Question

The allene molecule \(\mathrm{H}_{2} \mathrm{C}=\mathrm{C}=\mathrm{CH}_{2}\) is linear (the three \(\mathrm{C}\) atoms lie on a straight line). What are the hybridization states of the carbon atoms? Draw diagrams to show the formation of sigma bonds and pi bonds in allene.

Short answer

The hybridization states of the three carbon atoms, from left to right, are \(\mathrm{sp}^{2}\), \(\mathrm{sp}\), and \(\mathrm{sp}^{2}\) respectively. The \(\mathrm{sp}^{2}\) hybridized carbons form three sigma bonds while the \(\mathrm{sp}\) hybridized carbon forms two sigma bonds and two pi bonds. This configuration enables the molecule's linear structure.

Step-by-step solution

Count the sigma and pi bonds

Let's denote the carbon atoms with C1, C2 and C3 from left to right. C1 and C3 form one sigma bond with the other carbon atoms and two sigma bonds with two hydrogen, resulting in total 3 sigma bonds. C2 forms just two sigma bonds with C1 and C3 due to having two double bonds. Each double bond consists of one sigma bond and one pi bond, so C2 also forms two pi bonds.

Determine the hybridization states

The hybridization state of an atom is determined by the number of sigma bonds and lone pairs it forms. In our case, C1 and C3, each forms three sigma bonds and no lone pairs. Thus they are \(\mathrm{sp}^{2}\) hybridized (one s and two p orbitals hybridize to form three \(\mathrm{sp}^{2}\) orbitals). C2 forms two sigma bonds and no lone pairs, so it's \(\mathrm{sp}\) hybridized (one s and one p orbital hybridize to form two \(\mathrm{sp}\) orbitals).

Draw the sigma and pi bonds

The sigma bonds are formed by end-on-end overlapping of orbitals (a C1-H, a C3-H, and between the three carbons). The two pi bonds of C2 are parallel and formed by lateral overlap of two p orbitals with each other. It's crucial to note that as these pi bonds are perpendicular to each other, it leads to this molecule's linear structure.

Key concepts

Sigma and Pi Bonds

Understanding the nature of chemical bonds is vital for grasping the essence of molecular structures. In an allene molecule, like the one represented by \( \mathrm{H}_{2}C=C=CH_{2} \), sigma (\( \sigma \)) and pi (\( \pi \)) bonds play crucial roles in its formation and stability.

\( \sigma \) bonds are the strongest type of covalent chemical bonds and are formed by the head-to-head overlap of atomic orbitals. Each single bond in a molecule is a \( \sigma \) bond, which is responsible for the fundamental framework of the molecule. In allenes, each carbon atom forms \( \sigma \) bonds with its adjacent atoms—the end carbon atoms (C1 and C3) bonding with hydrogen and the central carbon atom, while the central carbon atom (C2) forms \( \sigma \) bonds with the other carbons only.

On the other hand, \( \pi \) bonds are formed by the side-to-side overlap of p-orbitals and are typically found in double and triple bonds alongside a single \( \sigma \) bond. These bonds allow for electron sharing over a larger area, which is crucial for the resonating structures seen in various organic molecules. The double bonds in the allene molecule contain one \( \sigma \) bond and one \( \pi \) bond each, creating a rigid planar structure. The perpendicular arrangement of the \( \pi \) bonds in allene leads to its unique linear geometry.

In summary, the allene molecule is constructed through a combination of \( \sigma \) bonds linking the atoms and \( \pi \) bonds contributing to the molecule's stability and shape.

Hybridization States

Diving deeper into the allene molecule's structural secrets, we examine the hybridization states of the carbon atoms, which explain how atoms share their electrons to form chemical bonds. Hybridization is the process of mixing atomic orbitals to create new orbitals, which result in the formation of specific molecular geometries.

In the allene molecule, the hybridization states are determined by the number of sigma bonds each carbon forms. The terminal carbon atoms (C1 and C3) form three \( \sigma \) bonds each, indicating an \( \mathrm{sp}^{2} \) hybridization—this involves one s orbital and two p orbitals combining to create three equivalent \( \sigma \) oriented orbital hybrids. This configuration allows planar arrangements, ensuring maximum overlap and therefore stronger bonds with adjacent atoms.

The central carbon (C2), with only two \( \sigma \) bonds, is \( \mathrm{sp} \) hybridized. This involves the combination of one s orbital with one p orbital to yield two equivalent linearly arranged \( \sigma \) oriented orbital hybrids. The absence of a third hybrid orbital in C2 accounts for its participation in two \( \pi \) bonds, which are essential for the molecule's linear structure.

In essence, the hybridization state of each carbon atom in an allene directs the spatial distribution of bonds around it, determining the molecule’s overall geometry and reactivity.

Molecular Geometry

The allene molecule showcases a fascinating aspect of chemical bonding—its molecular geometry. Geometry plays a crucial role in determining the molecule's physical and chemical properties. For \( \mathrm{H}_{2}C=C=CH_{2} \), the molecular geometry is linear; that is, the three carbon atoms all lie on a straight line.

The linear geometry of the allene molecule arises primarily from the \( \mathrm{sp} \) hybridization of the central carbon atom (C2), which aligns the two \( \sigma \) bonds in a straight line. The presence of perpendicular \( \pi \) bonds on plane, with respect to each other, as produced by the central carbon atom further enforces this linear arrangement.

The terminal carbons (C1 and C3), being \( \mathrm{sp}^{2} \) hybridized, would typically form a trigonal planar arrangement. However, in the context of the allene molecule, they conform to the straight-line structure enforced by the central carbon. The result is a molecule with a simple yet rigid framework which has direct implications on its reactivity. For instance, due to the linear structure and the rigidity conferred by the \( \pi \) bonds, allene can participate in unique chemical reactions.

Ultimately, analyzing molecular geometry helps in predicting the behavior of molecules in diverse chemical contexts, and allene is an excellent example of how electron sharing through sigma and pi bonds can lead to distinctive geometric formations.