Rotation Spectrum of a diatomic molecule (QM)

• joker_900
In summary, the conversation discusses the rotation spectrum of 12C16O and its relationship to the intra-nuclear distance and spring force constant. The classical mechanics equations for distance of O from the centre of mass and the molecule's moment of inertia are used to derive the frequency of the emitted photon. A correction to the formula is made, leading to the correct value.
joker_900

Homework Statement

In the rotation spectrum of 12C16O the line arising from the transition l = 4 -> 3 is at 461.04077GHz, while that arising from l = 36 -> 35 is at 4115.6055GHz. Show from these data that in a non-rotating CO molecule the intra-nuclear distance is s ~ 0.113 nm, and that the electrons provide a spring between the nuclei that has force constant ~1904Nm−1. Hence show that the vibrational frequency of CO should lie near 6.47×10^13Hz (measured value 6.43 × 1013 Hz). Hint: show from classical mechanics that the distance of O from the centre of mass is (3/7) s and that the molecule’s moment of inertia is (48/7) ms^2. Recall also the classical relation L = Iw.

Homework Equations

f = j h/((2pi)^2*I)

Where f is the frequency of the emitted photon and j(j+1) is the eigenvalue of J^2 (I think j is the same as l in this question?)

The Attempt at a Solution

I can derive the two classical bits in the hint and the above formula. I then tried taking one of the frequencies given, working out I using the above formula, and plugging it into the second classical expression to get s. It didn't work!

Thanks

It's not (2pi)^2 on the bottom, it's just 2pi. Plugging in the numbers with the modified formula gives the correct value.

1. What is the rotation spectrum of a diatomic molecule?

The rotation spectrum of a diatomic molecule is the spectrum of electromagnetic radiation that is emitted or absorbed by the molecule as it rotates. This spectrum is unique to each molecule and can provide valuable information about its structure and properties.

2. How is the rotation spectrum of a diatomic molecule measured?

The rotation spectrum of a diatomic molecule is typically measured using a technique called rotational spectroscopy. This involves passing a beam of electromagnetic radiation through a sample of the molecule and measuring the absorption or emission of specific wavelengths of light. The resulting spectrum can then be analyzed to determine the rotational energy levels of the molecule.

3. What factors affect the rotation spectrum of a diatomic molecule?

The rotation spectrum of a diatomic molecule is primarily determined by its moment of inertia, which is a measure of its mass distribution. Other factors that can affect the spectrum include the strength of the bond between the two atoms, the presence of any nearby electric or magnetic fields, and the temperature and pressure of the environment in which the molecule is located.

4. How does the rotation spectrum of a diatomic molecule differ from the vibration spectrum?

The rotation spectrum of a diatomic molecule is related to its rotational energy levels, while the vibration spectrum is related to its vibrational energy levels. In other words, the rotation spectrum measures the energy required to rotate the molecule, while the vibration spectrum measures the energy required to make the atoms within the molecule vibrate.

5. What applications does the rotation spectrum of a diatomic molecule have?

The rotation spectrum of a diatomic molecule has a wide range of applications in chemistry, physics, and engineering. It can be used to identify and characterize molecules, study chemical reactions and intermolecular interactions, and even determine the composition and temperature of distant objects in space. It is an important tool in fields such as atmospheric science, astrochemistry, and materials science.

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