Conductance of Electrolytes

Conductors of electricity. Materials fall into two classes. Electronic (metallic) conductors carry current by the flow of electrons; their conductance falls as temperature rises. Electrolytic conductors — solutions of salts, acids and bases, or molten salts — carry current by the movement of ions; their conductance rises with temperature and depends on ion concentration. Electrochemistry deals mainly with the second class.

Resistance and conductance. For any conductor, $R = \rho\frac{l}{A}$, where $\rho$ is the resistivity. The reciprocal of resistance is conductance $G = 1/R$, measured in siemens (S). The reciprocal of resistivity is the conductivity $\kappa$ (kappa): $\kappa = \frac{1}{\rho} = \frac{l}{A}\cdot\frac{1}{R} = G\cdot\frac{l}{A}$. The factor $\frac{l}{A}$ is the cell constant, $G^* $, fixed for a given conductivity cell. The unit of $\kappa$ is $\text{S m}^{-1}$ (often $\text{S cm}^{-1}$).

Conductivity, $\kappa$. Conductivity is the conductance of a solution held between two electrodes of unit area, one metre (or one cm) apart. It measures conductance per unit volume, so it falls on dilution — there are simply fewer ions in each unit volume.

Molar conductivity, $\Lambda_m$. To compare electrolytes fairly we use the molar conductivity, the conducting power of all the ions from one mole of electrolyte: $\Lambda_m = \frac{\kappa \times 1000}{C}$, where $C$ is the concentration in $\text{mol L}^{-1}$ and $\kappa$ is in $\text{S cm}^{-1}$, giving $\Lambda_m$ in $\text{S cm}^2\,\text{mol}^{-1}$. Unlike $\kappa$, molar conductivity rises on dilution, because the same one mole of ions spreads through more solution and the ions move more freely.

Variation with concentration. The way $\Lambda_m$ changes with dilution distinguishes the two kinds of electrolyte:

• Strong electrolytes (e.g. $\text{KCl}$, $\text{HCl}$, $\text{NaCl}$) are almost fully ionised. As concentration falls, $\Lambda_m$ increases only slightly and linearly with $\sqrt{C}$. Debye, Hückel and Onsager showed $\Lambda_m = \Lambda_m^0 - A\sqrt{C}$. Extrapolating the straight line to $C \rightarrow 0$ gives the limiting molar conductivity $\Lambda_m^0$.
• Weak electrolytes (e.g. $\text{CH}_3\text{COOH}$, $\text{NH}_4\text{OH}$) are only partly ionised. As the solution is diluted, the degree of dissociation rises sharply, so $\Lambda_m$ increases steeply and the curve shoots up near zero concentration — it cannot be extrapolated to find $\Lambda_m^0$ directly.

Kohlrausch's law of independent migration. At infinite dilution, where ions move independently of one another, the limiting molar conductivity is the sum of the individual ionic contributions: $\Lambda_m^0 = \nu_+\lambda_+^0 + \nu_-\lambda_-^0$, where $\lambda^0$ are the limiting molar conductivities of the ions and $\nu$ their numbers per formula unit.

Applications of Kohlrausch's law. (i) It gives $\Lambda_m^0$ for weak electrolytes indirectly — e.g. $\Lambda_m^0(\text{CH}_3\text{COOH}) = \Lambda_m^0(\text{CH}_3\text{COONa}) + \Lambda_m^0(\text{HCl}) - \Lambda_m^0(\text{NaCl})$. (ii) It gives the degree of dissociation $\alpha = \frac{\Lambda_m}{\Lambda_m^0}$ and hence the dissociation constant $K_a$ of weak acids.