First slide

Matter waves (DC Broglie waves)

Question

The ratio of de-Brogiie wavelengths of molecules of hydrogen and helium which are at temperature 27°C and 127°C respectivell is

Moderate

A

$\frac{1}{2}$
B

$\sqrt{\frac{3}{8}}$
C

$\sqrt{\frac{8}{3}}$
D

1

Solution

de-Broglie wavelength
$\begin{array}{l} λ = \frac{h}{{mv}_{rms}}, rms velocity of a gas particle at the given temperature (T) is given as \\ \frac{1}{2} {mv}_{rms}^{2} = \frac{3}{2} KT \Rightarrow v_{rms} = \sqrt{\frac{3 KT}{m}} \Rightarrow {mv}_{rms} = \sqrt{3 mKT} \\ ∴ λ = \frac{h}{{mv}_{rms}} = \frac{h}{\sqrt{3 mKT}} \\ \Rightarrow \frac{λ_{H}}{λ_{He}} = \sqrt{\frac{m_{He} T_{He}}{m_{H} T_{H}}} = \sqrt{\frac{4 (273 + 127)}{2 (273 + 27)}} = \sqrt{\frac{8}{3}} \end{array}$

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