A sample of hydrogen gas in its ground state is irradiated with photons of `10.2 eV` energies. The radiation from the above sample is used to irradiate two other sample of excited ionized `He^(+)` and excited ionized `Li^(2+)`, repectively. Both the ionized samples absorb the incident radiation.
How many spectral lines are obtained in the spectra of `He^(+)` ion?

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem of how many spectral lines are obtained in the spectra of the ionized helium \( \text{He}^+ \), we can follow these steps:

### Step 1: Understand the Energy Levels of \( \text{He}^+ \)
The energy levels of a hydrogen-like atom 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 \( \text{He}^+ \) (with \( Z = 2 \)):
\[ E_n = -\frac{2^2 \cdot 13.6 \, \text{eV}}{n^2} = -\frac{54.4 \, \text{eV}}{n^2} \]

### Step 2: Calculate the Energy Levels
Calculate the energy levels for the first few states:
- For \( n = 1 \):
\[ E_1 = -54.4 \, \text{eV} \]
- For \( n = 2 \):
\[ E_2 = -\frac{54.4}{2^2} = -13.6 \, \text{eV} \]
- For \( n = 3 \):
\[ E_3 = -\frac{54.4}{3^2} \approx -6.04 \, \text{eV} \]
- For \( n = 4 \):
\[ E_4 = -\frac{54.4}{4^2} = -3.4 \, \text{eV} \]

### Step 3: Determine the Absorption of Energy
The incident photon energy is \( 10.2 \, \text{eV} \). This energy can excite the electron in \( \text{He}^+ \) from a lower energy state to a higher energy state.

### Step 4: Identify Possible Transitions
To find the transitions, we need to determine which transitions correspond to an energy difference of \( 10.2 \, \text{eV} \):
- From \( n = 2 \) to \( n = 4 \):
\[ E_4 - E_2 = (-3.4) - (-13.6) = 10.2 \, \text{eV} \]

### Step 5: Calculate the Number of Spectral Lines
Once the electron is excited to \( n = 4 \), it can de-excite to lower energy levels. The possible transitions from \( n = 4 \) are:
- \( 4 \to 3 \)
- \( 4 \to 2 \)
- \( 4 \to 1 \)
- \( 3 \to 2 \)
- \( 3 \to 1 \)
- \( 2 \to 1 \)

The number of unique transitions can be calculated using the formula for the number of spectral lines:
\[ \text{Number of spectral lines} = \frac{n(n-1)}{2} \]
where \( n \) is the number of energy levels involved in the transitions. Here, \( n = 4 \) (from levels 1, 2, 3, and 4):
\[ \text{Number of spectral lines} = \frac{4(4-1)}{2} = \frac{4 \cdot 3}{2} = 6 \]

### Conclusion
Thus, the number of spectral lines obtained in the spectra of \( \text{He}^+ \) is **6**.

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