Why Pure Water Barely Conducts but Salt Water Does

Distilled water hardly conducts, yet add a pinch of salt and a circuit through it can light a bulb. The current is carried by moving ions, and how well a solution conducts depends on which ions it contains and how many.

This part covers conductivity and cell constant, how molar conductivity changes with concentration, Kohlrausch's law, and Faraday's laws of electrolysis.

**Resistivity $\rho$ measures how strongly a solution opposes current, conductivity $\kappa =\frac{1}{\rho }$ measures how well it conducts, the cell constant $G^{\ast }=\frac{l}{A}$ links resistance to conductivity through $\kappa =\frac{G^{\ast }}{R}$, and molar conductivity ${\Lambda }_m=\frac{\kappa }{c}$ is conductivity per mole of electrolyte.

Definitions and units:

- Resistance** $R=\rho \frac{l}{A}$, with $\rho$ in $\Omega$ m
- Conductivity $\kappa =\frac{1}{\rho }$, in S m${}^{-1}$
- Cell constant $G^{\ast }=\frac{l}{A}=\kappa R$, in m${}^{-1}$
- Molar conductivity ${\Lambda }_m=\frac{\kappa }{c}$, in S m${}^2$ mol${}^{-1}$ with $c$ in mol m${}^{-3}$

Worked example. A cell filled with $0.1$ mol L${}^{-1}$ KCl, of conductivity $1.29$ S m${}^{-1}$, has resistance $100\Omega$. Filled with $0.02$ mol L${}^{-1}$ KCl, its resistance is $520\Omega$:

$$G^{\ast }=1.29\times 100=129{\text{ m}}^{-1},\kappa =\frac{129}{520}\approx 0.248{\text{ S m}}^{-1}$$

$${\Lambda }_m=\frac{0.248}{0.02\times 1000}=1.24\times 10^{-2}{\text{ S m}}^2{\text{ mol}}^{-1}$$

An everyday example. A TDS meter used to check drinking water actually measures conductivity, since dissolved salts supply the ions.

The substance. **Convert mol L${}^{-1}$ to mol m${}^{-3}$ by multiplying by $1000$** — skipping this makes the molar conductivity a thousand times too large.

**On dilution, conductivity falls because each unit volume holds fewer ions, but molar conductivity rises; for strong electrolytes it rises gently and extrapolates to a limiting value ${\Lambda }_m^{\circ }$, while for weak electrolytes it rises steeply at low concentration as far more molecules dissociate.

Strong electrolytes** such as KCl:

- Almost fully dissociated at all concentrations
- ${\Lambda }_m={\Lambda }_m^{\circ }-A\sqrt{c}$ — a straight line against $\sqrt{c}$
- Dilution weakens attractions between ions, so they move more freely

Weak electrolytes such as ethanoic acid:

- Only slightly dissociated at ordinary concentrations
- ${\Lambda }_m$ rises sharply near zero concentration, so ${\Lambda }_m^{\circ }$ cannot be found by extrapolation

Limiting molar conductivity ${\Lambda }_m^{\circ }$ is the value as concentration approaches zero, when every ion moves independently.

Worked example. Suppose a strong electrolyte has ${\Lambda }_m=140$ S cm${}^2$ mol${}^{-1}$ at $\sqrt{c}=0.10$ and $146$ S cm${}^2$ mol${}^{-1}$ at $\sqrt{c}=0.05$:

$$A=\frac{146-140}{0.10-0.05}=120,{\Lambda }_m^{\circ }=140+120\times 0.10=152{\text{ S cm}}^2{\text{ mol}}^{-1}$$

An everyday example. Vinegar conducts far less than salt water of similar concentration, because ethanoic acid is a weak electrolyte and only a small fraction of its molecules form ions.

The substance. Conductivity and molar conductivity move in opposite directions on dilution — a favourite source of confusion.

**Kohlrausch's law states that the limiting molar conductivity of an electrolyte is the sum of the limiting molar conductivities of its cation and anion, ${\Lambda }_m^{\circ }={\nu }_{+}{\lambda }_{+}^{\circ }+{\nu }_{-}{\lambda }_{-}^{\circ }$, which gives ${\Lambda }_m^{\circ }$ for weak electrolytes indirectly and then their degree of dissociation, $\alpha =\frac{{\Lambda }_m}{{\Lambda }_m^{\circ }}$.

For a weak electrolyte**, combine strong electrolytes:

$${\Lambda }_m^{\circ }({\text{CH}}_3\text{COOH})={\Lambda }_m^{\circ }({\text{CH}}_3\text{COONa})+{\Lambda }_m^{\circ }(\text{HCl})-{\Lambda }_m^{\circ }(\text{NaCl})$$

Worked example. With ${\lambda }^{\circ }({\text{H}}^{+})=349.6$ and ${\lambda }^{\circ }({\text{CH}}_3{\text{COO}}^{-})=40.9$ S cm${}^2$ mol${}^{-1}$:

$${\Lambda }_m^{\circ }=349.6+40.9=390.5{\text{ S cm}}^2{\text{ mol}}^{-1}$$

Ethanoic acid at $0.001028$ M has ${\Lambda }_m=48.15$ S cm${}^2$ mol${}^{-1}$:

$$\alpha =\frac{48.15}{390.5}\approx 0.123,K_a=\frac{c{\alpha }^2}{1-\alpha }\approx 1.8\times 10^{-5}$$

An everyday example. Lemon juice tastes sharply sour but conducts only weakly, because its citric acid ionises only partly — exactly what $\alpha$ measures.

The substance. **${\lambda }^{\circ }$ for H${}^{+}$ is unusually large** because protons hop between neighbouring water molecules instead of moving bodily through the solution.

**Faraday's first law says the mass deposited is proportional to the charge passed, $m=\frac{MIt}{nF}$, and the second says equal charges deposit masses in the ratio of equivalent masses; the products depend on whether the electrolyte is molten or aqueous, since water itself can be oxidised or reduced.

Faraday's laws:

- First law** — $m=ZIt$, with $Z=\frac{M}{nF}$ and $F=96487$ C mol${}^{-1}$, the charge on one mole of electrons
- Second law — for the same charge, $\frac{m_1}{m_2}=\frac{E_1}{E_2}$, where $E=\frac{M}{n}$

Worked example. A current of $1.5$ A passes through CuSO${}_4$ solution for $10$ minutes:

$$Q=1.5\times 600=900\text{ C},m=\frac{63.5\times 900}{2\times 96487}\approx 0.296\text{ g of copper}$$

Predicting products:

- Molten NaCl — sodium at the cathode, chlorine at the anode
- Aqueous NaCl — hydrogen at the cathode, because water is reduced more easily than Na${}^{+}$; chlorine at the anode, because of overpotential
- **Aqueous CuSO${}_4$ with platinum electrodes — copper at the cathode, oxygen at the anode
- Aqueous CuSO${}_4$ with copper electrodes — the copper anode dissolves as copper deposits on the cathode

An everyday example. Electroplating a steel tap with nickel is electrolysis in action, and the charge passed decides the thickness of the coating.

The substance. Products are decided by competing electrode reactions**, not simply by which ions are present.