Question

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths $r=\frac{{\lambda }_1}{{\lambda }_2}$ is given by

• A. $r=\frac{1}{3}$
• B. $r=\frac{4}{3}$
• C. $r=\frac{2}{3}$
• D. $r=\frac{3}{4}$

Answer 1

The correct option is A $r=\frac{1}{3}$

Energy of photon of wavelength ${\lambda }_1,$
Kinetic energy $=\frac{hc}{\lambda }$
$I^{st}$ case
$\Delta E_1=-2E-(-E)=\frac{hc}{{\lambda }_1}$
$\frac{hc}{{\lambda }_1}=-E....\left(i\right)$
$I{I}^{st}$ case
energy of photon of wavelength ${\lambda }_2$
$\Delta E_2=-\frac{4}{3}E-(-E)=\frac{hc}{{\lambda }_2}....\left(ii\right)$
from equation (i) and (ii)
$\frac{-E}{3}=\frac{hc}{{\lambda }_2}$
$\frac{\frac{hc}{{\lambda }_2}}{\frac{hc}{{\lambda }_1}}=\frac{-\frac{E}{3}}{-E}$
$\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{1}{3}$