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

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Solution

Temperatureof the nitrogen molecule, T= 300 K
Atomicmass of nitrogen = 14.0076 u
Hence,mass of the nitrogen molecule, m= 2 × 14.0076 = 28.0152 u
But1 u = 1.66 × 10⁻²⁷kg
∴m= 28.0152 ×1.66 × 10⁻²⁷kg
Planck’sconstant, h= 6.63 × 10⁻³⁴Js
Boltzmannconstant, k= 1.38 × 10⁻²³J K⁻¹
Wehave the expression that relates mean kinetic energy of the nitrogen molecule with the root mean square speed as:
Hence, thede Broglie wavelength of the nitrogen molecule is given as:
Therefore,the de Broglie wavelength of the nitrogen molecule is 0.028 nm.

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