Consider the reactions `:`
`C_((s))+2H_(2(g)) rarr CH_(4(g)), DeltaH=-X kcal`
`C_((g))+4H_((g)) rarr CH_(4(g)),DeltaH-X_(1)kcal`
`CH_(4(g))rarr CH_(3(g))+H_((g)), DeltaH=+Ykcal`
The average bond energy of `C-H` bond is `:`

Options

A. `(X)/(4)` K cal. `mol^(-1)`
B. Y K cal. `mol^(-1s)`
C. `(X_(1))/(4)` K cal. `mol^(-1)`
D. `X_(1)` K cal. `mol^(-1)`

Explanation

To find the average bond energy of the C-H bond in methane (CH₄), we will analyze the given reactions step by step.

### Step 1: Understand the Reactions
We have three reactions with their respective enthalpy changes (ΔH):
1. \( C_{(s)} + 2H_{2(g)} \rightarrow CH_{4(g)}, \Delta H = -X \) kcal
2. \( C_{(g)} + 4H_{(g)} \rightarrow CH_{4(g)}, \Delta H = -X_1 \) kcal
3. \( CH_{4(g)} \rightarrow CH_{3(g)} + H_{(g)}, \Delta H = +Y \) kcal

### Step 2: Analyze the First Reaction
The first reaction indicates that solid carbon and hydrogen gas combine to form methane, releasing energy (exothermic reaction). The value of ΔH is -X, meaning that the formation of one mole of methane from solid carbon and hydrogen gas releases X kcal.

### Step 3: Analyze the Second Reaction
The second reaction shows the formation of methane from gaseous carbon and hydrogen. The enthalpy change is -X₁. This reaction also releases energy, but the amount may differ from the first reaction due to the different states of carbon.

### Step 4: Analyze the Third Reaction
The third reaction shows the dissociation of methane into methyl (CH₃) and a hydrogen atom. The enthalpy change is +Y, indicating that energy is absorbed when breaking the C-H bond in methane.

### Step 5: Relate the Reactions
To find the average bond energy of the C-H bond, we can relate the enthalpy changes:
- The formation of methane from its elements (first reaction) can be represented in terms of bond energies.
- The dissociation of methane (third reaction) indicates that breaking one C-H bond requires energy.

### Step 6: Calculate the Average Bond Energy
The average bond energy of one C-H bond in methane can be calculated using the enthalpy change associated with the dissociation of methane. Since there are 4 C-H bonds in one molecule of methane, the average bond energy (E) can be expressed as:
\[ E = \frac{Y + X_1}{4} \]
Where:
- Y is the energy required to break one C-H bond (from the third reaction).
- X₁ is the energy released when forming methane from gaseous carbon and hydrogen (from the second reaction).

### Final Answer
Thus, the average bond energy of the C-H bond in methane is:
\[ \text{Average bond energy of C-H bond} = \frac{X_1 + Y}{4} \text{ kcal} \]