Assuming the nitrogen molecule is moving with r.m.s.velocity at $400 \mathrm{~K}$, the de-Broglie wavelength of nitrongen molecule is close to :
(Given : nitrogen molecule weight : $4.64 \times 10^{-26} \mathrm{~kg}$, Boltzman constant : $1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$, Planck constant : $\left.6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s}\right)$
Correct Option: 1,
(1) Rms speed of gas molecule, $V_{r m s}=\sqrt{\frac{3 k T}{m}}$
de Broglie wavelength, $\lambda=\frac{h}{p}=\frac{h}{\sqrt{2 m k}}$
$\therefore \lambda=\frac{h}{\sqrt{2 m \times \frac{1}{2} m V_{r m s}^{2}}}=\frac{h}{\sqrt{m \times \frac{3}{2} k T}}=\frac{h}{\sqrt{3 m k T}}$
Substituting the respective values we get
$\lambda=\frac{6.63 \times 10^{-34}}{\sqrt{3 \times 4.64 \times 10^{-26} \times 1.38 \times 10^{-13} \times 400}}=0.24 \AA$