When we estimate $\Delta H^{\circ}$ from bond enthalpies we assume that all bonds of the same type (single, double, triple) between the same two atoms have the same energy, regardless of the molecule in which they occur. The purpose of this problem is to show you that this is only an approximation. You will need the following standard enthalpies of formation:
$$
\begin{array}{ll}
\mathrm{C}(\mathrm{g}) & \Delta H^{\circ}=716.7 \mathrm{~kJ} / \mathrm{mol} \\
\mathrm{CH}(\mathrm{g}) & \Delta H^{\circ}=596.3 \mathrm{~kJ} / \mathrm{mol} \\
\mathrm{CH}_{2}(\mathrm{~g}) & \Delta H^{\circ}=392.5 \mathrm{~kJ} / \mathrm{mol} \\
\mathrm{CH}_{3}(\mathrm{~g}) & \Delta H^{\circ}=146.0 \mathrm{~kJ} / \mathrm{mol} \\
\mathrm{H}(\mathrm{g}) & \Delta H^{\circ}=218.0 \mathrm{~kJ} / \mathrm{mol}
\end{array}
$$
(a) What is the average $\mathrm{C}-\mathrm{H}$ bond energy in methane, $\mathrm{CH}_{4} ?$
(b) Using bond enthalpies, estimate $\Delta H^{\circ}$ for the reaction
$$
\mathrm{CH}_{4}(\mathrm{~g}) \longrightarrow \mathrm{C}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{~g})
$$
(c) By heating $\mathrm{CH}_{4}$ in a flame it is possible to produce the reactive gaseous species $\mathrm{CH}_{3}, \mathrm{CH}_{2}, \mathrm{CH}$, and even carbon atoms, $\mathrm{C}$. Experiments give these values of $\Delta H^{\circ}$ for the reactions shown:
$$
\begin{array}{ll}
\mathrm{CH}_{3}(\mathrm{~g}) \longrightarrow \mathrm{C}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{~g})+\mathrm{H}(\mathrm{g}) & \Delta H^{\circ}=788.7 \mathrm{~kJ} \\
\mathrm{CH}_{2}(\mathrm{~g}) \longrightarrow \mathrm{C}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{~g}) & \Delta H^{\circ}=324.2 \mathrm{~kJ}
\end{array}
$$
$\mathrm{CH}(\mathrm{g}) \longrightarrow \mathrm{C}(\mathrm{g})+\mathrm{H}(\mathrm{g})$
$\Delta H^{\circ}=338.3 \mathrm{~kJ}$
For each of these reactions, draw a diagram similar to Figure $8.4$. Then calculate the average $\mathrm{C}-\mathrm{H}$ bond energy in $\mathrm{CH}_{5}, \mathrm{CH}_{2}$, and $\mathrm{CH}$. Comment on any trends you see.