 Matter waves (DC Broglie waves)
Question

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

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Solution

## de-Broglie wavelength $\mathrm{\lambda }=\frac{\mathrm{h}}{{\mathrm{mv}}_{\mathrm{rms}}}$ , rms velocity of a gas particle at the given temperature (T) is given as $\frac{1}{2}{\mathrm{mv}}_{\mathrm{rms}}^{2}=\frac{3}{2}\mathrm{kT}⇒{\mathrm{v}}_{\mathrm{rms}}=\sqrt{\frac{3\text{\hspace{0.17em}}\mathrm{kT}}{\mathrm{m}}}⇒{\mathrm{mv}}_{\mathrm{rms}}=\sqrt{3\text{\hspace{0.17em}}\mathrm{mk}\text{ }\mathrm{T}}$$\therefore \mathrm{\lambda }=\frac{\mathrm{h}}{{\mathrm{mv}}_{\mathrm{rms}}}=\frac{\mathrm{h}}{\sqrt{3\text{\hspace{0.17em}}\mathrm{mkT}}}$$⇒\frac{{\mathrm{\lambda }}_{\mathrm{H}}}{{\mathrm{\lambda }}_{\mathrm{He}}}=\sqrt{\frac{{\mathrm{m}}_{\mathrm{He}}{\mathrm{T}}_{\mathrm{He}}}{{\mathrm{m}}_{\mathrm{H}}{\mathrm{T}}_{\mathrm{H}}}}=\sqrt{\frac{4\text{\hspace{0.17em}}\left(273+127\right)}{2\text{\hspace{0.17em}}\left(273+27\right)}}=\sqrt{\frac{8}{3}}$

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