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Question
Starting with 𝑟 = `(ε_0h^2n^2)/(pimZe^2),` Show that the speed of an electron in nth orbit varies inversely to principal quantum number.
Solution
According to Bohr’s second postulate,
mrnvn = `"nh"/(2pi)`
∴ `"m"^2"v"_"n"^2"r"_"n"^2 = ("n"^2"h"^2)/(4pi^2)`
∴ `"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"^2"r"_"n"^2)`
Substituting, rn = `(ε_0"h"^2"n"^2)/(pi"m""Z""e"^2)` in above relation,
`"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"^2) xx ((pi"m"Z""e""^2)/(epsilon_0"h"^2"n"^2))^2`
= `("n"^2"h"^2)/(4pi^2"m"^2) xx (pi^2"m"^2"Z"^2"e"^4)/(epsilon_0^2"h"^4"n"^4)`
= `("Z"^2"e"^4)/(4epsilon_0^2"h"^2"n"^2)`
∴ `"v"_"n"^2 ∝ 1/"n"^2`
⇒ `"v"_"n" ∝ 1/"n"`
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