Finding Vibrational Spacing from Fluorescence and Absorption Spectre

[Reshi]

Homework Statement

The fluorescence spectrum of anthracene vapor in the figure shows peaks with maxima at 440 nm, 410 nm, 390 nm, and 370 nm, whereas the absorption spectrum has peaks at 360 nm, 345 nm, 330 nm, and 305 nm. Assuming that only one vibrational mode contributes to these progressions, determine the vibrational spacings in the ground and excited electronic states.

Homework Equations

B=h/8*pi^2*c*I where I is the moment of inertia.
nu=2B(J+1)

The Attempt at a Solution

I am not sure how to even start this problem. I was considering using the equations listed above to calculate the spacing but I don't know how to calculate I. I am lost and any help would be good. Thank you for reading.

 

[Simon Bridge]

Start from basics: what causes the absorption and florescence spectra?

 

[Reshi]

Spacing Between energy levels.
In a QM vibration model the spacing between levels is 2B. So if we know the wavelength a transition corresponds to we can find the spacing between the levels. There are two problems, I don't know this relation and I don't know how to calculate B for anthracene.

 

[Reshi]

Start from basics: what causes the absorption and florescence spectra?

Spacing Between energy levels.
In a QM vibration model the spacing between levels is 2B. So if we know the wavelength a transition corresponds to we can find the spacing between the levels. There are two problems, I don't know this relation and I don't know how to calculate B for anthracene.

 

[Simon Bridge]

Spacing Between energy levels.

How? What is the relationship between the spectra wavelengths and the spacing between the levels (and what is different between absorption and florescent spectra?)

In a QM vibration model the spacing between levels is 2B.

... isn't the spacing between levels also related to the wavelengths you have? Do you know anything else about the spacing of vibrational levels? i.e. how many vibrational modes are in the ground state?

 

[Reshi]

We are told to assume only one vibrational mode. I was wrong before. Since we assume one vibration mode we can approximate the energy level spacing as a particle in a box. The length of the box is 5micrometeres, as that is the length of the particle. I still don't know how to relate the wavelengths given, however.
This is a link to the particle in a box: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc2.html

 

[Simon Bridge]

You don't know how the spacing between energy levels gives rise to the wavelengths of the light absorbed or emmitted?

An electron makes a transition between two energy levels ... in florescence, a photon is released, in absorption, a photon supplies the needed energy.
The energy of a photon is related to it's wavelength. But if the same energy levels give rise to the absorption and emission spectra, then, shouldn't they be identical?

What is the difference between a vibrational mode and a vibrational spacing?

The idea, I think, is to use your understanding of energy levels to deduce the spacings from the data.
This will mean making educated guesses and checking/falsifying them.

Note:
Calculating the moment of inertia for anthracene would be pretty nasty - thought you could kinda do it in approximation: i.e. by treating the atoms as point masses in a fixed configuration and working out the principle rotation axes. But what kind of motion is a "moment of inertia" associated with?

 

[ehild]

Spacing Between energy levels.
In a QM vibration model the spacing between levels is 2B. So if we know the wavelength a transition corresponds to we can find the spacing between the levels. There are two problems, I don't know this relation and I don't know how to calculate B for anthracene.

2B is the spacing between the rotational levels. You need the vibrational energies, which are Ev=hf(v+0.5), with f the frequency of the vibrational mode.

The vibrational energies are superposed on the electronic energy. The possible electronic-vibrational energies of the molecule are E₀+hf₀(v+0.5) in the ground state and E_e+hf_e(w+0.5) in the excited electronic state. When absorbing or emitting an ultraviolet or visible photon, the molecule undergoes a transition between the ground state and the excited state. The energy of the photon is equal to the difference between the energy of the states.

In an absorption measurement, the gas is cool, its molecules are in the ground electronic state and also in the ground vibrational state, and absorbing a photon, they go into the excited electronic state and to the vibrational state with w=0, 1, 2, 3, 4, ...
When recording a fluorescence spectrum, the molecule is in its excited electronic state and vibrational ground state, and emitting a photon, it undergoes onto the ground electronic level and and any of the vibrational levels with v=0, 1, 2, 3, ...

You can calculate the energy of the photons from their wavelength. These energies are equal to the energy differences of the levels.


ehild