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Consider the hydrogen atom to be a proton embedded in a cavity of radius ao (Bohr radius) whose charge in neutralized by the addition of an electron to the cavity in vacuum, infinitely slowly. Estimate the average total energy of an electron in its ground state in a hydr5ogen atom as the work done in the above neutralization process. Also, if the magnitude of the average kinetic energy is half the magnitude of the average potential energy, find the average potential energy.

Simran Bhatia , 11 Years ago
Grade 11
anser 1 Answers
Aditi Chauhan
Work done while bringing an electron infinitely slowly from infinity to proton of radius a0 is given as follows
W = - e2/4πε0.a0
NOTE : This work done is equal to the total energy of an electron in its ground state in the hydrogen atom. At this stage, the electron is not moving and do not possess any K.E., so this total energy is equal to the potential energy.
T.E. = P.E + K.E. = e2/4πε0.a0 ….(i)
In order the electron to be captured by proton to form a ground state hydrogen atom it should also attain
K.E. = e2/8πε0a0
(it is given that magnitude of K.E. is half the magnitude of P.E. Note that P.E. is –ve and K.E is +ve)
∴ T.E. = P.E + K.E. = - e2/4πε0a0 + e2/8πε0a0
or T.E. = - e2/8πε0a0
P.E. = 2 * T.E. = 2 * -e2/8π ∈0 a0 or P.E. = -e2/8π ∈0 a0
Last Activity: 11 Years ago
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