Find out the percentage of the reactant molecules crosisng over the energy barrier at `325 K`.
Given: `Delta H_(325 K) = 0.12 kcal`,
`E_(a(b)) = 0.02 kcal`

Options

A. `80.65%`
B. `70.65%`
C. `60.65%`
D. `50.65%`

Explanation

To solve the problem of finding the percentage of reactant molecules crossing over the energy barrier at 325 K, we will follow these steps:

### Step 1: Identify Given Data
We have the following data:
- \( \Delta H_{325 K} = 0.12 \, \text{kcal} \)
- \( E_a(b) = 0.02 \, \text{kcal} \)

### Step 2: Calculate Activation Energy for Forward Reaction
Using the relationship:
\[ \Delta H = E_f - E_b \]
we can rearrange it to find the activation energy for the forward reaction (\(E_f\)):
\[ E_f = \Delta H + E_b \]
Substituting the values:
\[ E_f = 0.12 \, \text{kcal} + 0.02 \, \text{kcal} = 0.14 \, \text{kcal} \]

### Step 3: Convert Activation Energy to Calories
Since we will use the gas constant \(R\) in calories, we convert \(E_f\) from kilocalories to calories:
\[ E_f = 0.14 \, \text{kcal} \times 1000 = 140 \, \text{cal} \]

### Step 4: Use the Arrhenius Equation
The fraction of molecules (\(x\)) crossing over the energy barrier can be calculated using the Arrhenius equation:
\[ x = e^{-\frac{E_a}{RT}} \]
Where:
- \(E_a = E_f = 140 \, \text{cal}\)
- \(R = 2 \, \text{cal/(mol K)}\)
- \(T = 325 \, \text{K}\)

### Step 5: Calculate the Exponent
Substituting the values into the equation:
\[ x = e^{-\frac{140}{2 \times 325}} = e^{-\frac{140}{650}} = e^{-0.2153846154} \]

### Step 6: Calculate the Value of \(x\)
Calculating the exponent:
\[ x \approx e^{-0.2154} \approx 0.8063 \]

### Step 7: Convert Fraction to Percentage
To find the percentage of reactant molecules:
\[ \text{Percentage} = x \times 100 \approx 0.8063 \times 100 \approx 80.63\% \]

### Step 8: Final Answer
The percentage of the reactant molecules crossing over the energy barrier at 325 K is approximately **80.6%**.

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