# Vibrational Levels in Molecular Electronic State

• guyvsdcsniper
In summary, the conversation discusses the calculation of the vibrational levels of a molecule using various equations and the assumption that the motion can be treated as purely simple harmonic. The result is compared to a real molecule and it is found that the assumption leads to a much smaller number of levels than the real molecule has. This indicates that the assumption is not accurate and a different approach is needed to accurately calculate the vibrational levels of a molecule.

#### guyvsdcsniper

Homework Statement
In a diatomic molecule k (N/m) may be estimated from the dissociation energy and bond length by D.A. Let the bond length = 2*10^-10m and the dis. energy = 4.806*10^-19J.

Assuming the vibrational motion can be treated as purely simple harmonic, how many
vibrational levels would you expect to associated with this molecular electronic state if
the molecule were (a) hydrogen, and (b) oxygen ?
Relevant Equations
E=hw(n+1/2)
##K = \frac{N}{m} = \frac{3eV}{bondlength^2} = \frac{4.806*10^-19 J}{(2*10^-10)^2} = 12.015##

Then I know that ##H = \frac{1}{2}mωx^2 ## where ## k = mω^2 ## and also ##H=ħω(n+\frac{1}{2}) ##

Therefore, ## \frac{1}{2}kx^2 = ħω(n+\frac{1}{2})##

Solving for n, ## n = \frac{1/2kx^2}{ħω} - \frac{1}{2}##

for ω I used the fact the ##ω = \sqrt(\frac{k}{m}) ## where m was the reduced mass of H2.

So n should tell me the max vibrational levels. When plugging in all my parameters I get 18. I researched online and found this article, https://w.astro.berkeley.edu/~ay216/05/NOTES/Lecture18.pdf , which states
"c. The Electronic Levels - The ground state is X 1Σ+g because it has the quantum numbers: S = 0, I = 0, Λ = 0, & J = 0. It has a manifold of 30 vibrational levels, each with an infinite number of rotational states"

So I am thinking n must be 30 for my problem since the article states H2 has 30 vibrational levels.

I have found when plugging ~19 for K i get n=30

Any help with this problem?I feel like my k is correct since it checks out with the dimensional analysis, so maybe I am wrong somewhere else?

guyvsdcsniper said:
Assuming the vibrational motion can be treated as purely simple harmonic, how manySo I am thinking n must be 30 for my problem since the article states H2 has 30 vibrational levels.
The assumption you are asked to make will lead to a strong departure from reality (molecular bonds are anharmonic). Comparison to a real molecule is futile.

I haven't checked your calculation, but getting a smaller number of vibrational levels that the real molecule is a good sign.

guyvsdcsniper

## 1. What are vibrational levels in molecular electronic state?

Vibrational levels in molecular electronic state refer to the different energy levels that a molecule can occupy in its electronic state. These levels are determined by the vibrational motion of the molecule's atoms, which can be visualized as the stretching and bending of chemical bonds.

## 2. How are vibrational levels in molecular electronic state measured?

Vibrational levels in molecular electronic state are typically measured using spectroscopic techniques, such as infrared (IR) or Raman spectroscopy. These methods involve shining a beam of light onto a sample and measuring the absorption or scattering of specific wavelengths, which can provide information about the molecule's vibrational energy levels.

## 3. What factors influence the number of vibrational levels in a molecule's electronic state?

The number of vibrational levels in a molecule's electronic state is influenced by several factors, including the number of atoms in the molecule, the types of chemical bonds present, and the overall shape and symmetry of the molecule. Generally, larger and more complex molecules will have a higher number of vibrational levels.

## 4. How do vibrational levels in molecular electronic state affect a molecule's properties?

The vibrational levels in a molecule's electronic state can significantly affect its properties, such as its reactivity and stability. Vibrational energy can be transferred between different levels, leading to changes in the molecule's shape and bond strength. This can impact how the molecule interacts with other molecules and its overall behavior.

## 5. Can vibrational levels in molecular electronic state be manipulated?

Yes, vibrational levels in molecular electronic state can be manipulated through various methods, such as changing the temperature or applying an external force. This can result in changes in the molecule's properties and behavior, making it a useful tool in fields such as chemical synthesis and materials science.

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