Question 82: Dual Nature of Matter and Radiation

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 ground state to that in the excited state is

Accepted Answer

We know, $$\lambda  = {h \over {mv}}$$
and $$mvr = {{nh} \over {2\pi }}$$
$$ \Rightarrow mv = {{nh} \over {2\pi r}}$$
So, $$\lambda  = {h \over {nh}}2\pi r = 2\pi {r \over n}$$
$$ \Rightarrow \lambda  \propto {r \over n}$$
We can write, $${{{\lambda _g}} \over {{\lambda _e}}} = \left( {{{{r_g}} \over {{r_e}}}} ight)\left( {{{{n_e}} \over {{n_g}}}} ight)$$
$$ \Rightarrow {{{\lambda _g}} \over {{\lambda _e}}} = \left( {{{5.3 	imes {{10}^{ - 11}}} \over {84.8 	imes {{10}^{ - 11}}}}} ight)\left( {{4 \over 1}} ight) = {1 \over 4}$$
$$ \Rightarrow {{{\lambda _e}} \over {{\lambda _g}}} = 4$$

Answer

4

Answer

3

Answer

9

Answer

16

Dual Nature of Matter and Radiation

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