An electron in the ground state of the hydrogen atom has the orbital radius of $5.3 imes 10^{-11} \mathrm{~m}$ while that for the electron in third excited state is $8.48 imes 10^{-10} \mathrm{~m}$. The ratio of the de Broglie wavelengths of electron in the excited state to that in the ground…

Question

An electron in the ground state of the hydrogen atom has the orbital radius of $5.3 	imes 10^{-11} \mathrm{~m}$ while that for the electron in third excited state is $8.48 	imes 10^{-10} \mathrm{~m}$. The ratio of the de Broglie wavelengths of electron in the excited state to that in the ground state is

Aicharya · Accepted Answer

$\lambda=\frac{\mathrm{h}}{\mathrm{mv}}$
$\begin{aligned}
& \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \\ & \mathrm{mv}=\frac{\mathrm{nh}}{2 \pi \mathrm{r}} \\ & \lambda=\frac{2 \pi \mathrm{rh}}{\mathrm{nh}} \\ & \lambda \propto \frac{\mathrm{r}}{\mathrm{n}} \\ & \frac{\lambda_1}{\lambda_4}=\frac{\mathrm{r}_1 \mathrm{n}_4}{\mathrm{n}_1 \mathrm{r}_4}=\frac{5.3 \times 10^{-11} \times 4}{1 \times 84.8 \times 10^{-11}} \\ & \frac{\lambda_1}{\lambda_4}=\frac{1}{4}
\end{aligned}$

An electron in the ground state of the hydrogen atom has the orbital radius of $5.3 \times 10^{-11}…

Solution