Tell the relation between effective nuclear charge `(Z_(eff))` atomic number `(Z)` and shielding constant `(sigma)`. Explain it qualitatively.

Answer

To understand the relationship between effective nuclear charge (Z_eff), atomic number (Z), and shielding constant (σ), we can break it down step by step:

### Step 1: Define the Terms
- **Atomic Number (Z)**: This is the number of protons in the nucleus of an atom, which also equals the number of electrons in a neutral atom.
- **Shielding Constant (σ)**: This represents the extent to which other electrons in the atom shield a particular electron from the full positive charge of the nucleus. Electrons in inner shells can reduce the effective nuclear charge felt by outer shell electrons.
- **Effective Nuclear Charge (Z_eff)**: This is the net positive charge experienced by an electron in a multi-electron atom. It accounts for both the total positive charge of the nucleus and the shielding effect of other electrons.

### Step 2: Establish the Relationship
The relationship can be expressed mathematically as:
\[ Z_{eff} = Z - \sigma \]

Where:
- \( Z_{eff} \) is the effective nuclear charge,
- \( Z \) is the atomic number, and
- \( \sigma \) is the shielding constant.

### Step 3: Qualitative Explanation
1. **Understanding Shielding**: In multi-electron atoms, electrons repel each other due to their negative charge. Electrons in inner shells (closer to the nucleus) shield the outer electrons from the full attractive force of the positively charged nucleus. This means that outer electrons do not feel the full charge of the nucleus.

2. **Effective Nuclear Charge**: The effective nuclear charge is thus less than the actual nuclear charge (Z) because of this shielding effect. The greater the number of inner electrons, the larger the shielding constant (σ) becomes, leading to a lower effective nuclear charge for the outer electrons.

3. **Conclusion**: Therefore, the effective nuclear charge (Z_eff) is always less than the atomic number (Z) due to the presence of the shielding effect from other electrons. This relationship helps explain various properties of elements, such as ionization energy and atomic size.

### Summary
In summary, the effective nuclear charge (Z_eff) is calculated by subtracting the shielding constant (σ) from the atomic number (Z):
\[ Z_{eff} = Z - \sigma \]
This relationship illustrates how electrons experience a reduced nuclear charge due to the presence of other electrons.

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