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 given in the options and understand the energy levels involved.

### 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. **List the Given Transitions**:
The transitions provided in the question are:
- Option 1: 4f to 2s
- Option 2: 4d to 2p
- Option 3: 4p to 2s

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

- For the transition from 4d to 2p:
- \( E_{4d} = -0.85 \, \text{eV} \) (same as 4f)
- \( E_{2p} = -3.4 \, \text{eV} \) (same as 2s)
- Energy difference: \( \Delta E_{4d \to 2p} = E_{2p} - E_{4d} = -3.4 - (-0.85) = -2.55 \, \text{eV} \)

- For the transition from 4p to 2s:
- \( E_{4p} = -0.85 \, \text{eV} \) (same as 4f and 4d)
- Energy difference: \( \Delta E_{4p \to 2s} = E_{2s} - E_{4p} = -3.4 - (-0.85) = -2.55 \, \text{eV} \)

4. **Conclusion**:
All three transitions (4f to 2s, 4d to 2p, and 4p to 2s) have the same energy difference of -2.55 eV. Therefore, they will emit the same amount of energy.

5. **Final Answer**:
Since all transitions emit the same energy, any of the options can be considered correct. However, if we are to select one, we can say that:
- The correct option is any of the three options provided since they all emit the same maximum energy.