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/quantities are not mathematically related to each other, we will analyze each option given in the question step by step.

### Step 1: Analyze the first pair - Gibbs free energy and standard cell potential
- The relationship between Gibbs free energy (ΔG) and standard cell potential (E°cell) is given by the equation:
\[ \Delta G = -nFE^\circ_{\text{cell}} \]
- Here, ΔG is directly proportional to E°cell. Thus, they are mathematically related.

**Hint:** Look for equations that connect the two quantities to determine their relationship.

### Step 2: Analyze the second pair - Equilibrium constant and standard cell potential
- The relationship between the equilibrium constant (K) and standard cell potential (E°cell) can be derived from the Gibbs free energy equation:
\[ \Delta G = -2.303RT \log K \]
- By combining this with the first equation, we can show:
\[ -nFE^\circ_{\text{cell}} = -2.303RT \log K \]
- This indicates that E°cell is also mathematically related to K.

**Hint:** Check if the quantities can be expressed in terms of one another using known equations.

### Step 3: Analyze the third pair - Rate constant and activation energy
- The Arrhenius equation relates the rate constant (k) to the activation energy (E_a):
\[ k = A e^{-\frac{E_a}{RT}} \]
- This shows that the rate constant is mathematically related to the activation energy.

**Hint:** Identify if there is a formula that connects the two quantities.

### Step 4: Analyze the fourth pair - Rate constant and standard cell potential
- There is no direct mathematical relationship between the rate constant (k) and the standard cell potential (E°cell). They belong to different contexts in electrochemistry and kinetics.
- Therefore, this pair does not have a mathematical relationship.

**Hint:** Consider the context of each quantity to determine if they are related.

### Conclusion
The pair that is not mathematically related to each other is:
- **Rate constant and standard cell potential.**