`IE_(2)` for an element is inveriably higher than `IE_(1)` because

Options

A. It is difficult to remove electron from cation
B. The size of the cation is smaller than its atoms
C. `Z_(eff)` is more for cation
D. All the above

Explanation

To understand why the second ionization energy (IE₂) of an element is invariably higher than the first ionization energy (IE₁), we can break down the explanation into a few key steps:

### Step-by-Step Solution:

1. **Definition of Ionization Energy**:
- Ionization energy is the amount of energy required to remove an electron from an isolated, neutral gaseous atom. The first ionization energy (IE₁) refers to the energy needed to remove the first electron, while the second ionization energy (IE₂) is the energy required to remove a second electron from the resulting cation.

2. **Formation of Cation**:
- When the first electron is removed from an atom (let's call it atom A), it becomes a unipositive cation (A⁺). This process can be represented as:
\[ A(g) \rightarrow A^+(g) + e^- \]
- After this ionization, the number of electrons in the atom decreases.

3. **Change in Effective Nuclear Charge (Z_eff)**:
- The effective nuclear charge (Z_eff) is the net positive charge experienced by an electron in a multi-electron atom. When an electron is removed, the remaining electrons experience a greater attraction to the nucleus because there are fewer electrons to shield them from the nuclear charge.
- Thus, Z_eff increases after the first ionization.

4. **Increased Attraction and Decreased Size**:
- With the increase in Z_eff, the force of attraction between the nucleus and the remaining electrons becomes stronger. This results in a decrease in the atomic size of the cation (A⁺) compared to the neutral atom (A).

5. **Difficulty of Removing the Second Electron**:
- Because the remaining electrons are held more tightly by the nucleus (due to the increased Z_eff and decreased size), it becomes more difficult to remove a second electron from the cation (A⁺). This means that the energy required to remove the second electron (IE₂) is greater than that required to remove the first electron (IE₁).

6. **Conclusion**:
- Therefore, we conclude that the second ionization energy (IE₂) is invariably higher than the first ionization energy (IE₁) due to the increased effective nuclear charge, stronger attraction between the nucleus and the remaining electrons, and the smaller size of the cation compared to the neutral atom.

### Summary:
The second ionization energy is higher than the first because:
- The effective nuclear charge increases after the first electron is removed.
- The size of the cation is smaller than that of the neutral atom.
- The remaining electrons are held more tightly, making it harder to remove the second electron.