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General Chemistry--Unit 2

04/25/09

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The Solid State


�Before You Begin:

To master this material you need to understand kinetic molecular theory and know how to perform unit analysis.


 

 

 

snowflake

 

 

 

 

 

 

 

 

Properties of Solids

Solids have a definite shape, unlike liquids and gases. This is because the particles that make up the solid are locked in position with respect to one another. Because the particles are not free to move about the container, the solid has a shape of its own. At this point, some smarty-pants in the class usually volunteers that sugar is a solid that takes the shape of the sugar bowl. Yes, a collection of small solid bits will take the shape of the container. However, if you examine a single grain, it has a shape of its own, and it keeps that shape. In contrast, a drop of liquid will assume a roughly spherical shape due to surface tension, but it will drip, flow, or spread out into a disk shaped puddle, depending on its environment.

Like liquids, solids have a definite volume. Solids vary little with pressure because the particles are relatively close together. This is in contrast with gases, which are compressible. In a gas, the particles are very far apart, so they change volume radically as pressure changes.

 

Types of Solids With Respect to Degree of Organization

Crystalline solids have a regular repeating pattern. Amorphous solids, or non-crystalline solids, are randomly arranged. Examples are quartz and glass. Both are silicon dioxide, but the quartz has a hexagonal crystal structure. This makes it have a sidedness and a grain. When you strike a piece of quartz, it will break along a plane. In contrast, glass is quartz that has been heated until it melts, scrambling the crystal structure, then cooled quickly before the atoms can line up in a regular array.  Glass doesn’t have a grain. It can be cut in any direction even a circle. When it breaks, it breaks in random patterns rather than along lines of cleavage.

Types of Crystal Structures:

A unit cell is the smallest repeating pattern in a crystal structure. To identify a unit cell, examine the pattern and look for the smallest part of the pattern that, if duplicated and placed side by side, the pattern would continue without gaps or overlaps.

 

diagram of the concept of a unit cell using a tile pattern of jigsaw puzzle pieces

 

In the above example, if we duplicate the unit cell and place the two side by side, we get the same pattern. If we try the same exercise with a piece that isn’t a unit cell, the pattern has gaps or overlaps.

 

the same unit cell from the diagram above compared to sets of tiles that are not a unit cell

 

 

 

Common Unit Cells of Crystalline Solids

 

Simple Cubic:

The simplest unit cells are cubic (simplest as in easiest to draw and to visualize). In cubic unit cells, all sides are the same length and all planes are perpendicular. The first cubic unit cell is called the simple cubic. It has atoms at all the corners of the cube. The length of the unit cell is labeled a.

 

simple cubic unit cell diagram

 

In this diagram, the atoms are shown as small dots, but in actual unit cells, the atomic radius is half the length of the unit cell.

 

 

two dimensional representation of a simple cubic unit cell

 

The unit cell is the central square in the diagram above. Polonium is an example of a substance that forms a simple cubic solid. The corner of the unit cell is the center of the atom. The atoms are as close as possible, so the length of the unit cell is twice the non-bonding atomic radius.

The three orthogonal planes of the unit cell bisect each atom, so one eighth of any single atom is inside the unit cell. There are eight corners to a cube, each has one eighth of an atom. So each unit cell has eight one-eighths or one atom inside. To visualize this, look up at ceiling at one corner of the room. Visualize a ball centered at the corner. One wall cuts the ball in half; the other wall cuts the half in half leaving one quarter of the ball inside. The ceiling cuts the quarter in half, to give one eighth of the ball inside the room. Multiply this by eight to get 8(1/8) =1. So where are the rest of the parts of the eight atoms? They are inside other, adjacent unit cells.

 

space filled diagram of a simple cubic unit cell

 

 

 


4Concept Check: Polonium has a simple cubic crystal structure. If the unit cell is 3.34 Å long, what is the density of polonium?

 

Answer: Polonium has an atomic mass of 210 amu. One unit cell has one atom inside. It has a volume of a3. To find the density of polonium, find the mass of one atom and the volume of a unit cell and use the density formula.

