Energy levels $A, B, C$ of a certain atom correspond to increasing values of energy that is, $E_A

Energy levels $A, B, C$ of a certain atom correspond to increasing values of energy that is, $E_A < E_B < E_C$. If $\lambda_1, \lambda_2, \lambda_3$ are the wavelengths of radiations corresponding to the transitions $C$ to $B, B$ to $A$ and $C$ to $A$ respectively, as shown in figure. Which of the following statement is correct?

1. $\lambda_3=\lambda_1+\lambda_2$
2. $\lambda_3=\frac{\lambda_1 \lambda_2}{\lambda_1+\lambda_2}$
3. $\lambda_1+\lambda_2+\lambda_3=0$
4. $\lambda_3^2=\lambda_1^2+\lambda_2^2$

Solution

The given situation is shown in the following diagram,

Energy levels are related as
$E_A < E_B < E_C$
According to energy level diagram,
$\begin{aligned}
& & E_C-E_A & =\left(E_C-E_B\right)+\left(E_B-E_A\right) \\
\Rightarrow & & \frac{h c}{\lambda_3} & =\frac{h c}{\lambda_1}+\frac{h c}{\lambda_2} \\
\Rightarrow & & \frac{1}{\lambda_3} & =\frac{1}{\lambda_1}+\frac{1}{\lambda_2} \\
\Rightarrow & & \frac{1}{\lambda_3} & =\frac{\lambda_2+\lambda_1}{\lambda_1 \lambda_2} \\
\Rightarrow & & \lambda_3 & =\frac{\lambda_1 \lambda_2}{\lambda_1+\lambda_2}
\end{aligned}$

Asked in: AP EAMCET 2020 (22 Sep Shift 2)

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