In which of the following transition will the wavelength be minimum ?

Options

A. `n = 5 to n = 4`
B. `n = 4 to n = 3`
C. `n = 3 to n = 2`
D. `n = 2 to n = 1`

Answer

To determine which transition results in the minimum wavelength, we can follow these steps:

### Step 1: Understand the relationship between wavelength and energy
The wavelength (\( \lambda \)) of light emitted during an electronic transition is inversely related to the energy (\( E \)) of the transition. This relationship can be expressed using the formula:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light. From this, we can deduce that a higher energy transition corresponds to a shorter wavelength.

### Step 2: Identify the transitions
The given transitions are:
1. \( n = 5 \) to \( n = 4 \)
2. \( n = 4 \) to \( n = 3 \)
3. \( n = 3 \) to \( n = 2 \)
4. \( n = 2 \) to \( n = 1 \)

### Step 3: Calculate the energy differences for each transition
The energy levels in a hydrogen atom can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
The energy difference (\( \Delta E \)) between two levels \( n_i \) and \( n_f \) is given by:
\[ \Delta E = E_{n_f} - E_{n_i} = -\frac{13.6}{n_f^2} + \frac{13.6}{n_i^2} \]

Calculating the energy differences for each transition:
1. **For \( n = 5 \) to \( n = 4 \)**:
\[ \Delta E_{5 \to 4} = -\frac{13.6}{4^2} + \frac{13.6}{5^2} = -\frac{13.6}{16} + \frac{13.6}{25} \]
\[ = -0.85 + 0.544 = -0.306 \, \text{eV} \]

2. **For \( n = 4 \) to \( n = 3 \)**:
\[ \Delta E_{4 \to 3} = -\frac{13.6}{3^2} + \frac{13.6}{4^2} = -\frac{13.6}{9} + \frac{13.6}{16} \]
\[ = -1.51 + 0.85 = -0.66 \, \text{eV} \]

3. **For \( n = 3 \) to \( n = 2 \)**:
\[ \Delta E_{3 \to 2} = -\frac{13.6}{2^2} + \frac{13.6}{3^2} = -\frac{13.6}{4} + \frac{13.6}{9} \]
\[ = -3.4 + 1.51 = -1.89 \, \text{eV} \]

4. **For \( n = 2 \) to \( n = 1 \)**:
\[ \Delta E_{2 \to 1} = -\frac{13.6}{1^2} + \frac{13.6}{2^2} = -13.6 + 3.4 \]
\[ = -10.2 \, \text{eV} \]

### Step 4: Compare the energy differences
The transition with the maximum energy difference will correspond to the minimum wavelength. From our calculations:
- \( \Delta E_{5 \to 4} = -0.306 \, \text{eV} \)
- \( \Delta E_{4 \to 3} = -0.66 \, \text{eV} \)
- \( \Delta E_{3 \to 2} = -1.89 \, \text{eV} \)
- \( \Delta E_{2 \to 1} = -10.2 \, \text{eV} \)

### Step 5: Conclusion
The transition \( n = 2 \) to \( n = 1 \) has the highest energy difference, and thus it corresponds to the minimum wavelength. Therefore, the correct answer is:
**Option D: \( n = 2 \) to \( n = 1 \)**.