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Chapter 14: Problem 22

Calculate the values of the first five rotational energy levels of ethane, \(\mathrm{CH}_{3} \mathrm{CH}_{3}\), assuming it is in its energetically minimal staggered conformation. The moments of inertia for ethane are \(1.075 \times 10^{-46} \mathrm{~kg}-\mathrm{m}^{2}, 4.200 \times 10^{-46} \mathrm{~kg}-\mathrm{m}^{2}\), and \(4.200 \times 10^{-46} \mathrm{~kg} \cdot \mathrm{m}^{2}\).

Short Answer

Expert verified
First five energy levels: 0, \(1.036 \times 10^{-23}\) J, \(3.108 \times 10^{-23}\) J, \(6.18 \times 10^{-23}\) J, \(10.352 \times 10^{-23}\) J.

Step by step solution

01

Identify the Rotational Constant

The rotational energy levels for a symmetric top molecule like ethane can be calculated using the formula for the energy levels of a rigid rotor: \(E(J) = \frac{\hbar^2}{2I} J(J+1)\), where \(\hbar\) is the reduced Planck's constant \(1.0545718 \times 10^{-34} \text{ J s}\), \(I\) is the moment of inertia, and \(J\) is the rotational quantum number. For ethane, the moment of inertia to consider depends on the quantum state it lies in.
02

Calculate Energy for J=0

Let's calculate the first energy level with \(J=0\). Use the smallest moment of inertia \(I_A = 1.075 \times 10^{-46} \text{ kg m}^2\). Substitute into the energy formula: \(E(0) = \frac{\hbar^2}{2 \times 1.075 \times 10^{-46}} \times 0 \times 1 = 0 \text{ J}\). The energy at \(J=0\) is zero as expected.
03

Calculate Energy for J=1

For \(J=1\), again use \(I_A = 1.075 \times 10^{-46} \text{ kg m}^2\). Calculate \(E(1)\): \(E(1) = \frac{(1.0545718 \times 10^{-34})^2}{2 \times 1.075 \times 10^{-46}} \times 1 \times 2\). Simplifying gives \(E(1) \approx 1.036 \times 10^{-23} \text{ J}\).
04

Calculate Energy for J=2

For \(J=2\), use the same moment of inertia \(I_A\). Calculate \(E(2)\): \(E(2) = \frac{(1.0545718 \times 10^{-34})^2}{2 \times 1.075 \times 10^{-46}} \times 2 \times 3\). Simplifying gives \(E(2) \approx 3.108 \times 10^{-23} \text{ J}\).
05

Calculate Energy for J=3

For \(J=3\), calculate \(E(3)\): \(E(3) = \frac{(1.0545718 \times 10^{-34})^2}{2 \times 1.075 \times 10^{-46}} \times 3 \times 4\). Simplifying gives \(E(3) \approx 6.18 \times 10^{-23} \text{ J}\).
06

Calculate Energy for J=4

For \(J=4\), calculate \(E(4)\): \(E(4) = \frac{(1.0545718 \times 10^{-34})^2}{2 \times 1.075 \times 10^{-46}} \times 4 \times 5\). Simplifying gives \(E(4) \approx 10.352 \times 10^{-23} \text{ J}\).
07

Final Step: List the Values

Therefore, the first five rotational energy levels for ethane in its minimal conformation are: \(E(0) = 0\), \(E(1) \approx 1.036 \ldots \), \(E(2) \approx 3.108 \times 10^{-23}\), \(E(3) \approx 6.18 \times 10^{-23}\), and \(E(4) \approx 10.352 \times 10^{-23}\) J.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Top Molecule
A symmetric top molecule is a type of rigid rotor that has two or three equal moments of inertia. In the case of ethane, the molecule exhibits symmetry around one of its axes, resulting in two moments of inertia being equal. This symmetry simplifies the calculation of energy levels in rotational spectroscopy because it assumes the molecule spins about an axis that goes through the center of mass. For ethane, the moments of inertia are given as
  • 1.075 x 10-46 办驳路尘2
  • 4.200 x 10-46 办驳路尘2
  • 4.200 x 10-46 办驳路尘2
This 'symmetric top' model makes it easier to analyze rotational energy levels by using the reduced, symmetric nature of the rotational properties.
Moment of Inertia
The moment of inertia ( I ) is a key property when considering the rotational energy levels of molecules like ethane. It describes how mass is distributed relative to the axis of rotation, effectively acting as a measure of rotational resistance. Ethane has three moments of inertia due to its molecular structure. Calculating the rotational energy often involves choosing the smallest of these moments when considering different rotational states for simplicity.
  • Smaller moment: 1.075 x 10-46 办驳路尘2
  • Larger equal moments: 4.200 x 10-46 办驳路尘2
Using the particular moment of inertia helps in computing the exact energy levels by plugging values into the formulas. This distinction is essential for using correct parameters to derive precise rotational energy levels.
Rotational Constant
The rotational constant (B) is vital in analyzing rotational energy levels. It is directly related to the moment of inertia and inversely proportional to it. The rotational constant can be calculated using the formula:\[ B = \frac{\hbar^2}{2I} \]where \(\hbar\)is the reduced Planck's constant (1.0545718 x 10-34 J路s). Calculating \(B\)permits the prediction of energy levels for different values of the rotational quantum number (J).Ethane's specific rotational energies depend heavily on this constant. By finding \(B\)from the moment of inertia, it allows us to determine energy levels according to the formula:\[ E(J) = B J(J+1) \]Easy calculations are dependent on appropriate usage of these constants.
Rigid Rotor Model
The rigid rotor model is a simplifying assumption used widely in rotational spectroscopy to describe molecules in rotational motion. In this model, molecules are treated as rigid objects with fixed distances between atoms, implying that the atoms do not move relative to each other during rotation. This model is particularly useful for small or symmetric molecules like ethane. By assuming the molecule remains "rigid," calculations become manageable and allow exploration into rotational energy levels using established formulas. The rigid rotor model applies straightforward physics principles to compute rotational energies. These calculations provide insight into the molecular structure based on how they rotate, assisting in understanding interactions and dynamics within molecules.

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