Assuming the nitrogen molecule is moving with r.m.s. velocity at 400… | JEE Main 2020 (6 Sep, Shift 2)

Q115JEE Main 2020 · 6 SEP · SHIFT 2Medium

Matter Waves: de Broglie Wavelength

Assuming the nitrogen molecule is moving with r.m.s. velocity at 400 K, the de-Broglie wavelength of nitrogen molecule is close to:

(Given: nitrogen molecule weight: 4.64 $\times$ 10–26 kg,

Boltzmann constant: 1.38 $\times$ 10–23 J/K,

Planck constant: 6.63 $\times$ 10–34 J.s)

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Answer: (d)

$v_{rms} = \sqrt{3kT/m} \Rightarrow p = \sqrt{3mkT}$ , $\lambda = \dfrac{h}{\sqrt{3mkT}}$ .
$3mkT = 3 \times 4.64 \times 10^{-26} \times 1.38 \times 10^{-23} \times 400 \approx 7.68 \times 10^{-46}$ , so $\sqrt{3mkT} \approx 2.77 \times 10^{-23}$
$\lambda = \dfrac{6.63 \times 10^{-34}}{2.77 \times 10^{-23}} \approx 2.4 \times 10^{-11}\text{ m} = 0.24\ \mathring{A}$
So the answer is (d) 0.24 Å.

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