In a sample of hydrogen atoms, all the atoms exist in two energy levels A and B. A is the ground level and B is some higher energy level. These atoms absorb photons of energy 2.7 eV and attain a higher energy level C.After this, these atoms emit photons of six different energies. Some of these photon energies are higher than 2.7 eV, some equal to 2.7 eV and some are loss than 2.7 eV.
The principal quantum number corresponding to energy level C is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the principal quantum number corresponding to energy level C in the given hydrogen atom scenario, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy Levels**:
- We have two initial energy levels, A (ground state) and B (higher state). The atoms absorb a photon of energy 2.7 eV to reach a new energy level C.

2. **Photon Absorption**:
- The energy absorbed (2.7 eV) allows the electrons to transition from their initial energy levels (A or B) to the higher energy level C.

3. **Photon Emission**:
- After reaching energy level C, the atoms emit photons of six different energies. The emitted photon energies can be higher than, equal to, or less than the absorbed energy (2.7 eV).

4. **Using the Formula for Emission**:
- The number of different photon energies emitted when an electron transitions from a higher energy level (n) to lower energy levels is given by the formula:
\[ \text{Number of emissions} = \frac{1}{2} n (n - 1) \]
- Here, we know that the number of different emitted photon energies is 6.

5. **Setting Up the Equation**:
- We can set up the equation:
\[ \frac{1}{2} n (n - 1) = 6 \]
- Multiplying both sides by 2 gives:
\[ n(n - 1) = 12 \]

6. **Rearranging the Equation**:
- Rearranging the equation gives us:
\[ n^2 - n - 12 = 0 \]

7. **Factoring the Quadratic**:
- We can factor the quadratic equation:
\[ (n - 4)(n + 3) = 0 \]

8. **Finding the Solutions**:
- This gives us two potential solutions for n:
- \( n = 4 \)
- \( n = -3 \) (which is not a valid quantum number)

9. **Conclusion**:
- Therefore, the only valid solution is:
\[ n = 4 \]
- The principal quantum number corresponding to energy level C is 4.

### Final Answer:
The principal quantum number corresponding to energy level C is **4**.

---