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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) - Wave Nature of Matter

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.

Is there an error in this question or solution?

APPEARS IN

NCERT Physics Textbook for Class 12 Part 2
Chapter 11: Dual Nature of Radiation and Matter
Q: 19 | Page no. 408

Reference Material

Solution for 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) concept: Wave Nature of Matter. For the courses CBSE (Arts), CBSE (Commerce), CBSE (Science)
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