The potential energy of electron in fourth shell of `He^(oplus)` ion is `:-`

Options

A. `-(RCh)/(4)`
B. `-(RCh)/(2)`
C. `(RCh)/(2)`
D. `-(RCh)/(8)`

Explanation

To find the potential energy of an electron in the fourth shell of the He⁺ ion, we can follow these steps:

### Step 1: Understand the Formula for Energy
The energy of an electron in a hydrogen-like atom can be calculated using the formula:
\[ E = -\frac{R_H c Z^2}{n^2} \]
where:
- \( E \) is the energy of the electron,
- \( R_H \) is the Rydberg constant,
- \( c \) is the speed of light,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number (the shell number).

### Step 2: Identify the Values
For the He⁺ ion:
- The atomic number \( Z = 2 \) (since helium has 2 protons).
- We are interested in the fourth shell, so \( n = 4 \).

### Step 3: Substitute Values into the Energy Formula
Substituting the values into the energy formula:
\[ E = -\frac{R_H c (2^2)}{(4^2)} \]
\[ E = -\frac{R_H c \cdot 4}{16} \]
\[ E = -\frac{R_H c}{4} \]

### Step 4: Calculate the Potential Energy
The potential energy \( U \) is related to the energy \( E \) by the equation:
\[ U = 2E \]
Thus, substituting the expression for \( E \):
\[ U = 2 \left(-\frac{R_H c}{4}\right) \]
\[ U = -\frac{R_H c}{2} \]

### Final Answer
The potential energy of an electron in the fourth shell of the He⁺ ion is:
\[ U = -\frac{R_H c}{2} \]

---