The ionisation potential of hydrogen atom is -13.6 eV. An electron in the ground state of a hydrogen atom absorbs a photon of energy 12.75 eV. How many diggerent spectral lines can one expect when the electron make a downward transition

Options

A. 1
B. 4
C. 2
D. 6

Explanation

To solve the problem, we need to determine how many different spectral lines can be expected when an electron in the ground state of a hydrogen atom absorbs a photon of energy 12.75 eV and then makes downward transitions.

### Step-by-Step Solution:

1. **Identify the Ground State Energy Level**:
The ground state energy of a hydrogen atom is given as -13.6 eV. This means that the electron is in the n=1 state.

2. **Calculate the Energy after Absorption**:
When the electron absorbs a photon of energy 12.75 eV, we need to find the total energy of the electron after this absorption:
\[ E_{\text{final}} = E_{\text{ground}} + E_{\text{photon}} = -13.6 \, \text{eV} + 12.75 \, \text{eV} = -0.85 \, \text{eV} \]

3. **Determine the Energy Levels**:
The energy levels of the hydrogen atom are given by the formula:
\[ E_n = -\frac{13.6}{n^2} \, \text{eV} \]
We need to find the principal quantum number \( n \) for which the energy is -0.85 eV:
\[ -0.85 = -\frac{13.6}{n^2} \]
Rearranging gives:
\[ n^2 = \frac{13.6}{0.85} \approx 16 \]
Thus, \( n \approx 4 \). So, the electron can be in the n=4 state after absorbing the photon.

4. **Possible Transitions**:
The electron can transition from the n=4 state to lower energy levels (n=1, n=2, n=3). We need to find the number of different downward transitions:
- From n=4 to n=3
- From n=4 to n=2
- From n=4 to n=1
- From n=3 to n=2
- From n=3 to n=1
- From n=2 to n=1

5. **Count the Transitions**:
The possible transitions are:
- \( 4 \to 3 \)
- \( 4 \to 2 \)
- \( 4 \to 1 \)
- \( 3 \to 2 \)
- \( 3 \to 1 \)
- \( 2 \to 1 \)

This gives us a total of 6 different spectral lines.

### Final Answer:
The number of different spectral lines that can be expected when the electron makes downward transitions is **6**.