Step 1: Concept
Hybridization is a concept in valence bond theory that describes the mixing of atomic orbitals to form new hybrid orbitals. In methane ($\text{CH}_4$), carbon forms four equivalent bonds with hydrogen atoms, which requires a specific type of orbital hybridization.
Step 2: Meaning
The hybridization state of an atom determines the number and types of hybrid orbitals formed. For example, $\text{sp}^3$ hybridization involves one $s$ orbital and three $p$ orbitals mixing to form four equivalent sp$^3$ hybrid orbitals.
Step 3: Analysis
In methane ($\text{CH}_4$), carbon has four valence electrons. To form four single bonds with hydrogen atoms, the carbon atom undergoes $\text{sp}^3$ hybridization. This process involves one $2s$ orbital and three $2p$ orbitals mixing to create four equivalent sp$^3$ hybrid orbitals.
Each of these sp$^3$ hybrid orbitals then overlaps with a hydrogen 1s orbital, forming four $\sigma$ bonds. The tetrahedral geometry around the carbon atom is a direct result of the equal overlap between the sp$^3$ hybrid orbitals and the hydrogen atoms.
The other options can be ruled out:
$\text{sp}^2$ hybridization would involve one $s$ orbital and two $p$ orbitals, resulting in three equivalent sp$^2$ hybrid orbitals. This is not sufficient to form four bonds.
$\text{sp}$ hybridization involves one $s$ orbital and one $p$ orbital, resulting in two equivalent sp hybrid orbitals. Again, this is insufficient for forming four bonds.
$\text{dsp}^2$ hybridization would involve one $d$ orbital and two $p$ orbitals, which is not typical for carbon atoms.
Step 4: Conclusion
The correct hybridization of carbon in methane ($\text{CH}_4$) is sp$^3$.
Final Answer: (A)