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 \( \text{Li}^{2+} \) ion, we can follow these steps:

### Step 1: Understand the relationship between total energy and potential energy
The total energy (\( E \)) of an electron in a hydrogen-like atom can be expressed as:
\[ E = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number. For hydrogen, \( Z = 1 \) and \( n = 1 \), so:
\[ E_{\text{H}} = -13.6 \, \text{eV} \]

The potential energy (\( U \)) is related to the total energy by:
\[ U = 2E \]

### Step 2: Calculate the total energy for \( \text{Li}^{2+} \)
For \( \text{Li}^{2+} \), the atomic number \( Z = 3 \) and the electron is also in the ground state (\( n = 1 \)). Thus, we can calculate the total energy for \( \text{Li}^{2+} \):
\[ E_{\text{Li}^{2+}} = -\frac{3^2 \cdot 13.6 \, \text{eV}}{1^2} = -\frac{9 \cdot 13.6 \, \text{eV}}{1} = -122.4 \, \text{eV} \]

### Step 3: Calculate the potential energy for \( \text{Li}^{2+} \)
Using the relationship between total energy and potential energy:
\[ U = 2E_{\text{Li}^{2+}} = 2 \times (-122.4 \, \text{eV}) = -244.8 \, \text{eV} \]

### Conclusion
The potential energy of the electron in the ground state of the \( \text{Li}^{2+} \) ion is:
\[ U = -244.8 \, \text{eV} \]