Hybridization States of Carbon Atoms in Allene Molecule

To determine the hybridization states of the carbon atoms in the allene molecule ($\text{C}_3\text{H}_4$), we analyze its chemical structure:

$\text{H}_2\text{C}_{(1)}=\text{C}_{(2)}=\text{C}_{(3)}\text{H}_2$

The hybridization state of a carbon atom depends on the number of $\sigma$-bonds (or steric number) formed by that atom:

1. Terminal Carbon Atoms ($\text{C}_1$ and $\text{C}_3$):

   • Each terminal carbon atom is bonded to two hydrogen atoms via single ($\sigma$) bonds and to the central carbon atom via one $\sigma$-bond and one $\pi$-bond.
   • Total number of $\sigma$-bonds formed by $\text{C}_1$ (or $\text{C}_3$) $= 2 + 1 = 3$.
   • A steric number of $3$ corresponds to $sp^2$ hybridization.

2. Central Carbon Atom ($\text{C}_2$):

   • The central carbon atom is bonded to each of the two terminal carbon atoms through double bonds (each consisting of one $\sigma$-bond and one $\pi$-bond).
   • Total number of $\sigma$-bonds formed by $\text{C}_2 = 1 + 1 = 2$.
   • A steric number of $2$ corresponds to $sp$ hybridization.

Therefore, the types of hybridization present for the carbon atoms in allene are $sp$ and $sp^2$.

Correct Option: B