Energy levels A, B and C of a certain atom correspond to increasing values of energy ie, $\mathrm{E}

Question

Energy levels A, B and C of a certain atom correspond to increasing values of energy ie, $\mathrm{E}_{\mathrm{A}}<\mathrm{E}_{\mathrm{B}}<\mathrm{E}_{\mathrm{C}} . \lambda_{1}, \lambda_{2}$ and $\lambda_{3}$ are the wavelengths of radiations corresponding to transitions C to $\mathrm{B}, \mathrm{B}$ to A and C to A respectively. Of the following relations, the one which is correct is\

Professor PC Thomas Classes · Answer

From the energy level diagram as shown below

$$
\begin{equation*}
\mathrm{E}_{\mathrm{C}}-\mathrm{E}_{\mathrm{B}}=\frac{\mathrm{hc}}{\lambda_{1}} 	ag{1}
\end{equation*}
$$
$$
\begin{equation*}
\mathrm{E}_{\mathrm{B}}-\mathrm{E}_{\mathrm{A}}=\frac{\mathrm{hc}}{\lambda_{2}} 	ag{2}
\end{equation*}
$$
$$
\begin{equation*}
\mathrm{E}_{\mathrm{C}}-\mathrm{E}_{\mathrm{A}}=\frac{\mathrm{hc}}{\lambda_{3}} 	ag{3}
\end{equation*}
$$
(1)+(2) $\mathrm{E}_{\mathrm{C}}-\mathrm{E}_{\mathrm{A}}=\frac{\mathrm{hc}}{\lambda_{1}}+\frac{\mathrm{hc}}{\lambda_{2}}$

From (3) $\frac{h c}{\lambda_{3}}=\frac{h c}{\lambda_{1}}+\frac{h c}{\lambda_{2}}$
$\frac{1}{\lambda_{3}}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}$
$=\frac{\lambda_{2}+\lambda_{1}}{\lambda_{1} \lambda_{2}}$
$\lambda_{3}=\frac{\lambda_{1} \lambda_{2}}{\lambda_{1}+\lambda_{2}}$