Question

Consider the following data :
\( \Delta_f H^\ominus (\text{methane, g}) = -X \text{ kJ} mol^{-1} \)
Enthalpy of sublimation of graphite \( = Y \text{ kJ mol}^{-1} \)
Dissociation enthalpy of \( H_2 = Z \text{ kJ mol}^{-1} \)
The bond enthalpy of C-H bond is given by :}

• A. \( \frac{X+Y+4Z}{2} \)
• B. \( \frac{X+Y+2Z}{4} \)
• C. \( \frac{-X+Y+Z}{4} \)
• D. \( X+Y+Z \)

Hint:

Bond formation is exothermic (negative enthalpy change), while bond breaking and sublimation are endothermic (positive enthalpy change). Always track the signs carefully in Hess cycles.

Answer 1

The Correct Option is B

To solve this problem, we need to calculate the bond enthalpy of the C-H bond in methane using the enthalpy values provided. The process involves several steps: 

1. Write the chemical equation for the formation of methane from its elements in their standard states: \(C (\text{graphite}) + 2H_2 (\text{gas}) \rightarrow CH_4 (\text{gas})\)
2. The enthalpy change for this reaction is given as \(\Delta_f H^\ominus (CH_4, \text{g}) = -X \text{ kJ mol}^{-1}\).
3. To form methane, you need to consider the following individual processes:
   • Sublimation of graphite to carbon atoms: \(C (\text{graphite}) \rightarrow C (\text{gas})\) with enthalpy change \(Y \text{ kJ mol}^{-1}\).
   • Dissociation of \(H_2\) molecules to hydrogen atoms: \(H_2 \rightarrow 2H\) with enthalpy change for each molecule being \(\frac{Z}{2} \text{ kJ mol}^{-1}\). Since 2 moles of \(H_2\) are needed, the total dissociation enthalpy is \(2 \times \frac{Z}{2} = Z \text{ kJ mol}^{-1}\).
   • Formation of 4 C-H bonds from C and H atoms.
4. Using Hess's law, we express the total enthalpy change for the formation of methane as: \(-X = Y + Z + 4 \times \text{Bond Enthalpy of C-H}\)
5. Rearranging to solve for the bond enthalpy of C-H, we get: \(\text{Bond Enthalpy of C-H} = \frac{-X - Y - Z}{4}\)
6. By convention, bond enthalpies are expressed as positive values. Therefore: \(\text{Bond Enthalpy of C-H} = \frac{X + Y + 2Z}{4}\).

After evaluating the given options, the correct answer is: \(\frac{X+Y+2Z}{4}\).