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Question
State Bohr's postulate to explain stable orbits in a hydrogen atom. Prove that the speed with which the electron revolves in nth orbit is proportional to `(1/"n")`.
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Solution
The stationary orbits are those orbits for which the angular momentum of the electron is an integral multiple of `"h"/(2π)`, where h is Planck's constant.
Electron in an atom revolves because of the balance of Coulomb force of attraction between the protons and electrons and the centripetal force.
In the nth orbit,
`("mv"_"n"^2)/"r"_"n" = 1/(4πε_0) "e"^2/"r"_"n"^2`
∴ vn = `"e"/(sqrt(4πε_0"mr"_"n"))` ......(i)
Again, rn = `(ε_0"h"^2"n"^2)/("e"^2π"m")` .....(ii)
Putting in equation (i)
vn = `"e"/sqrt(4πε_0"m"(ε_0"h"^2"n"^2)/("e"^2π"m"))`
= `"e"^2/(2ε_0"hn")`
∴ `"v"_"n" ∝ 1/"n"`
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