(a) Using the Bohr's model, calculate the speed of the electron in a hydrogen atom in the n=1,2 and 3 levels. (b) Calculate the orbital period in each of these levels.

Explanation

(a) Let `v_(1)` be the orbital speed of the electron in a hydrogen atom in the ground state level `n_(1)=1` for charge (e) of an electron `v_(1)` is given by the relation,
`v_(1)=e^(2)/(n_(1)4pi in_(0)(h/(2pi)))=e^(2)/(2in_(0)h)`
Where, `e=1.6xx10^(-19)C`
`in_(0)` =Permittivity of free space `=8.85xx10^(-12)N^(-1)C^(2)m^(-2)`
h=Planck's constant `=6.62xx10^(-34)Js`
`:. v_(1)=((1.6xx10^(-19))^(2))/(2xx2xx8.85xx10^(-12)xx6.62xx10^(-34))`
`=1.09xx10^(6)m//s`
And, for `n_(3)=3` we can write the relation for the corresponding orbital speed as:
`v_(3)=e^(2)/(n_(3)2in_(0)h)`
`=(1.6xx10^(-19))^(2)/(3xx2xx8.85xx10^(-12)xx6.62xx10^(-34))`
`=7.27xx10^(5)m//s`
Hence, the speed of the electron in a hydrogen atom in `n=1, n=2 and n=3 is 2.18xx10^(6)m//s 1.09xx10^(6)m//s 7.27xx10^(5)m//s` respectively
(b) Let `T_(1)` be the orbital period of the electron when it is in level `n_(1)=1` Orbital period is related to orbital speed as:
`T_(1)=(2pir_(1))/v_(1)`
Where,
`r_(1)` = Radius of the orbit
`=(n_(1)^(2)h^(2)in_(0))/(pime^(2))`
h=Plack's constant `=6.62xx10^(-34)Js`
e=charge on a electron `=1.6xx10^(-19)C`
`in_(0)` =permittivity of free space `=8.85xx10^(-12)N^(-1)C^(2)m^(-2)`
m=Mass of an electron `=9.1xx10^(-31)kg`
`:. T_(1)=(2pir_(1))/v_(1)`
`=(2pixx(1)^(2)xx(6.62xx10^(-34))^(2)xx8.85xx10^(-12))/(2.18xx10^(6)xxpixx9.1xx10^(-31)xx(1.6xx10^(-19))^(2))`
`=15.27xx10^(-17)=1.52710^(-16)s`
For level `n_(2)=2` we can write the period as:
`T_(3)=(2pir_(3))/v_(3)`
Where, `r_(3)` = Radius of the electron in `n_(3)=3`
`=((n_(3))^(2)h^(2)in_(0))/(pime^(2))`
`:. T_(3)=(2pir_(3))/v_(3)`
`=(2pixx(3)^(2)xx(6.62xx10^(-34))^(2)xx8.85xx10^(-12))/(7.27xx10^(5)xxpixx9.1xx10^(-31)xx(1.6xx10^(-19))^(2))`
`=4.12 xx10^(-15)s`
Hence, the orbital period in each of these levels is `1.52xx10^(-16)s, 1.22xx10^(-15)s` and `4.12xx10^(-15)` s respectively