Iron has a density of 7.86 g / cm^3 and crystallizes in a bodycentered cubic lattice. Show that

Iron has a density of 7.86 g / cm^3 and crystallizes in a...

Key Concepts

Body-Centered Cubic Lattice

The body-centered cubic (BCC) lattice is a type of crystal structure in which atoms are located at each of the eight corners of a cube, with a single additional atom at the center of the cube. This arrangement results in a specific number of atoms per unit cell and distinctive geometric relationships between the lattice parameter (the edge length of the cube) and the atomic radius, which are essential for calculating properties like density and packing efficiency.

Atomic Packing Factor

The atomic packing factor (APF) is a measure of the fraction of space within a crystal lattice that is actually occupied by the atoms, considered as hard spheres. In a BCC lattice, although the atoms pack closely, only about 68% of the available volume is filled by the atoms, with the remaining space being void. This concept is critical in understanding the efficiency and density of different crystal structures.

Relationship Between Density, Mass, and Volume

The density of a material is related to the mass of atoms in a unit cell and the volume of that unit cell. For crystalline materials, knowing the number of atoms per unit cell, the atomic mass, and the density allows one to establish a relationship that can be used to solve for unknown parameters such as the lattice parameter or the atomic radius. This connection is fundamental in materials science for characterizing and predicting material behavior.

Atomic Radius Determination from Lattice Parameters

Determining the atomic radius from lattice parameters involves relating the geometric dimensions of the unit cell, derived from the crystal structure, to the size of the atoms. In the case of a BCC lattice, there is a specific relationship between the edge length of the cube and the atomic radius, which, along with density data, can be used to calculate the radius of the atoms. This approach is pivotal in correlating microscopic atomic characteristics with macroscopic material properties.

Transcript

00:01 We are given a value for the density of solid iron, and we are told that it is a body -centered cubic structure.

00:08 We first want to prove that for body -centered cubic structures, the volume that atoms occupy and the total volume is about 68 % of that total volume.

00:18 We also want to calculate what the value is for the atomic radius of iron.

00:25 So we are given the density, and we know that density is equal to the mass divided by the volume.

00:33 As part of proving that ratio, we need to find the total volume, which appears in that density equation.

00:40 Based on the given density and the fact that this is a body center cubic structure, we can calculate the total mass of iron, and we can rearrange this equation to solve for the total volume to be the total mass divided by the density.

00:57 So first, we need to develop an expression to determine the mass of iron that we have.

01:03 So the mass is equal to for body -centered cubic structures, we have a net total of two atoms on the face.

01:12 This is iron, and so that has a molar mass of about 55 .85 grams per mole.

01:22 Then we multiply this by the ratio of one mole is equal to 6 .022 times 10 to the power of 23 atoms.

01:40 When we multiply that all out and cancel appropriate units, we are left with final mass units and grams, which comes out to about 1 .85 times 10 to the negative 22 grams.

01:58 And now, along with the given value for the density, we can solve for the volume, from this rearranged form of the density equation.

02:07 So volume is equal to mass divided by density.

02:13 The mass we just solved for was 1 .85 times 10 to the power of negative 22 grams.

02:23 And the density was given in the problem to be 7 .86 grams per cubic centimeter.

02:35 So now we see that when we divide the stat, we are left with the total volume.

02:41 Units of cubic centimeters, and that comes out to about 2 .36 times 10 to the power of negative 23 cubic centimeters...

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