Chapter 1: The Solid State
Difference between Crystalline and Amorphous Solids
A solid state is simply one of the states of matter. One of the many different states of matter is solid. Solids have a distinct volume, mass, and shape. Solids differ from liquids and gases in that they exhibit unique characteristics. These solid states or shapes depending on how the particles are arranged in a specific or indeterminate geometry. There are, however, a few exceptions when it comes to the particles that make up the solid material. These particles are held together by powerful forces between them, irrespective of whether they are molecular, ionic, metallic, or covalent. Crystalline solids are made up of an array of particles that are uniformly arranged and kept together by intermolecular forces. On the other hand, the particles are not arranged in regular arrays in amorphous solids.
Solids are classified into two types based on the arrangement of constituent particles:
- Crystalline solid
- Amorphous Solid
Crystalline Solids
A crystalline solid has a well-arranged large small crystal. A crystal is an ordered arrangement of constituent particles (atoms, molecules, or ions).
Crystalline solid has a long-range order which means that there is a consistent pattern of particle arrangement that repeats itself on a regular basis across the entire crystal. Typical crystalline solids examples are sodium chloride and quartz.
Properties of Crystalline Solids
- Crystalline solids have a sharp melting point and begin to melt at a specific temperature.
- The shapes are well defined and also particle arrangements of crystalline solids are well-defined.
- Crystalline solid has cleavage property, which means that when cut with the edge of a sharp tool, they split into two pieces and the newly formed surfaces are smooth and plain.
- They have a distinct heat of fusion (amount of energy needed to melt a given mass of solid at its melting point).
- Crystalline solids are anisotropic. Anisotropic solids have physical properties, such as electrical resistance or refractive index, that differ when measured in different directions within the same crystal.
- True solids are crystalline solids.
Types of Crystalline Solids
Crystalline solids are classified into four types based on the nature of their intermolecular forces: molecular, ionic, metallic, and covalent solids. Let us now learn more about these classifications.
- Molecular Solids: In molecular solids, molecules are constituent particles. Molecular solids are further divided into three categories-
- Non-polar Molecular Solids – These are made up of either atoms, such as argon and helium, or molecules formed by non-polar covalent bonds, such as H2, Cl2, and I2. The atoms or molecules in these solids are held together by weak dispersion forces or London forces.
- Polar Molecular Solids – Polar covalent bonds form the molecules of substances such as HCl, SO2, and others. The molecules of these solids are held together by relatively stronger dipole-dipole interactions. These soft solids are electrically inactive. Examples of such solids include solid SO2 and solid NH3.
- Hydrogen-Bonded Molecular Solids – These solids’ molecules have polar covalent bonds between H and F, O, or N atoms. Strong hydrogen bonding holds molecules of solids like H2O together (Ice).
- Ionic Solids: Ions are the particles that make up ionic solids. Ionic solids are three-dimensional arrangements of cations and anions held together by strong electrostatic forces. These solids are naturally hard and brittle. Their melting and boiling points are both very high. Because the ions cannot move freely, they are electrical insulators in the solid state. When the ionic solid is molten or dissolved in water, the ions become free to move and conduct electricity.
- Metallic Solids: Metals are a well-organized collection of positive ions that are surrounded and held together by a sea of free electrons. These electrons are mobile and are distributed evenly throughout the crystal. Each metal atom adds one or more electrons to the sea of mobile electrons. Metals’ high electrical and thermal conductivity is due to these free and mobile electrons. These electrons flow through the network of positive ions when an electric field is applied.
- Covalent or Network Solids: The formation of covalent bonds between adjacent atoms throughout the crystal results in a wide range of nonmetal crystalline solids. They are also known as giant molecules. Because covalent bonds are strong and directional in nature, atoms are held very tightly in their positions. These solids are extremely hard and brittle. They have extremely high melting points and may decompose prior to melting. They are electrical insulators that do not conduct electricity. Diamond and silicon carbide are two well-known examples of such solids.
Amorphous Solids
Amorphous solids (Greek amorphous = no form) are made up of irregularly shaped particles. Short-range order exists in the arrangement of constituent particles (atoms, molecules, or ions) in such a solid. Only over short distances is a regular and periodically repeating pattern observed in such an arrangement.
Amorphous solids include gels, plastics, various polymers, wax, and thin films.
Properties of Amorphous solid
- Amorphous solids soften gradually over a temperature range and can be shaped into various shapes when heated.
- Amorphous solids are pseudo-solids or supercooled liquids, which means they move very slowly. If you look at the glass panes that are fixed to the windows of old buildings, you will notice that they are slightly thicker at the bottom than at the top.
- Amorphous solids have an irregular shape, indicating that the constituent particles do not have a definite geometry of arrangement.
- When amorphous solids are cut with a sharp edge tool, irregular surfaces are formed.
- Because of the irregular arrangement of the particles, amorphous solids do not have definite heat of fusion.
- Because of the irregular arrangement of particles, amorphous solids are isotropic in nature, which means that the value of any physical property would be the same along any direction.
Difference Between Crystalline Solids and Amorphous Solids
To help you see how crystalline and amorphous solids are not the same, here are some big differences between crystalline and amorphous:
Difference between Crystalline and Amorphous | |
Crystalline Solids | Amorphous Solids |
| A crystalline solid has well-arranged constituent particles. | Constituent particles of amorphous solids are not well arranged. |
| Crystalline solids are true solids. | Amorphous solids are pseudo-solids. |
| Crystalline solids are anisotropic. | Amorphous solids are isotropic. |
| Crystalline solids have a sharp melting point and begin to melt at a specific temperature. | Amorphous solids soften gradually over a temperature range and can be shaped into various shapes when heated. |
| The shapes are well defined and also particle arrangements of crystalline solids are well-defined. | The shape of amorphous solids is irregular and also particle arrangement is not well defined. |
| Crystalline solids when cut with the edge of a sharp tool, they split into two pieces and the newly formed surfaces are smooth and plain. | When amorphous solids are cut with a sharp edge tool, irregular surfaces are formed. |
This are some significant distinctions noted differences between amorphous and crystalline solids.
