Modern Physics Lab: Hydrogen Wavelengths & Energy Levels

[leonardthecow]

This question is based on a modern physics lab I'm working on, and it's conceptually killing me.

1. Homework Statement
For hydrogen, compare your measured wavelengths to the predicted wavelengths for hydrogen. Assuming the lower level is the same for all the lines you observed in the hydrogen spectrum, make an energy level diagram for hydrogen. Label the energy axis in electron volts.


2. Homework Equations /attempt at a solution
So I have measured wavelengths for hydrogen for a few orders of diffraction that come from exposing light emitted from hydrogen to a diffraction grating. What I'm really stumped on is how to relate the measured wavelengths for each order of diffraction to the "predicted wavelengths" made by the Bohr model. I know that Bragg's Law solved for the wavelengths give me the wavelengths for the lines for each order of diffraction, and I know that setting the difference between energy levels for hydrogen equal to the energy of a photon emitted of that energy difference can give me the wavelength for that photon. But how do these wavelengths relate to one another?

Thanks in advance

 

[ehild]

You need to study Bohr model for atomic spectra. http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

ehild

 

[leonardthecow]

I understand this and have been on that page many times, but what I still can't reconcile is how the diffraction orders relate to the energy levels. Is it simply that for, say, the second order diffraction line where n=2, where the wavelength would be given by

λ=dsinθ/2​

that this corresponds to the wavelength predicted by the Bohr model for n=2, where it would be given by

λ=2na₀​

I guess what my question boils down to is if the diffraction orders n=1,2,3... correspond to the energy levels for quantum numbers n=1,2,3..., or more simply if the variable "n" in Bragg's Law corresponds to the "n" used in the predictions for quantized energy levels and, consequently, for quantized wavelengths in the Bohr model. If not, then my question remains; how are these then related?

 

[radeculous]

BU PY351??
I found this link to be helpful:
http://www.colorado.edu/physics/2000/quantumzone/lines2.html

The simulation shows that as the electron jumps down energy levels, it produces a spectral emission line. Each line has a different wavelength, and therefore a different energy. The first order has all the lines of the "energy level diagram", and the second order is just another iteration of this. Each line correlates to one energy, which represents one transition between levels of the energy diagram. *Note that this could be a transition from non-adjacent levels.

 

[ehild]

The diffraction order "n" is not related to the quantum number "n" in the Bohr model. They are just integer numbers.

Determine the wavelength of the lines of different colour in the hydrogen spectrum. Try to fit the wavelengths into the Rydberg formula obtained from the Bohr model:

\frac{1}{\lambda}= R\left(\frac{1}{n_1 ^2} - \frac{1}{n_2 ^2} \right)

where λ is the wavelength, n₁ and n₂ are (small) positive integers and R=1.097 E7 m^-1.

What wavelengths did you get in the experiment?

ehild

 

[leonardthecow]

I wish I could have seen the different colors, but the data from the experiment came in the form of a graph of intensity versus angle from a central peak recorded on a computer.

Using again

λ=dsinθ/n​

I get wavelengths of 426.73 nm, 281.09 nm, 229.55 nm, and 806.93 nm; corresponding to peaks occurring on both sides of a central bright peak at average angular distances (radians) of 0.281, 0.374, 0.464, and 0.552, respectively.

To calculate the wavelengths above, I assumed that n=1, n=2, n=3, and n=4 for each peak. Was this incorrect? Should each peak be calculated using n=1 in Bragg's Law, since these are all "first order" spectral lines corresponding to four colors? If this is the case, the calculated wavelengths are closer to the Bohr model predictions for violet, blue, blue-green, and red lines. (Still pretty bad, though...not great data apparently.)

 

[radeculous]

Yes, it is incorrect to use n=1,2,3,4. It is most likely that ALL of your lines are first order, meaning in the equation you just provided, you use n=1.

 

[ehild]

What was the value of d (lattice parameter)?

ehild

 

[leonardthecow]

Thanks for clarifying the bragg's law issue!

And d is given as "about 1667 nm", calibrating it using a measurement for sodium gives about 1593 nm.

 

[ehild]

It is difficult to say what went wrong. Were there no lines at smaller angles?
I think the line at the greatest angle is of second order, and belongs to the same line of 410 nm as the first maximum.

ehild