Hydrogen atoms are excited from ground state to the principal quantum number 5. Number of spectral lines observed will be

Options

A. 5
B. 4
C. 10
D. 8

Answer

To determine the number of spectral lines observed when hydrogen atoms are excited from the ground state to the principal quantum number \( n = 5 \), we can use the formula for the number of spectral lines produced during transitions between energy levels.

### Step-by-Step Solution:

1. **Identify the Principal Quantum Number (n)**:
- The hydrogen atom is excited to the principal quantum number \( n = 5 \).

2. **Use the Formula for Spectral Lines**:
- The number of spectral lines \( N \) that can be observed when electrons transition between energy levels is given by the formula:
\[ N = \frac{n(n-1)}{2} \]
- Here, \( n \) is the principal quantum number to which the electron is excited.

3. **Substitute the Value of n**:
- Substitute \( n = 5 \) into the formula:
\[ N = \frac{5(5-1)}{2} \]

4. **Calculate the Expression**:
- First, calculate \( 5 - 1 = 4 \).
- Then, multiply \( 5 \times 4 = 20 \).
- Finally, divide by \( 2 \):
\[ N = \frac{20}{2} = 10 \]

5. **Conclusion**:
- The number of spectral lines observed when hydrogen atoms are excited from the ground state to the principal quantum number 5 is \( 10 \).

### Final Answer:
- Therefore, the correct option is \( 10 \) (Option 3).

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