The heat of combustion of decahydronaphthalene $\left(\mathrm{C}_{10} \mathrm{H}_{18}\right)\( is \)-6286 \mathrm{~kJ} / \mathrm{mol}$. The heat of combustion of naphthalene \(\left(\mathrm{C}_{10} \mathrm{H}_{8}\right)\) is \(-5157 \mathrm{~kJ} / \mathrm{mol}\). (In both cases \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\) are the products.) Using these data and data in Appendix \(\mathrm{C}\), calculate the heat of hydrogenation and the resonance energy of naphthalene.

First, we need to write the balanced combustion reactions of decahydronaphthalene and naphthalene, with CO2 and H2O as the products. Decahydronaphthalene: \[ C_{10}H_{18}(l) + \frac{21}{2}O_{2}(g) \rightarrow 10CO_{2}(g) +9H_{2}O(l) \] Naphthalene: \[ C_{10}H_{8}(s) + \frac{15}{2}O_{2}(g) \rightarrow 10CO_{2}(g) + 4H_{2}O(l) \]

Next, we need to write the balanced hydrogenation reaction for decahydronaphthalene to become naphthalene. \[ C_{10}H_{18}(l) \rightarrow C_{10}H_{8}(s) + 5H_{2}(g) \]

Using Hess's Law, let's calculate the heat of hydrogenation, which is the enthalpy change for the above hydrogenation reaction. \[ \Delta H_{\text{hydrogenation}} = \Delta H_{\text{combustion, naphthalene}} - \Delta H_{\text{combustion, decahydronaphthalene}} \] \[ \Delta H_{\text{hydrogenation}} = (-5157 \thinspace \mathrm{kJ/mol}) - (-6286 \thinspace \mathrm{kJ/mol}) \] \[ \Delta H_{\text{hydrogenation}} = 1129 \thinspace \mathrm{kJ/mol} \]

The resonance energy can be found by comparing the heat of hydrogenation of naphthalene to the heat of hydrogenation for an alternative structure, such as 1,3,5,7-cyclooctatetraene (COT), which has single and double bonds with no resonance. The heat of hydrogenation for 1,3,5,7-cyclooctatetraene is \(-460 \thinspace \mathrm{kJ/mol}\). Naphthalene has five double bonds, so we can estimate the heat of hydrogenation if it were a non-resonating structure: \[ \Delta H_{\text{non-resonating, estimated}} = 5 \times (-460 \thinspace \mathrm{kJ/mol}) = -2300 \thinspace \mathrm{kJ/mol} \] Now, we can find the resonance energy by comparing the heat of hydrogenation we found earlier to this estimated value: \[ \Delta H_{\text{resonance}} = \Delta H_{\text{non-resonating, estimated}} - \Delta H_{\text{hydrogenation}} \] \[ \Delta H_{\text{resonance}} = (-2300 \thinspace \mathrm{kJ/mol}) - (1129 \thinspace \mathrm{kJ/mol}) \] \[ \Delta H_{\text{resonance}} = -1171 \thinspace \mathrm{kJ/mol} \] The heat of hydrogenation of naphthalene is \(1129 \thinspace \mathrm{kJ/mol}\), and its resonance energy is \(-1171 \thinspace \mathrm{kJ/mol}\).