(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

. The problem statement, all variables and given/known data[/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?

2. Relevant 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

3. 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

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# Homework Help: Energy of electron in the hydrogen atom

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