9.5 Electrolyte Conductance, Ionic Mobility, Transport Numbers, and Concentration Cells

9.5 Electrolyte Conductance, Ionic Mobility, Transport Numbers, and Concentration Cells

Quick Answer: The TOS pairs kinetic theory with "solutions of electrolytes" and lists conductance quantities, ionic mobility, transport numbers, and the use of conductivity to find degrees of ionization and solubilities. The electrochemistry area adds concentration cells with and without transference. Know the definitions and units, Kohlrausch's two laws, and the standard calculations for weak-acid $K_a$ and sparingly soluble salt $K_{sp}$.

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Conductance Quantities

Cell constants are measured with a standard KCl solution. At 25 C, 0.0100 M KCl has $\kappa = 1.413 \times 10^{-3}$ S/cm. If a cell filled with it reads 1000 ohm, then $K_{cell} = \kappa R = 1.413\text{ cm}^{-1}$.

Units trap: to obtain $S\,cm^2\,mol^{-1}$, convert concentration to mol/cm$^3$: $\Lambda_m = 1000\,\kappa / c$ with $\kappa$ in S/cm and $c$ in mol/L.

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Kohlrausch's Laws

Concentration Dependence

Strong electrolytes follow Kohlrausch's law: $\Lambda_m = \Lambda_m^\circ - K\sqrt{c}$. Molar conductivity falls slightly with concentration because of interionic attraction (the relaxation and electrophoretic effects in Debye-Huckel-Onsager theory). Weak electrolytes fall steeply at low concentration because their degree of ionization changes, so their $\Lambda_m^\circ$ cannot be found by extrapolation.

Independent Migration of Ions

At infinite dilution each ion contributes independently: $\Lambda_m^\circ = \nu_+\lambda_+^\circ + \nu_-\lambda_-^\circ$.

For a weak acid, combine strong electrolytes:

$\Lambda^\circ(HOAc) = \Lambda^\circ(HCl) + \Lambda^\circ(NaOAc) - \Lambda^\circ(NaCl) = 390.5\ S\,cm^2\,mol^{-1}$

$H^+$ and $OH^-$ are unusually mobile because of the Grotthuss mechanism: protons hop along hydrogen-bonded water chains instead of moving through the solvent as a whole.

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Ionic Mobility and Transport Numbers

Ionic mobility is drift speed per unit field: $u = s/E$. It relates to ionic conductivity by $\lambda = zuF$. For $H^+$, $u = 349.6 / 96485 = 3.62 \times 10^{-3}\text{ cm}^2\,V^{-1}\,s^{-1}$.

The transport (transference) number is the fraction of current carried by an ion. For a 1:1 electrolyte at infinite dilution:

$t_+ = \frac{\lambda_+^\circ}{\lambda_+^\circ + \lambda_-^\circ}, \qquad t_+ + t_- = 1$

Transport numbers are measured by the Hittorf method (concentration changes in the anode and cathode compartments after passing known charge) or by the moving-boundary method (the speed of a visible boundary between two electrolytes sharing a common ion).

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Applications of Conductivity

Degree of Ionization and $K_a$ (Ostwald Dilution Law)

$\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}, \qquad K_a = \frac{c\,\alpha^2}{1 - \alpha}$

Worked example: For 0.0100 M acetic acid, $\Lambda_m = 16.2\ S\,cm^2\,mol^{-1}$, so $\alpha = 16.2/390.5 = 0.0415$ and $K_a = \frac{(0.0100)(0.0415)^2}{1 - 0.0415} = 1.80 \times 10^{-5}$.

Solubility of Sparingly Soluble Salts

A saturated solution is so dilute that $\Lambda_m \approx \Lambda_m^\circ$. Subtract the water's own conductivity first.

Worked example: Saturated AgCl at 25 C has $\kappa = 2.68 \times 10^{-6}$ S/cm, and the water used has $0.82 \times 10^{-6}$ S/cm.

• $\kappa(AgCl) = 1.86 \times 10^{-6}$ S/cm and $\Lambda_m^\circ = 61.9 + 76.3 = 138.2\ S\,cm^2\,mol^{-1}$
• $s = \frac{1000\,\kappa}{\Lambda_m^\circ} = \frac{1000 \times 1.86 \times 10^{-6}}{138.2} = 1.35 \times 10^{-5}$ mol/L
• $K_{sp} = s^2 = 1.8 \times 10^{-10}$

Conductometric Titrations

When HCl is titrated with NaOH, highly mobile $H^+$ is replaced by slow $Na^+$, so conductance falls to a minimum at equivalence and then rises as excess $OH^-$ accumulates. The endpoint is where the two straight lines cross. This method works for coloured or very dilute solutions where indicators fail.

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Concentration Cells With and Without Transference

A concentration cell has identical electrodes in solutions of different activity.

• Without transference (no liquid junction, for example two cells back to back or a salt bridge assumed ideal): for hydrogen electrodes in HCl, $E = \frac{2RT}{F}\ln\frac{a_{\pm,2}}{a_{\pm,1}}$.
• With transference (the two solutions in direct contact): ions also move across the junction. With electrodes reversible to the cation, $E_t = 2t_-\frac{RT}{F}\ln\frac{a_{\pm,2}}{a_{\pm,1}}$, so $E_t / E = t_-$. Measuring both emfs is a classic way to find transport numbers.

Worked example: Two HCl solutions differ tenfold in mean activity at 25 C, with $t_- = 0.179$. The cell with transference gives $E_t = 2(0.179)(0.05916) = 0.0212$ V. The electrodes alone would give 0.0592 V, so the liquid-junction potential is about $(t_+ - t_-)(0.05916) = 0.038$ V. A KCl salt bridge minimizes this error because its $t_+$ and $t_-$ are nearly equal.