The rate constant of a reaction becomes equal to the pre exponential factor when:

Options

A. the absolute temperature is zero
B. the activation energy is infinity
C. the absolute temperature is infinity
D. the activation energy is zero

Answer

To determine when the rate constant (k) of a reaction becomes equal to the pre-exponential factor (A), we can use the Arrhenius equation, which is given by:

\[ k = A e^{-\frac{E_a}{RT}} \]

where:
- \( k \) is the rate constant,
- \( A \) is the pre-exponential factor,
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant, and
- \( T \) is the absolute temperature.

### Step-by-Step Solution:

1. **Understand the Arrhenius Equation**: The equation shows that the rate constant \( k \) depends on the pre-exponential factor \( A \), the activation energy \( E_a \), the universal gas constant \( R \), and the temperature \( T \).

2. **Set \( k \) Equal to \( A \)**: We want to find the condition under which \( k = A \). Substituting this into the Arrhenius equation gives:

\[ A = A e^{-\frac{E_a}{RT}} \]

3. **Simplify the Equation**: We can divide both sides by \( A \) (assuming \( A \neq 0 \)):

\[ 1 = e^{-\frac{E_a}{RT}} \]

4. **Determine When the Exponential Equals One**: The exponential function equals 1 when its exponent is zero:

\[ -\frac{E_a}{RT} = 0 \]

5. **Solve for Conditions**: This implies:

\[ \frac{E_a}{RT} = 0 \]

Since \( R \) is a constant, the only way for this fraction to be zero is if \( E_a = 0 \) or \( T \) approaches infinity.

6. **Analyze the Conditions**:
- If \( E_a = 0 \), it suggests that there is no activation energy required for the reaction, which is not realistic for most reactions.
- If \( T \) approaches infinity, the term \( \frac{E_a}{RT} \) approaches zero, making the exponential term equal to 1.

7. **Conclusion**: Therefore, the rate constant \( k \) becomes equal to the pre-exponential factor \( A \) when the absolute temperature \( T \) approaches infinity.

### Final Answer:
The rate constant of a reaction becomes equal to the pre-exponential factor when the absolute temperature is infinite.