In which transition, one quantum of energy is emitted -
(a). `n=4 rarr n=2`
(b). `n=3 rarr n=1`
(c). `n=4 rarr n=1`
(d). `n=2 rarr n=1`

Options

A. `n=4 rarr n=2`
B. `n=3 rarr n=1`
C. `n=4 rarr n=1`
D. `n=2 rarr n=1`

Answer

To determine in which transition one quantum of energy is emitted, we need to analyze the transitions between energy levels in an atom, specifically in a hydrogen-like atom. The energy emitted during a transition can be calculated using the formula:

\[ E = -R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

where:
- \( E \) is the energy emitted,
- \( R_H \) is the Rydberg constant (approximately 13.6 eV for hydrogen),
- \( n_i \) is the initial energy level,
- \( n_f \) is the final energy level.

Let's analyze each option:

### Step 1: Analyze each transition

1. **Transition from \( n=4 \) to \( n=2 \)**:
\[ E = -R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right) = -R_H \left( \frac{1}{4} - \frac{1}{16} \right) = -R_H \left( \frac{4-1}{16} \right) = -R_H \left( \frac{3}{16} \right) \]
This transition emits energy.

2. **Transition from \( n=3 \) to \( n=1 \)**:
\[ E = -R_H \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = -R_H \left( 1 - \frac{1}{9} \right) = -R_H \left( \frac{9-1}{9} \right) = -R_H \left( \frac{8}{9} \right) \]
This transition emits energy.

3. **Transition from \( n=4 \) to \( n=1 \)**:
\[ E = -R_H \left( \frac{1}{1^2} - \frac{1}{4^2} \right) = -R_H \left( 1 - \frac{1}{16} \right) = -R_H \left( \frac{16-1}{16} \right) = -R_H \left( \frac{15}{16} \right) \]
This transition emits energy.

4. **Transition from \( n=2 \) to \( n=1 \)**:
\[ E = -R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -R_H \left( 1 - \frac{1}{4} \right) = -R_H \left( \frac{4-1}{4} \right) = -R_H \left( \frac{3}{4} \right) \]
This transition emits energy.

### Conclusion:
All transitions emit energy as they all involve moving from a higher energy level to a lower energy level. Therefore, the answer is that one quantum of energy is emitted in all the transitions listed in options (a), (b), (c), and (d).

### Final Answer:
All options (a), (b), (c), and (d) emit one quantum of energy.

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