Assuming the nitrogen molecule is moving with \(r.m.s.\) velocity at \(400\, K\), the de\(-\)Broglie wavelength of nitrogen molecule is close to\(......\mathring {A}\) (Given : nitrogen molecule weight : \(4.64 \times 10^{-26}\, kg ,\) Boltzman constant : \(1.38 \times 10^{-23}\, J / K ,\) Planck constant : \(\left.6.63 \times 10^{-34}\, J . s \right)\)

\(m \rightarrow\) mass of one molecule (in \(kg\) ) \(=\) \(\frac{\text { molar mass }}{ NA }\)

de-Broglie wavelenth,

\(\lambda=\frac{ h }{ mv }\)

given, \(v = v _{ rms }\)

\(\lambda=\frac{ h }{ m \sqrt{\frac{3 KT }{ m }}}\)

\(\lambda=\frac{ h }{\sqrt{3 K T m }}\)

\(=\frac{6.63\times 10^{-34}}{\sqrt{3 \times 1.38 \times 10^{-23} \times 400 \times\left(\frac{28 \times 10^{-3}}{6.023 \times 10^{-23}}\right)}}\)

\(\lambda=\frac{6.63 \times 10^{-11}}{2.77}=2.39 \times 10^{-11} m\)

\(\lambda=0.24\dot A\)