A sample of hydrogen gas in its ground state is irradiated with photons of `10.02 eV` energies. The radiation from the above sample is used to irradiate two other sample of excited ionized `He^(+)` and excited ionized `Li^(2+)`, respectively. Both the ionized samples absorb the incident radiation.
How many spectral lines are obtained in the spectra of `Li^(2+)`?

Options

A. 10
B. 15
C. 20
D. 17

Answer

To solve the problem of how many spectral lines are obtained in the spectra of \( \text{Li}^{2+} \), we will follow these steps:

### Step 1: Understand the Energy Levels
The energy levels for hydrogen-like ions can be calculated using the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

For \( \text{He}^{+} \) (Z = 2) and \( \text{Li}^{2+} \) (Z = 3), we will calculate the energy levels.

### Step 2: Determine the Energy Levels for \( \text{Li}^{2+} \)
For \( \text{Li}^{2+} \):
- The energy for \( n = 1 \):
\[ E_1 = -\frac{3^2 \cdot 13.6}{1^2} = -40.8 \, \text{eV} \]
- The energy for \( n = 2 \):
\[ E_2 = -\frac{3^2 \cdot 13.6}{2^2} = -10.2 \, \text{eV} \]
- The energy for \( n = 3 \):
\[ E_3 = -\frac{3^2 \cdot 13.6}{3^2} = -4.8 \, \text{eV} \]
- The energy for \( n = 4 \):
\[ E_4 = -\frac{3^2 \cdot 13.6}{4^2} = -2.55 \, \text{eV} \]
- The energy for \( n = 5 \):
\[ E_5 = -\frac{3^2 \cdot 13.6}{5^2} = -1.632 \, \text{eV} \]
- The energy for \( n = 6 \):
\[ E_6 = -\frac{3^2 \cdot 13.6}{6^2} = -1.14 \, \text{eV} \]

### Step 3: Determine the Transitions
The energy of the incident photons is \( 10.02 \, \text{eV} \). We need to find which transitions in \( \text{Li}^{2+} \) can occur that absorb this energy.

The possible transitions are:
- From \( n = 2 \) to \( n = 3 \) (absorbs \( 10.2 - 4.8 = 5.4 \, \text{eV} \))
- From \( n = 3 \) to \( n = 4 \) (absorbs \( 4.8 - 2.55 = 2.25 \, \text{eV} \))
- From \( n = 4 \) to \( n = 5 \) (absorbs \( 2.55 - 1.632 = 0.918 \, \text{eV} \))
- From \( n = 5 \) to \( n = 6 \) (absorbs \( 1.632 - 1.14 = 0.492 \, \text{eV} \))

### Step 4: Calculate the Number of Spectral Lines
For \( \text{Li}^{2+} \), the possible transitions from \( n = 2 \) to \( n = 6 \) are:
- \( 2 \to 3 \)
- \( 2 \to 4 \)
- \( 2 \to 5 \)
- \( 2 \to 6 \)
- \( 3 \to 4 \)
- \( 3 \to 5 \)
- \( 3 \to 6 \)
- \( 4 \to 5 \)
- \( 4 \to 6 \)
- \( 5 \to 6 \)

Counting these transitions gives us a total of 10 spectral lines.

### Final Answer
The number of spectral lines obtained in the spectra of \( \text{Li}^{2+} \) is **10**.

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