Question

A gas absorbs a photon of $300nm$ and then re-emits two photons to come back to its original state. One photon has a wavelength $900nm.$ The wavelength in nm of second photon is:

Answer 1

Energy of absorbed photon $\left(E_1\right)$, equals the energy of the two emitted photons $E_2$ & $E_3$ corresponding to wavelengths ${\lambda }_2(=900nm)$ & ${\lambda }_3$ respectively.

$E_1$=$E_2$+$E_3$
$\frac{hc}{{\lambda }_1}=\frac{hc}{{\lambda }_2}+\frac{hc}{{\lambda }_3}$
$\frac{hc}{300}=\frac{hc}{900}+\frac{hc}{{\lambda }_3}$
$\frac{1}{{\lambda }_3}=\frac{1}{300}-\frac{1}{900}=\frac{2}{900}=\frac{1}{450}$
$\therefore {\lambda }_3$ = 450 nm