Question

A certain gas absorbs photon of wavelength \(4.0 \times 10^{-7} \, m\) and emits radiation at two wavelengths. If one of the emissions occurs at \(7.5 \times 10^{-7} \, m\), what is the wavelength at which the second emission occurs?

• A. 650 nm
• B. 857 nm
• C. 700 nm
• D. 680 nm

Hint:

In multi-step emission, total energy is conserved. Always use \( \frac{1}{\lambda} \) relation for quick calculations in such problems.

Answer 1

The Correct Option is B

Step 1: Concept of energy conservation.
When a photon is absorbed and then emitted in multiple steps, total energy absorbed equals total energy emitted.
\[ E_{\text{absorbed}} = E_1 + E_2 \] Energy of photon is given by:
\[ E = \frac{hc}{\lambda} \]

Step 2: Writing the energy balance equation.
Let the second emitted wavelength be \(\lambda_2\).
\[ \frac{hc}{\lambda_{\text{absorbed}}} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2} \] Cancel \(hc\):
\[ \frac{1}{\lambda_{\text{absorbed}}} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} \]

Step 3: Substituting given values.
\[ \lambda_{\text{absorbed}} = 4.0 \times 10^{-7} \, m \] \[ \lambda_1 = 7.5 \times 10^{-7} \, m \] \[ \frac{1}{4.0 \times 10^{-7}} = \frac{1}{7.5 \times 10^{-7}} + \frac{1}{\lambda_2} \]

Step 4: Simplifying the equation.
\[ \frac{1}{4.0 \times 10^{-7}} = 2.5 \times 10^{6} \] \[ \frac{1}{7.5 \times 10^{-7}} = 1.33 \times 10^{6} \] \[ 2.5 \times 10^{6} = 1.33 \times 10^{6} + \frac{1}{\lambda_2} \]

Step 5: Solving for \(\frac{1}{\lambda_2}\).
\[ \frac{1}{\lambda_2} = 2.5 \times 10^{6} - 1.33 \times 10^{6} \] \[ \frac{1}{\lambda_2} = 1.17 \times 10^{6} \]

Step 6: Finding \(\lambda_2\).
\[ \lambda_2 = \frac{1}{1.17 \times 10^{6}} \] \[ \lambda_2 \approx 8.57 \times 10^{-7} \, m \]

Step 7: Converting to nm and final answer.
\[ \lambda_2 = 857 \, nm \] \[ \boxed{857 \, nm} \] Hence, the correct answer is option (B).