Question

Energy levels A,B and C of a certain atom correspond to increasing values of energy i.e., E_A<E_B<E_C. If \(\lambda_1,\lambda_2\) and \(\lambda_3\) are wavelengths of radiations corresponding to transitions C to B,B to A and C to A respectively, which of the following relations is correct ?

• A. \(\lambda_3\) = \(\lambda_1+\lambda_2\)
• B. \(\lambda_3\) = \(\frac{\lambda_1 \lambda_2}{\lambda_1+ \lambda_2}\)
• C. \(\lambda_1+\lambda_2\)\(+\lambda_3\) = 0
• D. \(\lambda_{23}\) = \(\lambda_{21}+\lambda_{22}\)

Answer 1

The Correct Option is B

 To solve the problem, we need to analyze the energy transitions and their corresponding wavelengths. Energy levels are given by \(E_A < E_B < E_C\), and the transitions are:

1. From level C to B with wavelength \(\lambda_1\).
2. From level B to A with wavelength \(\lambda_2\).
3. From level C to A with wavelength \(\lambda_3\).

According to the energy-wavelength relationship in spectroscopy, the energy of a photon emitted during a transition is related to the wavelength by the equation:

\(E = \frac{hc}{\lambda}\)

where \(h\) is Planck's constant and \(c\) is the speed of light. For each transition, the energy released is equal to the product of Planck's constant \(h\) and speed of light \(c\) divided by the corresponding wavelength. Therefore, the energy differences can be written as:

• \(E_C - E_B = \frac{hc}{\lambda_1}\)
• \(E_B - E_A = \frac{hc}{\lambda_2}\)
• \(E_C - E_A = \frac{hc}{\lambda_3}\)

Since \((E_C - E_A) = (E_C - E_B) + (E_B - E_A)\), we can equate these expressions:

\(\frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2}\)

Canceling \(hc\) from both sides, we get:

\(\frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2}\)

By taking the reciprocal of the entire equation, we derive the formula for \(\lambda_3\):

\(\lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2}\)

Therefore, the correct answer is the second option: \(\lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2}\).