Question

Europium crystallizes in a body-centered cubic lattice (the Eu atoms occupy only the lattice points). The density of Eu is \(5.26 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the unit cell edge length in picometers.

Short answer

The unit cell edge length is approximately 455 pm.

Step-by-step solution

Identify the Type of Crystal Lattice

Europium crystallizes in a body-centered cubic (BCC) lattice structure. In a BCC lattice, there is one atom at each corner of the cube and one atom in the center of the cube.

Calculate the Number of Atoms per Unit Cell

A BCC unit cell contains a total of 2 atoms. There are 8 corner atoms, each shared among 8 unit cells, contributing a total of 1 atom from the corners, plus the 1 atom in the center of the cube.

Determine the Molar Mass of Europium

The molar mass of europium, Eu, can be found from the periodic table: Molar mass of Eu = 151.96 g/mol.

Calculate the Volume of the Unit Cell

The density (\(\rho\)) formula is:\(\rho = \frac{mass}{volume}\).Given \(\rho = 5.26\, \text{g/cm}^3\) and using 2 atoms per unit cell:1 unit cell has \(2 \times \frac{151.96\, \text{g/mol}}{6.022\times10^{23}\, \text{atoms/mol}} = \text{mass of the unit cell in grams}\).From there, find:\[\text{Volume of unit cell} = \frac{\text{mass of unit cell}}{5.26 \text{g/cm}^3}\]

Calculate the Edge Length of the Unit Cell

The volume calculated in Step 4 is a cube:\(\text{Volume} = a^3\).Solve for the edge length \(a\) by taking the cube root of the volume:\(a = \sqrt[3]{\text{Volume}}\).Convert from cm to pm (1 cm = 10^10 pm).

Perform the Numerical Calculation

Plug in the numbers to find the volume.1. Calculate the mass of atoms in a unit cell:\( \frac{2 \times 151.96}{6.022 \times 10^{23}} = \text{mass in grams}\).2. Use\( \frac{\text{mass in grams}}{5.26} = \text{Volume in cm}^3\).3. Compute the cube root of the volume to find edge length in cm.4. Convert edge length from cm to pm by multiplying by\(10^{10}\).

Key concepts

Understanding the Unit Cell

A unit cell is the smallest repeating structure of a crystalline solid that, when stacked together, compromises the entire lattice of the crystal. For body-centered cubic (BCC) lattices like europium, the unit cell has a specific arrangement where you find one atom at each corner of the cube and one atom in the center.

• The corner atoms are each shared among eight adjacent cubes, so they contribute a total of one atom to the unit cell.
• The solitary atom in the center brings the total number of atoms per unit cell in the BCC structure to two.

The understanding of a unit cell is fundamental as it allows us to calculate both the mass and volume of the unit cell, which are crucial for further calculations.

The Process of Density Calculation

Density is a measure of how much mass is contained within a given volume. For a unit cell in a crystalline solid, density can be determined using the formula:
\[\rho = \frac{\text{mass}}{\text{volume}}\]
In the context of europium's BCC lattice:

• The mass of a unit cell can be determined by knowing how many atoms are present per unit cell and the molar mass of the element.
• The volume is typically derived from the cube's dimension, since the unit cell forms a cube.

By rearranging the density equation, it's possible to find the volume if the density and mass are known. This volume is key to finding the unit cell's edge length, another crucial step in understanding crystalline structures.

Exploring Molar Mass

Molar mass is the mass of a given substance (element or compound) divided by its amount of substance. It is expressed in g/mol. For europium:

• We find its molar mass, 151.96 g/mol, by consulting the periodic table.
• This value allows us to calculate the mass of europium atoms in a given quantity, like a unit cell, using Avogadro's number \(6.022 \times 10^{23}\) atoms/mol.

Understanding molar mass is vital in converting the unit cell's atom count into actual physical weight, which further allows the accurate calculation of density and volume, vital to uncovering properties of the Eu crystal lattice.

Conversion to Picometers

When working with unit cell dimensions in crystallography, converting measurements into picometers (pm) is often necessary for precision.

• One centimeter equals \(10^{10}\) picometers.
• After calculating the edge length of a unit cell in centimeters through volume, you convert it to picometers by multiplying the centimeter value by \(10^{10}\).

This conversion helps represent crystalline dimensions more accurately, facilitating a deeper understanding of the molecular structure and properties. Accurate measurement conversions are essential in physics and chemistry when working with the microscopic characteristics of materials.