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प्रश्न
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.
उत्तर
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 = `(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, vn = rn ω and rn = `(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`
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