Question

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.

Short answer

The heat of hydrogenation of naphthalene is \(1129 \thinspace \mathrm{kJ/mol}\), and its resonance energy is \(-1171 \thinspace \mathrm{kJ/mol}\).

Step-by-step solution

Write the balanced combustion reactions

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: \[ C_{10}H_{8}(s) + \frac{15}{2}O_{2}(g) \rightarrow 10CO_{2}(g) + 4H_{2}O(l) \]

Write the hydrogenation reaction

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

Use Hess's Law to determine the heat of hydrogenation

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

Calculate the resonance energy of naphthalene

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 \(-1171 \thinspace \mathrm{kJ/mol}\).

Key concepts

Heat of Combustion

The heat of combustion is the amount of heat released when a substance burns completely in the presence of oxygen. It's an important concept in thermodynamics and can be used to understand the energy content of fuels.
This is measured in kilojoules per mole (kJ/mol) and is specific to each substance. The heat of combustion for decahydronaphthalene is -6286 kJ/mol, while for naphthalene it is -5157 kJ/mol.
These numbers tell us how much energy these compounds can release when they are burned. By knowing these values, scientists and engineers can compare different fuels to understand which one might be more efficient or economically viable for specific applications.

• Complete combustion produces carbon dioxide and water.
• Heat of combustion is a crucial component in calculating energy changes in chemical reactions.

Hess's Law

Hess's Law is a principle that allows us to calculate enthalpy changes in multi-step reactions by breaking them into simpler steps. This is useful because directly measuring the enthalpy change of a complex reaction can be challenging.
Let's say you want to find the heat of hydrogenation of decahydronaphthalene converting into naphthalene. Hess's Law enables this by using known heat values from different reactions.
The law states that the change in enthalpy during a chemical reaction is the same regardless of the route taken, as long as the initial and final states are the same. Hence, we can calculate the heat of hydrogenation by subtracting the heat of combustion of decahydronaphthalene from that of naphthalene:

• \(\Delta H_{\text{hydrogenation}} = \Delta H_{\text{combustion, naphthalene}} - \Delta H_{\text{combustion, decahydronaphthalene}}\)
• This gives the result of 1129 kJ/mol.

This method saves a lot of hassle by using pre-existing data to piece together the larger picture.

Resonance Energy

Resonance energy is a measure of the stability that a compound gains from having multiple resonance structures. Naphthalene is highly stable due to its resonance, which means it has a lower energy structure than a similar molecule without resonance, such as a non-resonating structure of equal hydrogenation, like COT (1,3,5,7-cyclooctatetraene).
To estimate this, compare how much energy is released during hydrogenation for naphthalene versus a hypothetical non-resonating structure.
Here, naphthalene's calculated resonance energy is determined by the difference between its observed heat of hydrogenation and that of the non-resonating structure's estimated hydrogenation.

• Naphthalene's resonance energy comes out to be -1171 kJ/mol.
• This negative value highlights the extra stability gained from its resonance structures.

Hence, naphthalene's molecular structure allows it to distribute electron density more evenly, lowering the molecule's overall energy.

Enthalpy Change

Enthalpy change (\(\Delta H\)) is the heat content change at constant pressure and is an essential aspect of studying chemical reactions. It helps in understanding whether a reaction absorbs or releases heat, which is fundamental in determining the nature of the process.
In the context of thermochemistry, enthalpy changes can decide environmental and industrial applications of chemical substances.
The enthalpy change needs to be considered to configure the thermodynamics of hydrogenation and combustion. An exothermic reaction indicates heat release, whereas an endothermic process absorbs heat.

• In combustion reactions, a negative enthalpy value shows energy release into the surroundings.
• For our exercise, the calculated change in enthalpy for hydrogenation was +1129 kJ/mol, signifying it is an endothermic process.

Recognizing and calculating these changes help chemists design processes with optimal energy efficiencies.