Explain the hybridization and bond structure of the ethane $\left( C_2H_6 \right)$ molecule.

(N/A) $sp^3$ hybridization of carbon: The ground state of carbon is $[He] 2s^2 2p^2$. One electron from the $2s$ orbital is promoted to an empty $2p$ orbital to form the excited state of carbon $(C^*)$. The electronic configuration of $C^*$ is $[He] 2s^1 2p_x^1 2p_y^1 2p_z^1$.
The four half-filled orbitals of the excited carbon atom undergo $sp^3$ hybridization to form four equivalent $sp^3$ hybrid orbitals. These orbitals are arranged in a tetrahedral geometry with a bond angle of $109.5^{\circ}$ to minimize inter-electronic repulsion.
Bonding in $C_2H_6$: Each carbon atom uses its four $sp^3$ hybrid orbitals for bonding.
One $sp^3$ orbital from each carbon atom overlaps axially to form a $C-C$ sigma $(\sigma)$ bond. The remaining three $sp^3$ hybrid orbitals on each carbon atom overlap axially with the $1s$ orbital of three hydrogen atoms to form six $C-H$ sigma $(\sigma)$ bonds.
Thus,the ethane molecule contains a total of seven sigma $(\sigma)$ bonds ($1$ $C-C$ and $6$ $C-H$) and exhibits a tetrahedral geometry around each carbon atom.