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# Radius Ratio Rule: Ionic Model & Ionic Radius, Definition of Radius Ratio, Calculation of Limiting Radius Ratio Values, Limitations, Practice Problems, FAQs

There’s a school assignment, where the teacher has asked you to prepare a portrait of Dr. Abdul kalam. She wanted this portrait on an A3 sheet (29.7 x 42 cm) and the portrait should fit properly on that sheet.

How will you proceed?

Well, taking a proper measurements of the dimensions of the portrait is definitely going to help you in making your art look neat and clean. Else you have to adjust it and meanwhile, you will compromise the quality of your art. Taking a proper measurement won’t require much effort and can be done easily. Similarly, the 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.

Table of content

• Crystal Lattice
• Ionic Model & Ionic Radius
• Calculation of someLimiting Radius Ratio Values
• Limitations
• Practice Problems

## Crystal Lattice

Many metals,ionic and covalent compounds exhibit a regular three dimensional arrangement of a small unit cell. In these arrangements, each constituent atom/ ion/molecule is in close contact (touching) with a a fixed number of other constituents. The number in close contact is called coordination number.

The coordination number is decided by the relative sizes of the contacting spheres. Depending on the relative ratio of their radius some may a coordination number of 6, 8 or 12 coordination number.

This radius ration is quite appropriate in ionic solids where the anions are much bigger than the cations with the conditions that the cations and an anion should be in contact with each other.

## Ionic Crystal & Ionic Radius

• The cations and anions in an ionic crystal is held in position by the electrostatic interaction between them.
• Cations and anions charged spheres is responsible for the formation of bonding in an ionic crystal. The internuclear separation distance is the sum of the cationic and anionic radii.
• Ionic radius is helpful in the prediction of crystal structures including lengths of the axes, lattice parameters, etc.
• 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!

The ratio of the radius of a cation () to the radius of an anion () is known as the radius ratio (). The crystal structure of many ionic solids is related to their relative sizes. Using Pythagoras theorem the radius ratios suitable for all crystal structures can be calculated. From the radius ratio, the number of ions in contact with each other (coordination number) can also be calculated. Among crystals, coordination numbers of 3,4,6,8 are common.

Coordination Number 3- in 2D:

Consider an ionic compound with a formula . The large three cations say are in contact with each other on a table. There will be an empty space(hole or void) in between them. We can place a cation A in this void.

We have three choices-

1. Radius of A is much smaller than the void radius as in figure A. The cation cannot be in touch with all the cations at the same time and cannot form
2. Radius A is just equal to the void radius as in figure B. Here A is in contact with all three anions and can form . Coordination number of A cation is 3. Here the anions also are in contact with each other
3. Radius of A is greater than void radius s in figure C. Here A is in contact with all three anions and can form . Coordination number of A cation is 3. Here the anions are not in contact with each other

A limiting case arises when the are in contact with one another and each of them in contact with the cation also as in figure B. By simple geometry, this gives the ratio (radius /radius ) . This is the lower limit for a coordination number of three.

If the radius ratio is less than , then the positive ion is not in contact with the negative ions, and it rattles in the holes, and the structure is unstable.

The corresponding limiting radius ratio values for coordination numbers 2, 3, 4, 6, and 8 are:

The radius ratio is rarely 1in ionic solids; in which case the coordination number is 12.

## Calculation of some Limiting Radius Ratio Values

Coordination Number 3 (planar triangle)

The smaller positive ions with radius interacted with 3 larger negative ions with radius . Plainly , the angle is

Coordination Number 4 (tetrahedral)

Tetrahedral arrangement inscribed in a cube. The angle tetrahedral angle of .

XD is a perpendicular on YZ. Angle YXD= 54°44’

Let’s take reciprocal of the above form we get,

Hence,

Coordination Number 6 (octahedral)

Six negative ions of radius are in contact with a positive ion of radius . In the triangle XYP

Let’s take reciprocal of the above form we get,

Rearranging

Hence,

Coordination Number 8 (Cubical)

Let’s consider side of the cube as ‘a’ and the body diagonal as ‘d’

Length of side of the cube

Using Pythagoras theorem,

Now,

## Limitations

• The radius ratio rules can only be used to estimate the cation’s coordination number, not the anion’s.
• They can only be used successfully on crystals of the AB type, which have a cation and an anion in their formula. The rules cannot be applied well to A2B or AB2crystals.
• In actuality, the extremely high coordination number expected by these laws does not occur in practice.
• The rules are applicable only to crystals made up of ions, that too only when ionic radii are known accurately.

## Practice Problems

Q1. The radius ratio of caesium chloride is expected to be 0.945, what will be the expected structure?

1. Tetrahedral
2. Square planar
3. Body centered cubic
4. octahedral

Solution:

In case of caesium chloride,

If the value of is in between 0.732-0.999 then it will have body centered cubic structure having 8 coordination number.

Q2. If the radius ratio of a compound is , what will be the coordination number of the compound?

1. 2
2. 4
3. 3
4. 8

Solution:

If , then the coordination number of the compound will be 4.

Q3. The radius of the is 105 pm and that of ion is 191 pm. Predict the coordination number of .

1. 2
2. 4
3. 6
4. 8

Solution:radius ratio Since the radius ratio is in between 0.414 to 0.732, the coordination number of cation is 6.

Q4. Which of the following statement is not relevant?

1. The radius ratio rules can only be used to estimate the cation’s coordination number, not the anion’s.
2. They can only be used successfully on crystals of the AB type, which have a cation and an anion in their formula. The rules cannot be applied well to A2B or AB2 crystals.
3. In actuality, the extremely high coordination number expected by these laws does not occur in practice.
4. None of the above

Solution:There is certain limitations in applying radius ratio rules.

• The radius ratio rules can only be used to estimate the cation’s coordination number, not the anions.
• They can only be used successfully on crystals of the type, which have a cation and an anion in their formula. The rules cannot be applied well to or crystals.
• In actuality, the extremely high coordination number expected by these laws does not occur in practice.
• The rules are applicable only to crystals made up of ions, that too only when ionic radii are known accurately.

Q1. Which factor can influence the radius of an ion?
: The number of electrons and the size of the nucleus determine the size of an atom or ion. Atoms with more electrons often have bigger radii than those with fewer electrons.

Q2. Is radius of an ion remain constant in every condition?
Answer: Yes. The radius of an ion is half the internuclear distance between two such ions in contact with each other..

Q3. How can the radius ratio rule be used to predict an ionic solid's structure?