The activation energy for the reaction -
`H_(2)O_(2)toH_(2)O+(1)/(2)O_(2)`
is `18 K cal//mol` at 300 K. calculate the fraction of molecules of reactonts having energy equal to or greater than activation energy ?
Anti log `(-13.02)=9.36xx10^(-14)`

Options

A. `9.36xx10^(-14)`
B. `1.2xx10^(-12)`
C. `4.2xx10^(-16)`
D. `5.2xx10^(-15)`

Explanation

To solve the problem of calculating the fraction of molecules of reactants having energy equal to or greater than the activation energy for the reaction \( H_2O_2 \to H_2O + \frac{1}{2} O_2 \) with an activation energy \( E_a = 18 \, \text{kcal/mol} \) at \( T = 300 \, K \), we will use the Arrhenius equation.

### Step-by-Step Solution:

1. **Understand the Formula**:
The fraction of molecules \( F \) having energy equal to or greater than the activation energy is given by:
\[ F = e^{-\frac{E_a}{RT}} \]
where:
- \( E_a \) is the activation energy,
- \( R \) is the universal gas constant, and
- \( T \) is the temperature in Kelvin.

2. **Convert Activation Energy**:
Since the activation energy is given in kilocalories, we need to convert it to calories:
\[ E_a = 18 \, \text{kcal/mol} \times 1000 \, \text{cal/kcal} = 18000 \, \text{cal/mol} \]

3. **Identify the Gas Constant**:
The value of \( R \) in appropriate units (calories) is:
\[ R = 2 \, \text{cal/(mol K)} \]

4. **Substitute Values into the Formula**:
Now we can substitute the values into the fraction formula:
\[ F = e^{-\frac{18000}{2 \times 300}} \]

5. **Calculate the Exponent**:
First, calculate the denominator:
\[ 2 \times 300 = 600 \]
Now calculate the exponent:
\[ -\frac{18000}{600} = -30 \]

6. **Calculate the Fraction**:
Now we can compute \( F \):
\[ F = e^{-30} \]

7. **Use the Anti-logarithm**:
To express this in terms of a numerical value, we can use the approximation:
\[ e^{-30} \approx 10^{-30} \]
(This approximation is valid since \( e^{-x} \) can be estimated as \( 10^{-x \cdot \log_{10} e} \), and \( \log_{10} e \approx 0.434 \)).

8. **Final Result**:
Thus, the fraction of molecules of reactants having energy equal to or greater than the activation energy is:
\[ F \approx 10^{-30} \]