Question

For a hypothetical $H$-like atom , the wavelength in $A^o$ for the spectral lines for radiations from $n=p$ to $n=1$ are given by $\lambda =\frac{1500p^2}{p^2-1}$; where $p=2,3,\mathrm{4.....}$
The wavelengths of most energetic photon and least energetic photons (in $A^o$) during the transition are:

• A. $1500,2000A^o$
• B. $2000,1500A^o$
• C. $1000,2000A^o$
• D. $2000,1000A^o$

Answer 1

The correct option is A $1500,2000A^o$

$E_p=\frac{hc}{\lambda }$ and $\lambda =\frac{1500p^2}{p^2-1}$
$\frac{1}{\lambda }=\frac{1}{1500}[\frac{1}{1^2}-\frac{1}{p^2}]$
$\Delta E=\frac{hc}{1500}[\frac{1}{1^2}-\frac{1}{p^2}]\times {10}^{10}J$ $(1A^o={10}^{-10}m)$
$\Delta E=\frac{hc}{1500\times 1.6\times {10}^{-19}[\frac{1}{1^2}-\frac{1}{p^2}]}\times {10}^{10}eV$
$\Delta E=8.28[\frac{1}{1^2}-\frac{1}{p^2}]eV$
$E_p=-\frac{8.28}{p^2}eV$
Maximum $\lambda$ when electron jumps from $p=\infty$ to $1^{st}$ level
$E_p=8.28eV$
$\lambda =\frac{12375}{8.28}=1500A^o$
Minimum $\lambda$ when electron jumps from $p=2$ to $1^{st}$ level
$E_p=8.28\times \frac{3}{4}$
$\lambda =\frac{12375\times 4}{8.28\times 3}=2000A^o$