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=2`

Explanation

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

### Step 1: Understand the relationship between wavelength and energy
The wavelength (λ) is inversely proportional to the energy (E) of the transition. This means that as the energy gap between the energy levels increases, the wavelength decreases. The formula relating wavelength and energy is given by:

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

Where:
- \(h\) is Planck's constant,
- \(c\) is the speed of light,
- \(E\) is the energy of the transition.

### Step 2: Identify the transitions
The transitions given in the question are:
- A: From n=6 to n=4
- B: From n=4 to n=2
- C: From n=3 to n=1
- D: From n=2 to n=2 (no transition)

### Step 3: Calculate the energy difference for each transition
Using the formula for energy difference in the context of the Rydberg formula, we can express the energy difference as:

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

Where \(R\) is the Rydberg constant.

### Step 4: Calculate the wavelength for each transition
Using the modified formula for wavelength:

\[ \lambda = \frac{n_1^2 \cdot n_2^2}{n_2^2 - n_1^2} \]

#### Transition A: From n=6 to n=4
- \(n_1 = 4\), \(n_2 = 6\)

\[ \lambda_A = \frac{4^2 \cdot 6^2}{6^2 - 4^2} = \frac{16 \cdot 36}{36 - 16} = \frac{576}{20} = 28.8 \]

#### Transition B: From n=4 to n=2
- \(n_1 = 2\), \(n_2 = 4\)

\[ \lambda_B = \frac{2^2 \cdot 4^2}{4^2 - 2^2} = \frac{4 \cdot 16}{16 - 4} = \frac{64}{12} = 5.33 \]

#### Transition C: From n=3 to n=1
- \(n_1 = 1\), \(n_2 = 3\)

\[ \lambda_C = \frac{1^2 \cdot 3^2}{3^2 - 1^2} = \frac{1 \cdot 9}{9 - 1} = \frac{9}{8} = 1.125 \]

#### Transition D: From n=2 to n=2
This transition does not occur (no change in energy levels), so:

\[ \lambda_D = \infty \quad (\text{no transition}) \]

### Step 5: Compare the wavelengths
- \(\lambda_A = 28.8\)
- \(\lambda_B = 5.33\)
- \(\lambda_C = 1.125\)
- \(\lambda_D = \infty\)

### Conclusion
The transition with the minimum wavelength is from n=3 to n=1, which corresponds to option C.