Figure P 43.18 is a model of a benzene molecule. All atoms...

Figure P 43.18 is a model of a benzene molecule. All atoms lie in a plane, and the carbon atoms $\left(m_{\mathrm{C}}=1.99 \times 10^{-26} \mathrm{kg}\right)$ form a regular hexagon, as do the hydrogen atoms $\left(m_{\mathrm{H}}=\right.1.67 \times 10^{-27} \mathrm{kg} ) .$ The carbon atoms are 0.110 nm apart center to center, and the adjacent carbon and hydrogen atoms are 0.100 nm apart center to center. (a) Calculate the moment of inertia of the molecule about an axis perpendicular to the plane of the paper through the center point $O.$ (b) Determine the allowed rotational energies about this axis.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It is calculated by taking the sum over all point masses of the product of the mass and the square of its distance from the axis of rotation (I = ? m·r²). In the context of molecular models, such as benzene, the positions of the atoms relative to the rotation axis are used in this summation to obtain the total moment of inertia.

Rigid Rotor Model

The rigid rotor model treats a rotating molecule as a set of fixed masses connected rigidly, allowing the calculation of rotational properties assuming the distances between the masses remain constant. This model is essential in molecular quantum mechanics, where it provides a simplified context for understanding how molecules rotate and for calculating the associated energy levels.

Rotational Energy Quantization

In quantum mechanics, the allowed energy states of a rotating system are quantized. For a rigid rotor, the rotational energy levels are determined by the angular momentum quantum number, and the energy typically takes the form E = (?²/2I) multiplied by a function of the quantum number. In the case of rotation about a fixed axis (as with a planar molecule), the energy levels can be expressed in terms of the square of an integer multiple of ?, demonstrating that only discrete rotational energy values are permitted.

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