In a sample of three H-atoms , all in the 5th excited state, electrons make a transition to 1st excited state. The maximum number of different spectral lines observed will be :

Answer

To determine the maximum number of different spectral lines observed when electrons in three hydrogen atoms transition from the 5th excited state to the 1st excited state, we can follow these steps:

### Step 1: Understand the Energy Levels of Hydrogen
In a hydrogen atom, the energy levels are quantized and can be represented by the formula:
\[ E_n = -\frac{13.6 \, \text{eV}}{n^2} \]
where \( n \) is the principal quantum number (energy level).

### Step 2: Identify the Initial and Final States
In this case, the initial state is the 5th excited state, which corresponds to \( n = 6 \) (since the ground state is \( n = 1 \)). The final state is the 1st excited state, which corresponds to \( n = 2 \).

### Step 3: Calculate the Possible Transitions
Electrons can transition from the 5th excited state (n=6) to the 1st excited state (n=2) through various intermediate states. The possible transitions are:
- From \( n = 6 \) to \( n = 5 \)
- From \( n = 5 \) to \( n = 4 \)
- From \( n = 4 \) to \( n = 3 \)
- From \( n = 3 \) to \( n = 2 \)

### Step 4: Count the Number of Different Transitions
For each electron in the 5th excited state, it can transition to any of the lower states. The transitions can be represented as follows:
- \( 6 \rightarrow 5 \)
- \( 5 \rightarrow 4 \)
- \( 4 \rightarrow 3 \)
- \( 3 \rightarrow 2 \)

### Step 5: Determine the Number of Spectral Lines
The maximum number of different spectral lines can be calculated using the formula for the number of transitions:
\[ \text{Number of transitions} = \frac{n(n-1)}{2} \]
where \( n \) is the number of energy levels involved in the transition. Here, we have transitions from \( n = 6 \) to \( n = 2 \), which involves 5 energy levels (6, 5, 4, 3, and 2).

Thus, the number of transitions is:
\[ \frac{5(5-1)}{2} = \frac{5 \times 4}{2} = 10 \]

### Step 6: Consider the Number of Atoms
Since there are 3 hydrogen atoms, each can independently make these transitions. However, since we are interested in the maximum number of different spectral lines, we only need to consider the unique transitions.

### Final Answer
The maximum number of different spectral lines observed will be **10**.

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