How many maximum spectral lines are possible if electron is present in `4^(th)` shell and only two atom are present in sample ?

Options

A. 6
B. 4
C. 3
D. 2

Answer

To determine the maximum number of spectral lines possible when an electron is present in the 4th shell (n=4) and there are two atoms in the sample, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Spectral Lines for a Single Atom:**
The maximum number of spectral lines (transitions) for a single atom can be calculated using the formula:
\[ \text{Maximum Spectral Lines} = \frac{n(n-1)}{2} \]
where \( n \) is the principal quantum number (energy level).

2. **Substituting the Value of n:**
For the 4th shell, \( n = 4 \). We substitute this value into the formula:
\[ \text{Maximum Spectral Lines} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6 \]
Therefore, for a single atom in the 4th shell, a maximum of 6 spectral lines can be produced.

3. **Considering Two Atoms:**
Since we have two atoms in the sample, we need to consider the transitions that can occur between them. If both atoms are identical, the spectral lines will remain the same, which is 6.

4. **Considering Different Atoms:**
If the two atoms are different, we need to consider the possible transitions for each atom:
- Atom 1 can transition from n=4 to n=3, n=2, and n=1.
- Atom 2 can also transition from n=4 to n=3, n=2, and n=1.

5. **Counting Unique Transitions:**
Each atom can transition independently, but we need to account for unique transitions. The possible transitions for each atom can be counted as follows:
- From n=4 to n=3 (1 transition)
- From n=4 to n=2 (1 transition)
- From n=4 to n=1 (1 transition)

Thus, for each atom, there are 3 possible transitions.

6. **Total Spectral Lines:**
Since both atoms can independently transition, we can add the unique transitions:
- For Atom 1: 3 transitions
- For Atom 2: 3 transitions

However, since both atoms can produce the same transitions, we will only count unique transitions, which leads us back to the maximum of 6 unique spectral lines.

### Final Answer:
The maximum number of spectral lines possible when an electron is present in the 4th shell and there are two atoms in the sample is **6**.

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