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 step by step, we will analyze the given information and apply the Arrhenius equation.

### Step 1: Understand the Given Values
We are given:
- Rate constant (k) at 25°C = x
- Activation energy (Ea) = 10x kJ/mol
- Arrhenius parameter (A) = 2x s⁻¹

### Step 2: Convert Temperature to Kelvin
Since the Arrhenius equation uses temperature in Kelvin, we convert 25°C to Kelvin:
\[ T = 25 + 273.15 = 298.15 \, K \]

### Step 3: Write the Arrhenius Equation
The Arrhenius equation is:
\[ k = A \cdot e^{-\frac{E_a}{RT}} \]
Where:
- \( k \) = rate constant
- \( A \) = Arrhenius parameter
- \( E_a \) = activation energy
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin

### Step 4: Substitute Known Values at 298 K
Substituting the known values into the Arrhenius equation:
\[ k = 2x \cdot e^{-\frac{10x \times 1000}{8.314 \times 298}} \]
(Note: We convert \( E_a \) from kJ/mol to J/mol by multiplying by 1000.)

### Step 5: Analyze the Limit as Temperature Approaches Infinity
We need to find the value of the rate constant \( k \) as \( T \to \infty \):
As \( T \) approaches infinity, the term \( \frac{E_a}{RT} \) approaches 0. Thus:
\[ e^{-\frac{E_a}{RT}} \to e^0 = 1 \]

### Step 6: Determine the Rate Constant at Infinite Temperature
Substituting this back into the Arrhenius equation:
\[ k = A \cdot e^0 = A \]
Since \( A = 2x \), we have:
\[ k = 2x \]

### Step 7: Final Result
Thus, the value of the rate constant as \( T \to \infty \) is:
\[ k = 2x \]

### Step 8: Compare with Given Options
The question asks for the value of the rate constant at infinite temperature. Since \( A = 2x \), the final answer is:
\[ k = 2x \]