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Question
Use Bohr's postulate to prove that the radius of nth orbit in a hydrogen atom is proportional to n2.
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Solution
The necessary centripetal force for the rotation of an electron is supplied by the electrostatic force between the electron and the nucleus.
`"mv"^2/"r" = (1/(4πε_0))("e"^2/"r"^2)` ....[putting Z = 1]
Or, mv2 = `"e"^2/(4πε_0"r")` .....(i)
From Bohr's theory,
mvr = `"nh"/(2π)`
∴ v = `"nh"/(2π"mr")`
Putting in equation (i)
`"m"("nh"/(2π"mr"))^2 = "e"^2/(4πε_0"r")`
Or, r = `(ε_0"n"^2"h"^2)/(π"me"^2)`
In general,
rn = `(ε_0"n"^2"h"^2)/(π"me"^2)`
∴ `"rn" ∝ "n"^2`
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