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Crystalline and Amorphous Solids – FAQs
What causes solids to be rigid?
In solids all the constituent particles are strongly connected also the bonds between the atoms are very strong that’s why solids are rigid.
Why do solids have a specific volume?
Because of the rigidity of their structure, solids retain their volume. Interparticle forces are extremely strong. Furthermore, interparticle spaces are scarce and small. As a result, applying pressure to them will not change their volumes.
Ionic solids conduct electricity when molten but not when solid. Explain.
An ionic solids conduct electricity when molten because electrons are free and they can move from one point to another but in solid-state all the constituent particles are strongly connected so that electrons are not able to move to conduct electricity.
What are electrical conductors, malleable and ductile solids?
Metallic solids are electrical conductors, malleable and ductile. In metallic solids, there is a metallic bond.
Why is glass classified as a supercooled liquid?
Glass is considered a supercooled liquid because it exhibits some of the properties of liquids despite being an amorphous solid. It is, for example, slightly thicker at the bottom. This is only possible if it has flown like liquid, albeit very slowly.
Crystal Lattice and Unit Cell
In crystalline solids, their constituent particles have a definite arrangement in three dimensions. The positions of these particles in the crystal relative to each other are usually represented by points. The dispensation of these unendurable sets of points is called a space lattice. The positions occupied by atoms, ions, or molecules in a crystal lattice are the collar lattice points or lattice sites. This article will cover the basic understanding of space lattices, two-dimension lattices, and then three-dimensional lattices. Let’s start one by one.
Two Dimensional Lattices
A two-dimensional lattice is a stationary distribution of constituent particles (atoms, ions, or molecules) in the plane. There are five types of two-dimensional lattices. These are square rectangles, parallelograms, rhombi, and hexagonal lattices. These differ in the symmetry of the arrangement of the points. The hexagonal lattice has the most symmetrical arrangement of points while the parallelogram lattice has the least symmetric arrangement of points.
Two Dimensional Lattices
Because of the regularly repeating arrangement of points in a two-dimensional lattice, a small portion of the lattice needs to be described to fully specify it. For example, select four points in a two-dimensional lattice and join them to form a parallelogram. This smallest part is called the unit cell. An ideal lattice can be produced by repeatedly rotating the unit cell in the direction of its edges by a distance equal to the edge of the cell.
A unit cell gives the shape of the entire lattice. However, it can be noted that for any given lattice, unit cells can be selected in many different ways. This is because a lattice contains a very large number of atoms and many identical points can be found.
A cell with an interior point is called a focused unit cell. Those unit cells which do not have any internal point are called primitive unit cells.
Thus, a two-dimensional lattice can be described as specifying the unit cell by the lengths of the edges and the angles between them. Thus, the five two-dimensional lattices are:
- Square lattice
- Rectangular lattice
- Parallelogram lattice
- Rhombic Lattice
- Hexagonal Lattice
Three Dimensional Crystal Lattice
The constituent particles (atoms, ions, or molecules) in a crystalline solid have a definite three-dimensional arrangement. If the three-dimensional arrangement of the constituent particles in a crystal is represented diagrammatically by depicting each particle by points, the arrangement is called a crystal lattice or space lattice. A crystal lattice is a well-ordered arrangement of the constituent particles of a crystalline solid in three-dimensional space.
Three Dimensional Crystal Lattice
A three-dimensional crystal lattice is like a two-dimensional lattice. The set of repeated lattice points in a crystal lattice can be found by carefully analyzing the crystal lattice This undershot Ingeminate pattern is called a unit cell. A unit cell can be defined as the smallest three-dimensional repetitive part of the space lattice, which when repeated in different directions produces a complete crystal lattice.
Unit Cell
This smallest Ingeminate pattern represents the size of the entire crystal. A complete lattice can be produced by repeatedly rotating the unit cell in the direction of its edges from a distance equal to the edge of the cell. Crystals can be thought of as consisting of an infinite number of unit cells.
Crystal Lattice
A regular three-dimensional arrangement of points (ions, atoms, etc.) in space is called a crystal lattice. There are 14 possible three-dimensional lattices. These are called Bravais lattices.
Characteristics of Crystal Lattice
- Each point in the crystal lattice is called a lattice point or lattice site.
- Each point in the crystal lattice represents a constituent particle which can be an atom, a molecule (a group of atoms) or an ion.
- Three-dimensional distribution of lattice points representing a crystal lattice.
- The lattice points are joined by straight lines to bring out the geometry of the lattice.
Unit Cell
The unit cell is the smallest part of the crystal lattice, which when repeated in different directions produces the entire lattice, e.g. Primitive unit cells and concentrated unit cells.
Types of Unit Cells: Unit cells can be broadly divided into two categories,
- Primitive unit cell – In primitive unit cells, the constituent particles are present only at the corner positions of the unit cell whereas, in concentrated unit cells, one or more component particles are present at locations other than the corners.
- Centered unit cell Centered unit cell is classified as the body-centered unit cell (bcc), face-centered unit cell (fcc), end centered unit cell (ECC).
Parameters of a Unit Cell
A unit cell is characterized by:
- Its dimensions (length) are of form a, b, and c with three edges. These edges may or may not be mutually perpendicular.
- angles α, ß, and γ between the pair of edges. The angle α is between the edges b and c, angle ß is between the edges c and a and angle γ is between the edges a and b. Thus, a unit cell is characterized by six parameters, a, b, c, α, ß and γ. The complete crystal lattice can be obtained by expanding the unit cell in all three directions.
