The total energy of an electron in the ground state of hydrogen atom is `- 13.6 eV`. The potiential energy of an electron in the ground state of `Li ^(2+)` ion will be

Options

A. `122.4 eV`
B. `- 122.4 eV`
C. `244.8 eV`
D. `-244.8 eV`

Explanation

To find the potential energy of an electron in the ground state of the \( Li^{2+} \) ion, we can use the relationship between the total energy and potential energy in a hydrogen-like atom.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The total energy \( E \) of an electron in a hydrogen-like atom is given by:
\[ E = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number. For hydrogen, \( Z = 1 \) and for \( Li^{2+} \), \( Z = 3 \).

2. **Calculate Total Energy for \( Li^{2+} \)**:
For \( Li^{2+} \) in the ground state (where \( n = 1 \)):
\[ E = -\frac{3^2 \cdot 13.6 \, \text{eV}}{1^2} = -\frac{9 \cdot 13.6 \, \text{eV}}{1} = -122.4 \, \text{eV} \]

3. **Relate Total Energy to Potential Energy**:
The total energy \( E \) is related to the potential energy \( U \) by the equation:
\[ E = U + K \]
where \( K \) is the kinetic energy. For a hydrogen-like atom, the potential energy is twice the total energy:
\[ U = 2E \]

4. **Calculate Potential Energy**:
Using the relationship \( U = 2E \):
\[ U = 2 \times (-122.4 \, \text{eV}) = -244.8 \, \text{eV} \]

### Final Answer:
The potential energy of an electron in the ground state of the \( Li^{2+} \) ion is \( -244.8 \, \text{eV} \).