Given that the normal energy of the reactant and produce are 40J and 20 J respectively and threshold energy of the uncatalysed reaction is 120 J.If the rate of uncatalysed reaction at 400 K becomes equal to the rate of catalysed reaction at 300 K, then what will be the activation energy of the catalysed forward and backward reactions respectively ?

Options

A. 80 J, 60 J
B. 60 J , 80 J
C. 80 J , 100 J`
D. 50 J, 70 J

Explanation

To solve the problem, we need to find the activation energy for both the catalyzed forward and backward reactions based on the given information. Here’s how we can break it down step by step:

### Step 1: Identify Given Data
- Normal energy of reactants (E_r) = 40 J
- Normal energy of products (E_p) = 20 J
- Threshold energy of uncatalyzed reaction (E_threshold) = 120 J
- Temperature for uncatalyzed reaction (T1) = 400 K
- Temperature for catalyzed reaction (T2) = 300 K

### Step 2: Calculate Activation Energy for Uncatalyzed Reaction
The activation energy (E_a) for the uncatalyzed reaction can be calculated using the formula:
\[ E_a = E_{\text{threshold}} - E_r \]
Substituting the values:
\[ E_a = 120 \, \text{J} - 40 \, \text{J} = 80 \, \text{J} \]

### Step 3: Use the Rate Equality Condition
According to the problem, the rate of the uncatalyzed reaction at 400 K is equal to the rate of the catalyzed reaction at 300 K. We can use the Arrhenius equation in the form of:
\[ \frac{E_a}{T_1} = \frac{E_a'}{T_2} \]
Where:
- \( E_a \) = Activation energy of the uncatalyzed reaction
- \( E_a' \) = Activation energy of the catalyzed reaction

Substituting the known values:
\[ \frac{80 \, \text{J}}{400 \, \text{K}} = \frac{E_a'}{300 \, \text{K}} \]

### Step 4: Solve for \( E_a' \)
Cross-multiplying gives:
\[ E_a' = \frac{80 \, \text{J} \times 300 \, \text{K}}{400 \, \text{K}} \]
\[ E_a' = 60 \, \text{J} \]

### Step 5: Calculate Activation Energy of Catalyzed Forward Reaction
The activation energy for the catalyzed forward reaction can be calculated as:
\[ E_{a, \text{forward}} = E_a' + (E_p - E_r) \]
Where:
- \( E_p - E_r = 20 \, \text{J} - 40 \, \text{J} = -20 \, \text{J} \)

Thus:
\[ E_{a, \text{forward}} = 60 \, \text{J} + 20 \, \text{J} = 80 \, \text{J} \]

### Step 6: Calculate Activation Energy of Catalyzed Backward Reaction
The activation energy for the catalyzed backward reaction is simply:
\[ E_{a, \text{backward}} = E_a' \]
So:
\[ E_{a, \text{backward}} = 60 \, \text{J} \]

### Final Answer
The activation energies of the catalyzed forward and backward reactions are:
- Activation energy of forward reaction: **80 J**
- Activation energy of backward reaction: **60 J**