Explain the hybridization of carbon and the bond structure in the ethyne $(C_2H_2)$ molecule.

(N/A) In ethyne $(C_2H_2)$,each carbon atom undergoes $sp$ hybridization.
$1$. $sp$ Hybridization of Carbon:
In the excited state,the carbon atom has one $2s$ orbital and three half-filled $2p$ orbitals $(2p_x, 2p_y, 2p_z)$. The $2s$ and $2p_z$ orbitals undergo $sp$ hybridization to form two equivalent $sp$ hybrid orbitals,which are oriented linearly at an angle of $180^{\circ}$. The other two $2p$ orbitals ($2p_x$ and $2p_y$) remain unhybridized and are perpendicular to the axis of the $sp$ hybrid orbitals.
$2$. Bond Structure in Ethyne:
- The two carbon atoms are linked by a $C-C$ sigma $(\sigma)$ bond formed by the axial overlap of one $sp$ hybrid orbital from each carbon atom.
- The remaining $sp$ hybrid orbital on each carbon atom overlaps axially with the half-filled $1s$ orbital of a hydrogen atom to form a $C-H$ $\sigma$ bond.
- The unhybridized $2p_x$ and $2p_y$ orbitals on each carbon atom overlap laterally (sideways) with the corresponding $2p_x$ and $2p_y$ orbitals of the other carbon atom,forming two $\pi$ bonds.
- Thus,the ethyne molecule has a linear geometry with a triple bond $(C \equiv C)$ between the two carbon atoms.