Easy - determining n from orbital radius

In summary, the conversation is about determining the principle quantum number for an orbital of a given radius in the hydrogen atom. The formula used is Rn=(n^2)(Ao)/Z and the calculated value of n is 447.2135955, which is not a whole number and is abnormally large due to the approximate value of the radius. The corresponding excited state energy is also calculated using the formula E=-13.6Z^2/(n^2). The conversation then shifts to calculating the first excited state energies of He+, Li2+ and Be3+ and comparing them to the ground state energy of hydrogen. It is determined that the first excited state for these atoms is n=2.
  • #1
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Homework Statement


For the hydrogen atom, determine the principle quantum number corresponding to an orbital of radius Rn=0.01mm

Homework Equations


I'm using the following formula:

Rn=(n^2)(Ao)/Z

Where Rn is the orbital radius (0.01mm),
n is the principle quantum number (wanted)
Ao is a constant (0.5*10^-10)
and Z is the atomic number (for hydrogen =1)

The Attempt at a Solution


By converting Rn into metres and subbing in all the other values I obtain a value of n=447.2135955.

Obviously there are two discreptancies:
1) n is not a whole number
2) it is abnormally large

Can anybody show me where I have gone wrong?
 
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  • #2
i) it's not a whole number because 0.01mm is an approximate value. ii) It's abnormally large because an 0.01mm hydrogen atom is abnormally large. It doesn't seem out of line to me.
 
  • #3
Would I answer this with a rounded value then?

A lead on question asks: for this value of principle quantum number, find the correspongind excited state energy.

Thus E=-13.6Z^2/(n^2) becomes

E=-13.6/(447.2135955^2)=-6.8*10^-5 eV

(notice for this part I did not use the rounded value)
 
  • #4
There is no orbital at exactly 0.01mm. So yes, you round. It shouldn't matter much.
 
  • #5
I have another question, slightly different topic

-Calculate the first excited state energies of He+, Li2+ and Be3+ compare them with the ground state energy of hydrogen.

ground state means n=1, but how do i find n for the first excited state? does it differ for each of the 3 cases above?
 
Last edited:
  • #6
Yes, n=1 in all those cases. Look at your formula. What could change?
 
  • #7
atomic number is going to be different for each of those cases. But since I want to find the first excited state of He+... don't I need to use the relevant principle quantum number?

Not sure if I've missed something here, but how do I determine the principle quantum number (n) for the first excited state in the three above cases?
 
  • #8
They are all hydrogen-like one-electron atoms. The first excited state is n=2. Just like hydrogen. (Sorry if I said n=1 before, I missed the 'excited state'). The n's designate the state. n=1 is ground. The first state above ground is the first excited state. Must be n=2.
 
  • #9
cool, a lot clearer now!
 

1. How do you determine the value of n from an orbital radius?

The value of n can be determined using the Rydberg formula, which is given by 1/λ = R(1/n2 - 1/m2), where λ is the wavelength, R is the Rydberg constant, and n and m are the principal quantum numbers. By rearranging the formula and plugging in the known values of the orbital radius and other constants, we can solve for n.

2. What is the significance of determining n from an orbital radius?

Knowing the value of n allows us to understand the energy levels and electronic configurations of atoms. It helps us predict the spectral lines and transitions of atoms, which is essential in fields such as spectroscopy and quantum mechanics.

3. Can n be any value for a given orbital radius?

No, n can only take on specific values known as energy levels. These energy levels are represented by whole numbers starting from 1. The higher the value of n, the higher the energy level and the larger the orbital radius.

4. How does the orbital radius affect the value of n?

As mentioned earlier, the orbital radius increases as the value of n increases. This is because the energy level of an electron increases as it moves away from the nucleus. Therefore, a larger orbital radius corresponds to a higher energy level and a larger value of n.

5. Is the value of n the same for all elements?

No, the value of n varies for different elements and their respective electrons. Each element has a unique electronic configuration, which determines the energy levels and thus, the value of n for its electrons.

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