The entropies of `H_(2)` (g) and H (g) are 130.6 and 114.6 J `mol^(-1)K^(-1)` respectively at 298 K. Using the data given below calculate the bond energy of `H_(2)` (in kJ/mol) :
`H_(2)(g)rarr2H(g),DeltaG^(@)=406.6kJ`

Options

A. 377.2
B. 436
C. 425.5
D. 430.5

Explanation

To calculate the bond energy of \( H_2 \) using the given data, we will follow these steps:

### Step 1: Write the reaction and identify given data
The reaction we are considering is:
\[ H_2(g) \rightarrow 2H(g) \]
We are given:
- \( \Delta G^\circ = 406.6 \, \text{kJ} \)
- Entropy of \( H_2(g) = 130.6 \, \text{J mol}^{-1} \text{K}^{-1} \)
- Entropy of \( H(g) = 114.6 \, \text{J mol}^{-1} \text{K}^{-1} \)

### Step 2: Calculate the change in entropy (\( \Delta S \))
Using the formula for change in entropy:
\[ \Delta S = S_{\text{products}} - S_{\text{reactants}} \]
For our reaction:
\[ \Delta S = 2 \times S_H - S_{H_2} \]
Substituting the values:
\[ \Delta S = 2 \times 114.6 \, \text{J mol}^{-1} \text{K}^{-1} - 130.6 \, \text{J mol}^{-1} \text{K}^{-1} \]
Calculating:
\[ \Delta S = 229.2 \, \text{J mol}^{-1} \text{K}^{-1} - 130.6 \, \text{J mol}^{-1} \text{K}^{-1} = 98.6 \, \text{J mol}^{-1} \text{K}^{-1} \]

### Step 3: Convert \( \Delta S \) to kJ
Since \( \Delta G \) is given in kJ, we need to convert \( \Delta S \) to kJ:
\[ \Delta S = 98.6 \, \text{J mol}^{-1} \text{K}^{-1} = 0.0986 \, \text{kJ mol}^{-1} \text{K}^{-1} \]

### Step 4: Use the Gibbs free energy equation
The Gibbs free energy equation is:
\[ \Delta G = \Delta H - T \Delta S \]
Rearranging to find \( \Delta H \):
\[ \Delta H = \Delta G + T \Delta S \]
Substituting the values:
\[ \Delta H = 406.6 \, \text{kJ} + (298 \, \text{K} \times 0.0986 \, \text{kJ mol}^{-1} \text{K}^{-1}) \]

### Step 5: Calculate \( T \Delta S \)
Calculating \( T \Delta S \):
\[ T \Delta S = 298 \times 0.0986 = 29.4 \, \text{kJ} \]

### Step 6: Calculate \( \Delta H \)
Now substituting back into the equation:
\[ \Delta H = 406.6 \, \text{kJ} + 29.4 \, \text{kJ} = 436 \, \text{kJ} \]

### Final Answer
Thus, the bond energy of \( H_2 \) is:
\[ \Delta H = 436 \, \text{kJ/mol} \]

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