Which of the following transitions have minimum wavelength

Options

A. `n_4 to n_1`
B. `n_2 to n_1`
C. `n_4 to n_2`
D. `n_3 to n_1`

Explanation

To determine which transition has the minimum wavelength, we need to analyze the energy transitions of electrons in an atom. The relationship between energy (E), wavelength (λ), and the speed of light (c) is given by the equation:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( E \) is the energy of the photon emitted or absorbed during the transition,
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength.

### Step-by-Step Solution:

1. **Understanding Energy Levels**:
The energy levels of an electron in an atom are quantized. For hydrogen-like atoms, these levels are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

2. **Identifying the Transitions**:
The transitions we need to consider are:
- Transition from n=4 to n=1
- Transition from n=2 to n=1
- Transition from n=4 to n=2
- Transition from n=3 to n=1

3. **Calculating Energy Differences**:
The energy difference (\( \Delta E \)) for each transition can be calculated as:
\[ \Delta E = E_{initial} - E_{final} \]
- For \( n=4 \) to \( n=1 \):
\[ \Delta E_{4 \to 1} = E_4 - E_1 \]
- For \( n=2 \) to \( n=1 \):
\[ \Delta E_{2 \to 1} = E_2 - E_1 \]
- For \( n=4 \) to \( n=2 \):
\[ \Delta E_{4 \to 2} = E_4 - E_2 \]
- For \( n=3 \) to \( n=1 \):
\[ \Delta E_{3 \to 1} = E_3 - E_1 \]

4. **Determining Maximum Energy Gap**:
The transition with the maximum energy difference will correspond to the minimum wavelength. The transitions can be compared:
- \( n=4 \) to \( n=1 \) has the largest energy difference because it involves the highest energy level dropping to the lowest.
- Other transitions will have smaller energy differences.

5. **Conclusion**:
Since the transition from \( n=4 \) to \( n=1 \) has the maximum energy difference, it will correspond to the minimum wavelength. Therefore, the correct answer is:

**Answer**: Transition from \( n=4 \) to \( n=1 \) has the minimum wavelength.