Introduction:
The nonlinear C60 molecule, also known as buckminsterfullerene, is composed of 60 carbon atoms arranged in a spherical shape. It has a unique structure and exhibits various vibrational modes. Vibrational modes represent the different ways in which the atoms within a molecule can vibrate.

Determination of vibrational modes:
To determine the vibrational modes of a molecule, we need to consider its degrees of freedom. In a nonlinear molecule like C60, the degrees of freedom can be calculated using the formula:

Degrees of freedom = 3n - 6

Where n is the number of atoms in the molecule. In the case of C60, there are 60 carbon atoms, so the degrees of freedom would be:

Degrees of freedom = 3(60) - 6 = 174

This means that there are 174 vibrational modes in the C60 molecule.

Explanation:
To understand why there are 174 vibrational modes in the C60 molecule, we need to consider its structure and the types of motions that the atoms can undergo.

Types of Vibrational Modes:
The vibrational modes can be classified into three types:

1. Translational modes: These modes involve the movement of the entire molecule in space. However, for a molecule like C60, which is a large and heavy molecule, the translational modes can be considered negligible.

2. Rotational modes: These modes involve the rotation of the molecule around its center of mass. In the case of C60, the molecule has a spherical shape, and it exhibits various rotational modes. However, since C60 is a nonlinear molecule, it also exhibits vibrational-rotational coupling, which leads to additional vibrational modes.

3. Vibrational modes: These modes involve the vibration of the atoms within the molecule. The atoms can vibrate in different ways, such as stretching, bending, and twisting motions. In the case of C60, the presence of 60 carbon atoms arranged in a spherical shape gives rise to numerous vibrational modes.

Degrees of Freedom:
The degrees of freedom of a molecule represent the number of independent ways in which the molecule can move or vibrate. For a nonlinear molecule like C60, the degrees of freedom can be calculated using the formula mentioned earlier:

Degrees of freedom = 3n - 6

Where n is the number of atoms in the molecule. In the case of C60, with 60 carbon atoms, the calculation becomes:

Degrees of freedom = 3(60) - 6 = 174

This means that there are 174 independent vibrational modes in the C60 molecule.

Conclusion:
In conclusion, the nonlinear C60 molecule exhibits 174 vibrational modes. These modes arise due to the various ways in which the carbon atoms can vibrate within the spherical structure of the molecule. The calculation of degrees of freedom using the formula 3n - 6 helps determine the total number of vibrational modes.