Calculating Size Ratios for Energy Levels in Hydrogen Atom

[xiphoid]

Homework Statement

How many times larger is the spacing between the energy levels with n=3 and n=8 spacing between the energy levels with n=8 and n=9 for hydrogen atom?

Homework Equations

E_n=R_h[1/n_i²- 1/n_f²]

The Attempt at a Solution

From the above stated equation, i managed to calculate the ratio of their energies, however, I can even calculate λ or frequency v, but what about the ratio of their size?

 

[Simon Bridge]

When they say "larger spacing" they are talking about the energy spacing.

 

[xiphoid]

So, I need to calculate the ratio of the energies of the two given?
if yes then, I did this at first but the answer given, i.e. 14.82 times larger was not obtained

When they say "larger spacing" they are talking about the energy spacing.

 

[xiphoid]

I will try to upload my solution, and perhaps from there, you can help me out

 

[Simon Bridge]

You'd expect the 8-3 spacing to be bigger than the 8-9 spacing wouldn't you?

$$\Delta E_{8,3} = \frac{\frac{1}{9}-\frac{1}{64}}{\frac{1}{64}-\frac{1}{81}}\Delta E_{8,9}\approx 30\Delta E_{8,9}$$... looks like a factor of two got lost someplace.

If the first jump was 4-8 instead of 3-8 then the ratio would be 14.27 ... still too different.

The only other option for "size" would be differences in Bohr orbit radii.
Which would be dumb. So I'd say that either the model answer is in error or there is some aspect of the question not communicated someplace.

 

[AGNuke]

In Bohr Atomic Model, orbits are also known as energy levels, if I recall correct. Still, their ratio is absurd for answer.

 

[Simon Bridge]

AGNuke: not quite correct. The Bohr energy levels can be related to orbit radii in the planetary model. They are not different names for the same thing and you'll find physicists can get quite shirty about it.

OP will need to look at the context to see which is meant.
Or just do the math:-)

The orbit concept makes little sense here... but that may be the point of the exercise.