What Energy Levels Create the Spectral Lines of Element Fb?

[v_pino]

Homework Statement

A meteorite found in Antarctica contains a trace of a mysterious element. Its spectral lines are all multiples of a frequency f0. The multiples are:
1, 2, 3, 5, 6, 8, 10, 11, 13, 16, 18, 19, 21, 26, 29, 31, 32, 34, 42, 47, 50, 52, 53 .

The frequencies in bold have twice the intensity expected from the pattern of intensities of
the other lines. The investigating spectroscopists assume that these are cases in which two
distinct transitions have the same frequency.

(a) Find the simplest set of energy levels that will produce the frequency spectrum.
Assume that transitions between all levels occur, and that the energy levels (like those of
hydrogen) get closer together as the energy increases.

(b) Draw with care the energy-level diagram. To the right, draw the frequency spectrum,
scaled (as in Fig. 5) to the highest frequency.

(c) The symbol given to the new element is Fb. What is its name?

Homework Equations

$$hf=-13.6\left ( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right )eV$$

The Attempt at a Solution

If the spectral lines, like that of hydrogen, get smaller, how can I double the frequency? For example, assuming it is like hydrogen, the ground level is -13.6eV and the second is -3.4eV and the third is -1.51eV. If I take the transition of ground to first level as f0, then 2*f0 has to be from ground level to infinity.

How can I draw such spectral line?

Thanks

 

[anathema]

for your post! This is a very interesting problem and one that requires some careful thought and analysis. First, let's start with the given information:

- The spectral lines are all multiples of a frequency f0.
- The multiples are: 1, 2, 3, 5, 6, 8, 10, 11, 13, 16, 18, 19, 21, 26, 29, 31, 32, 34, 42, 47, 50, 52, 53.
- The frequencies in bold have twice the intensity expected from the pattern of intensities of the other lines.
- The investigating spectroscopists assume that these are cases in which two distinct transitions have the same frequency.

From this information, we can gather that the spectral lines are a result of transitions between energy levels in the atom. The fact that the multiples of f0 are all whole numbers and that some of the frequencies have twice the intensity expected indicates that there are two transitions that result in the same frequency. This suggests that there are two energy levels that are very close together, and the transitions between them produce the doubled frequency.

To find the simplest set of energy levels that will produce this frequency spectrum, we can start with the lowest energy level and work our way up. Let's assume that the ground state energy level is -13.6 eV, as in hydrogen. We can then use the equation given in the problem, hf=-13.6\left ( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right )eV, to find the energies of the other levels.

For the first transition (n=1 to n=2), we have hf=f0=-13.6\left ( \frac{1}{1^2} - \frac{1}{2^2} \right )=-10.2 eV. This gives us the energy of the second level as -3.4 eV.

For the second transition (n=1 to n=3), we have hf=2f0=-13.6\left ( \frac{1}{1^2} - \frac{1}{3^2} \right )=-11.77 eV. This gives us the energy of the third level as -1.51 eV.

For the third transition (n