Energy levels $A, B$ and $C$ of a certain atom corresponding to increasing values of energy i.e. $E_A \lt…

Energy levels $A, B$ and $C$ of a certain atom corresponding to increasing values of energy i.e. $E_A \lt E_B \lt E_C$. If $\lambda_1$, $\lambda_2 \& \lambda_3$ are the wavelengths of a photon corresponding to the transitions shown, then
  1. $\lambda_3=\lambda_1+\lambda_2$
  2. $\lambda_3=\frac{\left(\lambda_1+\lambda_2\right)}{\lambda_1 \lambda_2}$
  3. $\lambda_3^2=\lambda_1^2+\lambda_2^2$
  4. $\lambda_3=\frac{\lambda_1 \lambda_2}{\left(\lambda_1+\lambda_2\right)}$

Solution

$\left(E_C-E_A\right)=\left(E_C-E_B\right)+\left(E_B-E_A\right)$ $\begin{aligned} & \Rightarrow \frac{\mathrm{hc}}{\lambda_3}=\frac{\mathrm{hc}}{\lambda_1}+\frac{\mathrm{hc}}{\lambda_2} \\ & \Rightarrow \frac{1}{\lambda_3}=\frac{\lambda_1+\lambda_2}{\lambda_1+\lambda_2} \\ & \therefore \lambda_3=\frac{\lambda_1 \lambda_2}{\left(\lambda_1+\lambda_2\right)}\end{aligned}$

Asked in: AP EAMCET 2024 (23 May Shift 1)

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