(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)The velocity of electron in nth orbit in an atom is given by
`v_n=(Ze^2)/(2epsilon_0nh)`
For hydrogen , Z=1
`therefore v_n=((1.6xx10^(-19))^2)/(2xx8.854xx10^(-12)xxnxx6.63xx10^(-34))`
`=(2.18xx10^6)/n ms^(-1)`
For n=1,
`v_1=(2.18xx10^6)/1=2.18xx10^6 m s^(-1)`
For n=2 ,
`v_2=(2.18xx10^6)/2 =1.09xx10^6 m s^(-1)`
For n=3,
`v_3=(2.18xx10^6)/3`
`=0.727xx10^6 m s^(-1)`
`=7.27xx10^4 m s^(-1)`
(b) Since time period in `n^(th)` orbit is given by
`T_n=(2pir_n)/v_n`
`r_n=(epsilon_0n^2h^2)/(pi nZe^2)=r_0 n^2/Z`
where `r_0`=Bohr.s radius = `0.53xx10^(-10)` m
Also, Z=1 for hydrogen
`therefore T_n=(2pir_0n^2)/v_n`
For n=1,
`T_1=(2pir_0(1)^2)/v_1=(2xx3.14xx0.53xx10^(-10))/(2.18xx10^6)`
`=1.52xx10^(-16)` s
For n=2,
`T_2=(2pir_0 (2)^2)/v_2 =(2xx3.14xx0.53xx10^(-10)xx9)/(7.29xx10^5)`
`=41.1xx10^(-16)` s