The potential energy of the electron present in the ground state of ` Li^(2+)` ion is represented by :

Options

A. `(3e^(2))/(8pi in_(0)r)`
B. `-(3e^(2))/(8pi in_(0)r)`
C. `(3e^(2))/(4pi in_(0)r)`
D. `-(3e^(2))/(4pi in_(0)r)`

Explanation

To find the potential energy of the electron in the ground state of the \( \text{Li}^{2+} \) ion, we can use the formula derived from Bohr's model of the atom. Here’s a step-by-step solution:

### Step 1: Understand the Formula
The potential energy (PE) of an electron in a hydrogen-like atom is given by the formula:
\[ PE = -\frac{Z e^2}{4 \pi \epsilon_0 r} \]
where:
- \( Z \) is the atomic number,
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( r \) is the radius of the orbit.

### Step 2: Identify the Atomic Number
For the \( \text{Li}^{2+} \) ion, the atomic number \( Z \) is 3 because lithium has 3 protons.

### Step 3: Substitute the Values
Substituting \( Z = 3 \) into the potential energy formula, we get:
\[ PE = -\frac{3 e^2}{4 \pi \epsilon_0 r} \]

### Step 4: Analyze the Options
Now, we need to compare this expression with the given options to identify the correct one. The expression we derived is:
\[ PE = -\frac{3 e^2}{4 \pi \epsilon_0 r} \]

### Step 5: Choose the Correct Option
From the options provided:
- Option 1: Not matching
- Option 2: Not matching
- Option 3: Positive value, not matching
- Option 4: Matches our derived expression

Thus, the correct answer is:
\[ \text{Option D: } -\frac{3 e^2}{4 \pi \epsilon_0 r} \]

### Final Answer
The potential energy of the electron present in the ground state of the \( \text{Li}^{2+} \) ion is:
\[ -\frac{3 e^2}{4 \pi \epsilon_0 r} \]

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