Difference between Crystal Lattice and Unit Cell
Crystal lattice | Unit cell |
| It is a regular arrangement of constituent particles of crystalline solids in three-dimensional space. | It is the smallest repeating structural unit which when repeated in all three dimensions generates a crystal lattice. |
| It is made up of large numbers of unit cells. | It is the fundamental unit of crystal lattice having all the properties of the crystal. |
| The crystal lattice is defined in terms of the properties of the unit cell. | Unit cell defines all the fundamental properties of a crystalline solid. |
| A crystal lattice can be split up or broken into numerous individual unit cells. | A unit cell cannot be split or broken into smaller units without changing its properties. |
| It is a macroscopic system. | It is a microscopic system. |
| The crystal lattice can be isolated or prepared for analysis. | A unit cell cannot be isolated for analysis it is a hypothetical entity. |
Sample Questions
Question 1: What are the characteristics of this unit cell?
Answer:
The most convenient cell is the smallest unit cell that has perfect lattice symmetry.
- For square, rectangular, and parallelogram lattice, the unit cells chosen are square, rectangle, and parallelogram respectively.
- For a hexagonal lattice, the unit cell is a rhombus with an angle of 60°.
- For a rhombus lattice, a rectangular unit cell with one interior point is usually chosen.
Question 2: Which network solid is an exceptionally good conductor of electricity?
Answer:
Graphite, a network solid, is a good conductor of electricity.
Question 3. How are unit cell and space lattice related?
Answer:
The space lattice is obtained by repeating the unit cell in three dimensions. The spatial arrangement, stump, and density of the unit cell and space lattice are equivalent.
Question 3: Pick out the odd ones from the following sets: Sulphur, Argon, Solid Co2, Diamond, SIC, Quartz, BaO, Graphite
Answer:
Diamond because all other molecular solids and BaO because all are covalent solids.
Question 4: What is the difference between glass and quartz, while both are made of SiO4, tetrahedral? Under what conditions can quartz be converted to glass?
Answer:
Glass is an amorphous solid whereas quartz is a crystalline solid. When quartz melts and then rapidly cools, quartz turns into glass.
Question 5: If you break a piece of a cube, what difference would you expect to see in the behavior of glass and sodium chloride?
Answer:
Glass (an amorphous solid) will break irregularly, usually in a curved shape because its constituent molecules are not arranged in an ordered pattern. As a substitute, sodium chloride (tan ionic solid) will break all plane surfaces parallel to the faces of the cube because the planes of its constituent ions are parallel to the faces of the crystalline cube.
Calculate the Number of Particles per unit cell of a Cubic Crystal System
We are mostly surrounded by solids, which we use more frequently than liquids and gases. We require solids with a wide range of properties for various applications. These properties are determined by the nature of the constituent particles and the binding forces that exist between them. As a result, studying the structure of solids is critical.
Solids are classified into two types: crystalline and amorphous. This article will teach us more about crystalline solids. Crystalline solids are distinguished by a regular and repeating pattern of constituent particles. The three-dimensional arrangement of constituent particles in a crystal is known as crystal lattice if it is represented diagrammatically, with each particle depicted as a point.
The smallest portion of a crystal lattice that, when repeated in different directions, produces the entire lattice. Unit cells are three types:
- Primitive Unit Cells
- Body-centered Unit Cells
- Face-centered Unit Cells
Number of Particles in Unit Cells
We know that any crystal lattice is composed of a large number of unit cells and that each lattice point is occupied by one constituent particle (atom, molecule or ion). Let us now determine which portion of each article belongs to which unit cell. For the sake of simplicity, we will consider three types of cubic unit cells and assume that the constituent particle is an atom.
- Primitive Cubic Unit Cell
In Fig.(a), each small sphere represents only the centre of the particle occupying that position, not its actual size. Such structures are referred to as open structures. In open structures, particle arrangement is easier to follow. Figure (b) shows a space-filling representation of a unit cell with actual particle size.
Since each cubic unit cell has 8 atoms on its corners, the total number of atoms in one unit cell is-
Number of atoms in simple cubic unit cell = (Number of corners)×(Part of atom in each unit cell)
= (8)×(1/8)
= 1 atom.
- Body-Centered Cubic Unit Cell
A body-centered cubic (bcc) unit cell has an atom at each of its four corners, as well as one atom in the centre. It can be seen that the atom at the centre of the body belongs entirely to the unit cell in which it is found.
Thus in a body-centered cubic (bcc) unit cell atoms are present at all the corners and at the body centre of the unit cell.
Total number of atoms per unit cell = Atom at corners + Atom at body center
= (8 corners × 1/8 per corner atom) + (1 atom at body centre)
= 8 × 1/8 + 1
= 1 + 1
= 2 atoms.
Face-centered Cubic Unit Cell
A face-centered cubic (fcc) unit cell has atoms at all of the cube’s corners and in the centre of all of its faces. Each atom in the centre of the face is shared by two adjacent unit cells, and only half of each atom belongs to a unit cell.
Thus, a face-centered cubic (fcc) unit cell has atoms at all the corners and all the faces of the unit cell.
Total number of atoms per unit cell = Atoms at corners + Atoms at the face of the unit cell
= (8 corners × 1/8 per corner atom) + ( 6 faces × 1/2 per face atom)
= 8 × 1/8 + 6 × 1/2
= 1 + 3
= 4 atoms.
Solved Questions
Question 1: Solids are rigid why?
Answer:
Solids are rigid because the constituent particles of solid has fixed position and there is no relative motion between the particles.
Question 2: What are lattice point and their significance?
Answer:
The position of a specific constituent in the crystal lattice is denoted by the lattice point. This lattice point can be an atom, an ion, or a molecule. The shape of a crystalline solid is determined by the arrangement of the lattice points in space.
Question 3: Give an example of crystalline solids?
Answer:
Example of crystalline solids are salt (Sodium Chloride), Quartz, Diamond, etc.
