According to Bohr theory, which of the following transition in hydrogen atom will give rise to the least energetic proton?

Options

A. n=5 to n=3
B. n=6 to n=1
C. n=5 to n=4
D. n=6 to n=5

Explanation

To solve the question regarding which transition in a hydrogen atom will give rise to the least energetic photon according to Bohr's theory, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Energy Formula**: According to Bohr's theory, the energy of a photon emitted during a transition between two energy levels in a hydrogen atom can be calculated using the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number of the energy level.

2. **Identify the Transition**: The energy of the photon emitted during a transition from a higher energy level \( n_2 \) to a lower energy level \( n_1 \) is given by:
\[ E = E_{n_1} - E_{n_2} = -\frac{13.6}{n_1^2} + \frac{13.6}{n_2^2} \]
This can be simplified to:
\[ E = 13.6 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

3. **Determine the Least Energetic Transition**: To find the least energetic photon, we need to maximize \( n_1 \) and \( n_2 \) since energy is inversely proportional to \( n^2 \). The transition with the highest values of \( n_1 \) and \( n_2 \) will yield the least energy.

4. **Evaluate the Given Options**:
- Option 1: Transition from \( n=5 \) to \( n=3 \)
- Option 2: Transition from \( n=6 \) to \( n=1 \)
- Option 3: Transition from \( n=5 \) to \( n=4 \)
- Option 4: Transition from \( n=6 \) to \( n=5 \)

5. **Calculate \( n_1^2 \) and \( n_2^2 \) for Each Option**:
- Option 1: \( n_1 = 3, n_2 = 5 \) → \( \frac{1}{3^2} - \frac{1}{5^2} \)
- Option 2: \( n_1 = 1, n_2 = 6 \) → \( \frac{1}{1^2} - \frac{1}{6^2} \)
- Option 3: \( n_1 = 4, n_2 = 5 \) → \( \frac{1}{4^2} - \frac{1}{5^2} \)
- Option 4: \( n_1 = 5, n_2 = 6 \) → \( \frac{1}{5^2} - \frac{1}{6^2} \)

6. **Identify the Maximum Values**: The transition from \( n=6 \) to \( n=5 \) has the highest values for \( n_1 \) and \( n_2 \) (6 and 5), which means it will yield the least energetic photon.

7. **Conclusion**: Therefore, the transition that gives rise to the least energetic photon is from \( n=6 \) to \( n=5 \).

### Final Answer
The correct option is **Option 4: Transition from \( n=6 \) to \( n=5 \)**.