Question

The order of energies of energy levels $\text{A, B}$ and $\text{C}$ is $E_A<E_B<E_C$. If the wavelength corresponding to transition $\text{C}\to \text{B},\text{B}\to \text{A}\text{and}\text{C}\to \text{A}\text{are}{\lambda }_1,{\lambda }_2\text{and}{\lambda }_3$ respectively, then which of the following relation is correct ?

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

Answer 1

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


From energy conversation,

$E_{CB}+E_{BA}=E_{CA}$
$(E_C-E_B)+(E_B-E_A)=(E_C-E_A)$
$\frac{hc}{{\lambda }_1}+\frac{hc}{{\lambda }_2}=\frac{hc}{{\lambda }_3}$

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

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

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(D\right)$ is the correct answer.