Question 4: How many atoms are present in the face-centered cubic unit cell at the following position-
- at corners,
- at faces and
- at centre
Answer:
In face-centered cubic unit cell we know that atoms are present at all the corners and at all the face of unit cell.
- Number of atom at corners = 8 corner × 1/8 per corner atom = 8 × 1/8 = 1 atom at corner
- Number of atom at face = 6 face × 1/2 per face atom = 6×1/2 = 3 atoms at face of unit cell
- Number of atom at center = 0 (because in face-centered cubic unit cell no atom is present at center).
Question 5: What are the parameters that define a unit cell?
Answer:
The parameters that define a unit cell are as follows:
- unit cell dimensions along three edges: a, b, and c.
- the second parameter is the angles α, β and γ between the edges.
Close Packing in Crystals
In the formation of crystals, the constituent particles (atoms, ions, or molecules) are closely intertwined. A tightly packed arrangement is one in which maximum available space is occupied by leaving minimum free space. This corresponds to the condition of the maximum possible density. The closer the packing, the higher the stability of the packed system.
The majority of solids we come across are crystal solids. The arrangement of constituent particles in a precise configuration known as crystal lattices causes these crystalline structures to develop. The close packing of their atoms causes these formations to form. Let’s take a closer look at this.
Close Packing in Crystal
The constituent particles of a crystal can be of different sizes and therefore the method of closest packing of the particles will vary according to their size and shape. However, to understand why we can use uniform rigid spheres of equal size to represent atoms in metal as the closest packing of similar spheres.
In crystals, close packing refers to the efficient arrangement of constituent particles in the lattice. To further comprehend this packing, we must suppose that all particles (atoms, molecules, and ions) have the same spherical solid shape. As a result, the cubic shape of a lattice’s unit cell. There will always be some vacant spots in the cell when we stack the spheres. The arrangement of these spheres must be exceedingly effective in order to minimise these empty areas. To avoid empty spaces, the spheres should be positioned as near together as feasible.
The concept of Coordination Number is also connected. In a crystal lattice arrangement, the coordination number is the number of atoms that surround a centre atom. Ligancy is another name for it. As a result, there are three ways in which the constituent particles are tightly packed.
Close Packing in one Dimension:
There is an exclusive thoroughfare to arrange spheres in a one-dimensional densely packed structure in which the spheres are placed in a horizontal row touching each other. As shown in figure-
Close Packing of particles in one Dimension
As can be perceived, in this arrangement each sphere is in contact with its two neighbours. The number of proximate neighbours of a particle is called its coordination number. Hereby, in a one-dimensional densely packed arrangement, the coordination number is 2.
Close Packing in two Dimension:
Close-packed structures can be generated by placing rows of close-packed spheres. The rows can be joined in the following two ways concerning the first row to form a crystal plane.
- Square close packing or AAA… type arrangement in two dimensions- Linear arrangement of spheres in one direction is repeated in two dimensions i.e. more number of rows can be generated similar to a one-dimensional arrangement such that all spheres of different rows are aligned vertically as well as horizontally. If the first row is represented as an A-type arrangement, the packing described above is said to be AAA… type, since all rows are the same as the first row.
Close Packing of particles in two Dimension
A sphere is in contact with four spheres in a square closed packing. This type of packing is also called AAA… type arrangement in two dimensions.
Note: The space enclosed by four spheres is called a tetrahedral void.
- Hexagonal close-packing or ABABA… type arrangement in two dimensions- In this type of arrangement, the spheres of the second row are arranged in such a way that they fit into the depression of the first row. The second row is indicated as type B. The third row is arranged like the first row A, and the fourth row is arranged like the second row. i.e., the arrangement is depicted as ABAB… On comparing these two arrangements (AAAA…type and ABAB…type) we find that the closest arrangement is ABAB…type.
A sphere is in contact with six spheres in a hexagonal closed packing. This type of packing is also called ABABA… type arrangement in two dimensions.
Note: The space enclosed by six spheres is called an octahedral void.
Close Packing in three Dimension:
Three-dimensional packaging can be done by building up layers on a square pack and a hexagonal close-pack arrangement of the first layer.
- Three-dimensional close packing from two-dimensional square close-packed layers- This type of three-dimensional packing arrangement can be the AAAA type of two-dimensional arrangement is procured by restating it in three dimensions. Only polonium of all the metals in the periodic table crystallizes in a simple cubic pattern.
- Three-dimensional close packing from two-dimensional hexagonal close-packed layers- In this arrangement, the spheres in the first layer (type A) are separated slightly and the second layer is constituted by arranging the spheres in the depressions between the spheres in layer A. The third layer is a reiteration of the first. This arrangement ABABAB is go over again ubiquitously the crystal.
Difference between Hexagonal Close Packing and Cubic Close Packing
The key difference between Hexagonal Close Packing and Cubic Close Packing is listed below:
Hexagonal Close Packing | Cubic Close Packing |
|---|---|
| In HCP, the spheres of the third layer are placed on triangular-shaped tetrahedral voids of the second layer. | In CCP, the spheres of the third layer are placed on the octahedral voids of the second layer. |
| In HCP, the spheres of the third layer lie directly above the sphere of the first layer. | In CCP, the spheres of the third layer do not lie above the spheres of the first layer. |
| In HCP, the first and third layers are identical. | In CCP, the first and third layers are different. |
| In HCP, the first and fourth layers are different. | In CCP, the first and fourth layers are identical. |
| The arrangement of packing is ABAB type. | The arrangement of packing is ABCABC type. |
Sample Questions
Question 1. What is meant by “Coordination number”?
Answer:
The coordination number describes the number of nearest neighbours with which a given atom is in contact. In the case of ionic crystals, the coordination number of an ion in the crystal is the number of oppositely charged ions around that ion.
- The coordination number of the atom in the cubic closed pack structure is 12.
- In a body-centred cubic structure, the atom has 8 coordination numbers.
Question 2. What are interstitial compounds?
