Find the first excitation potential of `He^(+)` ion (a)Find the ionization potential of `Li^(++)` ion

Explanation

To solve the question, we will break it down into two parts as stated:

### Part A: Find the first excitation potential of `He^(+)` ion

1. **Understanding Excitation Potential**: The excitation potential is the energy required to move an electron from the ground state to the first excited state. For hydrogen-like atoms, the energy levels are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

2. **Identify the Atomic Number**: For the `He^(+)` ion, the atomic number \( Z = 2 \).

3. **Calculate the Ground State Energy (n=1)**:
\[ E_1 = -\frac{2^2 \cdot 13.6}{1^2} = -\frac{4 \cdot 13.6}{1} = -54.4 \, \text{eV} \]

4. **Calculate the First Excited State Energy (n=2)**:
\[ E_2 = -\frac{2^2 \cdot 13.6}{2^2} = -\frac{4 \cdot 13.6}{4} = -13.6 \, \text{eV} \]

5. **Find the First Excitation Potential**: The excitation potential is the difference between the first excited state and the ground state:
\[ \text{Excitation Potential} = E_2 - E_1 = -13.6 - (-54.4) = 40.8 \, \text{eV} \]

### Part B: Find the Ionization Potential of `Li^(++)` ion

1. **Understanding Ionization Potential**: The ionization potential is the energy required to remove the electron from the ground state of the atom.

2. **Identify the Atomic Number**: For the `Li^(++)` ion, the atomic number \( Z = 3 \).

3. **Calculate the Ionization Potential**: The ionization potential for hydrogen-like atoms is given by:
\[ \text{Ionization Potential} = Z^2 \cdot 13.6 \, \text{eV} \]
Substituting \( Z = 3 \):
\[ \text{Ionization Potential} = 3^2 \cdot 13.6 = 9 \cdot 13.6 = 122.4 \, \text{eV} \]

### Final Answers:
- **First Excitation Potential of `He^(+)`**: 40.8 eV
- **Ionization Potential of `Li^(++)`**: 122.4 eV