How Much Silver Coats a Spoon When Current Flows for One Hour?

A cell's emf depends not only on its electrodes but also on the concentrations of the ions around them, which is why its voltage drifts as it discharges. Run the process backwards with an external current, and electricity drives chemical change instead — plating ornaments, refining copper and extracting aluminium.

This lesson covers the Nernst equation, the link between emf, Gibbs energy and equilibrium, and electrolysis with Faraday's laws.

**The Nernst equation corrects the standard emf of a cell for the actual concentrations of its ions: at 298 K, $E_{\text{cell}}=E_{\text{cell}}^{\circ }-\frac{0.0591}{n}\log Q$.

The general form:**

$$E_{\text{cell}}=E_{\text{cell}}^{\circ }-\frac{2.303RT}{nF}\log Q$$

- n is the number of electrons transferred, F is the Faraday constant, 96 500 C mol${}^{-1}$, and Q is the reaction quotient
- At 298 K, $\frac{2.303RT}{F}=0.0591$ V
- Pure solids and liquids are left out of Q

For a single electrode, ${\mathrm{M}}^{\mathrm{n}+}+\mathrm{n}{\mathrm{e}}^{-}\to \mathrm{M}$:

$$E=E^{\circ }-\frac{0.0591}{n}\log \frac{1}{[{\mathrm{M}}^{\mathrm{n}+}]}$$

Worked example 1 — a Daniell cell. For $\mathrm{Zn}(\mathrm{s})|\mathrm{Z}{\mathrm{n}}^{2+}(0.10\mathrm{M})||\mathrm{C}{\mathrm{u}}^{2+}(0.0010\mathrm{M})|\mathrm{Cu}(\mathrm{s})$, with $E_{\text{cell}}^{\circ }=1.10$ V and n = 2:

$$E_{\text{cell}}=1.10-\frac{0.0591}{2}\log \frac{0.10}{0.0010}=1.10-0.02955\times 2=1.04\text{ V}$$

Worked example 2 — a single electrode. For copper in 0.010 M $\mathrm{C}{\mathrm{u}}^{2+}$, with $E^{\circ }=0.34$ V:

$$E=0.34-\frac{0.0591}{2}\log \frac{1}{0.010}=0.34-0.0591=0.281\text{ V}$$

What the equation shows:

- Raising the concentration of reactant ions raises the emf; raising product ions lowers it
- As a cell discharges, Q rises and the emf falls
- At equilibrium, $E_{\text{cell}}=0$ and the cell is flat

An everyday example. pH meters used to test soil and water samples read an electrode potential that shifts with hydrogen ion concentration exactly as the Nernst equation predicts.

The substance. A concentration cell has zero standard emf yet still gives a voltage — two identical electrodes in solutions of different concentration produce an emf entirely from the log term.

Electrolysis uses an external current to drive a non-spontaneous redox reaction, and Faraday's laws state that the mass of a substance deposited is proportional to the charge passed and, for the same charge, to its equivalent mass.

The electrolytic cell:

- The cathode is connected to the negative terminal, and reduction occurs there
- The anode is connected to the positive terminal, and oxidation occurs there
- Molten sodium chloride gives sodium at the cathode and chlorine at the anode

Faraday's first law:

$$m=ZIt=\frac{M}{nF}It$$

where Z is the electrochemical equivalent, I the current in amperes and t the time in seconds.

Faraday's second law. When the same charge passes through different electrolytes, the masses deposited are proportional to their equivalent masses: $\frac{m_1}{m_2}=\frac{E_1}{E_2}$.

Worked example 1 — silver plating. A current of 2.0 A flows for one hour, 3600 s, through silver nitrate solution:

$$Q=2.0\times 3600=7200\text{ C}m=\frac{108\times 7200}{1\times 96500}=8.06\text{ g}$$

Worked example 2 — the second law. The same charge passes through a copper sulphate cell in series. With equivalent masses of 108 for silver and 31.75 for copper:

$$m_{\mathrm{Cu}}=8.06\times \frac{31.75}{108}=2.37\text{ g}$$

Products depend on the electrolyte and electrodes:

- Aqueous sodium chloride gives hydrogen, not sodium, at the cathode, because water is reduced more easily than $\mathrm{N}{\mathrm{a}}^{+}$
- Copper sulphate solution with copper electrodes moves copper from anode to cathode, the basis of copper refining
- Dilute sulphuric acid with platinum electrodes releases oxygen at the anode

An everyday example. Silver-plated puja thalis and imitation jewellery are made by electroplating, with the object as the cathode in a solution of silver ions.

The substance. One faraday is the charge on one mole of electrons — it deposits one mole of silver but only half a mole of copper, because each copper ion needs two electrons.