(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.

Ans (a).
Let ν1 be the orbital speed of the electron in a hydrogen atom in the ground state level, n1 = 1. For charge (e) of an electron, ν1 is given by the relation,

ν1 = e²/n1 4π ε0 (h/2π) = e²/2 n1ε0h

Where, e = 1.6 x 10⁻¹⁹ C

ε0 = Permittivity of free space = 8.85 x 10⁻¹² N^-1 C² m²

h = Planck’s constant = 6.62 x 10⁻³⁴ Js

Therefore, ν1 = (1.6 x 10⁻¹⁹ )² /2 x (8.85 x 10⁻¹²) x (6.62 x 10⁻³⁴)

= 0.0218 x 10⁸ = 2.18 x 10⁶  m/s

For level n2= 2 ,we can write the relation for the corresponding orbital speed as:

ν2 = = e²/2 n2ε0h = (1.6 x 10⁻¹⁹ )² /2 x 2 x(8.85 x 10⁻¹²) x (6.62 x 10⁻³⁴)

=1.09 x 10⁶  m/s

For level n3= 3 ,we can write the relation for the corresponding orbital speed as:

ν3 = = e²/2 n3ε0h = (1.6 x 10⁻¹⁹ )² /2 x 3 x (8.85 x 10⁻¹²) x (6.62 x 10⁻³⁴)

=7.27 x 10⁵ m/s

Hence, the speed of the electron in a hydrogen atom in n = 1, n=2, and n=3 is 2.18 x 10⁶ m/s, 1.09 x 10⁶ m/s, 7.27 x 10⁵ m/s respectively.

Ans (b).

Let T1 be the orbital period of the electron when it is in level n1 = 1.

Orbital period is related to orbital speed as:

T1 = 2π r/ν1

Where,r1 = Radius of the orbit =( n1)²h²ε0/πme²

h = Planck’s constant = 6.62 x 10⁻³⁴ Js and

e = Charge on an electron = 1.6 x 10⁻¹⁹ C.

ε0 = Permittivity of free space = 8.85 x 10⁻¹² N⁻¹C² m⁻² 

m = Mass of an electron = 9.1 x 10⁻³¹ kg

Therefore, T1 = 2π r1/ν1

= 2π x (1)² x (6.62 x 10⁻³⁴)² x (8.85 x 10⁻¹²)/[(2.18 x 10⁶) x  π x ( 9.1 x 10⁻³¹) x (1.6 x 10⁻¹⁹)² ]

= 15.27 x 10⁻¹⁷ = 1.527 x 10⁻¹⁶ s

For level n2 = 2, we can write the period as:

T2 = 2π r2/ν2

Where, r2 = Radius of the electron in n2 = 2

= (n2)²h²ε0/πme²

Therefore,T2 =2π r2/ν2

=2π x (2)² x (6.62 x 10⁻³⁴)² x (8.85 x 10⁻¹²)/[(1.09 x 10⁶) x  π x ( 9.1 x 10⁻³¹) x (1.6 x 10⁻¹⁹)² ]

=1.22 x 10⁻¹⁵ s

For level n3 = 3, we can write the period as:

T3 = 2π r3/ν3

Where, r3 = Radius of the electron in n3 = 3

= (n3)²h²ε0/πme²

Therefore,T3 =2π r3/ν3

=2π x (3)² x (6.62 x 10⁻³⁴)² x (8.85 x 10⁻¹²)/[(7.27 x 10⁵) x  π x ( 9.1 x 10⁻³¹) x (1.6 x 10⁻¹⁹)² ]

=4.12 x 10⁻¹⁵ s

Hence, the orbital period in each of these levels is 1.527 x 10⁻¹⁶, 1.22 x 10⁻¹⁵ s, and 4.12 x 10⁻¹⁵ s respectively.