In a container a mixture is prepared by mixing of three samples of hydrogen helium ion `(He^(+))` and lithium ion `(Li^(2+)).` In sample , all the hydrogen atoms are in 1st excited state and all the `He^(+)` ions are in third excited state and all the `Li^(2+))` ions are in fifth excited state. Find the total number of spectral lines observed in the emission spectrum of such a sample when the electrons return back to the ground state.

Answer

To solve the problem of finding the total number of spectral lines observed in the emission spectrum of a mixture of hydrogen, helium ion (He⁺), and lithium ion (Li²⁺), we will follow these steps:

### Step 1: Analyze the Hydrogen Atom
- **Hydrogen (H)** is in the first excited state, which corresponds to the energy level \( n = 2 \).
- When the electron returns to the ground state (\( n = 1 \)), the transition is from \( n = 2 \) to \( n = 1 \).
- **Possible transitions:** There is only one transition: \( 2 \to 1 \).
- **Spectral lines from Hydrogen:** 1 line.

### Step 2: Analyze the Helium Ion (He⁺)
- **Helium ion (He⁺)** is in the third excited state, which corresponds to \( n = 4 \).
- Possible transitions from \( n = 4 \) to \( n = 1 \) can be calculated using the formula for the number of transitions:
\[ \text{Number of transitions} = \frac{n(n-1)}{2} \]
where \( n \) is the total number of energy levels (from \( n = 4 \) to \( n = 1 \)).

- Here, \( n = 4 \):
\[ \text{Number of transitions} = \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = 6 \]
- **Spectral lines from Helium ion:** 6 lines.

### Step 3: Analyze the Lithium Ion (Li²⁺)
- **Lithium ion (Li²⁺)** is in the fifth excited state, which corresponds to \( n = 6 \).
- Possible transitions from \( n = 6 \) to \( n = 1 \):
\[ \text{Number of transitions} = \frac{n(n-1)}{2} \]
where \( n = 6 \):
\[ \text{Number of transitions} = \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15 \]
- **Spectral lines from Lithium ion:** 15 lines.

### Step 4: Calculate Total Spectral Lines
- Now, we sum up the spectral lines from each species:
\[ \text{Total spectral lines} = \text{Lines from H} + \text{Lines from He⁺} + \text{Lines from Li²⁺} \]
\[ \text{Total spectral lines} = 1 + 6 + 15 = 22 \]

### Step 5: Adjust for Overlapping Transitions
- We need to account for overlapping transitions. The transitions \( 2 \to 1 \) from hydrogen, \( 4 \to 2 \) from helium, and \( 6 \to 3 \) from lithium will overlap.
- Since there are 2 overlapping transitions, we subtract these from the total:
\[ \text{Adjusted total} = 22 - 2 = 20 \]

### Final Answer
- The total number of spectral lines observed in the emission spectrum of the mixture is **20**.

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