Radius of the electron orbit in a Hydrogen atom

[Bolter]

Homework Statement

See question below

Relevant Equations

F = mv^2/r

I am really stuck on what to do here in this question

I have arrived at forming an equation to work out the radius of electron orbit from doing the following

However I do not know what to do next as I don't know what the value of n (quantum number) must be?

Any help would be really great! Thanks

 

[vela]

Derive an expression for the energy since you're provided the energy of the electron.

 

[Bolter]

Derive an expression for the energy since you're provided the energy of the electron.

I have tried doing that and get this

What would my total energy E be though? That is my confusion

 

[vela]

Try writing the energy in terms of $n$ instead of $r$. Then you can find the allowed energies by plugging in different values for $n$.

 

[Bolter]

Try writing the energy in terms of $n$ instead of $r$. Then you can find the allowed energies by plugging in different values for $n$.

Is this the expression that I need which I circled in my working for the total energy in terms of $n$?

How would I do this as I have a $v$ in my expression which I do not know? And it is $r$ that we are trying to find?

 

[vela]

You already found expressions for the kinetic and potential energy in terms of $r$, and you have $r$ in terms of $n$. Just put those together.

 

[Bolter]

You already found expressions for the kinetic and potential energy in terms of $r$, and you have $r$ in terms of $n$. Just put those together.

Thanks I have tried putting these together and have ended up with this expression

Now how do I make use of the 0.85eV and 12.75eV values given in the question to find radius?

 

[haruspex]

Shouldn't the energy in the excited state be -0.85eV? At zero it would become unbound.

 

[Bolter]

Shouldn't the energy in the excited state be -0.85eV? At zero it would become unbound.

I was thinking that too as usually energy level values are negative

Is it wrong to think that the total energy E would be 0.85eV + 12.75eV = 13.6 eV?

 

[Bolter]

I know that I shouldn't be looking at this but here is the energy level diagram for a hydrogen atom

Therefore I know that my energy En = –0.85eV is in the 4th energy level hence n = 4

Problem is that when I sub different integer values of $n$ in the equation that relates energy $E$ in terms of $n$ I keep getting zero on my calculator?

 

[haruspex]

I was thinking that too as usually energy level values are negative

Is it wrong to think that the total energy E would be 0.85eV + 12.75eV = 13.6 eV?

On reflection, I don't think the 0.85 matters. The key should be the 12.75eV difference between E₁ and E_n.
You have expressions for both of those from post #7.

 

[Bolter]

On reflection, I don't think the 0.85 matters. The key should be the 12.75eV difference between E₁ and E_n.
You have expressions for both of those from post #7.

Thank you, I used what I had in post #7 where I knew that energy in En = –0.85eV

Then from that I done:

So radius is 8.46 angstroms to 2 dp

 

[hutchphd]

Notice this is 4²=16 times the Bohr radius (=.529Angstrom) as it should be.

 

[Bolter]

Notice this is 4²=16 times the Bohr radius (=.529Angstrom) as it should be.

Right, so I should divide my 8.462... angstrom by 16 to get 0.529 angstrom

Also I should've showed the full question in the original post by showing the possible answers to choose from

Here it is

As you can see there isn't any 0.529 angstrom option here? Does this question have an error then? I thought option b would've been the correct answer.

 

[hutchphd]

You misunderstand. The answer you provide is correct. The Bohr radius is the size of the n=1 (ground) state. I was pointing out there was an easier route to your answer but all correct routes will provide the correct number!

 

[Bolter]

You misunderstand. The answer you provide is correct. The Bohr radius is the size of the n=1 (ground) state. I was pointing out there was an easier route to your answer but all correct routes will provide the correct number!

Oh I see, I understand what you had meant now, my bad