 

volume=a cubed=3.73EE-23 cm cubed
mass=210amu(1.66EE-24g/1amu)=3.5EE-22g
D=M/V=9.3g.cm cubed

 

 There are much more direct methods to find density, of course.

 


Face Centered Cubic

The face centered is another cubic unit cell. This unit cell has one atom at each corner and one atom centered on each face.

diagram of face centered cubic unit cell

 

As before, the corner atoms are bisected by three orthogonal planes, so each has one eighth of an atom inside the unit cell. On each face, an atom is bisected by one plane leaving half an atom inside the unit cell. A cube has eight corners and six faces, so there are 8(1/8) + 6(1/2) = 4 atoms inside the unit cell. Silver is an example of a substance that has a face centered cubic crystal structure.

 

space filled diagram of face centered cubit unit cell

The diagonal that stretches across a face has a length of four times the atomic radius. This line makes a right triangle with two sides of the face. We can use the Pythagorean Theorem to relate the length of the unit cell, a, to the atomic radius, r.     


                                                         

4Concept Check: Iridium has a face centered cubic crystal structure. If the unit cell is 3.83 Å, what is the non-bonding radius of an atom of iridium?

 

Answer: Use the Pythagorean Theorem to find the relationship between the unit cell length, a, and the atomic radius, r:

(4r)squared=a squared + a squared
2r=sq root(2a)
r=sq root (2)[a/2]
r=2.71EE-10m

 

 


Body Centered Cubic:

Body centered is another cubic unit cell. This unit cell has atoms at the eight corners of a cube and one atom in the center. Once again, the corner atoms are bisected by three orthogonal the planes leaving one-eighth of each atom inside. The central atom is also inside, so this unit cell contains two atoms. Nickel is an example of a substance that has a body centered cubic crystal structure.

 

diagram of body centered cubic unit cell

 

 

In this diagram, the atoms are shown as small dots. In the actual unit cell, the atom in the center is as close as possible to the atoms in the corners.

 

space fill diagram of body centered unit cell

 

The diagonal that stretches from the left upper front corner through the center of the unit cell to the right lower rear has a length of four times the atomic radius. This line forms a right triangle with a side and a diagonal across a face. We can use the Pythagorean Theorem to find the relationship between the length of a unit cell, a, and the atomic radius, r.

 

diagram of body centered unit cell that highlights the upper left corner to lower right corner atoms. These atoms touch, so the length of this diagonal is 4 times the atomic radius. The diagonal forms the hypotenuse of a right triangle whose sides the length of the unit cell (a) and square root of a.

 

 


4Concept Check:  Chromium has a body centered cubic crystal structure. If the density of 7.14g/cm3, what is the radius of the chromium atom?

Answer: Use the density and the mass of the chromium atom to find the volume of the unit cell and the unit cell length.

                                                                     

 

D=M/V
V=M/D=[51.996amu(1.66054EE-24g/1amu)]/(7.14g/cm cubed)
V=1.21EE-23cm cubed
a=cube root of V=2.2EE-10m

 

Use the Pythagorean Theorem to find the radius.

 

(4r)squared=a squared + [(square root of 2)a]sqared=3a squared
2r=(square root of 3)a
r=(sqare root f 3)a/2=1.9EE-10m

                                                                       

 


Hexagonal close pack

Another way of describing the arrangement of atoms is by identifying layers. In the hexagonal close pack, the base layer consists of six atoms in a hexagon around a central atom. The next layer is the same, but it is stacked above the first layer by having the atoms nestle in the spaces between atoms. The third layer is like the first, and so on. This structure allows the atoms to fill the available volume very effectively, much more so than the simple cubic structure. This is why very few atoms form solids with simple cubic unit cells but very many have the hexagonal close pack structure. cobalt is an example of a hexagonal close pack solid.

diagram of layers in the hexagonal close pack structure

 

 

The unit cell for the hexagonal close pack structure has hexagonal top and bottom and six rectangular sides. At the bottom of the unit cell there are six atoms at the corners of the hexagon and one in the center. In the middle there are three atoms from layer number two. At the top, there are six atoms at the corners of the hexagon and one in the center. The top and bottom layers are cut in half by the unit cell, so the top and bottom center faces have one half of an atom inside the unit cell. The corner atoms have one sixth of an atom inside the unit cell. The three center atoms are entirely inside.

unit cell of the hexagonal close pack structure

 

 

 

This means that a total of 12(1/6) + 2(1/2) + 3 = 6 atoms are inside the unit cell. In this diagram, the atoms are shown as small dots. In the actual unit cell, the atoms in the top and bottom layers are touching with the center layer lying in gaps between atoms.

