The atomic radius of iridium is 1.36 ÅA. The unit cell of iridium is a face-centered cube. Calcu

The atomic radius of iridium is 1.36 ÅA. The unit cell of...

Key Concepts

Density Calculation in Crystalline Solids

Density in crystalline solids is calculated using the formula that relates the mass contained in one unit cell to its volume. This involves determining the number of atoms per unit cell, converting atomic mass using Avogadro's number, and computing the unit cell volume from its geometric parameters.

Atomic Radius

The atomic radius is the measure of the size of an atom, typically defined as the distance from the center of the nucleus to the outer limit of the electron cloud. This concept is crucial in determining interatomic distances and relating microscopic dimensions to the overall structure of a material.

Face-Centered Cubic Structure

A face-centered cubic (FCC) structure is a common crystal lattice arrangement in which atoms are located at each of the corners and the centers of all the faces of the cube. This structure dictates specific geometric relationships within the unit cell, such as the number of atoms per cell and the relation between the atomic radius and the cell edge.

Unit Cell Parameter

The unit cell parameter is a key dimension of the smallest repeating unit in a crystal lattice. In crystals like the FCC, geometric relationships relate the atomic radius to the unit cell edge, enabling one to determine the volume occupied by the atoms within the crystal.

Avogadro's Number

Avogadro's number is a fundamental constant that gives the number of atoms or molecules in one mole of a substance. It is essential in converting between the microscopic scale of an individual atom and the macroscopic scale of grams per mole when calculating properties like density.

Transcript

00:03 In this question, we want to find the density of arridium, given the information about the unit cell.

00:13 So, aridium is a face -centered cubic unit cell.

00:19 So what it looks like is it's going to just be a cube, and then we're going to have atoms on each of the corners, and then on each of the faces we're going to have atoms.

00:37 So in a face -centric cubic unit cell, there are four atoms total.

00:44 So if we zoom in on one of the faces of this unit cell, we will see that it looks like that.

00:54 So the atoms are all touching.

00:56 And if we draw a line to form a right triangle, we see the hypotenuse is equal to 4r, because we go right through the center of the center atom, and then we end up in the center of the atoms on the corners.

01:21 And these atoms are all touching.

01:23 There's no space in between them, so the hypotenuse is 4r, and r is the radius of the atoms.

01:30 Then we can label the size as d or whatever variable we want, and then we can use the pythagorean theorem to solve for d because we know r and d and the sides of the cube are the same because it's a cube.

01:55 So knowing that, we then set up our formulas.

02:00 So d squared plus d squared is going to 4r squared.

02:03 Now we want to solve for d.

02:05 So convert antstroms to pico meters by multiplying by 100.

02:10 Then we will convert that to centimeters, knowing that there are 10 to the 12 piquemeters and 1 meter, there are 100 centimeters and one meter.

02:18 And then we get the length of the radius instead of meters.

02:25 Then we're just going to solve for d.

02:27 So if we solve for that algebraically, plugging in the value for r, we get this as a value of d, the side of the cube...