What minimum number of atoms lions should be present in a sample of H-like species, so that a maximum of 6 spectral lines can be produced of electronic transition from fifth excited state upto n=2 ?

Answer

To determine the minimum number of atoms required in a sample of hydrogen-like species to produce a maximum of 6 spectral lines from electronic transitions starting from the fifth excited state (n=6) down to n=2, we can follow these steps:

### Step 1: Understand the Electronic Transitions
The fifth excited state corresponds to n=6. The possible transitions from n=6 to n=2 are:
- n=6 to n=5
- n=6 to n=4
- n=6 to n=3
- n=6 to n=2

Additionally, we can have transitions from:
- n=5 to n=4
- n=5 to n=3
- n=5 to n=2
- n=4 to n=3
- n=4 to n=2
- n=3 to n=2

### Step 2: Calculate Maximum Spectral Lines
To find the maximum number of spectral lines, we can use the formula for the number of transitions possible from a given energy level:
\[ \text{Number of lines} = \frac{n(n-1)}{2} \]
where \( n \) is the number of energy levels involved.

In this case, we are considering transitions from n=6 down to n=2, which includes levels 6, 5, 4, 3, and 2 (5 levels):
\[ \text{Number of lines} = \frac{5(5-1)}{2} = \frac{5 \times 4}{2} = 10 \]
However, we only want to consider the transitions that can yield a maximum of 6 spectral lines.

### Step 3: Determine Minimum Number of Atoms
To achieve 6 spectral lines, we need to consider how many transitions can be produced by a single atom.

1. **One Atom**:
- An atom in n=6 can transition to n=5, n=4, n=3, and n=2 (4 lines).
- An atom in n=5 can transition to n=4, n=3, and n=2 (3 lines).
- An atom in n=4 can transition to n=3 and n=2 (2 lines).
- An atom in n=3 can transition to n=2 (1 line).

From one atom in n=6, we can get 4 lines. To get the remaining 2 lines, we need at least one more atom.

2. **Two Atoms**:
- The first atom can account for the transitions from n=6 to n=5, n=4, n=3, and n=2 (4 lines).
- The second atom can account for the transitions from n=5 to n=4 and n=3 (2 lines).

Thus, with 2 atoms, we can achieve a total of 6 spectral lines.

### Conclusion
The minimum number of atoms required in the sample to produce a maximum of 6 spectral lines from the fifth excited state down to n=2 is **2 atoms**.