Answer:
The presence of interstitial vacancies or interstitial sites plays an important role in the chemistry of transition metals. Transition metals can easily accommodate smaller non-metal atoms such as hydrogen, boron, carbon, and nitrogen due to the spaces between metal atoms. These compounds are called interstitial compounds.
Question 3. Define primitive Unit cells.
Answer:
Those unit cells whose constituent particles are present only at the corners are called primitive unit cells.
Question 4. What are crystalline solids anisotropic?
Answer:
Crystalline solids are anisotropic because the particles have different arrangements in different directions, some of their physical properties such as electrical resistance or refractive index show different values when measured in different directions in the same crystal.
Question 5. What is a distinguishing feature of metallic solids?
Answer:
Metallic solids are malleable, ductile, and good conductors of electricity in the solid-state as well as in the molten state.
Question 6. What is the coordination number of types of ions in a rock salt type crystal structure?
Answer:
The crystal structure of the rock salt type has a 6:6 coordination number for each type of ion. This means that in a NaCl crystal, each Na+ is surrounded by 6Cl ions and each CI– ion is surrounded by 6 Na+ ions.
Question 7. How many effective atoms are located at the edge centre of a unit cell in a sodium chloride crystal?
Answer:
1 atom at the edge is shared by 4 unit cells. Thus, the contribution of each atom at the edge = 1/4
The number of sodium ions present in the centre of the edge =12 × 1/4 = 3 atoms.
Question 8. Why is glass considered a supercooled liquid?
Answer:
Fluidity is the property of glass in that it looks like a liquid, so while glass is an amorphous solid, it hastened flow, although slowly. Therefore, it is called a supercooled liquid.
Packing Efficiency of Unit Cell
A crystal lattice is made up of a relatively large number of unit cells, each of which contains one constituent particle at each lattice point. A three-dimensional structure with one or more atoms can be thought of as the unit cell. Regardless of the packing method, there are always some empty spaces in the unit cell. so the question is, What Is Unit Cell Packing Efficiency? The packing fraction of the unit cell is the percentage of empty spaces in the unit cell that is filled with particles. In this article, we shall learn about packing efficiency.
Table of Content
- Packing Efficiency
- Packing Efficiency Formula
- Packing Fraction Formula
- Packing Efficiency of Metal Crystal in Simple Cubic Lattice
- Packing Efficiency of Metal Crystal in Body-centered Cubic Lattice
- Packing Efficiency of Metal Crystal in Face-centered Cubic Lattice
- Unit Cell Packing Efficiency
- Solved Examples of Packing Efficiency
Packing Efficiency
Packing efficiency is the fraction of a solid’s total volume that is occupied by spherical atoms.
Packing Efficiency is the proportion of a unit cell’s total volume that is occupied by the atoms, ions, or molecules that make up the lattice. It is the entire area that each of these particles takes up in three dimensions. It can be understood simply as the defined percentage of a solid’s total volume that is inhabited by spherical atoms.
Similar to the coordination number, the packing efficiency’s magnitude indicates how tightly particles are packed.
Packing Efficiency Formula
The formula for effective packing is,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
Packing Efficiency Determining Factors
The following elements affect how efficiently a unit cell is packed:
- The number of atoms in lattice structure’s
- Unit cell volume
- Atoms volume
Packing Efficiency can be evaluated through three different structures of geometry which are:
- Simple Cubic Lattice
- Body-Centered Cubic Lattice
- Face-Centered Cubic Lattice (or CCP or HCP Lattice)
Packing Fraction Formula
Packing Fraction Formula = Volume Occupied by all constituent particles / Total Volume of Unit Cell
Void Space Fraction: 1- Packing Fraction
Percentage of Void Space: 100 – Packing Efficiency
Packing Efficiency of Metal Crystal in Simple Cubic Lattice
The steps below are used to achieve Simple Cubic Lattice’s Packing Efficiency of Metal Crystal:
Step 1: Radius of sphere
In a simple cubic unit cell, spheres or particles are at the corners and touch along the edge. Below is an diagram of the face of a simple cubic unit cell.
It is evident that,
a=2r or r = a/2 …(Equation 1)
Where, r is the radius of atom and a is the length of unit cell edge.
Step 2: Volume of sphere:
Volume of a sphere = (4/3Ï€)(r3)
Substitution for r from equation 1 gives,
∴ Volume of one particle = (4/3Ï€)(a/2)3
∴ Volume of one particle = Ï€a3 / 6 …(Equation 2)
Step 3: Total volume of particles:
Simple cubic unit cells only contain one particle.
∴ Volume occupied by particle in unit cell = Ï€a3 / 6
Step 4: Packing Efficiency:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = ((Ï€a3 / 6) / a3) × 100
∴ Packing efficiency = 100Ï€ / 6
∴ Packing efficiency = (100 × 3.142) / 6
∴ Packing efficiency = 52.36 %
Therefore, in a simple cubic lattice, particles take up 52.36 % of space whereas void volume, or the remaining 47.64 %, is empty space.
Packing Efficiency of Metal Crystal in Body-centered Cubic Lattice
The steps below are used to achieve Body-centered Cubic Lattice’s Packing Efficiency of Metal Crystal
Step 1: Radius of sphere:
The corners of the bcc unit cell are filled with particles, and one particle also sits in the cube’s middle. The cube’s center particle hits two corner particles along its diagonal, as seen in the figure below.
The Pythagorean theorem is used to determine the particle’s (sphere’s) radius.
