Question

In general, the high-temperature limit for the rotational partition function is appropriate for almost all molecules at temperatures above the boiling point. Hydrogen is an exception to this generality because the moment of inertia is small due to the small mass of H. Given this, other molecules with \(\mathrm{H}\) may also represent exceptions to this general rule. For example, methane \(\left(\mathrm{CH}_{4}\right)\) has relatively modest moments of inertia \(\left(I_{A}=I_{B}=I_{C}=5.31 \times 10^{-40} \mathrm{g} \mathrm{cm}^{2}\right)\). a. determine \(B_{A}, B_{B},\) and \(B_{C}\) for this molecule. b. Use the answer from part (a) to determine the rotational partition function. Is the high-temperature limit valid? and has a relatively low boiling point of \(T=112 \mathrm{K}\)

Short answer

The rotational constants for methane are: \(B_A = B_B = B_C = B \approx 1.56\ \mathrm{cm^{-1}}\). The high-temperature limit of the rotational partition function is not valid for methane, as it predicts a partition function of around 4.56, which is much smaller than that of other molecules at temperatures above their boiling point.

Step-by-step solution

a. Determine the rotational constants B_A, B_B, and B_C for methane.

In order to determine the rotational constants B_A, B_B, and B_C for methane, we will use the formula: \(B = \frac{h}{8\pi^2Ic}\), where \(h\) is the Planck constant, \(I\) is the moment of inertia, and \(c\) is the speed of light in vacuum. For methane, the moments of inertia are equal, \(I_A = I_B = I_C = I = 5.31 \times 10^{-40}\ \mathrm{g\ cm^2}\). Applying the formula, we get \(B_A = B_B = B_C = B = \frac{h}{8\pi^2Ic}\) Plug in the values of constants \(h\) and \(c\): \(B = \frac{6.626 \times 10^{-34}\ \mathrm{J\ s}}{8\pi^2 (5.31 \times 10^{-40}\ \mathrm{g\ cm^2})(2.998 \times 10^{10}\ \mathrm{cm\ s^{-1}})}\) Now, after calculating, we get: \(B_A = B_B = B_C = B \approx 1.56\ \mathrm{cm^{-1}}\)

b. Determine the rotational partition function and validate the high-temperature limit.

We'll use the high-temperature limit approximation for the rotational partition function: \(q_\text{rot} \approx T^{3/2}(\frac{2\pi I_{\mathrm{rms}}}{h})^{3/2}\) We have \(I_{\mathrm{rms}} = \sqrt{\frac{I_A^2 + I_B^2 + I_C^2}{3}} = \sqrt{\frac{3I^2}{3}} = I = 5.31 \times 10^{-40}\ \mathrm{g\ cm^2}\). Hence, the expression becomes: \(q_\text{rot} \approx T^{3/2}(\frac{2\pi I}{h})^{3/2}\) Now, we plug in the values given in the problem: \(q_\text{rot} \approx (112\ \mathrm{K})^{3/2}(\frac{2\pi (5.31 \times 10^{-40}\ \mathrm{g\ cm^2})}{6.626 \times 10^{-34}\ \mathrm{J\ s}})^{3/2}\) After calculating, we get: \(q_\text{rot} \approx 4.56\) The high-temperature limit of the rotational partition function is not valid for methane, as it predicts a partition function of around 4.56, which is much smaller than that of other molecules at temperatures above their boiling point. This is due to the small moments of inertia of the hydrogen atoms in methane, as mentioned in the problem statement.

Key concepts

Rotational Constants

Rotational constants, denoted typically as B, are critical for understanding a molecule's rotational motion. They are directly related to the molecule's moment of inertia, and they help us describe the energy levels associated with its rotation. In quantum mechanics, these constants are used to quantify the energy levels of a diatomic or polyatomic molecule within the rigid rotor approximation.

The formula to calculate the rotational constant is given by
\[ B = \frac{h}{8\pi^2Ic} \]
where B represents the rotational constant in units of inverse centimeters (cm^-1), h is Planck's constant, I is the moment of inertia, and c is the speed of light in a vacuum. For symmetrical molecules like methane (CH₄), the rotational constants are the same for all axes (B_A = B_B = B_C). Understanding rotational constants provides us with essential information about a molecule's rotational spectra and is invaluable for interpreting infrared and microwave spectroscopy data.

High-Temperature Limit

In the realm of statistical mechanics, the high-temperature limit refers to an approximation used to simplify calculations of a molecule's rotational partition function when the temperature is significantly high. This approximation assumes that because of the elevated temperature, the energy spacing between rotational levels is so small that they can effectively be treated as a continuum.

The high-temperature limit approximation for the rotational partition function q_rot is given by:
\[ q_\text{rot} \approx T^{3/2}\left(\frac{2\pi I_{\mathrm{rms}}}{h}\right)^{3/2} \]
where T is the temperature, I_rms is the root mean square moment of inertia, and h is Planck's constant. This approximation is usually valid for temperatures above the boiling point of the molecule and fails when the moment of inertia is small, as in the case of hydrogen within molecules. The validity of this approximation can affect the accuracy with which we predict properties like heat capacity and spectral lines.

Moment of Inertia

The moment of inertia, often symbolized as I, is a fundamental concept in physics that measures the extent to which an object resists rotational acceleration about a particular axis. It depends not only on the mass of the object but also on how that mass is distributed with respect to the axis of rotation. For a given molecule, the moment of inertia is crucial in determining its rotational kinetic energy and rotational constants.

In the context of molecular rotations, the moment of inertia is calculated as I = m*r², where m is the mass and r is the distance from the axis of rotation. For polyatomic molecules like methane (CH₄), calculating the moment of inertia involves considering the geometry and mass distribution of all atoms. The smaller the moment of inertia, the higher the rotational constant, which means a molecule with negligible mass, like hydrogen, will have considerably different rotational spectra and partition functions when compared to heavier molecules.