In the emission line spectra of hydrogen atom, how many lines can be accounted for all possible electronic transitions from `5^(th)` energy level within the atom?

Options

A. 4
B. 5
C. 10
D. 20

Answer

To determine the number of emission lines in the hydrogen atom's spectrum due to electronic transitions from the 5th energy level, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Energy Levels**: In a hydrogen atom, electrons can occupy different energy levels, denoted by quantum numbers (n = 1, 2, 3, 4, 5, etc.). The 5th energy level corresponds to n = 5.

2. **Possible Transitions**: When an electron in the 5th energy level transitions to a lower energy level, it can move to any of the levels below it (n = 4, 3, 2, or 1).

3. **Counting the Transitions**:
- From n = 5 to n = 4
- From n = 5 to n = 3
- From n = 5 to n = 2
- From n = 5 to n = 1

This gives us **4 transitions** from the 5th level.

4. **Transitions from Lower Levels**:
- If the electron transitions from n = 4, it can go to:
- n = 3
- n = 2
- n = 1

This gives us **3 additional transitions**.

- If the electron transitions from n = 3, it can go to:
- n = 2
- n = 1

This gives us **2 additional transitions**.

- If the electron transitions from n = 2, it can go to:
- n = 1

This gives us **1 additional transition**.

5. **Total Transitions**: Now, we can sum all the possible transitions:
- Transitions from n = 5: 4
- Transitions from n = 4: 3
- Transitions from n = 3: 2
- Transitions from n = 2: 1

Total = 4 + 3 + 2 + 1 = **10 transitions**.

6. **Conclusion**: Therefore, the total number of lines that can be accounted for all possible electronic transitions from the 5th energy level in a hydrogen atom is **10**.

### Final Answer:
The number of emission lines in the hydrogen atom's spectrum from the 5th energy level is **10**.

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