The absorpotion transitions between the first and the fourth energy states of hydrogen atom are `3`. The emission transitions between these states will be

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem of finding the number of emission transitions between the first and the fourth energy states of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial and Final States**:
- The initial state (\(n_i\)) is the fourth energy level, so \(n_i = 4\).
- The final state (\(n_f\)) is the first energy level, so \(n_f = 1\).

2. **Use the Formula for Number of Transitions**:
- The number of possible transitions (spectral lines) can be calculated using the formula:
\[ \text{Number of transitions} = (n_i - n_f) \times (n_i - n_f + 1) / 2 \]

3. **Substitute the Values into the Formula**:
- Substitute \(n_i = 4\) and \(n_f = 1\) into the formula:
\[ \text{Number of transitions} = (4 - 1) \times (4 - 1 + 1) / 2 \]

4. **Calculate the Values**:
- Calculate \(n_i - n_f\):
\[ 4 - 1 = 3 \]
- Calculate \(n_i - n_f + 1\):
\[ 4 - 1 + 1 = 4 \]
- Now substitute these values back into the formula:
\[ \text{Number of transitions} = 3 \times 4 / 2 \]

5. **Final Calculation**:
- Perform the multiplication:
\[ 3 \times 4 = 12 \]
- Now divide by 2:
\[ 12 / 2 = 6 \]

6. **Conclusion**:
- Therefore, the number of emission transitions between the first and the fourth energy states of the hydrogen atom is **6**.

### Final Answer:
The number of emission transitions is **6**.

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