Question

Assuming the nitrogen molecule is moving with r.m.s. velocity at $400K$ , the de-Broglie wavelength of nitrogen molecule is close to: (Given : nitrogen molecule weight: $4.64\times 10^{-26kg}$, 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)

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