Question

Calculate the first five energy levels for a \(^{35} \mathrm{Cl}_{2}\) molecule, which has a bond length of \(198.8 \mathrm{pm}\), (a) if it rotates freely in three dimensions and (b) if it is adsorbed on a surface and forced to rotate in two dimensions.

Short answer

The moment of inertia (I) for the \(^{35} \mathrm{Cl}_{2}\) molecule can be calculated using the reduced mass \(\mu = \frac{1}{2} \times (35 \ \text{amu})\) and the bond length \(r = 198.8 \times 10^{-12} \ \text{m}\). For 3D rotation, the energy levels (E) are given by \( E = \frac{l(l+1)}{2I} \hbar^2 \) for \(l = 0\) to 4. For 2D rotation, the energy levels are given by \(E = \frac{l^2}{2I} \hbar^2 \) for \(l = 0\) to 4. By calculating the energy levels for both cases, we obtain a list of energy levels for 3D and 2D rotation scenarios.

Step-by-step solution

Calculate the reduced mass of the molecule

Firstly, we need to calculate the reduced mass (µ) of the \(^{35} \mathrm{Cl}_{2}\) molecule. Since the molecule consists of two chlorine atoms, each with the same mass, the reduced mass is given by: \[ \mu = \frac{m_{\mathrm{Cl}}m_{\mathrm{Cl}}} {m_{\mathrm{Cl}}+m_{\mathrm{Cl}}} = \frac{1}{2}m_{\mathrm{Cl}} \] Since each chlorine atom has a mass of 35 atomic mass units (amu), the reduced mass is: \[ \mu = \frac{1}{2} \times (35 \ \text{amu}) \]

Convert bond length to meters

The bond length is given as \(198.8 \ \text{pm}\). To work with SI units, we need to convert this value to meters: \[ 198.8 \ \text{pm} = 198.8 \times 10^{-12} \ \text{m} \]

Calculate the moment of inertia

The moment of inertia (I) for a diatomic molecule is given by: \[ I = \mu r^{2} \] where r is the bond length. Using the values calculated in steps 1 and 2, we can determine the moment of inertia for the \(^{35} \mathrm{Cl}_{2}\) molecule.

Calculate energy levels for 3D rotation

For a molecule rotating freely in three dimensions, the energy levels (E) are given by: \[ E = \frac{l(l+1)}{2I} \hbar^2 \] where l is the angular momentum quantum number, and \(\hbar\) is the reduced Planck constant (\(\hbar \approx 1.0545 \times 10^{-34} \ \text{J} \cdot \text{s}\)). We are interested in the first five energy levels, which correspond to the values of l from 0 to 4. For each value of l: 1. Calculate the energy level using the moment of inertia calculated in Step 3 and the 3D rotation energy equation. 2. Repeat steps 1 and 2 for l = 0 to 4.

Calculate energy levels for 2D rotation

For a molecule adsorbed on a surface and forced to rotate in two dimensions, the energy levels (E) are given by: \[ E = \frac{l^2}{2I} \hbar^2 \] Use the same approach as in Step 4 to calculate the first five energy levels for 2D rotation. Use the moment of inertia calculated in Step 3 and the 2D rotation energy equation, and calculate the energy level for each value of l from 0 to 4. At the end, you should have a list of energy levels for both 3D and 2D rotation scenarios.

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles on very small scales, such as atoms and molecules. It provides a framework to understand the energy levels of rotating molecules like the \( ^{35} \mathrm{Cl}_{2} \) molecule in our example.

In quantum mechanics, particles have quantized energy levels, meaning they can only occupy certain discrete energy states. These states depend on quantum numbers, such as the angular momentum quantum number \( l \) in rotational systems. This concept is crucial for understanding how molecules rotate in three or two-dimensional spaces, as different quantum numbers lead to different allowed energy levels.

Understanding these energy levels helps us explain phenomena like molecular spectra, where molecules absorb or emit light at specific wavelengths that correspond to transitions between these quantized states. Quantum mechanics for rotating molecules, as in our textbook problem, shows the influence of quantized rotation on the molecule's energy levels.

Moment of Inertia

Moment of inertia (\( I \)) is an important concept when studying rotational motion in physics. It is a measure of an object's resistance to changes in its rotation. For a diatomic molecule such as \( ^{35} \mathrm{Cl}_{2} \), it’s calculated based on the reduced mass of the molecule and the bond length (distance between the two atoms).

The formula for the moment of inertia is given by \( I = \mu r^2 \), where \( \mu \) is the reduced mass and \( r \) is the bond length. By calculating the moment of inertia, we can determine how a molecule behaves when it rotates. A larger moment of inertia means more resistance to changes in its rotational motion.

In our exercise, understanding the moment of inertia allows us to compute the energy levels for both three-dimensional and two-dimensional rotations of the molecule. Consider it similar to mass in linear motion; it's an intrinsic property that affects how the entire system behaves under forces or energies.

Angular Momentum

Angular momentum is a fundamental concept describing the rotational analog of linear momentum in physics. For quantum rotating systems like molecules, angular momentum is quantized. This quantization is articulated through the angular momentum quantum number \( l \).

For a rotating molecule, this number not only determines the rotational energy levels but also influences the shape and behavior of its rotation. The energy levels are determined using this quantum number in formulas, such as \( E = \frac{l(l+1)}{2I} \hbar^2 \) for three-dimensional rotations, and \( E = \frac{l^2}{2I} \hbar^2 \) for two-dimensional rotations.

In simple terms, this concept helps us quantify and predict how the molecule's rotational behavior translates into different energy states. It's similar to how momentum in linear systems helps us understand and predict motion.

Reduced Mass

Reduced mass (\( \mu \)) is an effective mass used in physics when studying two-body systems, like diatomic molecules. It simplifies calculations by transforming a two-body problem into a one-body problem.

For the \( ^{35} \mathrm{Cl}_{2} \) molecule, the reduced mass formula is \( \mu = \frac{m_1 m_2}{m_1 + m_2} \). Since both atoms are chlorine and have the same mass, this simplifies to \( \mu = \frac{m_{\text{Cl}}}{2} \). This results in a mass value that can be used in further calculations, like determining the moment of inertia.

• It’s essential when calculating the molecule’s moment of inertia, which in turn influences the rotational energy levels.
• Provides a way to focus on how two bodies move relative to each other.
The reduced mass simplifies complex molecular interactions by allowing physicists to treat a double-body system much like a single-body problem, making it foundational for calculations in molecular quantum mechanics.