Question

Starting with 𝑟 = `(ε_0h^2n^2)/(pimZe^2),` Show that the speed of an electron in n^th orbit varies inversely to principal quantum number. 

Answer 1

According to Bohr’s second postulate,

mr_nv_n = `"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, r_n = `(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"`