Energy & radius of an electron orbit

[SaintJude]

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Can someone please explain the answer to this question?

How does the energy and radius of an electron orbit in He+ compare to the energy and radius of an electron orbit in H, if each electrons has the same value of n?

 

[Pisiform]

I think Energy is proportional to Z^2/n^2 ... so Z for H is 1 and Z for He+ is 2

So, E of a He+ would be 4 times that of H

As far as radius is concerned it is proportional to n^2/Z^2

so for He+ radius would be 1/4 th that of H

someone confirm this though.

 

[SaintJude]

I think Energy is proportional to Z^2/n^2 ... so Z for H is 1 and Z for He+ is 2

So, E of a He+ would be 4 times that of H

As far as radius is concerned it is proportional to n^2/Z^2

so for He+ radius would be 1/4 th that of H

someone confirm this though.

Thanks. Qualitatively, you're right. But I think you may be off quantitatively about r (but I'm sure that equation would be given on the MCAT anyway so no worries!) The energy of the radius is actually proportional to n^2/Z. So the radius of the helium ion would be 1/2 of the radius of hydrogen.

Don't have the definitive answer though b/c this is from a Kaplan Lesson Book...where they don't include answers in the book 🙁

 

[SaintJude]

But I have a conceptual follow-up question

Looking at the atomic numbers and principal quantum number values used, the answer would have been the same if we had compared the He atom (not an ion) with a H.

Does this mean that the energy of an electron orbit in a He+ would be the same as that of a He atom (according to this Bohr model equation)?

 

[Pisiform]

But I have a conceptual follow-up question

Looking at the atomic numbers and principal quantum number values used, the answer would have been the same if we had compared the He atom (not an ion) with a H.

Does this mean that the energy of an electron orbit in a He+ would be the same as that of a He atom (according to this Bohr model equation)?

I think you can only apply for any atom/ion that has only one electron e.g. Li2+ and so on. This calculation from Bohr Model would not be applicable when more than one electron orbits the nucleus because it does not take into account the electrostatic forces that once electron exerts on another.

so He and H would have different values ... i dunno exactly how to calculate them

 

[chiddler]

But I have a conceptual follow-up question

Looking at the atomic numbers and principal quantum number values used, the answer would have been the same if we had compared the He atom (not an ion) with a H.

Does this mean that the energy of an electron orbit in a He+ would be the same as that of a He atom (according to this Bohr model equation)?

I don't think so because the equation is N^2 / Z^2 where Z is the nuclear charge. Nuclear charge of helium is larger than hydrogen. Nuclear charge defined as protons - the number of electrons between nucleus and electron in question. Since we're looking at the first electron in each atom, the only part that increases is number of protons while electrons between stays the same.

How did you guys find numbers to compare radii? Meaning how do you know radius will be 4x smaller?

 

[SaintJude]

Ah, as usual Pisiform is absolutely correct! and soooo right--that equation does only apply to the Bohr model...how could i get so confused?

To chiddler:
This is an equation that will likely be given on the MCAT if it appears and was given in the passage of my Kaplan physics test. No need to memorize 🙂