What is the value of rate constant k if the value of the activation energy Ea and the frequency factor A are 49 kJ / mol and 9 × `10^10 S^-1` respectively? (T = 313 K)

Options

A. 6 × `10^2 S^-1`
B. 9 × `10^2 S^-1`
C. 3 × `10^2 S^-1`
D. 6 × `10^-2 S^-1`

Answer

To find the value of the rate constant \( k \) using the Arrhenius equation, we can follow these steps:

### Step 1: Write down the Arrhenius equation
The Arrhenius equation relates the rate constant \( k \) to the activation energy \( E_a \), the frequency factor \( A \), and the temperature \( T \):
\[ k = A e^{-\frac{E_a}{RT}} \]
where:
- \( k \) = rate constant
- \( A \) = frequency factor
- \( E_a \) = activation energy
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin

### Step 2: Convert activation energy to the correct units
The activation energy \( E_a \) is given as 49 kJ/mol. We need to convert this to Joules per mole:
\[ E_a = 49 \, \text{kJ/mol} = 49 \times 10^3 \, \text{J/mol} = 49000 \, \text{J/mol} \]

### Step 3: Substitute the known values into the equation
Given:
- \( A = 9 \times 10^{10} \, \text{s}^{-1} \)
- \( R = 8.314 \, \text{J/(mol·K)} \)
- \( T = 313 \, \text{K} \)

Now we can substitute these values into the Arrhenius equation:
\[ k = 9 \times 10^{10} \, e^{-\frac{49000}{8.314 \times 313}} \]

### Step 4: Calculate the exponent
First, calculate the denominator:
\[ RT = 8.314 \times 313 \approx 2600.602 \, \text{J/mol} \]
Now calculate the exponent:
\[ -\frac{E_a}{RT} = -\frac{49000}{2600.602} \approx -18.84 \]

### Step 5: Calculate \( e^{-\frac{E_a}{RT}} \)
Now, calculate \( e^{-18.84} \):
\[ e^{-18.84} \approx 1.3 \times 10^{-9} \]

### Step 6: Calculate the rate constant \( k \)
Now substitute this back into the equation for \( k \):
\[ k = 9 \times 10^{10} \times 1.3 \times 10^{-9} \approx 1.17 \times 10^{2} \, \text{s}^{-1} \]

### Step 7: Finalize the result
Thus, the value of the rate constant \( k \) is approximately:
\[ k \approx 117 \, \text{s}^{-1} \approx 1.17 \times 10^{2} \, \text{s}^{-1} \]

### Summary
The value of the rate constant \( k \) is approximately \( 117 \, \text{s}^{-1} \).

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