Ionic Radius Ratio Calculator
Calculate ionic radius ratio with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Formula
Radius Ratio = r(cation) / r(anion)
The radius ratio is calculated by dividing the cation radius by the anion radius. Critical boundaries at 0.155, 0.225, 0.414, 0.732, and 1.0 determine the maximum coordination number that remains geometrically stable. The bond length equals the sum of ionic radii, and the lattice parameter can be derived from the geometry and bond length.
Worked Examples
Example 1: Sodium Chloride Structure
Problem: Calculate the radius ratio for NaCl (Na+ = 102 pm, Cl- = 181 pm) and predict its structure.
Solution: Radius ratio = r(Na+) / r(Cl-) = 102 / 181 = 0.5636\nRatio falls in 0.414 - 0.732 range\nPredicted: CN = 6, Octahedral, Rock Salt structure\nBond length = 102 + 181 = 283 pm\nThis matches the experimentally observed NaCl structure
Result: Ratio = 0.564 | CN = 6 | Rock Salt | Bond length = 283 pm
Example 2: Zinc Sulfide Structure
Problem: Predict the structure of ZnS (Zn2+ = 74 pm, S2- = 184 pm).
Solution: Radius ratio = 74 / 184 = 0.4022\nRatio falls in 0.225 - 0.414 range\nPredicted: CN = 4, Tetrahedral, Zinc Blende\nBond length = 74 + 184 = 258 pm\nZnS indeed forms the zinc blende (sphalerite) structure
Result: Ratio = 0.402 | CN = 4 | Zinc Blende | Bond length = 258 pm
Frequently Asked Questions
What is the ionic radius ratio and why is it important?
The ionic radius ratio is the ratio of the cation radius to the anion radius (r+/r-) in an ionic compound. It is a fundamental concept in solid-state chemistry that predicts the coordination number, crystal geometry, and structure type of ionic solids. The principle behind it is geometric: a certain minimum ratio is needed for the smaller cation to maintain contact with surrounding anions in a stable arrangement. If the cation is too small for its coordination environment, the anions would touch each other and repel, making the structure unstable. This rule was developed by Linus Pauling and remains a valuable first approximation for predicting ionic crystal structures.
What are the critical radius ratio boundaries?
The critical radius ratio boundaries correspond to the geometric limits where anions just touch each other around a central cation. Below 0.155, only linear coordination (CN=2) is stable. Between 0.155 and 0.225, three anions fit in a trigonal planar arrangement (CN=3). From 0.225 to 0.414, four anions arrange tetrahedrally (CN=4), as seen in zinc blende. The range 0.414 to 0.732 supports octahedral coordination (CN=6), exemplified by the rock salt structure. From 0.732 to 1.0, eight anions can surround the cation in a cubic arrangement (CN=8), as in cesium chloride. These boundaries are derived from simple geometric calculations of spheres in contact.
When does the radius ratio rule fail?
The radius ratio rule fails in several situations because it assumes perfectly spherical, non-polarizable ions with purely ionic bonding. Compounds with significant covalent character, such as AgI (which adopts the zinc blende structure despite its radius ratio predicting octahedral), deviate because directional covalent bonds prefer tetrahedral geometry. Highly polarizable ions like iodide can be distorted by small, highly-charged cations, changing effective radii. The rule also struggles with transition metal compounds where crystal field effects influence structure choice, and with layered structures like CdI2 where bonding is intermediate between ionic and covalent.
How do Shannon ionic radii differ from Pauling radii?
Shannon ionic radii (also called Shannon-Prewitt radii) and Pauling radii are two different systems for assigning sizes to ions. Pauling radii, published in 1927, were derived theoretically from quantum mechanical calculations and assign fixed radii regardless of coordination environment. Shannon radii, compiled from extensive X-ray crystallographic data in 1976, account for how ionic radius changes with coordination number and spin state. Shannon radii are generally preferred in modern chemistry because they are more accurate, coordination-number-specific, and available for many more ions. For example, Shannon lists Na+ as 102 pm in 6-coordination but 99 pm in 4-coordination, while Pauling gives a single value of 95 pm.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
What formula does Ionic Radius Ratio Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.