For a wonderful explanation of why nature uses the hexagonal close pack see “Hexagons in a Close-Packed World” by Pedro Gomez-Romero.

 

Types of Solids With Respect to Attractive Forces

Molecular Solids and Intermolecular Forces

Molecular solids are held together by intermolecular attractions, the same forces that affect liquids and non-ideal gases. The atomic/molecular structure of a substance dictates the types of intermolecular attractions it will have and whether or not it will be a solid at standard temperature and pressure.

 

Induced dipole forces

All substances have induced dipole forces. The bigger the particle is, the stronger the induced dipole forces (so the poorer the ideal gas and the better the solid). Iodine is an example of a solid held together by induced dipole forces.

 

Dipole-dipole forces

Compounds with polar molecule have induced dipole forces and dipole-dipole forces. These attractions tend to be stronger than induced dipole forces. If two molecules are of similar size, the more polar is more likely to be a solid at room temperature and pressure. However, this factor is not as important as size. Many relatively small yet polar compounds are liquids or gases at room temperature.

 

Hydrogen bonding

The most polar bonds, those between hydrogen and nitrogen, oxygen, and fluorine, have particularly strong dipole attractions called hydrogen bonds. If two molecules are of similar size, the one able to hydrogen bond is more likely to be a solid at room temperature and pressure. Once again, this factor is not as important as size in determining if a substance will be a solid.

 

Network Covalent Solids and ‘Intramolecular’ Forces

Some solids are held together by extensive systems of bonding rather than weak attractions between neighboring molecules. When the intramolecular forces are covalent bonds, these substances are called network covalent solids. Because covalent bonds are much stronger than intermolecular forces, network covalent solids are too rigid and too big to be liquids or gases. Network covalent substances tend to be strong, but their properties vary widely depending on the arrangement of the bonds. Some examples of network covalent solids are graphite and diamond, two of the allotropic forms of carbon. Graphite consists of sheets of carbon atoms bonded in hexagons, like chicken wire. Some lubricants are made of graphite because the molecular sheets slide with respect to one another, making them slippery. Pencil leads are made of graphite because the rough texture of the surface of a sheet of paper catches the sheets of carbon atoms as the pencil slides over the surface leaving a trail of graphite behind.

 

diagram of graphite

 

Graphite is soft and pliable because the molecular sheets are only weakly attracted to one another. Diamond, on the other hand, is very hard. In diamond, the carbon atoms are bonded together in a three dimensional tetrahedral network. Instead of layers of large molecules, a diamond is one big molecule. 

 

Ionic solids

Ionic solids are held together by electrostatic attractions between oppositely charged ions. These attractions are very strong. They extend in a three dimensional array so that, in order to get an ion to move with respect to its neighbors, you have to disrupt the location of the opposite charges. This makes ionic solids hard and brittle.

Different types of ions pack in different crystal forms in order to utilize space between ions. Sodium chloride has a simple cubic unit cell, which is normally very inefficient at using space. The chloride ions are so large compared to the sodium ions that the sodium ions fit in the spaces between the chloride ions.

 

 

unit cell of sodium chloride

 

Metallic Solids

Metallic bonds are fluid valence electrons surrounding stationary nuclei and core electrons. Like network covalent solids, metals can be thought of as one big molecule, so they are too large to be gases and only mercury is a liquid at room temperature. However, neighboring atoms can move without disrupting the metallic bonds. This allows metals to be malleable and ductile (they can be pounded into sheets and drawn into wires).

 

diagram of metallic bonds

 


 

 

 

 

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