An electron is in the first Bohr orbit of the hydrogen atom. (a) Find the speed of an electron, and (b) the time required for an electron to circle the nucleus. (Answer: (a) 2.19 ｘ 10⁶ m/s (b) 1.52 ｘ 10⁻¹⁶ s) 



Given:

Atom: Hydrogen

Energy level n = 1

∴ Plank's Constant = h =  6.62 ｘ 10⁻³⁴ m² kg s⁻¹

∴ Mass of the electron = m = 9.11 ｘ 10⁻³¹ kg

∴ Radius of 1st Bohr Orbit = r = 0.53 ｘ 10⁻¹⁰ m




To Find:


(a) Speed of electron in the first orbit = v₁ =  ?

(b) Time period = T₁ =  ?


Solution:

(a) Speed of electron in the first orbit = v₁ =  ?

The  general formula for finding the velocity of an electron in any orbit is 

 `\v_n` = `\frac{2Ï€ke²}{nh}`


by putting the corresponding values 

v₁ = `\frac{2 ｘ 3.1416 ｘ 9 ｘ 10⁹ N m² C⁻² ｘ (1.6 ｘ 10⁻¹⁹ C)²}{1 ｘ 6.62 ｘ 10⁻³⁴ m² kg s⁻¹}`

[N = kg m s⁻²]

v₁ = `\frac{56.5488ｘ 10⁹ kg m s⁻² m² C⁻² ｘ 2.56 ｘ 10⁻³⁸ C²}{ 6.62 ｘ 10⁻³⁴ m² kg s⁻¹}`

v₁ = `\frac{144.764928 ｘ 10⁻²⁹ kg s⁻² m³ }{ 6.62 ｘ 10⁻³⁴ m² kg s⁻¹}`

v₁ = 21.868ｘ 10⁵ m s⁻¹

or

v₁ = 2.187ｘ 10⁶ m s⁻¹ ---------------Ans.1



(b) Time period = T₁ =  ?

We know that 

S = v t

or 

t = `\frac {S}{v}`

where s = circumference of the orbit = 2Ï€r

t = `\frac {2Ï€r}{v}`

for n = 1 , t = T

T₁ = `\frac {2Ï€r₁}{v₁}`

by putting values

T₁ = `\frac {2 ｘ 3.1416 ｘ 0.53 ｘ 10⁻¹⁰ m}{2.187ｘ 10⁶ m s⁻¹}`

T₁ = `\frac {3.330096 ｘ 10⁻¹⁰ m}{2.187ｘ 10⁶ m s⁻¹}`

T₁ = 1.523 ï½˜ 10⁻¹⁶ s⁻¹ ------------Ans.2



************************************

Shortcut Links For 


1. Website for School and College Level Physics   
2. Website for School and College Level Mathematics  
3. Website for Single National Curriculum Pakistan - All Subjects Notes 

© 2022-Onwards by Academic Skills and Knowledge (ASK)     

Note:  Write me in the comment box below for any query and also Share this information with your class-fellows and friends.