In which case, effective nuclear charge is minimum ?

Options

A. `Be `
B. `Be^(2+)`
C. `Be^(3+)`
D. all have the same effective nuclear charge

Explanation

To determine in which case the effective nuclear charge (Z_eff) is minimum, we need to understand the concept of effective nuclear charge and how it relates to the number of electrons and protons in an atom.

### Step-by-Step Solution:

1. **Understanding Effective Nuclear Charge (Z_eff)**:
- Effective nuclear charge is the net positive charge experienced by an electron in a multi-electron atom. It is calculated using the formula:
\[ Z_{eff} = Z - S \]
where \( Z \) is the total number of protons (nuclear charge) and \( S \) is the shielding constant (the number of inner electrons that shield the outer electrons from the nucleus).

2. **Analyzing Beryllium (Be)**:
- Beryllium (Be) has an atomic number of 4, meaning it has 4 protons and, in its neutral state, 4 electrons.
- When we consider Beryllium ions:
- \( \text{Be}^{2+} \): This ion has 4 protons and 2 electrons. The effective nuclear charge will be higher because there are fewer electrons to shield the nucleus.
- \( \text{Be}^{3+} \): This ion has 4 protons and 1 electron. The effective nuclear charge is even higher due to the minimal shielding effect.

3. **Comparing Effective Nuclear Charge**:
- For neutral Beryllium (4 electrons), the effective nuclear charge is lower than for its ions because the presence of more electrons increases the shielding effect.
- As we remove electrons (as in \( \text{Be}^{2+} \) and \( \text{Be}^{3+} \)), the effective nuclear charge increases because there are fewer electrons to shield the nucleus.

4. **Conclusion**:
- The effective nuclear charge is minimum in the neutral Beryllium atom (Be) because it has the maximum number of electrons (4) that contribute to shielding the nucleus. The more electrons present, the more they shield each other from the nucleus, resulting in a lower effective nuclear charge.

### Final Answer:
The effective nuclear charge is minimum in the neutral Beryllium atom (Be).

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