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 10^{5} cm/sec. Calculate its kinetic energy and wavelength?

### Solution )

*m = 9.1 x 10 ^{-28} gm , v = 10^{5} cm/sec , h = 6.625 x 10 ^{-27} erg*

*Kinetic energy = mv ^{2} / 2*

*= 9.1 x 10 ^{-28} x 10^{5} x 10^{5} / 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.

**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.

**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 r^{3}/3 = 4 x π x 10^{-8} x 10^{-8} x 10^{-8} /3*

*volume of nucleus = 4 π r^{3}/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.

^{-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/ n_{1}^{2}) – ( 1/ n_{2}^{2})]

**λ = R**

_{H}[( 1/ n_{1}^{2}) – ( 1/ n_{2}^{2})]* λ = c / v*

**v = ****R _{H.} c [( 1/ n_{1}^{2}) – ( 1/ n_{2}^{2})]**

** = 109737 x 3 x 10 ^{10} **[( 1/ 1

^{2}) – ( 1/ 3

^{2})] =

** = 329211 x 10 ^{10} **[( 1/ 1) – ( 1/ 9)]

**v **= 2.92 x 10^{15} sec^{-1 } Ans.

**v**= 2.92 x 10

^{15}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 10^{7} m^{-1})

### Solution )

*1 / ***λ = R**_{H} [( 1/ n_{1}^{2}) – ( 1/ n_{2}^{2})]

**λ = R**

_{H}[( 1/ n_{1}^{2}) – ( 1/ n_{2}^{2})]*For Lyman series , n1 = 1*

*Given , n2 = 4*

*1 / ***λ = 1.1 x 10**^{7} [( 1/ 1^{2}) – ( 1/ 4^{2})]

**λ = 1.1 x 10**

^{7}[( 1/ 1^{2}) – ( 1/ 4^{2})]**1/λ = 1.03 x 10**^{7}

**1/λ = 1.03 x 10**

^{7}**λ = 0.9708 x 10**^{-7} metre Ans.

**λ = 0.9708 x 10**

^{-7}metre Ans.