Question

In 1,3 -butadiene, the observed length of the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond is \(1.48 \AA\), while the "standard" single bond distance is generally accepted to be \(1.54 \AA\). Explain this discrepancy and calculate the \(\pi\) bond order of the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) butadiene bond. (Hint, consider the resonance effects and the molecular orbital structure).

Short answer

The discrepancy in the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond length in 1,3-butadiene compared to the standard single bond distance is due to resonance effects and electron delocalization across the molecule, leading to intermediate bond lengths between single and double bonds. The π bond order for the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond in 1,3-butadiene is 0.5.

Step-by-step solution

1,3-butadiene is a molecule with the chemical structure: \(CH_{2} = CH - CH = CH_{2}\). There are alternating single (\(C - C\)) and double (\(C = C\)) bonds between the carbon atoms, making it a conjugated molecule. The molecular orbital theory can help us understand why the bond lengths are different from their standard values. #Step 2: Consider the resonance structures of 1,3-butadiene#

When we analyze the resonance structures of 1,3-butadiene, we can represent it in two different ways, with the double bonds located at different positions: Resonance Structure 1: \(CH_{2} = CH - CH = CH_{2}\) Resonance Structure 2: \(CH_{2} - CH = CH - CH_{2}\) The true structure of 1,3-butadiene is a resonance hybrid, which means a combination of these two structures. In this hybrid structure, the electrons from the double bonds are delocalized across the molecule, causing the actual bond lengths to be intermediate between single and double bonds. #Step 3: Calculate the number of π bond orders using resonance structures#

To calculate the π bond order, we can use the formula: π bond order = (number of π bonds in resonance structures) / (number of resonance structures). For the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond in the resonance hybrid of 1,3-butadiene: Resonance Structure 1: There is 1 π bond between \(\mathrm{C}_{2}\) and \(\mathrm{C}_{3}\). Resonance Structure 2: There is no π bond between \(\mathrm{C}_{2}\) and \(\mathrm{C}_{3}\). Hence, the π bond order for the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond is: π bond order = (1 + 0) / 2 = 0.5 #Step 4: Conclusion#

The discrepancy in the bond length of the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond in 1,3-butadiene compared to the standard single bond distance can be attributed to the resonance effects and the delocalization of electrons across the molecule, leading to intermediate bond lengths between single and double bonds. The π bond order for the \(\mathrm{C}_{2}-\mathrm{C}_{3}\) bond in 1,3-butadiene is calculated to be 0.5.

Key concepts

Resonance Structures in Organic Chemistry

Resonance structures are a way of describing the electronic structure of molecules where the bonding cannot be expressed by a single Lewis structure. In organic chemistry, resonance describes the delocalization of electrons within molecules, which means that electrons are not confined to a single bond or atom but are spread across multiple atoms.

In 1,3-butadiene, the conventional representation using double and single bonds does not accurately reflect the true electronic structure. Instead, two resonance structures depict the electron delocalization. Such delocalization tends to stabilize the molecule and leads to bond length discrepancies when compared with 'standard' bond distances.

By acknowledging the resonance effect, students can visualize the behavior of π electrons in conjugated systems and better understand molecular reactivity and properties. For instance, resonance is key to understanding the underlying reason why 1,3-butadiene has a shorter than expected C2-C3 bond length.

Conjugated Molecules

Conjugated molecules involve alternating single and double bonds, which allows the π electrons to be more mobile and spread over the molecule's length. In the context of 1,3-butadiene, this conjugation leads to a molecule where the electron density is shared over multiple carbon atoms, rather than localized to just one bond.

This electron sharing impacts the physical properties of the molecule, including the observed bond lengths. The C2-C3 bond length in 1,3-butadiene is shorter than a standard single bond because the electron density between these two carbons is greater than in a single bond, but less than a double bond. Understanding conjugation is crucial to predict the behavior and properties of organic molecules, including color, reactivity, and stability.

Molecular Orbital Theory

Molecular Orbital (MO) Theory provides a more nuanced approach to understanding chemical bonding in molecules like 1,3-butadiene. Unlike other models, MO Theory explains how atomic orbitals combine to form molecular orbitals, which can be occupied by electrons of the molecule.

The MO Theory enables chemists to visualize how π electrons are distributed across the molecule, which significantly affects its bond lengths and strength. Through the overlap of p-orbitals, electrons are delocalized in a molecular π orbital, which is spread across the carbons involved in the conjugated system. This results in partial double bond character in bonds that would otherwise be considered single bonds, such as the C2-C3 bond in 1,3-butadiene.

π Bond Order Calculation

The π bond order in a molecule like 1,3-butadiene can be calculated by looking at its resonance structures. The bond order provides an indication of the bond's strength and stability. In 1,3-butadiene, by considering both resonance structures, and the presence of a π bond, we determine a bond order of 0.5 for the C2-C3 bond.

This calculation is an elementary example of how bond order can be determined from a molecule's resonance structures. It offers a simplified way to understand the distribution of electrons in π bonds and helps students grasp why certain bonds in conjugated molecules have lengths that deviate from standard single or double bond lengths.

Being familiar with the concept of π bond order is essential for students as it applies to many aspects of advanced organic chemistry, including the analysis of UV-Vis spectra, the reactivity of compounds in electrophilic and nucleophilic reactions, and complicated bonding in transition metal complexes.