First slide

Matter waves (DC Broglie waves)

Question

The ratio of de-Broglie wavelengths of molecules of hydrogen and helium which are at temperature $27^{o} C$ and $127^{o} C$ respectively is $\sqrt{\frac{x}{y}}$ . Find x+y

Easy

A

10
B

11
C

8
D

3

Solution

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

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