Question

In Exercise 89 in Chapter 8, the Lewis structures for benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) were drawn. Using one of the Lewis structures, estimate \(\Delta H_{\mathrm{f}}^{\circ}\) for \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\) using bond energies and given that the standard enthalpy of formation of \(\mathrm{C}(g)\) is \(717 \mathrm{~kJ} / \mathrm{mol}\). The experimental \(\Delta H_{\mathrm{f}}^{\circ}\) value of \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\) is \(83 \mathrm{~kJ} / \mathrm{mol} .\) Explain the discrepancy between the experimental value and the calculated \(\Delta H_{\mathrm{f}}^{\circ}\) value for \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\)

Short answer

The calculated standard enthalpy of formation for benzene (C6H6) in the gas phase is -1068 kJ/mol, using the bond energies from one of the resonance structures. This value is significantly different from the experimental value of 83 kJ/mol. The discrepancy can be attributed to the fact that we only used one resonance structure for our calculations, and benzene's bond character is an intermediate between single and double bonds. This hybridization lowers the energy of the molecule, making it more stable than represented by a single Lewis structure, resulting in an underestimation of the actual enthalpy of formation.

Step-by-step solution

Lewis structure of benzene

First, let's recall the Lewis structure of benzene. Benzene has a hexagonal ring structure where each carbon atom is bonded to two other carbon atoms and one hydrogen atom. The carbon-carbon bonds alternate between single and double bonds. The resonance structure of benzene can be depicted as two alternating structures: Structure 1: C1=C2-C3=C4-C5=C6-C1 (single bond with C6) Structure 2: C1-C2=C3-C4=C5-C6=C1 (single bond with C1) In this solution, we will use Structure 1 for our calculations.

Calculating bond energies

Now, let's calculate the bond energy for each type of bond in the benzene molecule. We will multiply the bond energies by the number of each bond type in the molecule. We have: - 3 C=C (carbon-carbon double) bonds with bond energy of 614 kJ/mol - 3 C-C (carbon-carbon single) bonds with bond energy of 348 kJ/mol - 6 C-H (carbon-hydrogen) bonds with bond energy of 414 kJ/mol Calculating the total bond energy: Total bond energy = (3 x 614 kJ/mol) + (3 x 348 kJ/mol) + (6 x 414 kJ/mol) Total bond energy = 1842 + 1044 + 2484 kJ/mol = 5370 kJ/mol

Calculating ΔH𝑓° for C6H6(g)

We need to find the standard enthalpy of formation, ΔH𝑓°, for benzene (C6H6) in the gas phase. This can be calculated using bond energies: ΔH𝑓° = 6 x ΔH𝑓° of C(g) - Total bond energy Given that the standard enthalpy of formation of C(g) is 717 kJ/mol, we can calculate ΔH𝑓° for C6H6(g): ΔH𝑓° = (6 x 717 kJ/mol) - 5370 kJ/mol = 4302 kJ/mol - 5370 kJ/mol = -1068 kJ/mol

Comparing to the experimental value

Now that we have calculated the standard enthalpy of formation for benzene, we can compare it to the experimental value, which is given as 83 kJ/mol. Our calculated value is -1068 kJ/mol, which is significantly different from the experimental value of 83 kJ/mol. This discrepancy can be attributed to the fact that we used only one of the resonance structures of benzene for our calculations. In reality, the bond character of benzene is an intermediate between single and double bonds. This hybridization lowers the energy of the molecule, making it more stable than represented by a single Lewis structure. Therefore, using bond energies for only one of the resonance structures results in an underestimation of the actual enthalpy of formation.

Key concepts

Benzene

Benzene is a fundamental aromatic hydrocarbon with the chemical formula \(\mathrm{C}_6\mathrm{H}_6\). This compound is comprised of a hexagonal ring of six carbon atoms, with alternating double and single bonds between them. Each carbon atom is also bonded to a single hydrogen atom. This structure is highly symmetrical and forms the basis for the unique stability and chemical properties of benzene.
Benzene serves as a cornerstone in organic chemistry, influencing many related compounds. Its structure deviates from classical representations due to its resonance, leading to unique bonding properties. Understanding benzene helps us grasp the nature of aromatic rings and the role of resonance in molecular stability.
In calculations involving benzene, its structure plays a key role. Since the resonance alters typical single and double bond energy, more sophisticated methods beyond straightforward bond energy calculations are often needed to accurately determine properties like the enthalpy of formation.

Resonance Structures

Resonance structures are a way to represent molecules that cannot be adequately described by a single Lewis structure. In benzene, the electrons in the \(\pi\) bonds are delocalized over the entire ring, resulting in an intermediate bond length between that of a single and a double bond.
Benzene is commonly illustrated using two resonance structures where double and single bonds alternate. These structures are:

• Structure 1: Alternating double bonds (C1=C2-C3=C4-C5=C6-C1)
• Structure 2: The double bonds move one carbon over (C1-C2=C3-C4=C5-C6=C1)

In reality, neither of these resonance structures represents benzene alone. Instead, the true structure is a hybrid, with all bonds having equal character. This hybrid has a lower energy than either of the individual structures, giving benzene its stability and lower than expected enthalpy of formation.
This delocalization affects the calculations of its enthalpy, making benzene's energy much lower than what simple bond energy calculations suggest.

Bond Energies

Bond energies are the amounts of energy required to break one mole of a particular type of bond in a gaseous substance. They are critical for understanding the enthalpy of formation of molecules like benzene.
For benzene, the bond energies used in typical calculations include:

• Carbon-carbon double bonds (\(\mathrm{C=C}\): 614 kJ/mol
• Carbon-carbon single bonds (\(\mathrm{C-C}\): 348 kJ/mol
• Carbon-hydrogen bonds (\(\mathrm{C-H}\): 414 kJ/mol

In an attempt to estimate benzene's enthalpy of formation using these values, we sum the energy required to form each bond in benzene based on the chosen resonance structure.
However, the actual bonding in benzene is an average of these resonance structures, meaning that a simple calculation using these bond energies often results in significant discrepancies, as seen in the exercise provided. The true bond energy would be lower due to the resonance stabilization, explaining why calculated enthalpy values may differ from experimental ones.

Lewis Structures

Lewis structures are diagrams that depict the bonding between atoms of a molecule and the lone pairs of electrons that may exist. These structures help us understand how atoms are connected and share electrons within a molecule.
In benzene, the Lewis structure is somewhat challenging due to resonance. While each carbon in benzene is usually paired with a hydrogen atom, the carbon-carbon connections vary by structure due to their potential positions as single or double bonds. This complexity is why benzene is often shown with two separate structures, helping to illustrate resonance.
Drawing a Lewis structure involves:

• Counting the valence electrons of the molecule (for benzene, this is \(6 \times 4\) for carbon and \(6 \times 1\) for hydrogen)
• Arranging electrons to complete octets for carbon and complete pairs for hydrogen
• Identifying bond formation and representation in terms of single and double bonds

By using Lewis structures, chemists can predict molecular shapes, reactivity, and even properties like boiling points. Still, for benzene, they must account for resonance to accurately reflect its stability and unique characteristics.