The incorrect statement among the following is:

Options

A. the `IE_1` of Al is less than `IE_1` of Mg
B. the `IE_2` of Mg is greater than `IE_2` of Na
C. the `IE_1` of Na is less than `IE_1` of Mg
D. the `IE_3` of Mg is greater than `IE_2` of Al

Answer

To determine the incorrect statement among the given options regarding ionization energies of sodium, magnesium, and aluminum, we will analyze each statement step by step.

### Step 1: Understanding Ionization Energy
Ionization energy is defined as the minimum amount of energy required to remove an electron from an atom in its gaseous state. The first ionization energy refers to the energy needed to remove the first electron, the second ionization energy refers to the energy needed to remove the second electron, and so on.

### Step 2: Electronic Configurations
We need to write the electronic configurations of sodium (Na), magnesium (Mg), and aluminum (Al) to understand their ionization energies better:
- Sodium (Na): 1s² 2s² 2p⁶ 3s¹
- Magnesium (Mg): 1s² 2s² 2p⁶ 3s²
- Aluminum (Al): 1s² 2s² 2p⁶ 3s² 3p¹

### Step 3: Analyzing Each Statement
1. **Statement 1**: The first ionization energy of aluminum is less than that of magnesium.
- **Analysis**: In aluminum, we remove an electron from a p orbital (3p¹), while in magnesium, we remove from a filled s orbital (3s²). Since the 3s² electrons are closer to the nucleus and more stable, this statement is **correct**.

2. **Statement 2**: The second ionization energy of magnesium is greater than that of sodium.
- **Analysis**: The second ionization energy of magnesium involves removing an electron from a 3s orbital (3s¹), while for sodium, it involves removing an electron from a filled 2p orbital (2p⁶). The filled 2p orbital is more stable and requires more energy to remove an electron than the 3s electron from magnesium. Therefore, this statement is **incorrect**.

3. **Statement 3**: The first ionization energy of sodium is less than that of magnesium.
- **Analysis**: Sodium has one electron in its outermost shell (3s¹), while magnesium has two (3s²). It is easier to remove the single electron from sodium than to remove one from the stable 3s² of magnesium. Thus, this statement is **correct**.

4. **Statement 4**: The third ionization energy of magnesium is greater than the second ionization energy of aluminum.
- **Analysis**: The third ionization energy of magnesium involves removing an electron from a filled 2p orbital (2p⁶) after removing two electrons from 3s. The second ionization energy of aluminum involves removing an electron from a 3s orbital (3s²). The higher nuclear charge of magnesium (after losing two electrons) means it will require more energy to remove the third electron than it would for aluminum to lose its second electron. Hence, this statement is **correct**.

### Conclusion
The incorrect statement among the options is **Statement 2**: "The second ionization energy of magnesium is greater than that of sodium."