the electrical mobility \(\mathbf{u}_{j}^{\infty}\) is characteristic of the \(j\) ion, solvent, temperature and pressure. The superscript ‘\(\infty\)’ identifies that to all intents and purposes the \(j\) ion is in an infinitely dilute solution. However in a real solution, concentration \(\mathrm{c}_{i}\) in salt \(i\), the \(j\) ion is surrounded by an ‘ion atmosphere’ which has an electric charge equal in magnitude but opposite in sign to that on the \(j\) ion.
Electrophoretic Effect
The ion atmosphere is modelled as a series of shells, thickness \(\mathrm{dr}\) distance \(\mathrm{r}\) from the centre of the \(j\) ion. The electrical charge \(\mathrm{q}_{j}\) on a shell distance \(\mathrm{r}\) from the \(j\) ion is given by equation (c) [7].
\[q_{j}=4 \, \pi \, r^{2} \, \rho \, d r \nonumber \]
Here \(\rho\) is the electric charge density, measured in ‘coulombs per cubic metre’. As a result of the electric field gradient operating on the \(j\) ion the electric force \(\mathrm{F}\) expressed in newtons operating on this shell [8] is given by equation (d).
\[\mathrm{F}=4 \, \pi \, \mathrm{r}^{2} \, \rho \, \mathrm{E} \, \mathrm{dr} \nonumber \]
According to Stokes Law, a sphere having radius \(\mathrm{r}\) and moving with velocity \(v\) through a liquid having (shear) viscosity \(\eta\) is subject to a viscous resistance \(\mathrm{R}_{\eta}\), a force expressed in newtons and given by equation (e) [9].
\[\mathrm{R}_{\eta}=6 \, \pi \, \eta \, \mathrm{r} \, \mathrm{V} \nonumber \]
If the speed of the liquid stream increases by \(\mathrm{dv}\) when the radius of the shell defining the ion atmosphere increases by \(\mathrm{dr}\), the viscous resistance increases by \((6 \, \pi \, \eta \, r \, d v)\). If the motion of the \(j\) ion through the solution is steady, the increase in viscous resistance to movement of the \(j\) ion equals the electrical force (see equation d). Therefore
\[6 \, \pi \, \eta \, r \, d v=4 \, \pi \, r^{2} \, \rho_{j} \, E \, d r \nonumber \]
The charge density \(\rho_{j}\) is obtained by combining equations (u) and (x) of Topic 680. Thus [10],
\[\rho_{\mathrm{j}}=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right) \, \exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right)}{\left.\left.4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\right) 1+\kappa \, \mathrm{a}_{\mathrm{j}}\right)} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\kappa \, \mathrm{r})}{\mathrm{r}} \nonumber \]
Then with \(\ell^{-1}=\kappa\),
\[p_{j}=-\frac{z_{j} \, e \, \exp \left(a_{j} / \ell\right) \, \exp (-r / \ell)}{4 \, \pi \, \ell \,\left(a_{j}+\ell\right) \, r} \nonumber \]
From equation (f),
\[\mathrm{dv}=\frac{2}{3} \, \frac{\rho_{\mathrm{j}}}{\eta} \, \mathrm{E} \, \mathrm{r} \, \mathrm{dr} \nonumber \]
Hence
\[\mathrm{dv}=-\frac{2 \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \exp \left(\mathrm{a}_{\mathrm{j}} / \ell\right)}{12 \, \pi \, \eta \, \ell \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)} \, \mathrm{E} \, \exp (-\mathrm{r} / \ell) \, \mathrm{dr} \nonumber \]
Equation (j) is integrated between limits (i) \(r=\sigma\) to \(r = \infty\), and (ii) \(v = 0\) and \(v_{1}\) where \(v_{1}\) is the stream velocity of the solution outside the ion atmosphere of the \(j\) ion. Then
\[\mathrm{v}_{1}=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E} \, \exp \left(\mathrm{a}_{\mathrm{j}} / \ell\right)}{6 \, \pi \, \eta \, \ell \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)} \, \int_{\mathrm{a}_{\mathrm{j}}}^{\infty} \exp (-\mathrm{r} / \ell) \, \mathrm{dv} \nonumber \]
Hence,
\[\mathrm{v}_{1}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E}}{6 \, \pi \, \eta \,\left(\mathrm{a}_{\mathrm{j}}+\ell\right)} \nonumber \]
For dilute solutions, \(\mathrm{a}_{\mathrm{j}}<\ll \ell\) such that the stream velocity of the solution outside the ion atmosphere is given by equation (m)
Therefore [11]
\[\mathrm{v}_{1}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e} \, \mathrm{E}}{6 \, \pi \, \eta \, \ell} \nonumber \]
We shift the reference. The solvent does not physically move when we measure the electrical conductivity of a solution. Therefore the impact of the electrophoretic effect is to retard the \(j\)-ion in solution. The decrease in electrical mobility of the \(j\) ion is given by equation (n) [12].
