Question

Calculate the rotational partition function for \(\mathrm{SO}_{2}\) at \(298 \mathrm{K}\) where \(B_{A}=2.03 \mathrm{cm}^{-1}, B_{B}=0.344 \mathrm{cm}^{-1},\) and\(B_{C}=0.293 \mathrm{cm}^{-1}\)

Short answer

The rotational partition function (\(q_{rot}\)) for \(\mathrm{SO}_{2}\) at \(298 \mathrm{K}\) can be calculated using the given rotational constants: \(B_{A}=2.03\mathrm{cm}^{-1}\), \(B_{B}=0.344\mathrm{cm}^{-1}\), and \(B_{C}=0.293\mathrm{cm}^{-1}\). For a non-linear molecule like \(\mathrm{SO}_{2}\), the partition function is given by: \[q_{rot}=\frac{\pi^{\frac{1}{2}} (kT)^{\frac{3}{2}}}{\sigma\sqrt{B_{A}B_{B}B_{C}}}\] Using the symmetry number (\(\sigma\)) of 2, and converting the rotational constants to joules, plug in all values and compute to find that the rotational partition function for \(\mathrm{SO}_{2}\) at \(298\mathrm{K}\) is approximately 10.88.

Step-by-step solution

Understand the Rotational Partition Function

The rotational partition function (\(q_{rot}\)) is a measure of the number of accessible rotational energy levels for a molecule at a given temperature. For a linear molecule, the partition function is given by the following equation: \[q_{rot} = \frac{T}{\sigma \Theta_{rot}},\] where \(\sigma\) is the symmetry number, \(\Theta_{rot}\) is the characteristic rotational temperature, and \(T\) is the temperature. For a non-linear molecule, the partition function is given by a slightly more sophisticated equation: \[q_{rot}=\frac{\pi^{\frac{1}{2}} (kT)^{\frac{3}{2}}}{\sigma\sqrt{B_{A}B_{B}B_{C}}}\] Since \(\mathrm{SO}_{2}\) is a non-linear molecule, we will use this second equation to calculate its partition function.

Find the Symmetry Number

The symmetry number (\(\sigma\)) is a factor that accounts for the indistinguishability of equivalent orientations of the molecule due to symmetry. In the case of \(\mathrm{SO}_{2}\), the symmetry number is found to be 2.

Calculate the partition function

Given that \(T=298\mathrm{K}\), \(B_{A}=2.03\mathrm{cm}^{-1}\), \(B_{B}=0.344\mathrm{cm}^{-1}\), and \(B_{C}=0.293\mathrm{cm}^{-1}\), we can plug these values into the partition function equation for non-linear molecules: \[q_{rot}=\frac{\pi^{\frac{1}{2}} (kT)^{\frac{3}{2}}}{\sigma\sqrt{B_{A}B_{B}B_{C}}}\] Converting the rotational constants from \(\mathrm{cm}^{-1}\) to \(\mathrm{J}\) by multiplying by the energy conversion factor (1 cm⁻¹ = 1.98630 x 10⁻²³ J) gives us \(B_{A}=3.5296\times{10}^{-22}\mathrm{J}\), \(B_{B}=6.0062\times{10}^{-24}\mathrm{J}\), and \(B_{C}=5.1428\times{10}^{-24}\mathrm{J}\). Using the Boltzmann constant, \(k=1.380649\times10^{-23}\mathrm{J\cdot K}^{-1}\), and plugging in the values, we have: \[q_{rot}=\frac{\pi^{\frac{1}{2}} (1.380649\times10^{-23}\cdot298)^{\frac{3}{2}}}{2\sqrt{3.5296\times{10}^{-22}\cdot6.0062\times{10}^{-24}\cdot5.1428\times{10}^{-24}}}\] Now we can compute the values: \[q_{rot}=\frac{\pi^{\frac{1}{2}} (4.1076\times10^{-21})^{\frac{3}{2}}}{2\sqrt{1.0904\times{10}^{-69}}}\]

Final result

After evaluating the equation, we obtain: \[q_{rot}\approx 10.88\] The rotational partition function for \(\mathrm{SO}_{2}\) at \(298\mathrm{K}\) is approximately 10.88.

Key concepts

Rotational Energy Levels

In understanding the rotational partition function, it's important to first grasp the concept of rotational energy levels. These energy levels are the discrete states that molecules can take based on their rotation. Imagine a molecule spinning in space, and each way it spins corresponds to a different energy state.

For a molecule, the rotational energy levels are quantized, which means they can only take on certain specific values. This is similar to the idea of rungs on a ladder, where a molecule can 'stand' only on certain steps, not in between. These levels become important when calculating the partition function because they determine how many states a molecule can occupy at a given temperature.

• The energy associated with these levels depends on the molecule's moment of inertia, which is a measure of how "mass is distributed" around the rotation axis.
• Temperature also plays a big role in determining which energy levels are accessible. At higher temperatures, molecules have more kinetic energy to reach higher energy levels.
• The Boltzmann distribution helps in describing the likelihood of a molecule occupying a certain energy level.

Non-linear Molecules

The molecule \(\mathrm{SO}_{2}\) mentioned in the exercise is a non-linear molecule. But what does that mean? In molecular chemistry, non-linear molecules have atoms that are not arranged in a straight line. This complicates their rotational behavior compared to linear molecules, which is why non-linear molecules use a different equation for their rotational partition function.

Non-linear molecules have three different moments of inertia due to their unique structure, as opposed to linear molecules which have only one. This affects how they rotate and interact with energy.

• The rotational constants \(B_{A}, B_{B}, B_{C}\) given in the problem are related to these moments of inertia and are crucial for calculating the partition function.
• Each rotational constant corresponds to a different axis along which the molecule can rotate, hence explaining the equation’s dependence on \sqrt{B_A B_B B_C}\ in the partition function.
• Understanding the structure of non-linear molecules helps in visualizing how molecular symmetry and energy levels come into play.

Symmetry Number

The symmetry number \(\sigma\) is another key concept in calculating the rotational partition function. It's essentially a measure of how symmetrical a molecule is. This affects how many unique ways it can be oriented in space without looking the same.

The symmetry number corrects for identical orientations. For example, \(\mathrm{SO}_{2}\) has a symmetry number of 2, which means there are two ways you can rotate it in space to get an indistinguishable configuration.

• A higher symmetry number means fewer unique orientations, which lowers the partition function because there are fewer distinct states available for the molecule to occupy.
• For simple diatomic molecules, the symmetry number could be 1, but larger or more symmetrical molecules will have a larger symmetry number.
• Assessing symmetry requires a good understanding of molecular geometry and visualizing the rotations that preserve the molecule's appearance.

Boltzmann Constant

To fully appreciate the role of the Boltzmann constant \(k\) in the rotational partition function, we need to understand what it represents. This constant is fundamental in statistical mechanics, relating the average kinetic energy of particles in a gas with the temperature of the gas.

In the equation used for the rotational partition function, the Boltzmann constant adjusts the energy scale so that temperature can be incorporated properly into calculations in terms of energy.

• The constant \(k = 1.380649 \times 10^{-23} \, \text{J\cdot K}^{-1}\) provides a bridge between temperature in Kelvin and energy in Joules.
• It ensures that when temperature \(T\) is included in the equations, it correctly evaluates how energy states are populated according to the Boltzmann distribution.
• Understanding the Boltzmann constant helps link macroscopic measurements (like temperature) to microscopic phenomena (like rotational energy levels).