The compound \(\mathrm{C}_{3} \mathrm{H}_{4}\) has two double bonds. Describe its bonding and geometry, using a valence bond approach.

Valence Bond Theory explains how atoms in a molecule bond with each other. It begins with the idea that electrons reside in atomic orbitals of individual atoms and that chemical bonds form when the atomic orbitals overlap. The extent of the overlap dictates the strength of the bond. According to this theory, an atom can use its valence electrons to form bonds, and these valence electrons are present in the hybridized orbitals.

In the example of \(\mathrm{C}_{3} \mathrm{H}_{4}\), carbon atoms undergo sp² hybridization to form three equivalent sp² hybridized orbitals. These hybrid orbitals are used to form sigma bonds through head-on overlaps with orbitals of other atoms, which are typically stronger than other bond types due to the direct overlap. Valence Bond Theory helps us understand the makeup of molecules, their shapes, and how they are bonded.

In the molecular world, two main types of covalent bonds exist: sigma (\(\sigma\)) and pi (\(\pi\)) bonds. Sigma bonds are the first bonds formed between two atoms. They have a cylindrical shape about the bond axis, arising from the head-on overlap of atomic orbitals, such as the s-sp² overlap in hydrogen-carbon bonds or sp²-sp² in carbon-carbon bonds.

Pi bonds, on the other hand, result from the side-to-side overlap of p orbitals that are not used in sigma bonding. In the double bond of \(\mathrm{C}_{3} \mathrm{H}_{4}\), for instance, after the sigma bond has formed, the remaining unhybridized p orbitals on the carbon atoms overlap to form a pi bond. This bond is less strong since it does not involve direct overlap along the bond axis and allows for the existence of double and triple bonds in compounds.

While Valence Bond Theory illustrates how individual atoms bond, molecular geometry describes the three-dimensional arrangement of atoms within a molecule. The geometry is influenced by the electron pairs, both bonding and non-bonding, surrounding an atom. In sp² hybridization seen in a molecule such as \(\mathrm{C}_{3} \mathrm{H}_{4}\), hybrid orbitals create a trigonal planar arrangement around each carbon atom.

The bond angles in a trigonal planar geometry are approximately 120 degrees, causing the atoms to spread out as far as they can from each other, which reduces electron repulsion. This arrangement also affects the overall shape of the molecule and can give the molecule a linear or slightly bent appearance, depending on the double bonds' orientation.