Hydrogen-like ions energy levels

[CatWoman]

Homework Statement

An absorption line at 6966 eV is known to come from a hydrogen-like ion. Calculate the likely atomic number of the element and hence identify it. (The ionization energy of hydrogen is 13.6eV.)

Homework Equations

not sure

The Attempt at a Solution

Using E=(13.6Z^2)/n^2 eV where n=1 => 6966=13.6Z^2
So Z=22.6.
I thought I should get a precise whole number and neither Z=22or 23 are hydrogen-like ions, so I think I must have done this wrong.

 

[quantumdude]

I wouldn't worry too much about not getting an integer. The data that you were given carries an uncertainty with it. I would just round to the nearest integer.

Here's a thought though: How do you know that the initial state is n=1? The problem doesn't say that.

 

[CatWoman]

I just assumed n=1, and hoped it would come out ok. Not very good I know. But I couldn't find anything better to use. Thanks for your help.

 

[Redbelly98]

Besides the initial state, there should be a final state--and an additional value n₂--involved also.

 

[CatWoman]

I think that is another formula - the Rydberg formula, which came before Bohr's formula (which I am using) which proves the Rydberg - but I may be wrong. So i don't think I need to use that one? many thanks

 

[diazona]

I think that is another formula - the Rydberg formula, which came before Bohr's formula (which I am using) which proves the Rydberg - but I may be wrong. So i don't think I need to use that one? many thanks

Yep, Bohr's formula can be used to derive the Rydberg formula. Whereas Bohr's equation (the one you used in your initial post) gives you the energy of each level, the Rydberg formula gives you the difference in energy between different energy levels. You will need to subtract the energy of level n₁ from the energy of level n₂ (this is what Redbelly98 is alluding to).

 

[CatWoman]

I can't see how the rydberg formula would help here as there would be too many unknowns - the rydberg constant and atomic number for the unknown element, and I would still have to make an assumption about the energy levels involved in the absorption.

 

[Redbelly98]

It's fairly likely that one of the levels involved is the ground state (n1 = 1). That will leave just two unknowns, Z and n2. Try different n2's, and see if any produce a Z that is close to an integer.

 

[CatWoman]

don't I need to know the rydberg constant for the unknown element too?

 

[Redbelly98]

The Rydberg constant is a constant, and is the same for all elements.

 

[CatWoman]

I thought there is a specific one for hydrogen and different ones for each element? It says this here: http://scienceworld.wolfram.com/physics/RydbergConstant.html
Have I misunderstood - please can you explain?

 

[diazona]

Wikipedia has a better explanation: http://en.wikipedia.org/wiki/Rydberg_formula

 

[Redbelly98]

Ah, okay, the different nuclear mass will make a small difference in the Rydberg "constant". But the difference is at most 0.05%.

 

[CatWoman]

Ok, that makes sense now, many thanks. However, having tries a few n values for energy level transitions I cannot get anything values for Z that look viable. Not sure what I'm doing wrong. I will probably just hand in my answer as is and see what they wanted me to do - many thanks everyone. I have understood it all a lot more from your help anyway!

 

[Redbelly98]

Couple more thoughts:

1. Are you using consistent units? The given information is in terms of energy, while the Rydberg constant uses 1/wavelength (to my knowledge). In that case you would have to convert the given energy into 1/λ.

2. You haven't shown us the actual equation you are using. If there were something wrong with the equation, we would have no way of knowing or helping you with that.

 

[CatWoman]

Hi, I used the Rydberg formula 1/λ=E/hc=RZ^2 (1/(n_1^2 )-1/(n_2^2 )) and used the same R for the hydrogen and unknown atom. First I assumed the transitions were the same which leads to E_unknown/E_H =Z^2 and gives Z=22.6, as before using the Bohr formula. Then I tried different n values and didn't find anything better.

 

[Redbelly98]

E/hc=RZ^2 (1/(n_1^2 )-1/(n_2^2 ))

I'll suggest dividing this equation by R on both sides, to get

E / (Rhc) = Z² (1/n1² - 1/n2²)​

That way, the E/(Rhc) can be thought of as a single "constant" for this problem (since E is given). Once you have the value for E/(Rhc), try n1=1 and see what Z is for n2 = 2, 3, 4, etc.

And remember, E, R, h and c must be expressed in units that are consistent with one another.

If it still doesn't work out, post the value of E/(Rhc) that you are using.

EDIT: and just use R for hydrogen.