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Question
Derive an expression for the radius of the nth Bohr orbit for the hydrogen atom.
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Solution
Let, me = mass of electron,
−e = charge on electron,
rn = radius of nth Bohr’s orbit,
+e = charge on nucleus,
vn = linear velocity of electron in nth orbit,
Z = number of electrons in an atom,
n = principal quantum number.
∴ The angular momentum = mevnrn
According to second postulate.
mevnrn = `"nh"/("2"pi)` ...(i)
From Bohr’s first postulate,
Centripetal force = Electrostatic force
∴ `("m"_"e""v"_"n"^2)/"r"_"n" = "Ze"^2/(4piepsilon_0"r"_"n"^2)`
∴ `"v"_"n"^2 = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")` ....(ii)
Squaring equation (i) we get
`"m"_"e"^2 "v"_"n"^2 "r"_"n"^2 = ("n"^2"h"^2)/(4pi^2)`
∴ `"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2)` ….(iii)
Equating equations (ii) and (iii), we get,
`("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2) = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")`
∴ rn = `("n"^2"h"^2epsilon_0)/(pi"m"_"e""Z"_"e"^2)`
This is the required expression for radius of nth Bohr orbit of the electron.
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