In triangle EFD,
b2 = a2 + a2
∴ b2 = 2a2
∴ b = √2 a
According to the Pythagoras theorem, now in triangle AFD,
c2 = a2 + b2
∴ c2 = a2 + 2a2
∴ c2 = 3a2
∴ c = √3 a
We can rewrite the equation as since the radius of each sphere equals r.
c = 4r
√3 a = 4r
∴ r = √3/4 a …(Equation 1)
Step 2: Volume of sphere:
Volume of sphere particle = 4/3 πr3. Substitution for r from equation 1, we get
∴ Volume of one particle = 4/3 Ï€(√3/4 a)3
∴ Volume of one particle = 4/3 Ï€ × (√3)3/64 × a3
∴ Volume of one particle = √3 Ï€a3 / 16
Step 3: Total volume of particles:
Unit cell bcc contains 2 particles. Hence,
volume occupied by particles in bcc unit cell = 2 × ((2√3 Ï€a3) / 16)
∴ volume occupied by particles in bcc unit cell = √3 Ï€a3 / 8 …(Equation 2)
Step 4: Packing Efficiency:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = (√3 Ï€a3 / 8a3) × 100
∴ Packing efficiency = 68 %
As a result, atoms occupy 68 % volume of the bcc unit lattice while void space, or 32 %, is left unoccupied.
Packing Efficiency of Metal Crystal in Face-centered Cubic Lattice
The steps below are used to achieve Face-centered Cubic Lattice’s Packing Efficiency of Metal Crystal:
Step 1: Radius of sphere
The corner particles are expected to touch the face ABCD’s central particle, as indicated in the figure below.
According to Pythagoras Theorem, the triangle ABC has a right angle.
AC2 = AB2 + BC2
∴ AC2 = a2 + a2
∴ AC2 = 2a2
∴ AC = √2 a …(Equation 1)
We can rewrite the equation as since the radius of each sphere equals r.
AC = 4r …(Equation 2)
From equation 1 and 2, we get
√2 a = 4r
∴ r = a / 2√2 …(Equation 3)
Step 2: Volume of sphere:
Volume of sphere particle = 4/3 πr3. Substitution for r from equation 3, we get
∴ Volume of one particle = 4/3 Ï€(a / 2√2)3
∴ Volume of one particle = 4/3 Ï€a3 × (1/2√2)3
∴ Volume of one particle = Ï€a3 / 12√2
Step 3: Total volume of particles:
Unit cell bcc contains 4 particles. Hence,
volume occupied by particles in FCC unit cell = 4 × Ï€a3 / 12√2
∴ volume occupied by particles in FCC unit cell = Ï€a3 / 3√2
Step 4: Packing Efficiency:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = Ï€a3 / 3√2 a3 × 100
∴ Packing efficiency = 74 %
As a result, particles occupy 74% of the entire volume in the FCC, CCP, and HCP crystal lattice, whereas void volume, or empty space, makes up 26% of the total volume.
Importance of Efficient Packing
Packing Efficiency is important as:
- The object’s sturdy construction is shown through packing efficiency.
- It shows various solid qualities, including isotropy, consistency, and density.
- Numerous characteristics of solid structures can be obtained with the aid of packing efficiency.
Following table lists the packing efficiency of several solid architectures
| Unit Cell | Relation between a and r | Number of Atoms in Unit Cell | Coordination Number of atoms | Packing efficiency | Free space |
|---|---|---|---|---|---|
| SCC | 0.5000a | 1 | 6 | 52.4% | 47.6% |
| BCC | 0.4330a | 2 | 8 | 68% | 32% |
| FCC | 0.3535a | 4 | 12 | 74% | 26% |
Unit Cell Packing Efficiency
A unit cell is defined as a three-dimensional structure containing one or more atoms, featuring inherent voids despite the packing. The proportion of the volume occupied by these atoms or particles relative to the total volume of the cell is known as the packing fraction. This fraction, when expressed as a percentage, indicates the packing efficiency of the unit cell, detailing how much of the cell’s space is actually filled by particles.
Lattices consist of numerous unit cells, with each lattice point typically occupied by a particle. These unit cells, even with packing, contain void spaces. Among various lattice types, cubic closed-packed (ccp) and hexagonal closed-packed (hcp) structures are noted for their high packing efficiency of 74%, meaning 74% of their volume is filled. In contrast, simple cubic lattices show a lower packing efficiency of 52.4%, and body-centered cubic lattices (bcc) have a packing efficiency of 68%.
Solved Examples of Packing Efficiency
Example 1: Calculate the total volume of particles in the BCC lattice.
Solution:
Unit cell bcc contains 2 particles. Hence,
volume occupied by particles in bcc unit cell = 2 × ((2√3 Ï€a3) / 16)
∴ volume occupied by particles in bcc unit cell = √3 Ï€a3 / 8
Example 2: Calculate Packing Efficiency of Face-centered cubic lattice.
Solution:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = Ï€a3 / 3√2 a3 × 100
∴ Packing efficiency = 74 %
Example 3: Calculate Packing Efficiency of Simple cubic lattice.
Solution:
We have,
Packing efficiency = (Volume occupied by particles in unit cell / Total volume of unit cell) × 100
∴ Packing efficiency = ((Ï€a3 / 6) / a3) × 100
∴ Packing efficiency = 100Ï€ / 6
∴ Packing efficiency = (100 × 3.142) / 6
∴ Packing efficiency = 52.36 %
Example 4: Calculate the volume of spherical particles of the body-centered cubic lattice.
Solution:
Volume of sphere particle = 4/3 Ï€r3. Substitution for r from r = √3/4 a, we get
∴ Volume of one particle = 4/3 Ï€(√3/4 a)3
∴ Volume of one particle = 4/3 Ï€ × (√3)3/64 × a3
∴ Volume of one particle = √3 Ï€a3 / 16
Packing Efficiency – FAQs
What is Face Centered Unit Cell?
In a face centered unit cell the corner atoms are shared by 8 unit cells. Additionally, it has a single atom in the middle of each face of the cubic lattice. Two unit cells share these atoms in the faces of the molecules.
What role does packing efficiency play?
Roles of Packing efficiency:
- The structure of the solid can be identified and determined using packing efficiency.
