Q.Ionic radii vary in (Note: more than one of the given options may be correct.)

Concept: Effective Nuclear Charge

The ionic radius depends on how tightly the nucleus holds the outermost electrons. The effective nuclear charge $Z_{\text{eff}}$ is the net positive charge experienced by an electron after accounting for shielding by inner electrons:

where $Z$ is the atomic number and $\sigma$ is the screening constant.

Reasoning:

1. A higher $Z_{\text{eff}}$ pulls electrons closer to the nucleus, decreasing the ionic radius. The relationship is inversely proportional: $r \propto \frac{1}{Z_{\text{eff}}}$.

2. Greater screening (larger $\sigma$) reduces $Z_{\text{eff}}$, allowing the electron cloud to expand. Thus ionic radius increases directly with screening effect: $r \propto \sigma$. …

The size of an ion is determined by how tightly the nucleus can hold its electrons. Two competing forces govern this: the nucleus pulling electrons inward, and inner electrons pushing outer electrons away through repulsion.

Effective nuclear charge ($Z_{\text{eff}}$) is the net positive charge experienced by an electron after accounting for shielding by other electrons. When $Z_{\text{eff}}$ increases, the nucleus grips the electron cloud more tightly, shrinking the ion. When screening increases, inner electrons better shield outer ones from the nucleus, allowing the ion to expand.

Let me work through each relationship:

1. Relationship with effective nuclear charge

The effective nuclear charge is given by:

where $Z$ is the actual nuclear charge and $\sigma$ is the screening constant.

For hydrogen-like systems, the radius follows:

For many-electron ions, while the exact relationship is more complex, the fundamental inverse dependence remains: as $Z_{\text{eff}}$ increases, the electron cloud contracts. The radius varies as $r \propto \frac{1}{Z_{\text{eff}}}$, not $\frac{1}{Z_{\text{eff}}^2}$.

2. Relationship with screening effect

The screening constant $\sigma$ measures how effectively inner electrons shield outer electrons from the nuclear charge. From the definition of $Z_{\text{eff}}$:

Rearranging the radius relationship:

When screening increases (larger $\sigma$), the denominator decreases, so the radius increases. This is a direct relationship: more screening → larger radius.

However, the relationship is not simply $r \propto \sigma$ because $Z$ is also present. The correct statement is that radius increases with screening effect, making it a direct proportion in the qualitative sense tested here.

3. Why not the square relationships? …