Equilibrium Constant from Nernst Equation — Revision Notes

⚡ 30-Second Revision

Nernst Equation (general): — $E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q$
At Equilibrium: — $E_{cell} = 0$ and $Q = K_c$
Relationship at any T: — $E^\circ_{cell} = \frac{RT}{nF} \ln K_c$
Relationship at 298 K: — $E^\circ_{cell} = \frac{0.0592}{n} \log K_c$
Solving for $K_c$ at 298 K: — $\log K_c = \frac{n E^\circ_{cell}}{0.0592} \implies K_c = 10^{\left(\frac{n E^\circ_{cell}}{0.0592}\right)}$
Constants: — $R = 8.314\,\text{J mol}^{-1}\text{K}^{-1}$, $F = 96485\,\text{C mol}^{-1}$
'n': — Number of electrons transferred in balanced redox reaction.

2-Minute Revision

The equilibrium constant ($K_c$) for a redox reaction in a galvanic cell is fundamentally linked to its standard cell potential ($E^\circ_{cell}$). This connection arises from the Nernst equation. At equilibrium, a galvanic cell ceases to produce electrical energy, meaning its cell potential ($E_{cell}$) becomes zero.

Simultaneously, the reaction quotient ($Q$) reaches its equilibrium value, which is $K_c$. Substituting these conditions into the Nernst equation, $E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q$, leads to the crucial relationship: $E^\circ_{cell} = \frac{RT}{nF} \ln K_c$.

For calculations at $298\,\text{K}$ ($25^\circ\text{C}$), this simplifies to $E^\circ_{cell} = \frac{0.0592}{n} \log K_c$. To find $K_c$, you rearrange this to $\log K_c = \frac{n E^\circ_{cell}}{0.0592}$ and then take the antilog ($10^x$).

Remember to correctly determine 'n', the number of electrons transferred in the balanced redox reaction. A positive $E^\circ_{cell}$ corresponds to $K_c > 1$, indicating a spontaneous, product-favored reaction.

5-Minute Revision

To quickly revise the 'Equilibrium Constant from Nernst Equation' for NEET, focus on the core derivation and its practical application. Start with the Nernst equation: $E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q$.

Understand that at equilibrium, two key conditions are met: the cell potential $E_{cell}$ drops to zero (no more useful work can be done), and the reaction quotient $Q$ becomes equal to the equilibrium constant $K_c$.

Substituting these into the Nernst equation gives $0 = E^\circ_{cell} - \frac{RT}{nF} \ln K_c$. Rearranging this yields the fundamental thermodynamic relationship: $E^\circ_{cell} = \frac{RT}{nF} \ln K_c$.

This equation is valid at any temperature $T$ (in Kelvin). For NEET, most problems are at $298\,\text{K}$ ($25^\circ\text{C}$), where the constant $\frac{RT}{F}$ simplifies to approximately $0.02569\,\text{V}$, and converting $\ln$ to $\log$ ($2.

303 \log x = \ln x$) gives the commonly used form:$E^\circ_{cell} = \frac{0.0592}{n} \log K_c$.

Key Steps for Problems:

1
Balance the redox reaction: — Determine 'n', the total number of electrons transferred. This is often the trickiest part.
2
Identify given values: — $E^\circ_{cell}$, $K_c$, or $n$, and temperature.
3
Choose the correct formula: — Use the $0.0592$ form for $298\,\text{K}$, or the $RT/nF$ form for other temperatures.
4
Rearrange and calculate: — Solve for the unknown. If calculating $K_c$, remember to take the antilog ($10^x$).

Example: If $E^\circ_{cell} = +0.60\,\text{V}$ for a reaction with $n=2$ at $298\,\text{K}$, find $K_c$. $\log K_c = \frac{n E^\circ_{cell}}{0.0592} = \frac{2 \times 0.60}{0.0592} = \frac{1.20}{0.0592} \approx 20.27$ $K_c = 10^{20.27} \approx 1.86 \times 10^{20}$.

Remember that a positive $E^\circ_{cell}$ implies a spontaneous reaction and $K_c > 1$ (products favored), while a negative $E^\circ_{cell}$ implies a non-spontaneous reaction and $K_c < 1$ (reactants favored). Always double-check your 'n' value and calculator operations, especially for logarithms and antilogarithms.

Prelims Revision Notes

The relationship between the equilibrium constant ($K_c$) and the standard cell potential ($E^\circ_{cell}$) is a crucial concept for NEET. This relationship is derived from the Nernst equation under equilibrium conditions. At equilibrium, the net cell potential ($E_{cell}$) of a galvanic cell becomes zero, as the cell can no longer perform useful electrical work. Simultaneously, the reaction quotient ($Q$) becomes equal to the equilibrium constant ($K_c$).

Key Formulae:

1
General Relationship (any temperature T): — $E^\circ_{cell} = \frac{RT}{nF} \ln K_c$

* $R = 8.314\,\text{J mol}^{-1}\text{K}^{-1}$ (Gas constant) * $T = \text{Temperature in Kelvin}$ * $n = \text{Number of moles of electrons transferred in the balanced redox reaction}$ * $F = 96485\,\text{C mol}^{-1}$ (Faraday's constant)

1
Simplified Relationship (at $298\,\text{K}$ or $25^\circ\text{C}$): — $E^\circ_{cell} = \frac{0.0592}{n} \log K_c$

* This is the most commonly used form in NEET problems. The constant $0.0592$ comes from $\frac{2.303 RT}{F}$ at $298\,\text{K}$.

How to Calculate $K_c$ from $E^\circ_{cell}$ (at $298\,\text{K}$):

1
Rearrange the simplified formula: $\log K_c = \frac{n E^\circ_{cell}}{0.0592}$
2
Calculate the value of $\log K_c$.
3
Take the antilogarithm: $K_c = 10^{\left(\frac{n E^\circ_{cell}}{0.0592}\right)}$

How to Calculate $E^\circ_{cell}$ from $K_c$ (at $298\,\text{K}$):

1
Calculate $\log K_c$.
2
Substitute into the formula: $E^\circ_{cell} = \frac{0.0592}{n} \log K_c$

Important Points to Remember:

'n' value: — Always correctly determine the number of electrons transferred by balancing the redox reaction. This is a frequent source of error.
Temperature: — Use the $0.0592$ constant ONLY at $298\,\text{K}$. For other temperatures, use the general formula with $T$ in Kelvin and $\ln K_c$.
**Sign of $E^\circ_{cell}$ and $K_c$:**

* If $E^\circ_{cell} > 0$, then $K_c > 1$ (reaction is spontaneous, products are favored). * If $E^\circ_{cell} < 0$, then $K_c < 1$ (reaction is non-spontaneous, reactants are favored). * If $E^\circ_{cell} = 0$, then $K_c = 1$ (reaction is at equilibrium under standard conditions, which is rare).

Relationship with $\Delta G^\circ$: — $\Delta G^\circ = -nFE^\circ_{cell} = -RT \ln K_c$. All three quantities ($E^\circ_{cell}$, $K_c$, $\Delta G^\circ$) indicate the spontaneity and extent of a reaction under standard conditions.

Vyyuha Quick Recall

Nice Electrons Really Need Standard Temperature (Nernst): $E^\circ_{cell} = \frac{0.0592}{n} \log K_c$ (at 298K). Remember 'n' is for 'Number of electrons transferred'.