Question

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let ${\lambda }_n$, ${\lambda }_g$ be the de Broglie wavelength of the electron in the $n^{th}$ state and the ground state respectively. Let ${\Lambda }_n$ be the wavelength of the emitted photon in the transition from the $n^{th}$ state to the ground state. For large n, (A, B are constants).

• A. ${\Lambda }_n^2\approx A+B{\lambda }_n^2$
• B. ${\Lambda }_n^2\approx \lambda$
• C. ${\Lambda }_n\approx A+\frac{B}{{\lambda }_n^2}$
• D. ${\Lambda }_n\approx A+B{\lambda }_n$

Answer 1

The correct option is C ${\Lambda }_n\approx A+\frac{B}{{\lambda }_n^2}$

$h_c=En-Eg$

$\frac{h_c}{An}=\frac{z^{4}m{e}^{4}4\pi a_0^2}{8h{ϵ}_0}[\frac{1}{{\lambda }_n^2}-1]$

$\frac{1}{An}=CONSTANT[\frac{1}{{\lambda }_n^2}-1]$

$\frac{1}{An}=\frac{P}{{\lambda }_n^2}+Q$

so $An=\frac{Q{\lambda }_n^2}{{\lambda }_n^2+1}$

hence $An=A+\frac{B}{{\lambda }_n^2}$