A hydrogen like atom in ground st6ate abserbs n photon having the same energy and its emit exacity n photon when electron transition tekes placed .Then the energy of the absorbed photon may be

Options

A. `91.8 eV`
B. `40.8 eV`
C. `48.4 eV`
D. `54.4 eV`

Explanation

To solve the problem of a hydrogen-like atom in the ground state absorbing \( n \) photons and emitting \( n \) photons during electron transitions, we need to analyze the energy levels and the transitions involved.

### Step-by-Step Solution:

1. **Understanding the Energy Levels**:
- The energy levels of a hydrogen-like atom are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
- For hydrogen (\( Z = 1 \)), the energy levels are:
- \( E_1 = -13.6 \, \text{eV} \) (ground state)
- \( E_2 = -3.4 \, \text{eV} \)

2. **Calculating the Energy Difference**:
- The energy difference between the ground state (\( n=1 \)) and the first excited state (\( n=2 \)) is:
\[ \Delta E = E_2 - E_1 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \]

3. **Photon Absorption and Emission**:
- When the atom absorbs \( n \) photons, it must absorb enough energy to excite the electron from \( n=1 \) to \( n=2 \). The energy of each absorbed photon must therefore be:
\[ E_{\text{photon}} = \frac{\Delta E}{n} = \frac{10.2 \, \text{eV}}{n} \]

4. **Condition for Integer Values**:
- For the energy of the absorbed photon to be valid, \( \frac{10.2 \, \text{eV}}{n} \) must yield an integer value. This means \( n \) must be a divisor of \( 10.2 \).

5. **Finding Divisors of 10.2**:
- The possible integer values for \( n \) that satisfy this condition are those that divide \( 10.2 \). The integer divisors of \( 10.2 \) are \( 1, 2, 3, 6, \) and \( 10 \).

6. **Conclusion**:
- Therefore, the energy of the absorbed photon can be expressed as:
\[ E_{\text{photon}} = \frac{10.2 \, \text{eV}}{n} \quad \text{for } n = 1, 2, 3, 6, 10 \]

### Final Answer:
The energy of the absorbed photon may be \( \frac{10.2 \, \text{eV}}{n} \) where \( n \) is an integer divisor of \( 10.2 \).