A cell's voltage, ΔG° and $K$ all measure how far a reaction wants to go, so each can be computed from the others. Standard potentials are often the easiest of the three to look up.
Key idea
Why it works
Worked examples
Worked example: Zinc and copper
- Two electrons move per zinc atom: $n = 2$.
- $$\Delta G^\circ = -(2)(96\,485)(1.10) = -2.12 \times 10^5\ \text{J} = -212\ \text{kJ}$$
- $\log K = \dfrac{(2)(1.10)}{0.0592} = 37.2$, so $K = 10^{37.2} = 1.5 \times 10^{37}$. The reaction goes to completion.
Answer: $\Delta G^\circ = -212\ \text{kJ}$; $K \approx 1.5 \times 10^{37}$.
Worked example: Finding n
- Aluminum gives 3 electrons, taken by 3 silver ions: $n = 3$.
- $$\Delta G^\circ = -(3)(96\,485)(2.46) = -7.12 \times 10^5\ \text{J} = -712\ \text{kJ}$$
Answer: $-712\ \text{kJ}$.
Worked example: A small voltage
- $\log K = \dfrac{(1)(0.10)}{0.0592} = 1.69$.
- $K = 10^{1.69} = 49$. Even a tenth of a volt favors products noticeably.
Answer: $K \approx 49$.
| $E^\circ$ | $\Delta G^\circ$ | K |
|---|---|---|
| positive | negative | > 1, products favored |
| zero | zero | = 1 |
| negative | positive | < 1, reactants favored |
One consequence of $\Delta G^\circ = -nFE^\circ$: when two half-reactions are combined into a new half-reaction, potentials do not simply add, because the electrons do not cancel. Add the free energies instead. For example, $\mathrm{Fe^{3+} + e^- \to Fe^{2+}}$ ($+0.77\ \text{V}$) and $\mathrm{Fe^{2+} + 2e^- \to Fe}$ ($-0.45\ \text{V}$) give $\mathrm{Fe^{3+} + 3e^- \to Fe}$ with $E^\circ = \dfrac{1(0.77) + 2(-0.45)}{3} = -0.04\ \text{V}$, not $0.32\ \text{V}$.
Common mistake