Question

Transitions between three energy levels in a particular atom give rise to three spectral lines of wavelengths, in increasing magnitudes $\lambda _{1}, \, \lambda _{2}$ and $\lambda _{3.}$ Which one of the following equations correctly relates $\lambda _{1}, \, \lambda _{2}$ and $\lambda _{3}$ ?

• A. $\lambda _{1}=\lambda _{2}-\lambda _{3}$
• B. $\lambda _{1}=\lambda _{3}-\lambda _{2}$
• C. $\frac{1}{\lambda _{1}}=\frac{1}{\lambda _{2}}+\frac{1}{\lambda _{3}}$
• D. $\frac{1}{\lambda _{2}}=\frac{1}{\lambda _{3}}+\frac{1}{\lambda _{1}}$

Answer 1

Answer: (C)

Let the three energy levels be $E_{1},E_{2}$ , and $E_{3}$ . The wavelength $\lambda _{1},\lambda _{2}$ and $\lambda _{3}$ of the spectra lines corresponding to the three energy transitions are depicted as shown in figure.

$E=hf=\frac{hc}{\lambda }$ or $E \propto \frac{1}{\lambda }$
(given $\lambda _{1} < \lambda _{2} < \lambda _{3}$ )
Thus, for the three wavelengths, we have
$\mathrm{E}_3-\mathrm{E}_2=\frac{\mathrm{hc}}{\lambda_3} \ldots$ (i)
$\mathrm{E}_2-\mathrm{E}_1=\frac{\mathrm{hc}}{\lambda_1} \ldots(ii)$
$\mathrm{E}_3-\mathrm{E}_1=\frac{\mathrm{hc}}{\lambda_1} \ldots$ (iv)
Now, $E_{3}-E_{1}=\left(E_{3} - E_{2}\right)+\left(E_{2} - E_{1}\right)$

$\Rightarrow \frac{hc}{\lambda _{1}}=\frac{hc}{\lambda _{3}}+\frac{hc}{\lambda _{2}}\Rightarrow \frac{1}{\lambda _{1}}=\frac{1}{\lambda _{2}}+\frac{1}{\lambda _{3}}$