In which of the following pairs, the constants/quantities are not mathematically related to each other?

Options

A. Gibbs free energy and standard cell potential
B. Equilibrium constant and standard cell potential
C. Rate constant and activation energy
D. Rate constant and standard cell potential

Explanation

To determine which pairs of constants or quantities are not mathematically related to each other, we will analyze each pair and identify the relationships based on established equations in electrochemistry.

### Step-by-Step Solution:

1. **Identify the pairs of quantities**: We need to analyze the following pairs:
- Gibbs free energy (ΔG°) and standard cell potential (E°)
- Equilibrium constant (K) and standard cell potential (E°)
- Rate constant (k) and activation energy (Ea)
- Rate constant (k) and standard cell potential (E°)

2. **Analyze the first pair: Gibbs free energy (ΔG°) and standard cell potential (E°)**:
- The relationship is given by the equation:
\[ \Delta G° = -nFE° \]
- Here, ΔG° is related to E° through the number of electrons (n) and Faraday's constant (F). Therefore, these two quantities are mathematically related.

3. **Analyze the second pair: Equilibrium constant (K) and standard cell potential (E°)**:
- From the Nernst equation, we can derive:
\[ E° = \frac{RT}{nF} \ln K \]
- At equilibrium, E° becomes zero, leading to the relationship:
\[ E° = \frac{0.0591}{n} \log K \]
- This indicates that K is related to E°. Thus, these two quantities are also mathematically related.

4. **Analyze the third pair: Rate constant (k) and activation energy (Ea)**:
- According to the Arrhenius equation:
\[ k = Ae^{-\frac{E_a}{RT}} \]
- This shows a direct relationship between the rate constant (k) and activation energy (Ea), meaning they are mathematically related.

5. **Analyze the fourth pair: Rate constant (k) and standard cell potential (E°)**:
- There is no established equation that connects the rate constant (k) directly with the standard cell potential (E°). These two quantities operate in different contexts (kinetics vs. thermodynamics) and do not have a mathematical relationship.

6. **Conclusion**: The pair that is not mathematically related is **Rate constant (k) and standard cell potential (E°)**. Therefore, the correct answer is option D.

### Final Answer:
**D. Rate constant (k) and standard cell potential (E°)** are not mathematically related.