The second ionization potential of elements is invariable higher than first ionization potential because:

Options

A. The size of cation is smaller than its atom
B. it is easier to remove electron from cation
C. ionization is an endothermic process
D. none of these

Answer

To understand why the second ionization potential of elements is invariably higher than the first ionization potential, we can break down the explanation into several steps:

### Step-by-Step Solution:

1. **Definition of Ionization Potential**:
- Ionization potential (or ionization energy) is the energy required to remove an electron from an atom or ion in its gaseous state. The first ionization potential (I1) refers to the energy needed to remove the first electron, while the second ionization potential (I2) is the energy required to remove a second electron after the first has already been removed.

**Hint**: Remember that ionization potential is related to the energy needed to remove electrons from an atom.

2. **Formation of Cation**:
- When the first electron is removed from a neutral atom (let's call it A), it becomes a positively charged ion (cation), denoted as A⁺. For example, if A has 10 electrons, after removing one, it will have 9 electrons.

**Hint**: Consider how the removal of an electron changes the charge of the atom.

3. **Change in Electronic Configuration**:
- After the first ionization, the electronic configuration of A changes. For instance, if the original configuration was 1s² 2s² 2p⁶ (for a neutral atom), after removing one electron, it becomes 1s² 2s² 2p⁵ for the cation A⁺.

**Hint**: Think about how the electronic structure changes when an electron is removed.

4. **Effect of Nuclear Charge**:
- In a neutral atom, the electrons experience a certain effective nuclear charge due to the attraction from the nucleus. When one electron is removed, the remaining electrons in the cation experience a greater effective nuclear charge because there are fewer electrons to shield them from the nucleus.

**Hint**: Reflect on how the balance of attractive forces changes when electrons are removed.

5. **Size of the Cation**:
- The size of the cation (A⁺) is smaller than that of the neutral atom (A). This is because the same nuclear charge is acting on fewer electrons, leading to a stronger attraction between the nucleus and the remaining electrons.

**Hint**: Consider how the size of an atom changes when it loses electrons.

6. **Higher Energy Requirement for Second Ionization**:
- Since the cation has a smaller radius and the remaining electrons are held more tightly by the nucleus, more energy is required to remove the second electron (I2) compared to the first electron (I1). Thus, I2 is invariably higher than I1.

**Hint**: Think about why it would be harder to remove an electron from a cation than from a neutral atom.

### Conclusion:
The second ionization potential is invariably higher than the first because the removal of the first electron creates a cation with a smaller radius and a higher effective nuclear charge acting on the remaining electrons, making it more difficult to remove the second electron.