Consider the reactions given below .On the basis of these reactions ,Find out which of the algebraic relations given in options (a) to(d) is correct?
`(p)C(g)+4H(g)toCH_(4)(g), Delta_(r)H=xKJ mol^(-1)`
`(Q)C("graphite")+2H_(2)(g)to CH_(4)(g),`
`Delta_(r)H=yKJmol^(-1)`

Options

A. `x = y`
B. `x = 2y`
C. `x gt y`
D. `x lt y`

Explanation

To solve the problem, we need to analyze the two given reactions and their corresponding enthalpy changes (ΔH).

### Given Reactions:
1. **Reaction (p)**:
\[ C(g) + 4H(g) \rightarrow CH_4(g), \quad \Delta_rH = x \text{ kJ/mol} \]

2. **Reaction (Q)**:
\[ C(\text{graphite}) + 2H_2(g) \rightarrow CH_4(g), \quad \Delta_rH = y \text{ kJ/mol} \]

### Step 1: Understand the Reactions
- In reaction (p), carbon is in the gaseous state (C(g)), and hydrogen is also in the gaseous state (4H(g)). This reaction directly forms methane (CH₄) from its atomic components.
- In reaction (Q), carbon is in the solid state (C(graphite)), and hydrogen is in the molecular form (H₂). This reaction requires the conversion of solid carbon to gaseous carbon and the dissociation of hydrogen molecules into atomic hydrogen before forming methane.

### Step 2: Analyze Enthalpy Changes
- For reaction (p), since both reactants are already in their gaseous state, the enthalpy change (ΔH = x) reflects the energy required to form methane directly from its atomic components.
- For reaction (Q), additional energy is required to convert solid carbon to gaseous carbon and to dissociate H₂ into atomic hydrogen. Therefore, the enthalpy change (ΔH = y) for this reaction will be greater than that for reaction (p) because of the extra energy needed to break the bonds in solid carbon and molecular hydrogen.

### Step 3: Establish the Relationship Between x and y
- Since the reaction (Q) requires additional energy to convert the reactants to their gaseous atomic states, we can conclude that:
\[ x > y \]

### Step 4: Check the Options
Now, we need to check which of the provided options corresponds to our conclusion that \( x > y \).

- **Option (a)**: \( x = y \) (Incorrect)
- **Option (b)**: \( x < y \) (Incorrect)
- **Option (c)**: \( x > y \) (Correct)
- **Option (d)**: \( x \geq y \) (Incorrect)

### Conclusion
The correct relation between the enthalpy changes for the two reactions is:
\[ \text{Option (c): } x > y \]