An atom absorbs a photon of wavelength 500   nm and emits another photon of wavelength 600   nm .…

The energy of a photon is given by $E=\frac{hc}{\lambda }$.

It is given that ${\lambda }_1=500 \mathrm{nm}$, ${\lambda }_2=600 \mathrm{nm}$.

The net energy absorbed is 

$\Delta E=\frac{hc}{{\lambda }_1}-\frac{hc}{{\lambda }_2}=\frac{hc}{{10}^{-9}}(\frac{1}{500}-\frac{1}{600})$

$=\frac{6.6\times {10}^{-34}\times 3\times {10}^8\times 100}{500\times 600\times {10}^{-9}}$

$=\frac{6.6\times 3}{30}\times {10}^{-19} \mathrm{J}$

$1 \mathrm{eV}=1.6\times {10}^{-19} \mathrm{J}$

$=\frac{6.6\times 3}{30\times 1.6}eV$

$=4125\times {10}^{-4} \mathrm{eV}$