A gas absorbs a photon of `310nm` and emits three photons.
If energy of emitted photons is in the ratio `1:2:1` then select correct statement(s) :

Options

A. The wave length of emitted photons is in the ratio `1: 2:1`
B. The wave length of emitted photons is in the ratio `2:1:2`
C. The wave length of emitted photons is in the ratio `1240 nm`.
D. The wavelength of least energatic photon is `1310 nm`.

Explanation

To solve the problem step by step, we will analyze the absorption and emission of photons by the gas, using the given information about the wavelengths and energy ratios.

### Step 1: Understand the Absorption of Photon
The gas absorbs a photon of wavelength \( \lambda_1 = 310 \, \text{nm} \).

### Step 2: Calculate the Energy of the Absorbed Photon
The energy of a photon can be calculated using the formula:
\[ E = \frac{12400}{\lambda} \quad (\text{where } \lambda \text{ is in nm}) \]
For the absorbed photon:
\[ E_1 = \frac{12400}{310} \, \text{eV} \]

### Step 3: Determine the Energy of Emitted Photons
The gas emits three photons with energies in the ratio \( 1:2:1 \). Let the energies of the emitted photons be \( E_1, E_2, E_3 \) such that:
\[ E_1 : E_2 : E_3 = 1 : 2 : 1 \]
This implies:
\[ E_1 = E_3 \quad \text{and} \quad E_2 = 2E_1 \]

### Step 4: Express Total Energy of Emitted Photons
The total energy of the emitted photons can be expressed as:
\[ E_{\text{total}} = E_1 + E_2 + E_3 = E_1 + 2E_1 + E_1 = 4E_1 \]

### Step 5: Set Up the Energy Conservation Equation
Since the energy absorbed must equal the total energy emitted:
\[ E_1 = 4E_1 \]
Substituting the expression for \( E_1 \):
\[ \frac{12400}{310} = 4E_1 \]

### Step 6: Solve for \( E_1 \)
From the equation:
\[ E_1 = \frac{12400}{310 \times 4} = \frac{12400}{1240} = 10 \, \text{eV} \]

### Step 7: Calculate the Wavelengths of Emitted Photons
Using the energy values, we can calculate the wavelengths of the emitted photons:
- For \( E_1 \):
\[ E_1 = \frac{12400}{\lambda_1} \implies \lambda_1 = \frac{12400}{E_1} = \frac{12400}{10} = 1240 \, \text{nm} \]
- For \( E_2 \):
\[ E_2 = 2E_1 = 20 \, \text{eV} \implies \lambda_2 = \frac{12400}{20} = 620 \, \text{nm} \]
- For \( E_3 \):
\[ E_3 = E_1 = 10 \, \text{eV} \implies \lambda_3 = 1240 \, \text{nm} \]

### Step 8: Write the Ratio of Wavelengths
The wavelengths of the emitted photons are:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 1240 : 620 : 1240 \]
This simplifies to:
\[ 2 : 1 : 2 \]

### Conclusion
The correct statement based on the calculations is that the wavelengths of the emitted photons are in the ratio \( 2 : 1 : 2 \).