What will be the number of spectral lines in infrared region when electron transition occur from `n=7` to `n=2` in hydrogen atom `:`

Options

A. `5`
B. `10`
C. `15`
D. Zero

Explanation

To determine the number of spectral lines in the infrared region when an electron transitions from \( n = 7 \) to \( n = 2 \) in a hydrogen atom, we can follow these steps:

### Step 1: Identify the relevant energy levels
The energy levels of the hydrogen atom are denoted by \( n \), where \( n \) is a positive integer. The transition occurs from \( n = 7 \) to \( n = 2 \). However, we are interested in the spectral lines that fall within the infrared region.

### Step 2: Determine the lower energy level for infrared
The infrared region of the hydrogen spectrum corresponds to transitions that end at \( n = 3 \) or lower. Therefore, we need to consider transitions that start from \( n = 7 \) and can go down to \( n = 3 \).

### Step 3: Calculate the number of transitions
To find the number of spectral lines, we can use the formula for the number of lines produced by transitions between energy levels:
\[ \text{Number of spectral lines} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]
where \( n_2 \) is the higher energy level and \( n_1 \) is the lower energy level.

### Step 4: Assign values to \( n_2 \) and \( n_1 \)
In this case:
- \( n_2 = 7 \) (the upper level)
- \( n_1 = 3 \) (the lower level for infrared transitions)

### Step 5: Substitute the values into the formula
Now, substituting the values into the formula:
\[ \text{Number of spectral lines} = \frac{(7 - 3)(7 - 3 + 1)}{2} \]
\[ = \frac{(4)(5)}{2} \]
\[ = \frac{20}{2} = 10 \]

### Conclusion
Thus, the number of spectral lines in the infrared region when an electron transitions from \( n = 7 \) to \( n = 2 \) is **10**.

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