Question:
(a) Using the Bohrs 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.
Answer:
Let, electron of hydrogen atom at ground state n1=1
v1=e2/n14πε0(h/2π)= e2/2ε0h
where, e = 1.6 x 10-19C, ε0= 8.85 x 10-12 /NC2/m2, h = 6.62 x 10-34 Js
then,
v1=e2/2ε0h
= (1.6 x 10-19)2/ 2x2x8.85 x 10-12 x 6.62 x 10-34
= 1.09 x 106 m/s
for n3 = 3,
then,
v1=e2/ n32ε0h
= (1.6 x 10-19)2/ 3x2x8.85 x 10-12 x 6.62 x 10-34
= 7.27 x 106 m/s
(b) Let T1 = orbital period of electron, n1=1
Then, Orbital period is related to orbital speed
T1 = 2πr1/v1 where, Radius of the orbit (r1) = n12 h2 ε0/πme2
Where, e = 1.6 x 10-19C, ε0= 8.85 x 10-12 /NC2/m2, h = 6.62 x 10-34 Js
Mass of an electron (m) = 9.1 x10-31 Kg
Therefore,
T1 = 2πr1/v1
= 2xπ x (1)2 x (6.62 x 10-34)2 x 8.85 x 10-12/2.18x106x π x 9.1 x10-31 x(1.6 x 10-19)2
= 15.27 x 10-27
= 1.52 x 10-16 s
For level n2=2
T2 = 2πr2/v2
= 2xπ x (2)2 x (6.62 x 10-34)2 x 8.85 x 10-12/1.09x106x π x 9.1 x10-31 x(1.6 x 10-19)2
= 1.22 x 10-15 s
For level n3=3
T3 = 2πr3/v3
= 2xπ x (3)2 x (6.62 x 10-34)2 x 8.85 x 10-12/7.27x106x π x 9.1 x10-31 x(1.6 x 10-19)2
= 4.12 x 10-15 s
Hence, the orbit period in each level is 1.527 x 10-16 s, 1.22 x 10-15 s, 4.12 x 10-15 s respectively.
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