In ground state of `He^(oplus)` ion how much energy is supplied to
electron so that second line of balmer series is obtained in its
spectrum `:-`

Options

A. `10.2`eV
B. `48.36`eV
C. `51`eV
D. None of these

Explanation

To find the energy supplied to an electron in the ground state of the He⁺ ion to obtain the second line of the Balmer series, we can follow these steps:

### Step 1: Understand the Balmer Series
The Balmer series corresponds to transitions where the final energy level (nf) is 2. The second line of the Balmer series corresponds to a transition from n=4 to n=2.

### Step 2: Identify the Formula for Energy Levels
The energy levels of a hydrogen-like atom (like He⁺) can be calculated using the formula:
\[ E = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where:
- \( Z \) is the atomic number (for He⁺, \( Z = 2 \))
- \( n \) is the principal quantum number.

### Step 3: Calculate the Energy for Initial and Final States
1. **Initial State (nf = 4)**:
\[ E_i = -\frac{Z^2 \cdot 13.6}{n_i^2} = -\frac{2^2 \cdot 13.6}{4^2} = -\frac{4 \cdot 13.6}{16} = -3.4 \, \text{eV} \]

2. **Final State (ni = 2)**:
\[ E_f = -\frac{Z^2 \cdot 13.6}{n_f^2} = -\frac{2^2 \cdot 13.6}{2^2} = -\frac{4 \cdot 13.6}{4} = -13.6 \, \text{eV} \]

### Step 4: Calculate the Energy Difference
The energy required to transition from n=4 to n=2 is given by:
\[ \Delta E = E_f - E_i = (-13.6) - (-3.4) = -13.6 + 3.4 = -10.2 \, \text{eV} \]

### Step 5: Conclusion
The energy supplied to the electron to achieve the transition and obtain the second line of the Balmer series is:
\[ \Delta E = 10.2 \, \text{eV} \]

### Final Answer
The energy supplied to the electron is **10.2 eV**.

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