In which of the following transition, the wavelength will be minimum `:`

Options

A. n = 6 to n = 4
B. n = 4 to n = 2
C. n = 3 to n = 1
D. n = 2 to n = 1

Answer

To determine which transition will result in the minimum wavelength, we can use the Rydberg formula for hydrogen-like atoms:

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

Where:
- \(\lambda\) is the wavelength,
- \(R\) is the Rydberg constant,
- \(n_f\) is the final energy level,
- \(n_i\) is the initial energy level.

Since the wavelength \(\lambda\) is inversely proportional to the term \(\left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\), we need to maximize this term to minimize the wavelength.

### Step-by-Step Solution:

1. **Identify the transitions**: We are given several transitions with different \(n_f\) and \(n_i\) values. We need to calculate \(\frac{1}{n_f^2} - \frac{1}{n_i^2}\) for each option.

2. **Calculate for each option**:
- **Option 1**: \(n_f = 4\), \(n_i = 6\)
\[ \frac{1}{n_f^2} - \frac{1}{n_i^2} = \frac{1}{4^2} - \frac{1}{6^2} = \frac{1}{16} - \frac{1}{36} \]
Finding a common denominator (which is 144):
\[ = \frac{9}{144} - \frac{4}{144} = \frac{5}{144} \approx 0.0347 \]

- **Option 2**: \(n_f = 4\), \(n_i = 2\)
\[ \frac{1}{n_f^2} - \frac{1}{n_i^2} = \frac{1}{2^2} - \frac{1}{4^2} = \frac{1}{4} - \frac{1}{16} \]
Common denominator (16):
\[ = \frac{4}{16} - \frac{1}{16} = \frac{3}{16} = 0.1875 \]

- **Option 3**: \(n_f = 3\), \(n_i = 1\)
\[ \frac{1}{n_f^2} - \frac{1}{n_i^2} = \frac{1}{1^2} - \frac{1}{3^2} = 1 - \frac{1}{9} \]
Common denominator (9):
\[ = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \approx 0.8889 \]

- **Option 4**: \(n_f = 2\), \(n_i = 1\)
\[ \frac{1}{n_f^2} - \frac{1}{n_i^2} = \frac{1}{1^2} - \frac{1}{2^2} = 1 - \frac{1}{4} \]
Common denominator (4):
\[ = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} = 0.75 \]

3. **Compare the results**:
- Option 1: \(0.0347\)
- Option 2: \(0.1875\)
- Option 3: \(0.8889\) (maximum)
- Option 4: \(0.75\)

4. **Conclusion**: The maximum value of \(\frac{1}{n_f^2} - \frac{1}{n_i^2}\) occurs in Option 3, which corresponds to the minimum wavelength. Therefore, the transition with the minimum wavelength is:

**Answer**: Option 3.