The rate constant , the activation energy and Arrhenius parameter of a chemical reaction at `25^@C` are x, 10x kJ/ mol and `2x s^(-1)` . Value of rate constant as `T rarr oo` is

Options

A. `xs^(-1)`
B. `2x s^(-1)`
C. `oo`
D. `10x s^(-1)`

Answer

To solve the problem, we need to find the value of the rate constant \( k \) as the temperature \( T \) approaches infinity. We are given the following parameters:

- Rate constant \( k = x \)
- Activation energy \( E_a = 10x \) kJ/mol
- Arrhenius parameter \( A = 2x \) s\(^{-1}\)

### Step-by-Step Solution:

1. **Understand the Arrhenius Equation**:
The Arrhenius equation relates the rate constant \( k \) to the temperature \( T \) and the activation energy \( E_a \):
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( A \) is the Arrhenius parameter (or frequency factor),
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/mol K} \)),
- \( T \) is the temperature in Kelvin.

2. **Consider the Limit as \( T \to \infty \)**:
As the temperature \( T \) approaches infinity, the term \( \frac{E_a}{RT} \) approaches zero because \( R \) is a constant and \( T \) is increasing indefinitely. Therefore:
\[ \lim_{T \to \infty} \frac{E_a}{RT} = 0 \]

3. **Evaluate the Exponential Term**:
Substituting this limit into the Arrhenius equation, we have:
\[ k = A e^{-\frac{E_a}{RT}} \to A e^0 = A \]
Since \( e^0 = 1 \), we find that:
\[ k = A \]

4. **Substitute the Given Value of \( A \)**:
We know from the problem statement that the Arrhenius parameter \( A = 2x \) s\(^{-1}\). Therefore:
\[ k = 2x \, \text{s}^{-1} \]

5. **Conclusion**:
The value of the rate constant \( k \) as \( T \to \infty \) is:
\[ k = 2x \, \text{s}^{-1} \]

### Final Answer:
The value of the rate constant as \( T \to \infty \) is \( 2x \, \text{s}^{-1} \).

---