In a reaction carried out at 400K, 0.01% of the total number of collisions is effective. The energy of activation of the reaction is

Options

A. 13.3 kJ/mol
B. 23.5kJ/mol
C. 3.2kJ/mol
D. 30.6kJ/mol

Answer

To find the energy of activation (Ea) for the reaction, we can follow these steps:

### Step 1: Write down the given data
- Temperature (T) = 400 K
- Fraction of effective collisions = 0.01%

### Step 2: Convert the fraction of effective collisions to decimal form
0.01% can be expressed as:
\[ \text{Fraction} = \frac{0.01}{100} = 0.0001 = 10^{-4} \]

### Step 3: Use the formula for the fraction of effective collisions
The fraction of effective collisions can be expressed using the Arrhenius equation:
\[ \text{Fraction} = e^{-\frac{E_a}{RT}} \]
Where:
- \(E_a\) = activation energy
- \(R\) = universal gas constant = 8.314 J/(K·mol)
- \(T\) = temperature in Kelvin = 400 K

### Step 4: Set up the equation
From the previous step, we have:
\[ 10^{-4} = e^{-\frac{E_a}{RT}} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ \ln(10^{-4}) = -\frac{E_a}{RT} \]

### Step 6: Simplify the left side
Using the property of logarithms:
\[ \ln(10^{-4}) = -4 \ln(10) \approx -4 \times 2.303 = -9.212 \]

### Step 7: Substitute the values into the equation
Now we can substitute this back into our equation:
\[ -9.212 = -\frac{E_a}{(8.314)(400)} \]

### Step 8: Rearrange to solve for \(E_a\)
Rearranging gives:
\[ E_a = 9.212 \times (8.314)(400) \]

### Step 9: Calculate \(E_a\)
Calculating the right-hand side:
\[ E_a = 9.212 \times 3325.6 \approx 30635.42 \text{ J/mol} \]

### Step 10: Convert to kJ/mol
To convert from J/mol to kJ/mol:
\[ E_a = \frac{30635.42}{1000} \approx 30.63542 \text{ kJ/mol} \]

### Final Answer
Thus, the energy of activation \(E_a\) is approximately:
\[ \boxed{30.63 \text{ kJ/mol}} \]