Atomic structure numerical part-1

Atomic structure

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Atomic structure- numerical

Question 1) An electron (m = 9.1 x 10^-28 gm ) is moving at a velocity of 10⁵ cm/sec. Calculate its kinetic energy and wavelength?

Solution )

m = 9.1 x 10^-28 gm , v = 10⁵ cm/sec , h = 6.625 x 10 ^-27 erg

Kinetic energy = mv² / 2

= 9.1 x 10^-28 x 10⁵ x 10⁵ / 2

K E = 4.55 x  10^-18 erg Ans.

λ = h / mv = 6.625 x 10 ^-27  / 9.1 x 10^-28 x 10 ⁵

λ = 0.728 x 10^-4 cm -1 Ans.

Question 2) The energies of electron in second and third orbits in hydrogen atom are -5.42 x 10^-12 ergs and – 2.41 x 10^-12 ergs. Calculate the wavelength of radiation emitted when the electron in a hydrogen atom undergoes a transition from third orbit to second orbit ?

Solution )

E1 = – 5.42 x 10^-12 ergs

E2 = – 2.41  x 10^-12 ergs

Δ E = E2 – E1 = – 2.41  x 10^-12 – (- 5.42  x 10^-12)

Δ E = 3.01 x 10 ^-12 ergs

λ = hc / ΔE

h = 6.625 x 10^-27 , c = 3 x 10 ¹⁰ cm /sec

λ = 6.625 x 10^-27 x 3 x 10 ¹⁰ / 3.01 x 10 ^-12

wavelength =  6.60 x 10^-5 cm Ans.

Question 3) Atomic radius is of the order of 10^-8 cm and nuclear radius is of the order of 10^-13 cm. Calculate what fraction of atom is occupied by nucleus ?

Solution )

Radius  of atom = 10^-8 cm

Radius  of nucleus = 10^-13 cm

volume of atom = 4 π r³/3 = 4 x π x 10^-8 x 10^-8 x 10^-8 /3

volume of nucleus =  4 π r³/3 = 4 x π x 10^-13 x 10^-13 x 10^-13 /3

volume of nucleus / volume of atom = [4 x π x 10^-13 x 10^-13 x 10^-13 /3]/ [4 x π x 10^-8 x 10^-8 x 10^-8 /3]

fraction of volume of nucleus in volume in atom = 10^-15 Ans.

Question 4) Calculate the frequency of the spectral line emitted when electron in n = 3 in H-atom de-excites to ground state.

( R_H = 109737 cm^-1)

Solution )

n1 = 1 (ground state)

n2 = 3

1 / λ =  R_H [( 1/ n₁²) – ( 1/ n₂²)]

λ = c / v

v = R_H. c [( 1/ n₁²) – ( 1/ n₂²)]

   =  109737 x 3 x 10¹⁰ [( 1/ 1²) – ( 1/ 3²)] = 

   = 329211 x 10¹⁰ [( 1/ 1) – ( 1/ 9)]

v = 2.92 x 10¹⁵ sec^-1  Ans.

Question 5) Calculate the wavelength of radiations emitted producing a line in Lyman series ,when an electron falls from fourth stationary state in H- atom. (R_H = 1.1 x 10⁷ m^-1)

Solution )

1 / λ =  R_H [( 1/ n₁²) – ( 1/ n₂²)]

For Lyman series , n1 = 1

Given , n2 = 4

1 / λ =  1.1 x 10⁷  [( 1/ 1²) – ( 1/ 4²)] 

1/λ =  1.03 x 10⁷

λ = 0.9708 x 10^-7 metre Ans.