Supposing the energy of the energy levels in the hydrogen atom is given under
Energy level K L M N..
n=1 n=2 n=3 n=4=n=`infty`
Total energy of an at n =1 is 864 a.u the ecitation energy needed to raise the electron from M level to `n= infty` would be :

Options

A. 192
B. 96
C. 188
D. 384

Answer

To solve the problem, we need to calculate the excitation energy required to raise an electron from the M level (n=3) to the ionization level (n=∞) in a hydrogen atom. We will use the formula for the energy difference between two energy levels in a hydrogen atom.

### Step-by-Step Solution:

1. **Identify the Energy Levels**:
- The energy levels are defined as follows:
- K level (n=1)
- L level (n=2)
- M level (n=3)
- N level (n=4)
- Ionization level (n=∞)

2. **Given Energy at n=1**:
- The total energy of the hydrogen atom at n=1 is given as \( E_1 = -864 \, \text{a.u.} \).

3. **Use the Energy Formula**:
- The energy of an electron in a hydrogen atom at a given level \( n \) can be expressed as:
\[ E_n = \frac{E_0}{n^2} \]
- Where \( E_0 \) is the energy at n=1. Given \( E_1 = -864 \, \text{a.u.} \), we can find \( E_0 \) as:
\[ E_0 = -864 \, \text{a.u.} \]

4. **Calculate the Energy at M Level (n=3)**:
- Using the formula:
\[ E_3 = \frac{-864}{3^2} = \frac{-864}{9} = -96 \, \text{a.u.} \]

5. **Calculate the Energy at Ionization Level (n=∞)**:
- At ionization level (n=∞), the energy is:
\[ E_{\infty} = 0 \, \text{a.u.} \]

6. **Calculate the Excitation Energy**:
- The excitation energy required to move from the M level (n=3) to the ionization level (n=∞) is given by the difference in energy:
\[ \text{Excitation Energy} = E_{\infty} - E_3 = 0 - (-96) = 96 \, \text{a.u.} \]

### Final Answer:
The excitation energy needed to raise the electron from the M level to \( n = \infty \) is **96 a.u.**.