Question

Using the expression for the radius of orbit for the Hydrogen atom, show that the linear speed varies inversely to the principal quantum number n the angular speed varies inversely to the cube of principal quantum number n. 

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 = `(epsilon_0"h"^2"n"^2)/(pi"m""Ze"^2)` in above relation,

`"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"^2) xx ((pi"m""Ze"^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"`

Expression for angular speed:

Since, v_n = r_n ω and r_n = `(epsilon_0"h"^2"n"^2)/(pi"m"_"e""e"^2)`

∴ `omega = ("v"_"n")/"r"_"n" = (("e"^2)/(2epsilon_0"h")) 1/"n"/(epsilon_0"h"^2"n"^2)/(pi"m"_"e""e"^2)`

∴ `omega = "e"^2/(2epsilon0"hn") xx (pi"m"_"e""e"^2)/(epsilon_0"h"^2"n"^2) = ((pi"m"_"e""e"^4)/(2epsilon_0^2"h"^3)) 1/"n"^3`

⇒ `omega ∝ 1/"n"^3`