Radius Ratio Rule is not much of a big topic in the chapter “The Solid State”. However, it plays a very important role in the determination of a stable structure in an ionic crystal. It also helps in the determination of the arrangement of the ions in the crystal structure. Let us study this radius-ratio rule in detail and how it affects the stability and arrangement of a structure.

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## Ionic Model & Ionic Radius

The electrostatic interaction between charged spheres is responsible for the formation of bonding in an ionic model. The determination of the sizes of the ionic radius is possible by the internuclear separation of the separate contributions of the anion from cation.

The radius of one ion is calculated on the basis of a standard ion (by assuming the value of one ion). The standard ion is oxide ion which helps in the determination of the other ions. This is because an oxide ion occurs in combination with many different elements. Moreover, an oxide ion is comparatively unpolarizable. Therefore, the significant change in the size is negligible on the basis of the counterion present.

Ionic radius is helpful in the prediction of crystal structures including lengths of the axes, lattice parameters, etc. However, this prediction is possible provided the values of the radius of the ions are taken from the same origin or same reference ion. This is important for achieving the correct relative sizes.

It is essential to understand that ionic radius differs on the basis of coordination number. The increase in coordination number results in the ions moving further away from the central ion to fit more ions. Therefore, increase in coordination number will increase the interionic separation and decrease the short ranged repulsion. This, in turn, will allow the electrons present on the central ion to expand thereby increasing the size of the central ion.

Therefore, we can conclude that ionic radius will increase with an increase in coordination number. The sizes of ions help in the prediction of the structure which will form during the combination of ions. The prediction is of the ions in the structure is done by Radius Ratio or Radius Ration rule. How? Let us understand!

**Browse more Topics under The Solid State**

- General Introduction
- Crystalline and Amorphous Solids
- Space Lattice or Crystal Lattice and Unit Cell
- Number of Particles in Unit Cells
- Close Packing in Crystals
- Tetrahedral and Octahedral Voids
- Density of a Cubic Crystal
- Imperfections or Defects in a Solid
- Electrical Properties of Solids
- Magnetic Properties of Solids

## The Radius Ratio Rule

If we consider an array of anions present in the form of cubic close packing, the tetrahedral holes and the octahedral holes will vary in the sizes. Therefore, the cations will occupy the voids only if they are enough space to accommodate them. This prediction of whether the ions will be able to hold the cations can be done on the basis of Radius Ratio.

Ionic Crystals comprises many cations and anions. We know that anions are larger in size and surround the smaller cations. They are arranged in space such that anions and cations touch each other and produce maximum stability.

This stability of the ionic crystals can be explained on the basis of radius ratio. Therefore, radius ratio is the ratio of cation to the ratio of an anion. Here, Ratio of cation= r, Ratio of anion = R. Thus, Radius ratio = (r/R). Limiting radius ratio helps in expressing the range of radius ratio.

### Definition of the Radius Ratio Rule

Radius Ratio refers to as the ration of smaller ionic radius (cation) by the ratio of larger ionic radius (anion). Hence, Radius ration ρ = r_{s}/r_{l}.

### Importance of the Radius Ratio Rule

This rule helps in the determination of arrangement of ions in various types of crystal structures. It also helps to determine the stability of an ionic crystal structure. For instance, larger cations will fill the larger voids like cubic sites whereas smaller cations will fill the smaller voids such as tetrahedral sites.

It is also possible to predict the coordination number of any compound. Hence, the radius ratio rule helps in determining the structure of ionic solids.

### Examples of Radius Ratio Rule

The ratio of radii of the ions can affect the arrangement of ions in a crystal. Moreover, the limiting ratio has to be greater than 0.414, (radius ratio greater than 0.414) to fit an octahedral arrangement of anions. In this formation, cations will be able to accommodate 6 anions.

However, radius ration in between 0.225 to 0.414 will be able to fit into tetrahedral voids in the crystal lattice thereby preferring tetrahedral coordination and above 0.414 will prefer octahedral coordination. For example, if we consider an ion zinc sulfide, the radius ration will be

Therefore, zinc ion will favour tetrahedral voids in the closely arranged lattice of sulfide ions. However, in case of larger cations such as caesium, the radius ratio is larger than the limit of the coordination number of 6. Hence, the caesium ions will fit cubic sites so the coordination number will increase to 8 in the chloride ions lattice.

Below Table demonstrates the relationship between radius ratio (limiting ratio) and coordination number.

## Effect of Difference in Size of Ions on Arrangement

Below are the three diagrams which explain how the different sizes of ions will affect the arrangement as well as the stability of the ions.

Figure 1 represents M^{+} ion in contact with X^{–} ions and the fit of the first layer are “just correct.” However, Figure 2 the M^{+} ion in contact with X^{–} ions is smaller. Thus, the fit of the first layer of arrangement is poor. Moreover, if the cations move closer to the anion, the crowding of the arrangement will increase which in turn would lead to repulsion among each other.

This will further increase the energy of the system. Furthermore, if the ions try to hold each other apart then it will not be near to the opposite M^{+} ion. Figure 3 represents the arrangement of M^{+} ion and X^{–} ions where M^{+} ion is too large. Therefore, the X^{–} will be far away and not feel the X^{–} – X^{–} repulsion.

Therefore, in the case similar to figure 2 and figure 3, the crystal packing of MX will be different. Thus, if we take the example of compounds, then consider figure 1 as the lattice of NaCl. We ca relate figure 3 to the structure of CsCl where the M^{+} ion is large and it can accommodate more X^{–} ions.

Similarly, in Figure 2 M^{+} ion is very small so it can accommodate only four negative ions and compounds such as ZnS can be the example of such a case.

## Solved Example for You

Question: If a solid “A^{+}B^{–}” has a structure similar to NaCl. Consider the radius of anion as 250 pm. Find the ideal radius of the cation in the structure. Is it possible to fit a cation C^{+} of radius 180 pm in the tetrahedral site of the structure “A^{+}B^{–}”? Explain your answer

Solution: If the A^{+}B^{–} structure is similar to Na^{+}Cl^{–} ion then we know that six Cl^{–} ions will surround Na^{+} and vice versa. Therefore, Na^{+} ion fits into the octahedral void. Therefore the limiting ratio for an octahedral site is 0.414

Thus, limiting radius ratio= r/R= 0.414

From the question, R=250 pm

Therefore, r=0.414R= 0.414 × 250pm

Hence, r= 103.5pm

So the ideal radius ration of cation will be 103.5pm or A^{+} = 103.5 pm

From the above table, we know r/R for a tetrahedral site is 0.225

Therefore, r/R= 0.225

Or, r= 0.225R = 0.225 × 250pm = 56.25 pm

Thus, the ideal radius for the cation in the given structure will be 56.25 pm for a tetrahedral site. However, we know the radius of C^{+} is 180 pm. This means that the radius of C^{+ }is much larger than 56.25 pm. Therefore, it is not possible to fit cation C^{+} in a tetrahedral site.

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