In a H-like sample, electrons make transition from `4^(th)` excited state upto `2^(nd)` state. Then ,

Options

A. `10` different spectral lines are observed
B. `6` different spectral lines are observed
C. number of lines belonging to the balmer series is `3`
D. Number of lines belonging to paschen series is `2`

Explanation

To solve the problem of electron transitions in a hydrogen-like atom, we need to determine the number of spectral lines produced when electrons transition from the 4th excited state to the 2nd state. Here’s a step-by-step solution:

### Step 1: Identify the states
In a hydrogen-like atom, the nth excited state corresponds to the principal quantum number \( n \). The 4th excited state means \( n = 5 \) (since the ground state is \( n = 1 \), the first excited state is \( n = 2 \), and so on). The 2nd state corresponds to \( n = 2 \).

- **n1 (initial state)** = 2 (2nd state)
- **n2 (final state)** = 5 (4th excited state)

### Step 2: Calculate the number of spectral lines
The formula to calculate the number of spectral lines (N) produced when electrons transition between two energy levels is given by:

\[ N = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} \]

Substituting the values of \( n_1 \) and \( n_2 \):

\[ N = \frac{(5 - 2)(5 - 2 + 1)}{2} \]

### Step 3: Simplify the expression
Now, simplify the expression:

\[ N = \frac{(3)(3 + 1)}{2} = \frac{(3)(4)}{2} = \frac{12}{2} = 6 \]

### Step 4: Conclusion
Thus, the number of different spectral lines observed when electrons transition from the 4th excited state to the 2nd state is **6**.

### Step 5: Identify the series
Out of these spectral lines, we can identify which ones belong to the Balmer series. The Balmer series corresponds to transitions where the final state \( n \) is 2. The relevant transitions from \( n = 5 \) to \( n = 2 \) will produce spectral lines:

- Transition from \( n = 5 \) to \( n = 2 \) (522 nm)
- Transition from \( n = 4 \) to \( n = 2 \) (422 nm)
- Transition from \( n = 3 \) to \( n = 2 \) (322 nm)

Thus, there are **3 lines** in the Balmer series.

### Final Answer
- The total number of spectral lines is **6**.
- The number of lines belonging to the Balmer series is **3**.

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