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What is the De Broglie Wavelength of a Nitrogen Molecule in Air at 300 K? Assume that the Molecule is Moving with the Root-mean Square Speed of Molecules at this Temperature. (Atomic Mass of Nitrogen = 14.0076 U) - CBSE (Science) Class 12 - Physics

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Question

What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root-mean square speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u)

Solution

Temperature of the nitrogen molecule, T = 300 K

Atomic mass of nitrogen = 14.0076 u

Hence, mass of the nitrogen molecule, m = 2 × 14.0076 = 28.0152 u

But 1 u = 1.66 × 10−27 kg

m = 28.0152 ×1.66 × 10−27 kg

Planck’s constant, h = 6.63 × 10−34 Js

Boltzmann constant, k = 1.38 × 10−23 J K−1

We have the expression that relates mean kinetic energy `(3/2 kT)` of the nitrogen molecule with the root mean square speed `(V_(nms))` as:

`1/2 mv_"rms"^2 = 3/2 kT`

`v_"rms" = sqrt((3kT)/m)`

Hence, the de Broglie wavelength of the nitrogen molecule is given as:

`lambda = h/(mv_"rms") = h/sqrt(3mkT)`

`= (6.63 xx 10^(-34))/sqrt(3xx28.0152 xx 1.66 xx 10^(-27) xx 1.38 xx 20^(-23)  xx  300 )`

` = 0.028  xx 10^(-9) m`

= 0.028 nm

Therefore, the de Broglie wavelength of the nitrogen molecule is 0.028 nm.

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 NCERT Solution for Physics Textbook for Class 12 (2018 to Current)
Chapter 11: Dual Nature of Radiation and Matter
Q: 19 | Page no. 408

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Solution What is the De Broglie Wavelength of a Nitrogen Molecule in Air at 300 K? Assume that the Molecule is Moving with the Root-mean Square Speed of Molecules at this Temperature. (Atomic Mass of Nitrogen = 14.0076 U) Concept: Wave Nature of Matter.
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