\[-\left(\Delta \mathrm{u}_{\mathrm{j}}\right)_{\mathrm{clectrpbor}}=-\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{6 \, \pi \, \eta \, \ell} \nonumber \]
Relaxation Effect
In the limit of infinite dilution, a given \(j\)-ion proceeds through an aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\) under the influence of an applied electric field gradient. The impediment to its progress arises from the solvent molecules. However in a real salt solution, the \(j\) ion is surrounded by its ion atmosphere which has an electric charge equal in magnitude and opposite on sign. In the absence of an applied electric field the ion atmosphere is spherically symmetric about the \(j\) ion. In a real solution, the migrating ion is not at the centre of the ion atmosphere, the latter therefore retarding the migrating ion. This retardation is called the relaxation effect on the grounds that the build-up of the ion atmosphere preceeding the ion and the decay in the wake of the ion is characterised by a relaxation time.
The relaxation effect can be understood in terms of irreversible thermodynamics. Thus the flow of cations and anions in opposite directions are coupled. The stronger the coupling the greater is the retardation of the migrating ions. The first treatment of this coupling of flows and forces was developed by Onsager who published a reasonably successful description of the impact of this coupling on ionic mobilities. The analogue of equation (n) describing the relaxation effect takes the following form [13] where \(\mathrm{w}\) is a correction factor depending on the type of electrolyte [14].
\[-\left(\Delta u_{j}\right)=\frac{e^{3} \, w \, u_{j}^{\infty}}{24 \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, k \, T \, \ell} \nonumber \]
Here \(\ell\) is the radius of the ion atmosphere surrounding the \(j\) ion ; equation (p) where the concentration of \(j\) ions \(\mathrm{c}_{j}\) is expressed in \(\mathrm{mol dm}^{-3}\).
\[\ell=\frac{10^{3} \, 4 \, \pi \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}{8 \, \pi \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{I}} \nonumber \]
For dilute solutions
\[I=(0.5) \, \sum_{i=1}^{i=j} c_{i} \, z_{i}^{2} \nonumber \]
Molar Conductivity
In summary two retarding effects, electrophoretic and relaxation, mean that the molar conductivity of a given aqueous salt solution is less than the molar conductivity of the corresponding solution at infinite dilution, \(\Lambda^{\infty}\). The outcome is the famous Debye-Huckel-Onsager Equation for molar conductivities. For a 1:1 salt (e.g. \(\mathrm{KBr}\)) in aqueous solution at \(298.15 \mathrm{~K}\) and ambient pressure, the molar conductivity \(\Lambda\) is given by equation (r) [15,16].
\[\Lambda=\Lambda^{\infty}-\left(0.229 \, \Lambda^{\infty}+60.2\right) \,\left(c_{j} / c_{r}\right)^{1 / 2} \nonumber \]
Footnotes
[1] L. Onsager, Physik. Z.,1926, 27 ,388.
