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
${\mathrm{E}}_3-{\mathrm{E}}_2=\frac{\mathrm{hc}}{{\lambda }_3}\dots$ (i)
${\mathrm{E}}_2-{\mathrm{E}}_1=\frac{\mathrm{hc}}{{\lambda }_1}\dots (ii)$
${\mathrm{E}}_3-{\mathrm{E}}_1=\frac{\mathrm{hc}}{{\lambda }_1}\dots$ (iv)
Now, $E_3-E_1=\left(E_3-E_2\right)+\left(E_2-E_1\right)$

$\Rightarrow \frac{hc}{{\lambda }_1}=\frac{hc}{{\lambda }_3}+\frac{hc}{{\lambda }_2}\Rightarrow \frac{1}{{\lambda }_1}=\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_3}$