What is the energy of the electron in `He^(+)` in ground state?

Answer

To find the energy of the electron in the ground state of the helium ion \( \text{He}^+ \), we can use the formula for the energy of an electron in a hydrogen-like atom:

\[ E = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2} \]

where:
- \( E \) is the energy of the electron,
- \( Z \) is the atomic number,
- \( n \) is the principal quantum number (which is 1 for the ground state).

### Step 1: Identify the atomic number \( Z \)
For the helium ion \( \text{He}^+ \):
- The atomic number \( Z \) of helium is 2.

### Step 2: Identify the principal quantum number \( n \)
Since we are looking for the ground state:
- The principal quantum number \( n \) is 1.

### Step 3: Substitute the values into the formula
Now we can substitute \( Z \) and \( n \) into the energy formula:

\[ E = -\frac{13.6 \, \text{eV} \cdot (2)^2}{(1)^2} \]

### Step 4: Calculate \( Z^2 \) and \( n^2 \)
Calculating \( Z^2 \) and \( n^2 \):
- \( Z^2 = 2^2 = 4 \)
- \( n^2 = 1^2 = 1 \)

### Step 5: Substitute the squared values into the formula
Now substituting these values back into the equation:

\[ E = -\frac{13.6 \, \text{eV} \cdot 4}{1} \]

### Step 6: Perform the multiplication
Calculating the energy:

\[ E = -13.6 \, \text{eV} \cdot 4 = -54.4 \, \text{eV} \]

### Final Answer
Thus, the energy of the electron in \( \text{He}^+ \) in the ground state is:

\[ E = -54.4 \, \text{eV} \]

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