Why a Car Battery Can Be Recharged but a Torch Cell Cannot

Pure water barely conducts, yet add a pinch of salt and a current flows — ions carry the charge. The same electrochemistry powers the cell in a torch, the battery in a car and the fuel cells in spacecraft, and it also quietly eats away iron gates and bridges.

This lesson covers molar and equivalent conductivity with Kohlrausch's law, common batteries and fuel cells, and the electrochemical basis of corrosion.

Molar conductivity is the conductivity of a solution divided by its molar concentration, equivalent conductivity is the conductivity divided by its normality, and both rise on dilution — while Kohlrausch's law builds their limiting values from contributions of the individual ions.

$${\Lambda }_m=\frac{\kappa \times 1000}{C}{\Lambda }_{eq}=\frac{\kappa \times 1000}{N}$$

with conductivity $\kappa$ in S cm${}^{-1}$ and concentration in mol L${}^{-1}$, giving ${\Lambda }_m$ in S cm${}^2$ mol${}^{-1}$.

Worked example. A 0.020 M KCl solution has $\kappa =0.00277$ S cm${}^{-1}$:

$${\Lambda }_m=\frac{0.00277\times 1000}{0.020}=138.5{\text{ S cm}}^2{\text{ mol}}^{-1}$$

Variation with concentration:

- Conductivity decreases on dilution, because each cm${}^3$ holds fewer ions
- Molar conductivity increases on dilution, because ions move more freely and weak electrolytes ionise more
- Strong electrolytes follow ${\Lambda }_m={\Lambda }_m^{\circ }-A\sqrt{C}$, so ${\Lambda }_m^{\circ }$ is found by extrapolating the straight line to zero concentration
- Weak electrolytes rise steeply at low concentration and cannot be extrapolated

Kohlrausch's law. At infinite dilution, each ion contributes independently: ${\Lambda }_m^{\circ }={\nu }_{+}{\lambda }_{+}^{\circ }+{\nu }_{-}{\lambda }_{-}^{\circ }$.

Worked example — acetic acid. From limiting molar conductivities of 426.2 for HCl, 91.0 for sodium acetate and 126.4 for NaCl:

$${\Lambda }_m^{\circ }(\mathrm{C}{\mathrm{H}}_3\mathrm{COOH})=426.2+91.0-126.4=390.8{\text{ S cm}}^2{\text{ mol}}^{-1}$$

If dilute acetic acid has ${\Lambda }_m=48.2$ S cm${}^2$ mol${}^{-1}$, its degree of dissociation is $\alpha =\frac{48.2}{390.8}=0.123$.

An everyday example. TDS meters used to check drinking water actually measure conductivity, since more dissolved ions mean a higher reading.

The substance. Conductivity falls on dilution while molar conductivity rises — there is no contradiction, because molar conductivity is counted per mole, and each mole of ions works more efficiently when spread out.

Primary cells such as the dry cell use reactions that cannot be reversed, secondary cells such as the lead storage battery are recharged by reversing their reaction, and fuel cells produce electricity continuously as long as fuel and oxidant are supplied.

Dry cell (Leclanche cell):

- A zinc container is the anode, and a carbon rod surrounded by manganese dioxide is the cathode, in a moist paste of ammonium chloride and zinc chloride
- Anode: $\mathrm{Zn}\to \mathrm{Z}{\mathrm{n}}^{2+}+2{\mathrm{e}}^{-}$; cathode: $\mathrm{Mn}{\mathrm{O}}_2+\mathrm{N}{\mathrm{H}}_4^{+}+{\mathrm{e}}^{-}\to \mathrm{MnO}(\mathrm{OH})+\mathrm{N}{\mathrm{H}}_3$
- Gives about 1.5 V and cannot be recharged

Lead storage battery:

- A lead anode and a lead dioxide cathode dip in dilute sulphuric acid
- On discharge: $\mathrm{Pb}+\mathrm{Pb}{\mathrm{O}}_2+2{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\to 2PbS{\mathrm{O}}_4+2{\mathrm{H}}_2\mathrm{O}$
- Sulphuric acid is used up, so the density of the acid falls as the battery runs down
- Recharging with an external current reverses the reaction
- Each cell gives about 2 V, and six in series make a 12 V battery

Hydrogen-oxygen fuel cell:

- Hydrogen and oxygen bubble through porous carbon electrodes containing a catalyst, in concentrated sodium hydroxide
- Anode: $2{\mathrm{H}}_2+4O{\mathrm{H}}^{-}\to 4{\mathrm{H}}_2\mathrm{O}+4{\mathrm{e}}^{-}$; cathode: ${\mathrm{O}}_2+2{\mathrm{H}}_2\mathrm{O}+4{\mathrm{e}}^{-}\to 4O{\mathrm{H}}^{-}$
- Overall: $2{\mathrm{H}}_2+{\mathrm{O}}_2\to 2{\mathrm{H}}_2\mathrm{O}$, with no polluting products
- It converts chemical energy directly into electricity, skipping the heat-engine stage of a power station, so it is highly efficient

An everyday example. The battery in a car or a home inverter is a lead storage battery, recharged while the engine runs or while mains power is available.

The substance. A fuel cell never goes flat like a battery — its reactants are not stored inside, so it keeps working as long as hydrogen and oxygen are fed in.

Corrosion is the slow electrochemical destruction of a metal; in rusting, different spots on the same piece of iron act as the anode and cathode of a tiny cell, with moisture carrying dissolved ions as the electrolyte.

Mechanism of rusting:

- Anode — iron is oxidised at one spot: $2Fe\to 2F{\mathrm{e}}^{2+}+4{\mathrm{e}}^{-}$, with $E^{\circ }=-0.44$ V
- Cathode — electrons travel through the metal to another spot, where oxygen is reduced: ${\mathrm{O}}_2+4{\mathrm{H}}^{+}+4{\mathrm{e}}^{-}\to 2{\mathrm{H}}_2\mathrm{O}$, with $E^{\circ }=+1.23$ V
- Overall: $2Fe+{\mathrm{O}}_2+4{\mathrm{H}}^{+}\to 2F{\mathrm{e}}^{2+}+2{\mathrm{H}}_2\mathrm{O}$, with $E_{\text{cell}}^{\circ }=1.23-(-0.44)=1.67$ V
- The $\mathrm{F}{\mathrm{e}}^{2+}$ ions are further oxidised by air to hydrated iron(III) oxide, $\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3\cdot \mathrm{x}{\mathrm{H}}_2\mathrm{O}$, which is rust

Prevention:

- Barrier protection — painting, oiling or greasing to keep out air and water
- Galvanising — coating with zinc, which is oxidised in preference to iron
- Sacrificial protection — attaching blocks of magnesium or zinc to ship hulls and buried pipelines
- Alloying, as in stainless steel, which forms a protective chromium oxide film

An everyday example. Iron balcony grilles repainted before the monsoon last far longer, because the paint keeps out the water that rusting needs.

The substance. Rust does not protect iron the way aluminium oxide protects aluminium — rust is porous and flakes away, exposing fresh metal to further attack.