[2] L. Onsager, Trans. Faraday Soc.,1927, 23 ,341.
[3] L. Onsager and R. M. Fuoss, J. Phys. Chem.,1932,36,2689.
[4] H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 1950, 2nd edn. Revised and enlarged
[5] N. K. Adam, Physical Chemistry, Oxford, 1956.
[6]
\[\begin{aligned}
&v_{j}=\left[\mathrm{m} \mathrm{s}^{-1}\right] \\
&u_{j}=\left[\mathrm{m} \mathrm{s}^{-1}\right] /[\mathrm{V} \mathrm{m}
\end{aligned} \nonumber \]
[7] \(\mathrm{q}_{\mathrm{j}}=[1] \,[1] \,\left[\mathrm{m}^{2}\right] \,\left[\mathrm{Cm}^{-3}\right] \,[\mathrm{m}]=[\mathrm{C}]\)
[8] \(4 \, \pi \, r^{2} \, \rho \, E \, d r=[1] \,[1] \,\left[\mathrm{m}^{2}\right] \,\left[\mathrm{C} \mathrm{m}^{-3}\right] \,\left[\mathrm{V} \mathrm{m}^{-1}\right] \,[\mathrm{m}] =\left[\mathrm{J} \mathrm{m}^{-1}\right]=[\mathrm{N}]\)
[9] \(\mathrm{R}_{\eta}=[1] \,[1] \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}] \,\left[\mathrm{m} \mathrm{s}^{-1}\right]=\left[\mathrm{kg} \mathrm{m} \mathrm{s}^{-22}\right]=[\mathrm{N}]\)
[10] \(\rho_{j}=\frac{[1] \,[\mathrm{C}] \,[1]}{[1] \,[1] \,\left[\mathrm{Fm}^{-1}\right] \,[1] \,[1]} \, \frac{\left[\mathrm{Fm}^{-1}\right] \,[1] \,[\mathrm{m}]^{2} \,[1]}{[\mathrm{m}]} =\left[\mathrm{C} \mathrm{m}^{-3}\right]\)
[11] \(\mathrm{v}_{1}=\frac{[\mathrm{l}] \,[\mathrm{C}] \,\left[\mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]}{[\mathrm{l}] \,[\mathrm{l}] \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}]} \mathrm{v}_{1}=\frac{[\mathrm{l}] \,[\mathrm{C}] \,\left[\mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]}{[\mathrm{l}] \,[\mathrm{l}] \,\left[\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\right] \,[\mathrm{m}]}
[12] \(\Delta \mathrm{u}_{\mathrm{j}}=\frac{\left[\mathrm{m} \mathrm{s}^{-1}\right]}{\left[\mathrm{V} \mathrm{} \mathrm{m}^{-1}\right]}=\left[\mathrm{m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}\right]\)
\(-\left(\Delta \mathrm{u}_{\mathrm{j}}\right)_{\text {relax }}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{[1] \,\left[\mathrm{Fm}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{m}]} \, \mathrm{u}_{\mathrm{j}}^{\infty}\)
[13] \(=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{[\mathrm{F}] \,[\mathrm{J}]}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-2}\right] \,\left[\mathrm{kg} \mathrm{m} \mathrm{m}^{2} \mathrm{~s}^{-2}\right]} \, \mathrm{u}_{\mathrm{j}}^{\infty}\)
\(=[1] \, \mathrm{u}_{\mathrm{j}}^{\infty}\)
[14] For an advanced treatment, see J. O’M. Bockris and A.K.N Reddy, Modern Electrochemistry: Ionics, Plenum Press, New York, 2nd. edn.,1998, chapter 4.
[15] P. W. Atkins, Physical Chemistry, Oxford University Press, 1982, 2nd. edn., p.900.
[16] \(\Lambda=\left[\Omega^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\right] ; 0.229=[1] ; 60.2=\left[\Omega^{-1} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}\right]\)