Question

Energy levels I, II, III of a certain atom correspond to increasing values of energy i.e., $E_I<E_{II}<E_{III}$. If ${\lambda }_1$,${\lambda }_2$,${\lambda }_3$ be the wavelengths of radiations corresponding to the transitions III to II, II to I and III to I respectively, which of the following relations is correct ?

• A. ${\lambda }_1={\lambda }_2+{\lambda }_3$
• B. ${\lambda }_3=\frac{{\lambda }_1{\lambda }_2}{{\lambda }_1+{\lambda }_2}$
• C. ${\lambda }_1+{\lambda }_2+{\lambda }_3=0$
• D. $\lambda \frac{2}{3}=\lambda \frac{2}{1}+\lambda \frac{2}{2}$

Answer 1

Answer: (B)

${\lambda }_3=\frac{{\lambda }_1{\lambda }_2}{{\lambda }_1+{\lambda }_2}$

The sum of the energies for transitions III to II and II to I is equal to the energy for transition III to I. In other words, the energies and frequencies are additive whereas wavelength is inversely proportional to the frequency. So the reciprocals of the wavelengths are additive.

Thus, $\frac{1}{{\lambda }_3}=\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_1}$

or

${\lambda }_3=\frac{{\lambda }_1{\lambda }_1}{{\lambda }_1+{\lambda }_2}$