Question

When the hydrogen atom in going from $n=5$ to $n=1$ state, its recoil speed is nearly

Answer 1

When the hydrogen atom is going from $n=5$ to $n=1$ state its recoil speed is nearly

Solution

We know that, momentum is given by equation

$P=\frac{h}{\lambda }\Rightarrow m\times V=\frac{h}{\lambda }V=\frac{h}{\lambda \times m}\to \left(1\right)$

For hydrogen atom emits photon during transition from $n=5$ to $n=1$

$\frac{1}{\lambda }=R(\frac{1}{n_1^2}-\frac{1}{n_2^2})$

$R=$ Rydberg's constant $=1.097\times {10}^7{m}^{-1}$

$\Rightarrow \frac{1}{\lambda }=R(\frac{1}{1^2}-\frac{1}{5^2})\lambda =\frac{25}{24\times 1.097\times {10}^7}=948\times {10}^{-8}m$

Substituiting in equation $\left(1\right)$ we get,

mass of proton$=1.6\times {10}^{-27}kg$

$\Rightarrow v=\frac{6.62\times {10}^{-34}{m}^{2}Kg/s}{1.\times {10}^{-27}\times 9.48\times {10}^{-8}}=4.36m/s$