Question

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: $4.64 \times 10^{-26 kg }$, Boltzman constant : $1.38 \times 10^{-23} \, J / K$, Planck constant: $6.63 \times 10^{-34} J .s)$

• A. $0.34\,\mathring{A}$
• B. $0.24 \,\mathring{A}$
• C. $0.20\,\mathring{A} $
• D. $0.44\,\mathring{A}$

Answer 1

Answer: (B)

$v_{\text{ rms }}=\sqrt{\frac{3 KT }{ m }}$
$m \to$ mass of one molecule (in kg) $=\frac{\text { molar mass }}{\text { NA }}$
de-Broglie wavelenth,
$\lambda=\frac{ h }{ mv }$
given, $v = v _{\text{ rms }}$
$\lambda=\frac{ h }{ m \sqrt{\frac{3 KT }{ m }}}=\frac{ h }{\sqrt{3 KTm }}$
$=\frac{6.63 \times 10^{-34}}{\sqrt{3 \times 1.38 \times 10^{-23} \times 400 \times\left(\frac{28 \times 10^{-3}}{6.023 \times 10^{-23}}\right)}}$
$\lambda=\frac{6.63 \times 10^{-11}}{2.77}=2.39 \times 10^{-11}\, m$
$\lambda=0.24 \,\mathring{A}$