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

Options

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

Answer

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 (E) of an electron in a hydrogen atom is given by the formula:
\[ E = -\frac{13.6 \, Z^2}{n^2} \text{ eV} \]
where \( Z \) is the atomic number (which is 1 for hydrogen) and \( n \) is the principal quantum number.

2. **Determine Energy Levels**: The energy levels are inversely proportional to the square of the principal quantum number \( n \). This means that as \( n \) increases, the energy \( E \) becomes less negative (or increases).

3. **Calculate Energy Difference for Transitions**: The energy difference (\( \Delta E \)) between two energy levels \( n_1 \) and \( n_2 \) can be expressed as:
\[ \Delta E = E_{n_1} - E_{n_2} = -\frac{13.6}{n_1^2} + \frac{13.6}{n_2^2} \]
This can be rewritten as:
\[ \Delta E = 13.6 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

4. **Identify the Transition with Least Energy**: To find the transition that gives rise to the least energetic photon, we need to look for the transition where \( n_1 \) is the lowest and \( n_2 \) is the highest. This is because a larger \( n_2 \) will result in a smaller \( \Delta E \) (less energetic photon).

5. **Evaluate the Given Options**: Among the options provided, identify the transitions and their corresponding \( n_1 \) and \( n_2 \) values. The transition with the highest \( n_2 \) will yield the least energetic photon.

6. **Conclusion**: Based on the analysis, the transition with the highest \( n_2 \) value will produce the least energetic photon. If option A has the highest \( n_2 \) value among the given options, then option A is the correct answer.

### Final Answer:
The transition in the hydrogen atom that will give rise to the least energetic photon is **option A**.