The correct answer is option 'C', which is 48.

Naphthalene is a hydrocarbon compound with the molecular formula C10H8. To determine the number of normal modes of vibration in naphthalene, we need to consider its molecular structure and the number of degrees of freedom.

1. Molecular Structure:
Naphthalene consists of two benzene rings fused together. Each benzene ring contains six carbon atoms and six hydrogen atoms. The carbon atoms are sp2 hybridized and form a planar hexagonal structure with alternating single and double bonds. The two benzene rings are connected by a single bond between two carbon atoms.

2. Degrees of Freedom:
The degrees of freedom refer to the number of independent ways in which a molecule can move or vibrate. In the case of naphthalene, there are three types of motion to consider:

a) Translational motion: Naphthalene can move as a whole in three dimensions. Therefore, it has three translational degrees of freedom.

b) Rotational motion: Naphthalene can rotate around its center of mass in three different axes. Hence, it has three rotational degrees of freedom.

c) Vibrational motion: Naphthalene can vibrate internally, and these vibrations can be described as normal modes. To determine the number of vibrational degrees of freedom, we can use the formula:

Degrees of freedom = 3N - 6

Where N is the total number of atoms in the molecule.

3. Calculation:
Naphthalene has a total of 10 carbon atoms and 8 hydrogen atoms, giving a total of 18 atoms. Therefore, the number of vibrational degrees of freedom can be calculated as follows:

Degrees of freedom = 3(18) - 6 = 54

However, we need to consider that naphthalene is a planar molecule, and it exhibits some symmetry. The vibrational modes can be classified into different symmetry classes based on the symmetry operations of the molecule. In the case of naphthalene, there are certain vibrational modes that are degenerate due to the symmetry of the molecule. These degenerate modes should be counted only once.

4. Symmetry Considerations:
Naphthalene belongs to the D2h point group, which has several symmetry elements, including a C2 rotation axis perpendicular to the plane of the molecule and two mirror planes. The vibrational modes can be classified into the following symmetry classes:

a) Ag: Vibrations symmetric with respect to all symmetry operations.
b) B1g: Vibrations symmetric with respect to the C2 rotation axis but antisymmetric with respect to the mirror planes.
c) B2g: Vibrations symmetric with respect to one mirror plane but antisymmetric with respect to the other.
d) B3g: Vibrations symmetric with respect to the two mirror planes but antisymmetric with respect to the C2 rotation axis.
e) Au: Vibrations antisymmetric with respect to all symmetry operations.
f) Bu: Vibrations antisymmetric with respect to the C2 rotation axis but symmetric with respect to the mirror planes.

Based on the symmetry considerations, the number of normal modes of vibration in naphthalene is reduced. The final count is:

Number of normal modes = Number of unique symmetry classes = 48

Therefore, the correct answer is option 'C' (