Radius Ratio Rule: Limiting Values and Geometry
The radius ratio rule is a key concept in solid-state chemistry used to predict coordination numbers and crystal geometry in ionic compounds. In Class 12 board examinations and competitive entrance tests like JEE Main, NEET, and CUET, radius ratio numericals appear regularly. Questions test your ability to determine limiting ratios from geometry, check whether a cation fits stably inside an interstitial void, and predict crystal lattice types.
Limiting Radius Ratio and Lattice Stability
Ionic solids are held together by electrostatic forces. For maximum lattice stability, a cation surrounds itself with as many anions as possible while minimizing anion-anion repulsion. A stable lattice requires surrounding anions to touch the central cation without overlapping.
The radius ratio is the ratio of cation radius (r+) to anion radius (r-). When a cation is large, surrounding anions touch it comfortably without mutual contact. As cation size decreases, surrounding anions draw closer until they touch edge-to-edge. This geometric boundary defines the limiting radius ratio. If the cation shrinks further, it rattles loosely in the void, forcing a transition to a lower coordination number.
Limiting Values and Geometries
Simple geometry yields exact limiting radius ratio thresholds for each coordination geometry:
Radius Ratio Definition:
ρ = r+ / r- (dimensionless ratio, with r+ and r- in identical units such as pm or Å)
Limiting Geometric Values:
1. Coordination Number 3 (Planar Triangular): limiting ratio = (2 / √3) - 1 ≈ 0.155
2. Coordination Number 4 (Tetrahedral): limiting ratio = √(1.5) - 1 ≈ 0.225
3. Coordination Number 6 (Octahedral): limiting ratio = √2 - 1 ≈ 0.414
4. Coordination Number 8 (Cubic): limiting ratio = √3 - 1 ≈ 0.732
When the radius ratio falls within a specific range, the crystal adopts the corresponding geometry:
• 0.155 to 0.225: Coordination number 3, planar triangular geometry (e.g., B2O3).
• 0.225 to 0.414: Coordination number 4, tetrahedral void geometry (e.g., ZnS).
• 0.414 to 0.732: Coordination number 6, octahedral void geometry (e.g., NaCl, MgO).
• 0.732 to 1.000: Coordination number 8, body-centred cubic coordination (e.g., CsCl).
Geometric Derivation of Limiting Ratios
For an octahedral void (CN = 6), four anions form a square of side 2r-, with the central cation touching all four. The diagonal length is 2(r+ + r-). By Pythagoras' theorem:
[2(r+ + r-)]2 = (2r-)2 + (2r-)2 = 8(r-)2
Taking the square root gives 2(r+ + r-) = 2√2 · r-, so r+ + r- = √2 · r-.
Dividing by r- gives (r+ / r-) + 1 = √2 ≈ 1.4142, yielding r+ / r- = √2 - 1 = 0.414.
For 8-fold cubic coordination (CN = 8), eight anions sit at cube corners of edge a = 2r-. The body diagonal passes through the central cation: body diagonal = √3 · a = 2√3 · r- = 2(r+ + r-). Dividing by 2r- yields r+ / r- = √3 - 1 = 1.732 - 1 = 0.732.
Worked Numerical Problems
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CBSE Class 12, State Boards, JEE Main, NEET, CUET-UG
- Topic: Radius Ratio Rule, Limiting Values and Void Geometry
- Difficulty: Intermediate
Question: An ionic solid MX crystallises in a close-packed arrangement. The ionic radius of cation M+ is 95 pm and that of anion X- is 181 pm. Calculate the radius ratio r+ / r-, predict the coordination number of the cation, and identify the expected structural arrangement.
Solution step by step:
Step 1: Calculate the radius ratio.
r+ / r- = 95 pm / 181 pm = 0.52486, which rounds to 0.525.
Step 2: Compare the ratio with standard limiting ranges.
Because 0.414 ≤ 0.525 < 0.732, this ratio lies within the octahedral range.
Step 3: State the structural prediction.
The coordination number of M+ is 6. The cation occupies octahedral voids in a face-centred cubic lattice of X- ions, forming a rock-salt (NaCl-type) structure.
Verification by second route (Boundary Check):
Minimum cation radius for octahedral void = 0.4142 × 181 pm = 74.97 pm.
Maximum cation radius for octahedral void = 0.7320 × 181 pm = 132.49 pm.
Because 95 pm lies strictly between 74.97 pm and 132.49 pm, octahedral coordination (CN = 6) is stable.
Answer: 0.525 (Coordination number = 6, Octahedral / NaCl type)
Question: The oxide anion O2- has an ionic radius of 140 pm. Calculate the minimum radius a divalent metal cation M2+ must have to fit stably into an 8-fold cubic coordination lattice. Also calculate the minimum radius required for the cation to fit into a 4-fold tetrahedral void.
Solution step by step:
Step 1: Determine the limiting condition for 8-fold cubic coordination.
For cubic coordination (CN = 8), limiting radius ratio is r+ / r- = 0.732.
Minimum cation radius r+,min = 0.732 × 140 pm = 102.48 pm, which rounds to 102.5 pm.
Step 2: Determine the limiting condition for 4-fold tetrahedral coordination.
For tetrahedral coordination (CN = 4), limiting radius ratio is r+ / r- = 0.225.
Minimum cation radius r+,min = 0.225 × 140 pm = 31.50 pm.
Verification by second route (Geometric First Principles):
For cubic: r+ = (√3 - 1) × 140 pm = (1.73205 - 1) × 140 pm = 0.73205 × 140 pm = 102.49 pm.
For tetrahedral: r+ = [√(1.5) - 1] × 140 pm = (1.22474 - 1) × 140 pm = 0.22474 × 140 pm = 31.46 pm ≈ 31.5 pm.
Both computation methods yield matching limiting radii.
Answer: Minimum radius for cubic (CN = 8) is 102.5 pm; minimum radius for tetrahedral (CN = 4) is 31.5 pm
Common Mistakes and Practical Limitations
1. Inverting the ratio: Dividing anion radius by cation radius (r- / r+) is a frequent error. The rule is strictly defined as cation radius divided by anion radius (r+ / r-).
2. Assuming 100% ionic character: The rule assumes rigid spherical ions. Partial covalency and polarisability (Fajans' rules) cause deviations. For example, transition metal halides like CuCl adopt tetrahedral structures despite larger radius ratios.
3. Confusion between void ranges: Do not confuse tetrahedral voids (0.225 to 0.414) with octahedral voids (0.414 to 0.732).
4. Misunderstanding the lower boundary: If r+ / r- drops below the limiting value, the cation rattles in the void without touching all surrounding anions, forcing transition to a lower coordination number.
Exam Relevance and Question Patterns
| Examination | Question Types Tested | Key Skills Assessed |
|---|---|---|
| CBSE and State Boards | Numerical problems and derivations | Radius ratio calculation, octahedral void derivation, and crystal type matching. |
| JEE Main | Single-choice MCQs and integer numericals | Void boundary limits, maximum interstitial radius, and void occupancy. |
| NEET | Direct formula-based MCQs | Identifying coordination numbers from ionic radii and standard ratio ranges. |
| CUET-UG | Conceptual matching and multiple choice | Linking limiting ratios to crystal lattices and deviations from covalency. |
Turn this concept into marks: solve previous-year CSIR-NET questions and take a timed quiz.
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