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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)

Answer:

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

Therefore, 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 (vrms) as:

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


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