Energy of electron in the hydrogen atom

[kihr]

. Homework Statement [/b]

In a hydrogen atom the electron and proton are bound at a distance of about 0.53A.
(a) Assuming the zero of potential energy at infinite separation of the electron from the proton, what is the minimum work required to free the electron?
(b) What would the answer to (a) be if the zero of potential energy is taken at 1.06A?

Homework Equations

Potential energy of the electron at a distance r from the proton = U= -9*10^9 e^2/r

Kinetic energy of electron K = mv^2/2 = 9*10^9*e^2/2r

The Attempt at a Solution

(a) After substituting the relevant values, we get
U = -27.2 eV
K = 13.6 eV

Hence total energy of the electron = U + K = -13.6 eV
Work required to free the electron = Energy at infinite separation (i.e. zero total energy) minus energy at r=0.53A
= 0 -(-13.6)
= 13.6 eV
This answer matches with that given in the book.

(b) When the zero of potential energy is taken at r=1.06A, the potential energy of the electron at r=0.53A has to be calculated by integrating dr/r^2 from r=1.06A to r=0.53A.
This gives U= -9*10^9*e^2*(1.06 - 0.53)/1.06*0.53
= -13.6 eV (after substituting the relevant values).
K remains unchanged at 13.6 eV, as the kinetic energy of the electron at r=0.53A is not dependent on the zero reference of potential energy.
Therefore the total energy of the electron = 13.6 - 13.6 = 0

This implies that the electron is free from the proton at r = 0.53A under the new zero reference of potential energy.

I have reached upto this point, but cannot figure out how to proceed further to calculate the work done to free the electron from the proton. The answer given in the book is 13.6 eV.

I need some help to solve part (b) of the question

 

[vela]

You generally choose where to set the zero of potential energy to make the calculations easier. It has no effect on the actual physics of the situation. If it takes 13.6 eV to free the electron with the zero set at infinity, it will take 13.6 eV wherever you place the zero of potential energy.

An unbound or free electron is one that has enough energy to get infinitely far away, so regardless of where the zero is set, you find, in your words, "Energy at infinite separation ... minus energy at r=0.53A." When you choose the zero of potential energy to be at infinity, E=0 at infinity because U=0 and K=0. If you place the zero somewhere else, then E at infinity won't be zero because U is no longer zero at infinity. It won't matter because E at r=0.53A also changed, so the difference is still 13.6 eV.

 

[kihr]

Consider this:
At r=1.06A K=6.8 eV and U=0 {as per the new zero reference of P.E. as in case (b)}. Since E=0 at r=0.53A, the electron gains 6.8 eV of energy in moving from 0.53A to 1.06A. This means that when the electron is separated from its initial orbit to the orbit where U=0 it acquires 6.8 eV of energy in case (b), whereas it acquires 13.2 eV of energy in case (a). Actually the grey area in my mind is that under the reference of zero PE as at case (b) the electron is already free (unbound) at r=0.53A since its total energy is zero. So where does the question of any further work to be done to free it arise? Maybe I need some more clarity on this score. Thanks.

 

[vela]

"E=0" isn't synonymous with "unbound". That's only true when you set the zero of potential energy at infinity.

 

[kihr]

I could not quite understand that. When the electron is unbound it is free, i.e. its total energy is zero. This is precisely its state at r=0.53A in case (b).

 

[vela]

Unbound means it has enough energy to get infinitely far away. If you set the zero of potential energy at infinity, the electron is unbound when E=0 because it can just reach infinity where U=0 with K=0. If you use a convention where U=100 eV at infinity, then the electron has to have at least 100 eV of total mechanical energy to reach infinity and therefore be unbound.

 

[kihr]

I think now I understand it barring for the sign of work to be done. At r=infinity U = -13.6eV in case (b) {electron moved from r=1.06A corresponding to zero PE to r=infinity}. Since K=0 at infinity, total energy of the electron at r=infinity is -13.6eV. Therefore in order to free the electron, i.e. is to remove it from r=0.53A to r=infinity, the work required to be done = -13.6 - 0 (final energy minus initial energy). The only problem with this is the sign of the work done which does not tally with the answer. Maybe you could provide a clue for this. Thanks.

 

[vela]

How did you calculate U at infinity?

 

[kihr]

U=-k*e^2 {Integ dr/r^2} (limits of integration from r=1.06A (where PE is zero) to r=infinity). Sorry for the strange mode of representation as I do not have a compatible keyboard!

 

[vela]

You have an extra minus sign in there.

 

[kihr]

I see the error. In case (b) U= +13.6 eV at r=infinity and NOT -13.6 eV as I had earlier calculated. Thanks a lot for your assistance.