Radius Ratio Rules — Revision Notes

⚡ 30-Second Revision

Radius Ratio ($r_c/r_a$): — Cation radius / Anion radius.
Purpose: — Predicts Coordination Number (CN) and geometry of ionic solids.
Stability Condition: — Cation touches all anions; anions don't touch each other (or just touch in limiting case).
Ranges & Geometries:

* $< 0.155$: CN=2, Linear * $0.155 - 0.225$: CN=3, Trigonal Planar * $0.225 - 0.414$: CN=4, Tetrahedral * $0.414 - 0.732$: CN=6, Octahedral * $0.732 - 1.000$: CN=8, Cubic

Key Examples: — NaCl (Octahedral, CN=6), CsCl (Cubic, CN=8), ZnS (Tetrahedral, CN=4).
Limitation: — Assumes rigid, non-polarizable spheres; deviations occur due to covalent character, polarization.

2-Minute Revision

The Radius Ratio Rule is a crucial tool in solid-state chemistry for predicting the coordination number (CN) and geometric arrangement of ions in an ionic crystal. It's defined as the ratio of the cation radius ($r_c$) to the anion radius ($r_a$).

The fundamental principle for stability is that the cation must be in contact with all its surrounding anions, and the anions should not touch each other (or just touch in the limiting case). If the cation is too small, anions repel, leading to instability and a transition to a lower CN.

Key radius ratio ranges to remember are:

$0.155 - 0.225$: CN=3, Trigonal Planar
$0.225 - 0.414$: CN=4, Tetrahedral
$0.414 - 0.732$: CN=6, Octahedral
$0.732 - 1.000$: CN=8, Cubic

For example, if $r_c/r_a = 0.5$, it falls in the $0.414 - 0.732$ range, predicting CN=6 and an octahedral structure, typical of NaCl. If $r_c/r_a = 0.8$, it falls in the $0.732 - 1.000$ range, predicting CN=8 and a cubic structure, like CsCl. Remember that this rule is an approximation based on ideal spherical ions and may have limitations due to factors like ion polarization or covalent character.

5-Minute Revision

The Radius Ratio Rule is a predictive guideline for determining the most stable coordination number (CN) and geometry of ionic crystals. It's calculated as the ratio of the cation radius ($r_c$) to the anion radius ($r_a$).

The core idea is that for a stable structure, the central cation must touch all its surrounding anions to maximize electrostatic attraction. Simultaneously, the anions themselves should not touch each other, as this would lead to repulsive forces and destabilize the lattice.

The 'limiting radius ratio' for each geometry is the minimum $r_c/r_a$ where the cation touches all anions, and the anions just touch each other.

Key Ranges and Geometries:

1
Trigonal Planar (CN=3): — $0.155 le r_c/r_a < 0.225$. Example: Boron in $B_2O_3$.
2
Tetrahedral (CN=4): — $0.225 le r_c/r_a < 0.414$. Example: $Zn^{2+}$ in ZnS (Zinc Blende).
3
Octahedral (CN=6): — $0.414 le r_c/r_a < 0.732$. Example: $Na^+$ in NaCl (Rock Salt).
4
Cubic (CN=8): — $0.732 le r_c/r_a < 1.000$. Example: $Cs^+$ in CsCl.

Worked Example: Consider an ionic compound where $r_c = 90,\text{pm}$ and $r_a = 200,\text{pm}$.

1
Calculate radius ratio: $r_c/r_a = 90/200 = 0.45$.
2
Compare to ranges: $0.45$ falls within $0.414 - 0.732$.
3
Prediction: CN=6, Octahedral geometry.

Important Considerations:

If the calculated $r_c/r_a$ falls below the lower limit for a specific CN, the structure will likely adopt a lower coordination number to maintain stability (e.g., if $r_c/r_a = 0.200$, it's too small for tetrahedral, so it will be trigonal planar).
The rule is an approximation. Factors like ion polarizability, partial covalent character, and temperature/pressure can cause deviations from predicted structures. However, it remains a highly effective first-order prediction tool for NEET.

Prelims Revision Notes

The Radius Ratio Rule is a critical concept for NEET, predicting the coordination number (CN) and geometry of ionic solids. It's calculated as $r_c/r_a$, where $r_c$ is cation radius and $r_a$ is anion radius. The rule's validity hinges on two conditions for stability: the cation must touch all surrounding anions, and the anions should not touch each other (or just touch in the limiting case). If the cation is too small, anions repel, leading to instability and a shift to a lower CN.

Key Radius Ratio Ranges to Memorize:

CN=2 (Linear): — $r_c/r_a < 0.155$
CN=3 (Trigonal Planar): — $0.155 le r_c/r_a < 0.225$
CN=4 (Tetrahedral): — $0.225 le r_c/r_a < 0.414$
CN=6 (Octahedral): — $0.414 le r_c/r_a < 0.732$
CN=8 (Cubic): — $0.732 le r_c/r_a < 1.000$

Important Examples for NEET:

NaCl (Rock Salt): — $r_{Na^+}/r_{Cl^-}$ is typically in the $0.414-0.732$ range, leading to CN=6 (octahedral).
CsCl (Cesium Chloride): — $r_{Cs^+}/r_{Cl^-}$ is typically in the $0.732-1.000$ range, leading to CN=8 (cubic).
ZnS (Zinc Blende): — $r_{Zn^{2+}}/r_{S^{2-}}$ is typically in the $0.225-0.414$ range, leading to CN=4 (tetrahedral).

Conceptual Points:

Limiting Radius Ratio: — This is the minimum ratio where the cation just fits, touching all anions, and the anions just touch each other. Below this, the structure is unstable.
Assumptions: — Ions are rigid, non-polarizable spheres; bonds are purely ionic.
Limitations: — Deviations occur due to covalent character, ion polarization, and non-spherical shapes. Temperature and pressure also affect structures.

Strategy for MCQs:

1
Calculate $r_c/r_a$ accurately.
2
Match the calculated value to the correct range. Be precise with boundary values.
3
For conceptual questions, recall the assumptions, stability conditions, and limitations of the rule.
4
Associate common compounds with their known structures and CNs.

Vyyuha Quick Recall

To remember the radius ratio ranges and their coordination numbers:

'Little Tigers Often Catch Cats'

Linear (CN=2): $< 0.155$
Trigonal Planar (CN=3): $0.155 - 0.225$
Tetrahedral (CN=4): $0.225 - 0.414$
Octahedral (CN=6): $0.414 - 0.732$
Cubic (CN=8): $0.732 - 1.000$

(Note: The 'T' for Tetrahedral is the second 'T' in 'Tigers', and 'C' for Cubic is the first 'C' in 'Catch'. The last 'C' for 'Cats' can be for Close-packed if $r_c/r_a = 1.000$, CN=12, but this is less common for NEET.)