Calculate heat of formation of isoprene using bond energy data.
`5C(s) + 4H_(2)(g) rarr H_(2)C = underset(CH_3)(|)(C ) - HC =CH_(2)`
Given `C - H = 98.8 kcal`
`H - H = 104 kcal`
C – C= 83 kcal
C = C = 147 kcal and `C(s) rarr C(g) = 171 kcal`

Options

A. `29 kcal`
B. `14.7 kcal`
C. `20.6 kcal`
D. `18.6 kcal`

Explanation

To calculate the heat of formation of isoprene using bond energy data, we will follow these steps:

### Step 1: Write the Reaction
The formation of isoprene from its elements can be represented as:
\[ 5C(s) + 4H_2(g) \rightarrow H_2C = CH(CH_3) - HC = CH_2 \]

### Step 2: Identify the Bond Energies
We are given the following bond energies:
- C-H = 98.8 kcal
- H-H = 104 kcal
- C-C = 83 kcal
- C=C = 147 kcal
- Sublimation of C(s) to C(g) = 171 kcal

### Step 3: Calculate the Bond Energy of Reactants
For the reactants:
1. **Sublimation of Carbon**:
- 5 moles of solid carbon to gaseous carbon:
\[ 5 \times 171 \text{ kcal} = 855 \text{ kcal} \]

2. **Dissociation of Hydrogen**:
- 4 moles of H₂ to H atoms:
\[ 4 \times 104 \text{ kcal} = 416 \text{ kcal} \]

**Total Bond Energy of Reactants**:
\[ \text{Total} = 855 \text{ kcal} + 416 \text{ kcal} = 1271 \text{ kcal} \]

### Step 4: Calculate the Bond Energy of Products
For the products (isoprene):
1. **C=C bonds**: There are 2 C=C bonds:
\[ 2 \times 147 \text{ kcal} = 294 \text{ kcal} \]

2. **C-C bonds**: There is 1 C-C bond:
\[ 1 \times 83 \text{ kcal} = 83 \text{ kcal} \]

3. **C-H bonds**: There are 8 C-H bonds:
\[ 8 \times 98.8 \text{ kcal} = 790.4 \text{ kcal} \]

**Total Bond Energy of Products**:
\[ \text{Total} = 294 \text{ kcal} + 83 \text{ kcal} + 790.4 \text{ kcal} = 1167.4 \text{ kcal} \]

### Step 5: Calculate the Heat of Formation
Using the formula:
\[ \Delta H_f = \text{Bond Energy of Reactants} - \text{Bond Energy of Products} \]
Substituting the values:
\[ \Delta H_f = 1271 \text{ kcal} - 1167.4 \text{ kcal} = 103.6 \text{ kcal} \]

### Step 6: Final Result
The heat of formation of isoprene is approximately:
\[ \Delta H_f \approx 103.6 \text{ kcal} \]