- Consistency, density, and isotropy are some of the effects. How well an element is bound can be learned from packing efficiency.
- Chemical, physical, and mechanical qualities, as well as a number of other attributes, are revealed by packing efficiency.
How effective are SCC, BCC, and FCC at packing?
The packing efficiency of different solid structures is as follows.
- Simple Cubic (SCC) – 52.4%
- Body Centered (BCC) – 68%
- Face Centered (FCC) – 74%
How effective is packing?
Packing efficiency is the proportion of a given packing’s total volume that its particles occupy. In the crystal lattice, the constituent particles, such as atoms, ions, or molecules, are tightly packed. They have two options for doing so: cubic close packing (CCP) and hexagonal close packing (HCP). The complete amount of space is not occupied in either of the scenarios, leaving a number of empty spaces or voids.
What are the factors of packing efficiency?
The following elements affect how efficiently a unit cell is packed:
- Number of atoms in the lattice structure
- Unit cell volume
- Atoms volume
Imperfections or Defects in a Solid
Matter can exist in broadly three states named solids, liquids, and gases. Solids are those substances that have short intermolecular forces between them that keep molecules (atoms or ions) closely packed. They have definite mass, volume, and shape. Their intermolecular forces are strong and intermolecular distances are short. They are rigid and incompressible.
Solids can be classified as crystalline or amorphous based on the nature of order present in the arrangement of their respective constituent particles. A crystalline solid is a large number of small crystals. These crystals have a definite characteristic geometrical shape. The arrangement of particles inside these crystals is ordered and is repeated on all three dimensions.
Imperfections or Defects in Solids
Though crystalline solids are arranged in short-range and long-range order, the crystals are not perfect. A solid usually consists of a large collection of small crystals. These small crystals have defects or imperfections in them.
Defects occur in crystals when the crystallization process takes place at a very fast or moderate rate. Single crystals are formed when the crystallization process takes place at an extremely slow rate. We consider the defects to be irregularities in the arrangement of constituent particles. Defects are considered to be of two types: Point defects and Line defects.
Point defects are the deviations or irregularities from the model arrangement around a point or an atom in a crystalline substance, whereas line defects are the irregularities or deviations from the model structuring arrangement in complete rows of different lattice points.
These irregularities are called crystal defects. Point Defects are of three types:
- Stoichiometric Defects
- Impurity Defects
- Non-Stoichiometric Defects
Stoichiometric Defects
Stoichiometric Defects are basically the point defects that don’t disturb the stoichiometry of the given solid. Stoichiometry is the relationship between the given quantities of reactants and respective products before, during, and following the chemical reactions. Stoichiometric Defects are also called intrinsic or thermodynamic defects.
- Vacancy Defect
In the crystals, when some lattice sites are left vacant, the crystal is said to have a vacancy defect. This causes a decrease in the density of the given substance. This defect can also form when the substance is heated.
Vacancy Defect
- Interstitial Defect
In the crystals, when some constituent particles occupy an interstitial site, the crystal is said to have an interstitial defect. Contrary to vacancy defect, this causes an increase in the density of the given substance.
Interstitial Defect
Note: Vacancy and Interstitial Defects are observed in non-ionic solids. Ionic solids need to maintain electric neutrality. Thus, they do not have simple Vacancy and Interstitial defects but have Frenkel and Schottky Defects.
- Frenkel Defect
This defect is present in ionic solids. The smaller ion (mostly the cation) gets dislocated from its normal site and gets moved to an interstitial site. This shift in ion creates a vacancy defect at the original site and an interstitial defect at the new location. Due to this, the Frenkel defect is also called the dislocation defect. It doesn’t affect the density of the solid as the ion is still present in the structure. This defect is found mostly in ionic substances where there is a large difference between the size of the ions. For example, ZnS, AgCl, AgBr, and AgI show Frenkel defects as Zn2+ and Ag+ ions are small in size.
Characteristic of Frenkel Defect
- This defect occurs exclusively when the cations are smaller than the anions. There are no changes in chemical attributes as well.
- The Frenkel defect also has no effect on the density of the solid, hence both the volume and mass of the solid are conserved.
- Substances retain their electrical neutrality in such instances.
- As the like charge ions get closer together, the dielectric constant rises.
- Because of the presence of unoccupied lattice sites, materials with Frenkel defects exhibit conductivity and diffusion in the solid state.
- The Frenkel flaw reduces the lattice energy and stability of crystalline solids. This flaw has an impact on the chemical characteristics of ionic compounds.
- The solid’s entropy rises.
Frenkel Defect
- Schottky Defect
The Schottky defect is like a vacancy defect in the ionic solids. The number of mission cations and anions are always constant in this defect to uphold electric neutrality. Just like a simple vacancy defect, the Schottky defect also ends up decreasing the density of the given substance. The number of defects in ionic solids is pretty significant. This defect can be observed in ionic solids having cations and anions of similar sizes. For example, KCL, NaCl, CsCl, and AgBr.
Characteristics of Schottky Defect
- The size difference between cation and anion is negligible.
- Both cation and anion depart the solid crystal.
- Atoms are also permanently ejected from the crystal.
- In most cases, two positions are created.
- The density of the solid decreases significantly.
Schottky Defect
Exception: AgBr shows both Frenkel and Schottky defects.
Impurity Defects
In impurity defects, foreign components or impurities replace the position of existing ions. For example, if molten NaCl is crystallized in the presence of SrCl2, some of the sites of Na+ ions can be occupied by Sr2+ ions. To maintain electric neutrality, each Sr2+ ion will replace two Na+ ions. Sr2+ ion occupies the site of one Na+ ion and the other Na+ ion site remains vacant. Thus, the number of vacancies created is also equal to the number of Sr2+ ions present.
Other examples of impurity defects are CdCl2 and AgCl.
