Which of the following transition will emit maximum energy in hydrogen atom ?

Options

A. `4f to 2s`
B. `4d to 2p`
C. `4p to 2s`
D. all have same enegy

Answer

To determine which transition will emit maximum energy in a hydrogen atom, we need to analyze the transitions provided in the options and understand the concept of energy levels in hydrogen.

### Step-by-Step Solution:

1. **Identify the Energy Levels**:
In a hydrogen atom, the energy levels are determined by the principal quantum number (n). The energy of an electron in a hydrogen atom is given by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number.

2. **Calculate Energy Levels for Each Transition**:
- For the transition from **4f to 2s**:
- \( n = 4 \): \( E_4 = -\frac{13.6}{4^2} = -\frac{13.6}{16} = -0.85 \, \text{eV} \)
- \( n = 2 \): \( E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV} \)
- Energy emitted: \( E_{emit} = E_2 - E_4 = -3.4 - (-0.85) = -3.4 + 0.85 = -2.55 \, \text{eV} \)

- For the transition from **4d to 2p**:
- The energy calculations will be the same as above since 4d is also in the 4th energy level.
- Energy emitted: \( E_{emit} = -2.55 \, \text{eV} \)

- For the transition from **4p to 2s**:
- Again, the energy calculations will be the same as above since 4p is also in the 4th energy level.
- Energy emitted: \( E_{emit} = -2.55 \, \text{eV} \)

3. **Compare the Energy Emitted**:
Since all transitions (4f to 2s, 4d to 2p, and 4p to 2s) involve a transition from the same higher energy level (n=4) to the same lower energy level (n=2), they will all emit the same amount of energy.

4. **Conclusion**:
Therefore, the correct answer is:
**Option D: All have the same energy.**