A sample consisting of Hydrogen atoms in the ground state is excited by monochromatic radiation of energy 12.75 eV. If we were to observe the emission spectrum of this sample, then the number of spectral lines observed, will be

Options

A. 3
B. 6
C. 10
D. 15

Answer

To solve the problem, we need to determine the number of spectral lines that will be observed when hydrogen atoms in the ground state are excited by monochromatic radiation of energy 12.75 eV.

### Step-by-Step Solution:

1. **Identify the Ground State Energy Level:**
The ground state of hydrogen corresponds to the energy level \( n = 1 \). The energy of this level is given by:
\[ E_1 = -13.6 \, \text{eV} \]

2. **Determine the Energy of the Excited State:**
The energy of the radiation used to excite the hydrogen atoms is 12.75 eV. When this energy is absorbed, the electron transitions from the ground state to a higher energy level. The energy of the excited state can be calculated as:
\[ E_{\text{excited}} = E_1 + 12.75 \, \text{eV} = -13.6 \, \text{eV} + 12.75 \, \text{eV} = -0.85 \, \text{eV} \]
This energy corresponds to the energy level \( n = 4 \) in the hydrogen atom.

3. **Identify Possible Transitions:**
From the excited state \( n = 4 \), the electron can transition back to lower energy levels. The possible transitions are:
- \( n = 4 \) to \( n = 3 \)
- \( n = 4 \) to \( n = 2 \)
- \( n = 4 \) to \( n = 1 \)
- \( n = 3 \) to \( n = 2 \)
- \( n = 3 \) to \( n = 1 \)
- \( n = 2 \) to \( n = 1 \)

4. **Count the Number of Spectral Lines:**
The number of spectral lines can be calculated using the formula:
\[ \text{Number of spectral lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]
where \( n_2 \) is the upper energy level (4) and \( n_1 \) is the lower energy level (1). Thus:
\[ n_2 = 4, \quad n_1 = 1 \]
Substituting these values into the formula:
\[ \text{Number of spectral lines} = \frac{(4 - 1)(4 - 1 + 1)}{2} = \frac{3 \times 4}{2} = 6 \]

5. **Conclusion:**
The total number of spectral lines observed in the emission spectrum will be 6.

### Final Answer:
The number of spectral lines observed will be **6**.