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CBSE Class 12 Physics 2017 Delhi Set 1 Paper

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Question : 8 of 26
Marks: +1, -0
Find out the wavelength of the electron orbiting in the ground state of hydrogen atom.
Solution:  
Calculation of wavelength of electron in ground state:
Radius of ground state of hydrogen atom = 0.53A˚=0.53×10−10m0.53 \text{Å} = 0.53 \times 10^{-10} \mathrm{m}
According to de Broglie relation 2πr=nλ2 \pi r = n \lambda
For ground state     n=1\;\; n = 1
2×3.14×0.53×10−10  =1×λ2 \times 3.14 \times 0.53 \times 10^{-10} \; = 1 \times \lambda
∴    λ  =3.32×10−10m  \therefore \;\; \lambda \; = 3.32 \times 10^{-10} \mathrm{m} \;
  =3.32A˚  \; = 3.32 \text{\AA} \;
Alternatively
Velocity of electron, in the ground state, of hydrogen atom
=2.18×10−6m/s= 2.18 \times 10^{-6} \mathrm{m} / \mathrm{s}
Hence momentum of revolving electron
p  =mvp \; = m v
  =9.1×10−31×2.18\; = 9.1 \times 10^{-31} \times 2.18 ×10−6kgm/s\times 10^{-6} \mathrm{kg} \mathrm{m} / \mathrm{s}
λ  =  hp\lambda \; = \; \frac{h}{p} =  6.63×10−349.1×10−31×2.18×106m= \; \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \times 2.18 \times 10^6} \mathrm{m}
  =3.32A˚\; = 3.32 \text{\AA}
[Note : Also accept the following answer:
Let λn\lambda_n be the wavelength of the electron in the nthn^{\text{th}} orbit, we then have
2πrn=nλ2 \pi r_n = n \lambda
For ground state n=1n = 1
2πrn=λ2 \pi r_n = \lambda
( r=r0r = r_0 is the radius of the ground state)
[Alternatively
λn=  hmvn\lambda_n = \; \frac{h}{m v_n}
and vn=v0v_n = v_0 (velocity of electron in ground state)
λ=  hmvn\lambda = \; \frac{h}{m v_n}
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