Impurity Defect
Non-Stoichiometric Defects
A large number of nonstoichiometric inorganic solids are known which contain the constituent elements in a non-stoichiometric ratio due to defects in their crystal structures. These defects are of two types:
- Metal excess defect
- Metal deficiency defect
- Metal Excess Defect
Metal excess defect due to anionic vacancies – this defect is observed in alkali halides like NaCl and KCl. In this defect, crystals have an excess of cation because the anion present combines with an external ion to form another compound. For example, when NaCl crystals are heated in presence of sodium vapors, the Cl– ions diffuse and form NaCl with the sodium vapor. Thus, the crystals now have an extra Na+ ion present. This anionic site is occupied by an unpaired electron. We call this site an F-center. This word is derived from the German word Farbenzenter for color center). This gives off a yellow color to the NaCl crystals. The color appears from the excitation of these electrons when they absorb energy from the visible light falling on the crystals. Just like the above cases, excess Lithium makes LiCl crystals pink and excess Potassium makes KCl crystals violet (or lilac).
Consequences of metal excess defect
The existence of free electrons in crystals with metal excess flaws causes them to be coloured. Because of the presence of free electrons, crystals with metal excess flaws conduct electricity and are semiconductors. Because the electric transport is mostly accomplished by “excess” electrons, these are referred to as n-type (n for negative) semiconductors. Because of the existence of unpaired electrons at lattice sites, crystals with metal excess defects are often paramagnetic. When the crystal is exposed to white light, the trapped electron absorbs some of it in order to be stimulated from the ground state to the excited state. Color is created as a result of this. These are known as F-centres. Positive ion vacancies accompany such surplus ions. These vacancies serve the same purpose as anion vacancies in trapping electrons. V-centres are the colour centres that result from this process.
F-center in a crystal
The metal excess defect is caused by the presence of extra cations at the interstitial sites – at room temperature, Zinc oxide is white. When it gets heated, it loses oxygen and turns from white to yellow.
ZnO ⇢ Zn2+ + ½O2 + 2e– (in presence of heat)
As you can see in the above equation, there is now an excess of Zinc in the crystal and its formula becomes Zn1+xO. The extra Zn2+ ions get shifted to interstitial sites and the respective electrons to neighboring interstitial sites.
- Metal Deficiency Defect
As we know, it is tough to prepare several solids in stoichiometric composition. These solids have less amount of metal as compared to stoichiometric proportion.
A perfect example of this type is FeO which is mostly found with a composition of Fe0.95O. It may range from Fe0.93O to Fe0.96O. In the crystals of FeO, there is a loss of positive charge due to the missing Fe2+ cations. The loss of charge is covered by the presence of the Fe3+ ions to make the substance electrically stable.
Sample Problems
Question 1: Which defects decrease the density of the substance? Explain those defects.
Answer:
Density of substance decreases in Vacancy defect and Schottky Defect.
In the crystals, when some lattice sites are left vacant, the crystal is said to have a vacancy defect. This causes a decrease in the density of the given substance. This defect can also form when the substance is heated.
Schottky defect is basically a vacancy defect in given ionic solids. The number of mission cations and anions are always constant in this defect to uphold electric neutrality. Just like a simple vacancy defect, the Schottky defect also ends up decreasing the density of the given substance. The number of defects in ionic solids is pretty significant. This defect can be observed in ionic solids having cations and anions of similar sizes. For example, KCL, NaCl, CsCl and AgBr.
Question 2: What is the dislocation defect? Give examples.
Answer:
Frenkel defect is present in ionic solids. The smaller ion (mostly the cation) gets dislocated from its normal site and gets moved to an interstitial site. This shift in ion creates a vacancy defect at the original site and an interstitial defect at the new location. Due to this, the Frenkel defect is also called the dislocation defect. It doesn’t affect the density of the solid as the ion is still present in the structure. This defect is found mostly in ionic substances where there is a large difference between the size of the ions. For example, ZnS, AgCl, AgBr, and AgI show Frenkel defects as Zn2+ and Ag+ ions are small in size.
Question 3: Which compound has both Frenkel and Schottky defects?
Answer:
Silver Bromide (AgBr) is the compound that shows both Frenkel and Schottky defects.
Question 4: Explain how defects are different in non-ionic and ionic solids.
Answer:
Vacancy Defect – In the crystals, when some lattice sites are left vacant, the crystal is said to have a vacancy defect. This causes a decrease in the density of the given substance. This defect can also form when the substance is heated.
Interstitial Defect – In the crystals, when some constituent particles occupy an interstitial site, the crystal is said to have an interstitial defect. Contrary to vacancy defect, this causes an increase in the density of the given substance.
Vacancy and Interstitial Defects are observed in non-ionic solids. Ionic solids need to maintain electric neutrality. Thus, they do not have simple Vacancy and Interstitial defects but have Frenkel and Schottky Defects.
Question 5: Why does NaCl acquire yellow color when subjected to excess sodium?
Answer:
When NaCl crystals are heated in presence of sodium vapours, the Cl– ions diffuse and form NaCl with the sodium vapour. Thus, the crystals now have an extra Na+ ion present. This anionic site is occupied by an unpaired electron. We call this site an F-center. This word is derived from the German word Farbenzenter for colour centre). This gives off a yellow colour to the NaCl crystals. The colour appears from the excitation of these electrons when they absorb energy from the visible light falling on the crystals. Just like the above cases, excess Lithium makes LiCl crystals pink and excess Potassium makes KCl crystals violet (or lilac).
Question 6: What are line defects?
Answer:
Line defects are the irregularities or deviations from the model structuring arrangement in complete rows of different lattice points.
Question 7: What are thermodynamic defects? Mention its types.
Answer:
Stoichiometric Defects are basically the point defects that don’t disturb the stoichiometry of the given solid. Stoichiometry is the relationship between the given quantities of reactants and respective products before, during and following the chemical reactions. Stoichiometric Defects are also called intrinsic or thermodynamic defects.
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