Question

Following are lengths for a series of \(\mathrm{C}-\mathrm{C}\) single bonds. Propose an explanation for the differences in bond lengths. \begin{tabular}{|lc|} \hline Structure & Length of \(\mathrm{C}-\mathrm{C}\) Single Bond \((\mathrm{pm})\) \\ \hline \(\mathrm{CH}_{3}-\mathrm{CH}_{3}\) & \(153.7\) \\ \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{3}\) & \(151.0\) \\ \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}_{2}\) & \(146.5\) \\ \(\mathrm{HC} \equiv \mathrm{C}-\mathrm{CH}_{3}\) & \(145.9\) \\ \hline \end{tabular}

Short answer

Short Answer: The differences in carbon-carbon single bond lengths can be explained by the variation in hybridization of the carbon atoms involved in the bond formation. Carbon atoms with greater s-character (such as sp hybridized carbon atoms) are closer to the nucleus, resulting in shorter and stronger bonds. This trend is observed in the bond length data provided, with bond lengths ranging from 153.7 pm for an sp3-sp3 bond to 145.9 pm for an sp-sp3 bond.

Step-by-step solution

Examine the given molecular structures

For each molecule in the table, pay attention to its geometry and chemical bond types, particularly the double and triple bonds. - \(\mathrm{CH}_{3}-\mathrm{CH}_{3}\): Ethane molecule with a single bond between two carbon atoms - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{3}\): Propene molecule with a double bond between two carbon atoms and a single bond between the other carbon atoms - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}_{2}\): 1,3-butadiene molecule with conjugated double and single bonds in an alternating pattern - \(\mathrm{HC} \equiv \mathrm{C}-\mathrm{CH}_{3}\): Propyne molecule with a triple bond between two carbon atoms and a single bond between the other carbon atoms

Analyze hybridization of carbon atoms

The hybridization of carbon atoms can help explain the differences in bond lengths. Analyze the hybridization of the carbon atoms in each molecule. - \(\mathrm{CH}_{3}-\mathrm{CH}_{3}\): Both carbon atoms are \(\mathrm{sp^3}\) hybridized - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{3}\): The carbon atom bonded to the hydrogen atom with a double bond (\(\mathrm{CH}_{2}\)) is \(\mathrm{sp^2}\) hybridized, while the other carbon atom (\(\mathrm{CH}_{3}\)) is \(\mathrm{sp^3}\) hybridized - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}_{2}\): All carbon atoms involved in the double bonds are \(\mathrm{sp^2}\) hybridized - \(\mathrm{HC} \equiv \mathrm{C}-\mathrm{CH}_{3}\): The carbon atom bonded to the hydrogen atom with a triple bond (\(\mathrm{HC}\)) is \(\mathrm{sp}\) hybridized, while the other carbon atom (\(\mathrm{CH}_{3}\)) is \(\mathrm{sp^3}\) hybridized

Understand the influence of hybridization on bond lengths

With the hybridization identified, we can now understand how it influences bond lengths. The greater the s-character of a hybridized orbital, the shorter and stronger the bond will be. - \(\mathrm{CH}_{3}-\mathrm{CH}_{3}\): Bond between two \(\mathrm{sp^3}\) hybridized carbon atoms has a bond length of \(153.7\,\mathrm{pm}\) - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{3}\): Bond between an \(\mathrm{sp^2}\) and an \(\mathrm{sp^3}\) hybridized carbon atoms has a bond length of \(151.0\,\mathrm{pm}\) - \(\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}_{2}\): Bond between two \(\mathrm{sp^2}\) hybridized carbon atoms has a bond length of \(146.5\,\mathrm{pm}\) - \(\mathrm{HC} \equiv \mathrm{C}-\mathrm{CH}_{3}\): Bond between an \(\mathrm{sp}\) and an \(\mathrm{sp^3}\) hybridized carbon atoms has a bond length of \(145.9\,\mathrm{pm}\)

Propose an explanation

The differences in carbon-carbon single bond lengths can be attributed to the differences in hybridization of the carbon atoms forming the bonds. Carbon atoms with greater s-character (such as \(\mathrm{sp}\) hybridized carbon atoms) are closer to the nucleus, resulting in shorter and stronger bonds. This trend is observed in the bond length data provided. In summary, the observed differences in carbon-carbon single bond lengths can be explained by the variation in hybridization of the carbon atoms involved in the bond formation.

Key concepts

Chemical Bond Hybridization

Understanding chemical bond hybridization is key to grasping why different molecules exhibit varied carbon-carbon single bond lengths. Hybridization is a concept in chemistry where atomic orbitals mix to form new hybrid orbitals, which in turn influence molecular bonding and shape. Each type of hybridization, such as sp, sp², and sp³, has a different percentage of 's' and 'p' character, which affects the electron density around the atomic nuclei and results in different bond lengths.

For instance, an sp³ hybridized carbon, which is prevalent in alkanes like ethane, has a bond length typically around 153.7 pm. This is because the 's' character is diluted among four hybrid orbitals. In contrast, an sp² hybridized carbon, found in alkenes such as propene, has a bond length of about 151.0 pm due to a higher 's' character compared to sp³ hybridization, but lesser than in an sp hybridized carbon. The highest 's' character is found in the sp orbital, as seen in alkynes like propyne, resulting in the shortest bond length of 145.9 pm.

The 's' character's influence on bond strength and length is also an important aspect. The more 's' character present within the bond, the stronger and shorter the bond tends to be. This results from the electrons being held closer to the nucleus, creating a tighter and more robust electron pair bond between the atoms.

Molecular Geometry

The molecular geometry of a compound is another essential factor contributing to the length of its carbon-carbon single bonds. The geometry, determined by the spatial arrangement of the atoms, is a direct consequence of the hybridization state of the atoms involved. sp³ hybridization results in a tetrahedral geometry, where bond angles are approximately 109.5 degrees, as seen in ethane. sp² hybridization leads to a trigonal planar geometry, with bond angles close to 120 degrees, which is the case for the carbon atoms in propene and 1,3-butadiene. Finally, sp hybridization gives rise to a linear molecular geometry with bond angles of approximately 180 degrees, which occurs in alkynes like propyne.

Differences in molecular geometry can cause variations in bond lengths due to changes in the angles at which the orbitals overlap to form bonds. Tighter bond angles can lead to more s-character in the bond, thus causing shorter bond lengths. Conversely, wider bond angles tend to result in longer bonds due to the more prominent p-character.

Bond Strength and Length

The relationship between bond strength and length is inversely proportional; stronger bonds are typically shorter. The reason behind this is rooted in the attraction between the bonding electrons and the nuclei of the bonded atoms. Stronger bonds are formed when the electrons are closer to the nuclei, hence shortening the bond length.

An sp³-sp³ bond, like in ethane, is longer and weaker compared to an sp-sp³ bond, as present in propyne, due to the higher p-character of the orbitals in the former, which positions the bonding electrons further from the atomic nucleus. This concept explains the variance in bond lengths given in the problem statement. It's crucial for students to recognize that as the s-character increases within a bond, the electrons are drawn closer to the nucleus, creating a stronger, shorter bond.

To summarize, chemical bond hybridization, molecular geometry, and bond strength and length are interrelated concepts that together help explain the differences in bond lengths of various carbon compounds. By examining these factors individually and how they interact with one another, students can develop a deeper understanding of molecular structure at the atomic level.