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

Explanation

To determine which transition results in the minimum wavelength, we can use the Rydberg formula for hydrogen-like atoms, which is given by:

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

Where:
- \(\lambda\) is the wavelength,
- \(R\) is the Rydberg constant,
- \(Z\) is the atomic number,
- \(n_1\) and \(n_2\) are the principal quantum numbers of the two energy levels involved in the transition, with \(n_2 > n_1\).

### Step-by-Step Solution:

1. **Identify the transitions and their quantum numbers**:
- Option A: \(n_1 = 2\), \(n_2 = 5\)
- Option B: \(n_1 = 3\), \(n_2 = 4\)
- Option C: \(n_1 = 2\), \(n_2 = 3\)
- Option D: \(n_1 = 1\), \(n_2 = 2\)

2. **Calculate the wavelength for each transition**:
- **Option A**: Transition from \(n = 5\) to \(n = 2\)
\[ \frac{1}{\lambda_A} = R \left( \frac{1}{2^2} - \frac{1}{5^2} \right) = R \left( \frac{1}{4} - \frac{1}{25} \right) = R \left( \frac{25 - 4}{100} \right) = R \left( \frac{21}{100} \right) \]
\[ \lambda_A = \frac{100}{21R} \]

- **Option B**: Transition from \(n = 4\) to \(n = 3\)
\[ \frac{1}{\lambda_B} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{16 - 9}{144} \right) = R \left( \frac{7}{144} \right) \]
\[ \lambda_B = \frac{144}{7R} \]

- **Option C**: Transition from \(n = 3\) to \(n = 2\)
\[ \frac{1}{\lambda_C} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{9 - 4}{36} \right) = R \left( \frac{5}{36} \right) \]
\[ \lambda_C = \frac{36}{5R} \]

- **Option D**: Transition from \(n = 2\) to \(n = 1\)
\[ \frac{1}{\lambda_D} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \]
\[ \lambda_D = \frac{4}{3R} \]

3. **Compare the wavelengths**:
- \(\lambda_A = \frac{100}{21R}\)
- \(\lambda_B = \frac{144}{7R}\)
- \(\lambda_C = \frac{36}{5R}\)
- \(\lambda_D = \frac{4}{3R}\)

4. **Determine the minimum wavelength**:
- To find the minimum wavelength, we need to compare the values obtained:
- \(\lambda_A \approx 4.76\)
- \(\lambda_B \approx 20.57\)
- \(\lambda_C \approx 7.2\)
- \(\lambda_D \approx 1.33\)

From the calculations, we find that the minimum wavelength occurs for **Option D**: Transition from \(n = 2\) to \(n = 1\).

### Final Answer:
The transition with the minimum wavelength is from \(n = 2\) to \(n = 1\).