The increasing order of effective nuclear charge in `Na, AI, Mg` and `Si` atoms

Options

A. `Na lt Mg lt SI lt AI`
B. `Na lt Mg lt AI lt Si`
C. `Mg lt Na lt AI lt Si`
D. `Na = Mg = Al = Si`

Answer

To determine the increasing order of effective nuclear charge (Z_eff) for the atoms Na, Al, Mg, and Si, we will follow these steps:

### Step 1: Understand the Concept of Effective Nuclear Charge
The effective nuclear charge (Z_eff) is calculated using the formula:
\[ Z_{eff} = Z - \sigma \]
where:
- \( Z \) is the atomic number of the element.
- \( \sigma \) is the screening constant, which accounts for the shielding effect of the inner electrons.

### Step 2: Write the Electronic Configurations
- **Sodium (Na)**: \( 1s^2 2s^2 2p^6 3s^1 \) (Z = 11)
- **Magnesium (Mg)**: \( 1s^2 2s^2 2p^6 3s^2 \) (Z = 12)
- **Aluminium (Al)**: \( 1s^2 2s^2 2p^6 3s^2 3p^1 \) (Z = 13)
- **Silicon (Si)**: \( 1s^2 2s^2 2p^6 3s^2 3p^2 \) (Z = 14)

### Step 3: Calculate the Screening Constant (σ)
Using Slater's rules, we will calculate σ for each atom:

1. **Sodium (Na)**:
- Electrons in the same shell (3s): 0 (1 electron)
- Electrons in the n-1 shell (2s, 2p): 8 (2 from 2s and 6 from 2p) → Contribution = \( 8 \times 0.85 = 6.8 \)
- Electrons in the n-2 shell (1s): 2 → Contribution = \( 2 \times 1 = 2 \)
- Total σ = \( 6.8 + 2 = 8.8 \)

\[ Z_{eff} = 11 - 8.8 = 2.2 \]

2. **Magnesium (Mg)**:
- Electrons in the same shell (3s): 0 (2 electrons)
- Electrons in the n-1 shell (2s, 2p): 8 → Contribution = \( 8 \times 0.85 = 6.8 \)
- Electrons in the n-2 shell (1s): 2 → Contribution = \( 2 \times 1 = 2 \)
- Total σ = \( 6.8 + 2 = 8.8 \)

\[ Z_{eff} = 12 - 8.8 = 3.2 \]

3. **Aluminium (Al)**:
- Electrons in the same shell (3s, 3p): 0 (3 electrons)
- Electrons in the n-1 shell (2s, 2p): 8 → Contribution = \( 8 \times 0.85 = 6.8 \)
- Electrons in the n-2 shell (1s): 2 → Contribution = \( 2 \times 1 = 2 \)
- Total σ = \( 6.8 + 2 + 0.35 \times 2 = 9.5 \)

\[ Z_{eff} = 13 - 9.5 = 3.5 \]

4. **Silicon (Si)**:
- Electrons in the same shell (3s, 3p): 0 (4 electrons)
- Electrons in the n-1 shell (2s, 2p): 8 → Contribution = \( 8 \times 0.85 = 6.8 \)
- Electrons in the n-2 shell (1s): 2 → Contribution = \( 2 \times 1 = 2 \)
- Total σ = \( 6.8 + 2 + 0.35 \times 3 = 9.5 \)

\[ Z_{eff} = 14 - 9.5 = 4.5 \]

### Step 4: Compare the Effective Nuclear Charges
Now that we have calculated the effective nuclear charges:
- Sodium (Na): \( Z_{eff} = 2.2 \)
- Magnesium (Mg): \( Z_{eff} = 3.2 \)
- Aluminium (Al): \( Z_{eff} = 3.5 \)
- Silicon (Si): \( Z_{eff} = 4.5 \)

### Step 5: Write the Increasing Order
The increasing order of effective nuclear charge is:
\[ \text{Na} < \text{Mg} < \text{Al} < \text{Si} \]

### Final Answer
Thus, the increasing order of effective nuclear charge in Na, Al, Mg, and Si atoms is:
\[ \text{Na} < \text{Mg} < \text{Al} < \text{Si} \]

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