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?

• A

  $\lambda_3=\lambda_1+\lambda_2$
• Correct answer
  B

  $\lambda_3=\frac{\lambda_1 \lambda_2}{\lambda_1+\lambda_2}$
• C

  $\lambda_1+\lambda_2+\lambda_3=0$
• D

  $\lambda_3^2=\lambda_1^2+\lambda_2^2$

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,
$\begin{aligned}<br />& & E_C-E_A & =\left(E_C-E_B\right)+\left(E_B-E_A\right) \\<br />\Rightarrow & & \frac{h c}{\lambda_3} & =\frac{h c}{\lambda_1}+\frac{h c}{\lambda_2} \\<br />\Rightarrow & & \frac{1}{\lambda_3} & =\frac{1}{\lambda_1}+\frac{1}{\lambda_2} \\<br />\Rightarrow & & \frac{1}{\lambda_3} & =\frac{\lambda_2+\lambda_1}{\lambda_1 \lambda_2} \\<br />\Rightarrow & & \lambda_3 & =\frac{\lambda_1 \lambda_2}{\lambda_1+\lambda_2}<br />\end{aligned}$

Hence, the correct answer is option: 'B'.