Question

Use average bond enthalpies (Table 8.4) to estimate \(\Delta H\) for the atomization of benzene, \(\mathrm{C}_{6} \mathrm{H}_{6}\) : $$ \mathrm{C}_{6} \mathrm{H}_{6}(g) \longrightarrow 6 \mathrm{C}(g)+6 \mathrm{H}(g) $$ Compare the value to that obtained by using \(\Delta H_{f}^{\circ}\) data given in Appendix \(C\) and Hess's law. To what do you attribute the large discrepancy in the two values?

Short answer

The large discrepancy between the enthalpy change for the atomization of benzene using average bond enthalpies (\(\Delta H_{atomization} = 4566 \, \text{kJ/mol}\)) and using Hess's law and enthalpy of formation data (\(\Delta H_{atomization} = + 49 \,\text{kJ/mol}\)) can be attributed to the fact that average bond enthalpies are an approximation and do not take into account the specific molecular environment. In benzene, resonance stabilization leads to a more stable molecule, which means that the actual enthalpy change of atomization would be higher than what is estimated using average bond enthalpies. The actual enthalpy change obtained using standard enthalpy of formation data and Hess's law is a more accurate depiction of the energy required to break the bonds in benzene and convert it into atoms.

Step-by-step solution

Calculate the total bond enthalpies of the bonds in benzene

Using the average bond enthalpies given in Table 8.4, we can find the sum of bond enthalpies for the bonds present in benzene, which has 6 C-C bonds and 6 C-H bonds: - Average bond enthalpy of C-C bond: 348 kJ/mol - Average bond enthalpy of C-H bond: 413 kJ/mol Total bond enthalpies in benzene: \((6 \times 348) + (6 \times 413) = 2088 + 2478 = 4566 \, \text{kJ/mol}\)

Estimate the enthalpy change for the atomization reaction

The enthalpy change for the atomization reaction of benzene, \(\Delta H_{atomization}\), is the total bond enthalpies in benzene: \(\Delta H_{atomization} = 4566 \, \text{kJ/mol}\)

Determine the enthalpy change using standard enthalpy of formation data and Hess's law

Using Appendix C and Hess's law, we can determine the enthalpy change for the atomization reaction. Standard enthalpy of formation of benzene, \(\Delta H_{f}^{\circ}(\text{C}_{6}\text{H}_{6}) = + 49\, \text{kJ/mol}\) (We don't need the enthalpies of formation of the elemental carbon and hydrogen, because both of them are in their standard states and their enthalpy of formation values are zero.) Using Hess's law, we can write: \(\Delta H_{atomization} = \Delta H_{f}^{\circ}(\text{C}_{6}\text{H}_{6}) - \Delta H_{f}^{\circ}(\text{6C}) - \Delta H_{f}^{\circ}(\text{6H}) \) Since the enthalpies of formation of the elemental carbon and hydrogen are zero, the equation becomes: \(\Delta H_{atomization} = \Delta H_{f}^{\circ}(\text{C}_{6}\text{H}_{6}) = + 49 \,\text{kJ/mol}\)

Compare the two values

We have found two values for the enthalpy change for the atomization of benzene: From Step 2, using average bond enthalpies: \(\Delta H_{atomization} = 4566 \, \text{kJ/mol}\) From Step 3, using Hess's law and enthalpy of formation data: \(\Delta H_{atomization} = + 49 \,\text{kJ/mol}\) There is a large discrepancy between these two values.

Discuss the reasons for the discrepancy

The large discrepancy between the two values can be attributed to the fact that average bond enthalpies are an approximation and do not take into account the specific molecular environment. In benzene, resonance stabilization leads to a more stable molecule, which means that the actual enthalpy change of atomization would be higher than what is estimated using average bond enthalpies. The actual enthalpy change obtained using standard enthalpy of formation data and Hess's law is a more accurate depiction of the energy required to break the bonds in benzene and convert it into atoms.

Key concepts

Average Bond Enthalpies

Average bond enthalpies are values that provide an approximation of the energy needed to break one mole of a specific type of bond in a gaseous molecule. These values are averages because they are calculated from a variety of molecules containing that bond, thus they may not perfectly match the energy required in any one particular molecule. In the case of benzene, using average bond enthalpies helps estimate the total energy needed to atomize the molecule.
However, this approach is oversimplified as it doesn't consider specific molecular environments, like resonance in benzene, which can stabilize the structure and alter the actual energy required.
To illustrate:

• The average bond enthalpy for a C-C bond is 348 kJ/mol.
• The average bond enthalpy for a C-H bond is 413 kJ/mol.

These numbers are useful for rough calculations but they may yield different results compared to more tailored methods such as Hess's law and enthalpy of formation data.

Hess's Law

Hess's law is a principle that states the total enthalpy change for a chemical reaction is the same, no matter how many steps the reaction is broken into. This law is based on the conservation of energy, which means energy cannot be created or destroyed. Instead, it can be transferred or converted.
In practice, Hess's law enables chemists to calculate the enthalpy change of difficult reactions by using simpler steps with known enthalpies.
For benzene atomization, by knowing the standard enthalpy of formation for benzene, we can estimate the enthalpy change required for its atomization using Hess's law.
Because elemental carbon and hydrogen are in their standard states with formation enthalpies of zero, the enthalpy change for atomizing benzene is directly equivalent to the negative of its formation enthalpy:

1. It introduces the concept that the exact pathway does not matter, simplifying complex problems.
2. It allows for more accurate enthalpy calculations by incorporating additional thermal data.

This makes Hess's law particularly valuable when experimentation is difficult or dangerous.

Enthalpy of Formation

The enthalpy of formation, ext( extH_{f}^{ circ}), is the heat change when one mole of a compound forms from its elements in their standard states. This value is significant as it provides insight into the stability and energy content of substances.
These values can be referenced from standardized tables and are essential for more precise calculations in chemical thermodynamics.
Using benzene as an example, its formation enthalpy is given as 49 kJ/mol. This implies that forming one mole of benzene from elemental carbon and hydrogen requires the absorption of 49 kJ of energy.
When assessing reactions, enthalpy of formation data allows chemists to better predict how much energy is involved compared to using rough estimates like average bond enthalpies.

• They reflect the actual conditions of a reaction profile.
• They help overcome inaccuracies inherent in generalized calculations.
• They incorporate specific molecular interactions often overlooked by average values.

Overall, for reactions like benzene atomization, enthalpy of formation data often gives a more realistic and accurate depiction than average bond enthalpies.