According to the Bohr Theory, the third from the red end corresponds to which one of the following transitions in the hydrogen atom will give rise to the least energetic photon ?

Options

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

Explanation

To solve the problem of determining which transition in the hydrogen atom corresponds to the least energetic photon according to Bohr's theory, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Formula**:
The energy of the photon emitted during a transition between two energy levels in a hydrogen atom is given by the formula:
\[ \Delta E = 13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
where \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels, respectively.

2. **Identify the Transitions**:
The problem states that we need to evaluate the following transitions:
- Option 1: Transition from \( n_2 = 6 \) to \( n_1 = 1 \)
- Option 2: Transition from \( n_2 = 5 \) to \( n_1 = 4 \)
- Option 3: Transition from \( n_2 = 6 \) to \( n_1 = 5 \)
- Option 4: Transition from \( n_2 = 5 \) to \( n_1 = 3 \)

3. **Calculate Energy for Each Transition**:
- **Option 1 (6 to 1)**:
\[ \Delta E_1 = 13.6 \left( \frac{1}{1^2} - \frac{1}{6^2} \right) = 13.6 \left( 1 - \frac{1}{36} \right) = 13.6 \left( \frac{35}{36} \right) = \frac{476}{36} \approx 13.2 \, \text{eV} \]

- **Option 2 (5 to 4)**:
\[ \Delta E_2 = 13.6 \left( \frac{1}{4^2} - \frac{1}{5^2} \right) = 13.6 \left( \frac{1}{16} - \frac{1}{25} \right) = 13.6 \left( \frac{25 - 16}{400} \right) = 13.6 \left( \frac{9}{400} \right) = \frac{122.4}{400} \approx 0.306 \, \text{eV} \]

- **Option 3 (6 to 5)**:
\[ \Delta E_3 = 13.6 \left( \frac{1}{5^2} - \frac{1}{6^2} \right) = 13.6 \left( \frac{1}{25} - \frac{1}{36} \right) = 13.6 \left( \frac{36 - 25}{900} \right) = 13.6 \left( \frac{11}{900} \right) = \frac{149.6}{900} \approx 0.166 \, \text{eV} \]

- **Option 4 (5 to 3)**:
\[ \Delta E_4 = 13.6 \left( \frac{1}{3^2} - \frac{1}{5^2} \right) = 13.6 \left( \frac{1}{9} - \frac{1}{25} \right) = 13.6 \left( \frac{25 - 9}{225} \right) = 13.6 \left( \frac{16}{225} \right) = \frac{217.6}{225} \approx 0.968 \, \text{eV} \]

4. **Compare Energies**:
Now, we compare the energies calculated:
- Option 1: \( \approx 13.2 \, \text{eV} \)
- Option 2: \( \approx 0.306 \, \text{eV} \)
- Option 3: \( \approx 0.166 \, \text{eV} \)
- Option 4: \( \approx 0.968 \, \text{eV} \)

5. **Identify the Least Energetic Transition**:
The transition that results in the least energetic photon is the one with the smallest energy value. From our calculations, Option 3 (transition from \( n = 6 \) to \( n = 5 \)) has the least energy of \( \approx 0.166 \, \text{eV} \).

### Final Answer:
The transition corresponding to the least energetic photon is **Option 3: Transition from 6 to 5**.