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. $\frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_3}-\frac{1}{{\lambda }_2}$

Answer 1

The correct option is C $\frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_3}$

Let the three energy levels be $E_1,E_2$ and $E_3$. The wavelengths ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ of the spectral lines corresponding to the three energy transitions are depicted as shown in Fig.

$E=hf=\frac{h}{\lambda }$ or $E\propto \frac{1}{\lambda }$
(given ${\lambda }_1<{\lambda }_2<{\lambda }_3$)
Thus, for the three wavelengths, we have
$\{\begin{array}{c}{E}_3-E_2=\frac{h}{{\lambda }_3}\left(i\right)\\ E_2-E_1=\frac{h}{{\lambda }_2}\left(ii\right)\\ E_3-E_1=\frac{h}{{\lambda }_1}\left(iii\right)\end{array}$
Now, $E_3-E_1=(E_3-E_2)+(E_2-E_1)$
$\Rightarrow \frac{h}{{\lambda }_1}=\frac{h}{{\lambda }_3}+\frac{h}{{\lambda }_2}\Rightarrow \frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_3}$