In a sample of `H-` atom electrons make transition from `5^(th)` excited state to ground state, producing all possible types of photons, then number of lines in infrared region are

Options

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

Explanation

To solve the problem of determining how many spectral lines in the infrared region are produced when electrons in a hydrogen atom transition from the 5th excited state to the ground state, we can follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen atom, the energy levels are quantized and can be represented by the principal quantum number \( n \). The ground state corresponds to \( n = 1 \), the first excited state to \( n = 2 \), the second excited state to \( n = 3 \), and so on. The 5th excited state corresponds to \( n = 6 \).

### Step 2: Calculate Possible Transitions
When an electron transitions from a higher energy level to a lower one, it can do so in multiple ways. The number of possible transitions from the 5th excited state (\( n = 6 \)) to the ground state (\( n = 1 \)) can be calculated using the formula:

\[ \text{Number of lines} = \frac{n(n-1)}{2} \]

where \( n \) is the principal quantum number of the initial state. In this case, \( n = 6 \):

\[ \text{Number of lines} = \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15 \]

### Step 3: Identify the Series of Spectral Lines
The transitions will produce spectral lines that fall into different series based on the final energy level:

1. **Lyman Series**: Transitions to \( n = 1 \) (ultraviolet region)
2. **Balmer Series**: Transitions to \( n = 2 \) (visible region)
3. **Paschen Series**: Transitions to \( n = 3 \) (infrared region)
4. **Brackett Series**: Transitions to \( n = 4 \) (infrared region)
5. **Pfund Series**: Transitions to \( n = 5 \) (infrared region)

### Step 4: Count the Lines in Each Series
From the 5th excited state (\( n = 6 \)), the possible transitions are:

- From \( n = 6 \) to \( n = 1 \): 1 line (Lyman series)
- From \( n = 6 \) to \( n = 2 \): 1 line (Balmer series)
- From \( n = 6 \) to \( n = 3 \): 1 line (Paschen series)
- From \( n = 6 \) to \( n = 4 \): 1 line (Brackett series)
- From \( n = 6 \) to \( n = 5 \): 1 line (Pfund series)

Now, we can also consider the transitions from \( n = 5 \) and \( n = 4 \) to the lower levels:

- From \( n = 5 \) to \( n = 3 \): 1 line (Paschen series)
- From \( n = 5 \) to \( n = 4 \): 1 line (Brackett series)
- From \( n = 4 \) to \( n = 3 \): 1 line (Paschen series)

### Step 5: Total Lines in the Infrared Region
The lines in the infrared region come from the Paschen, Brackett, and Pfund series. The transitions that contribute to the infrared region are:

- Paschen series: \( n = 6 \to 3 \), \( n = 5 \to 3 \)
- Brackett series: \( n = 6 \to 4 \), \( n = 5 \to 4 \)
- Pfund series: \( n = 6 \to 5 \)

Counting these, we find:

- **Paschen series**: 2 lines
- **Brackett series**: 2 lines
- **Pfund series**: 1 line

Thus, the total number of lines in the infrared region is:

\[ 2 + 2 + 1 = 5 \]

### Final Answer
The number of lines in the infrared region is **6**.