Hydrogen atoms are excited from ground state of the principle quantum number `4`. Then the number of spectral lines observed will be

Options

A. 3
B. 6
C. 5
D. 2

Answer

To find the number of spectral lines observed when hydrogen atoms are excited to the principal quantum number \( n = 4 \), we can follow these steps:

### Step 1: Understand the Quantum States
The hydrogen atom has quantized energy levels defined by the principal quantum number \( n \). The ground state corresponds to \( n = 1 \), and the excited states are \( n = 2 \), \( n = 3 \), and \( n = 4 \).

### Step 2: Identify the Excited State
In this problem, the hydrogen atom is excited to the state where \( n = 4 \). This means the electron can transition down to any of the lower energy levels: \( n = 1 \), \( n = 2 \), or \( n = 3 \).

### Step 3: Calculate the Number of Spectral Lines
The number of spectral lines that can be observed when an electron transitions from a higher energy level to lower energy levels can be calculated using the formula:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]
where \( n \) is the principal quantum number of the excited state.

### Step 4: Substitute the Value
For our case, since the electron is excited to \( n = 4 \):
\[ \text{Number of spectral lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} \]

### Step 5: Simplify the Calculation
Calculating the above expression:
\[ \frac{4 \times 3}{2} = \frac{12}{2} = 6 \]

### Conclusion
Thus, the number of spectral lines observed when hydrogen atoms are excited from the ground state to the principal quantum number \( n = 4 \) is **6**.

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