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 will have the minimum wavelength, we can use the formula for the wavelength of emitted light during electronic transitions in an atom, 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,
- \(n_1\) is the principal quantum number of the lower energy level,
- \(n_2\) is the principal quantum number of the higher energy level.

### Step-by-Step Solution:

1. **Identify the Transitions**: We need to evaluate the transitions given in the options. Let's assume the options are:
- A: \(n_1 = 4\), \(n_2 = 5\)
- B: \(n_1 = 3\), \(n_2 = 4\)
- C: \(n_1 = 2\), \(n_2 = 3\)
- D: \(n_1 = 1\), \(n_2 = 2\)

2. **Calculate \(\frac{1}{\lambda}\) for Each Transition**:
- For Option A (\(n_1 = 4\), \(n_2 = 5\)):
\[ \frac{1}{\lambda_A} = RZ^2 \left( \frac{1}{4^2} - \frac{1}{5^2} \right) = RZ^2 \left( \frac{1}{16} - \frac{1}{25} \right) = RZ^2 \left( \frac{25 - 16}{400} \right) = \frac{9RZ^2}{400} \]
- For Option B (\(n_1 = 3\), \(n_2 = 4\)):
\[ \frac{1}{\lambda_B} = RZ^2 \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = RZ^2 \left( \frac{1}{9} - \frac{1}{16} \right) = RZ^2 \left( \frac{16 - 9}{144} \right) = \frac{7RZ^2}{144} \]
- For Option C (\(n_1 = 2\), \(n_2 = 3\)):
\[ \frac{1}{\lambda_C} = RZ^2 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = RZ^2 \left( \frac{1}{4} - \frac{1}{9} \right) = RZ^2 \left( \frac{9 - 4}{36} \right) = \frac{5RZ^2}{36} \]
- For Option D (\(n_1 = 1\), \(n_2 = 2\)):
\[ \frac{1}{\lambda_D} = RZ^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = RZ^2 \left( 1 - \frac{1}{4} \right) = RZ^2 \left( \frac{3}{4} \right) = \frac{3RZ^2}{4} \]

3. **Compare the Values of \(\frac{1}{\lambda}\)**:
- \(\frac{1}{\lambda_A} = \frac{9RZ^2}{400}\)
- \(\frac{1}{\lambda_B} = \frac{7RZ^2}{144}\)
- \(\frac{1}{\lambda_C} = \frac{5RZ^2}{36}\)
- \(\frac{1}{\lambda_D} = \frac{3RZ^2}{4}\)

4. **Determine the Minimum Wavelength**:
The transition with the highest value of \(\frac{1}{\lambda}\) will correspond to the minimum wavelength. Comparing the fractions:
- \(\frac{3RZ^2}{4} > \frac{5RZ^2}{36} > \frac{7RZ^2}{144} > \frac{9RZ^2}{400}\)

Therefore, the transition with the minimum wavelength is **Option D**.

### Final Answer:
The transition with the minimum wavelength is **Option D**.