Vibrational Levels in Molecular Electronic State

AI Thread Summary
The discussion focuses on calculating the vibrational levels of the H2 molecule using the formula for vibrational energy levels. The user derives the spring constant (K) and attempts to find the maximum vibrational levels (n), initially calculating it to be 18. However, they reference an article stating that H2 has 30 vibrational levels, leading to confusion about their calculations. A key point raised is that treating vibrational motion as purely harmonic may not accurately reflect real molecular behavior, as molecular bonds are typically anharmonic. The conclusion suggests that obtaining a lower number of vibrational levels than the actual molecule is a positive indication of the calculations.
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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?
 
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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.
 
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