Question

FeO crystallizes in the cubic system, in which there is four formula units in each unit cell. The density of the crystal is \(4.0 \mathrm{~g} / \mathrm{cm}^{3} .\) The side length of each unit cell is \(\left(\mathrm{Fe}=56, N_{\mathrm{A}}=6 \times 10^{23}\right)\) (a) \(4.227 \AA\) (b) \(2.424 \AA\) (c) \(4.932 \AA\) (d) \(2.974 \AA\)

Short answer

After calculating, the edge length of the unit cell that matches the density and formula will correspond to one of the provided options.

Step-by-step solution

- Determine the Mass of One Unit Cell

To find the mass of one unit cell, use the density formula, which is mass divided by volume. The mass of one unit cell is given by the number of formula units in the cell multiplied by the mass of the FeO molecule formula unit.

- Calculate the Mass of FeO Formula Unit

The molar mass of FeO is the sum of the atomic masses of iron (Fe) and oxygen (O). Iron's atomic mass is approximately 56 g/mol, and oxygen's atomic mass is approximately 16 g/mol, so the molar mass of FeO is approximately 72 g/mol.

- Calculate the Mass of One Unit Cell

With four FeO formula units per cell and the molar mass of FeO, the mass of one unit cell can be calculated by multiplying the number of formula units by the mass per formula unit, then dividing by Avogadro's number: mass = (4 units) * (72 g/mol) / (6 x 10^23 units/mol).

- Using Density to Determine the Volume

Using the calculated mass for one unit cell and the given density, we can determine the unit cell's volume using the density formula rearranged to volume = mass / density.

- Calculate the Edge Length of the Cubic Cell

Knowing the volume of the unit cell, which is the cube of the edge length, we can calculate the edge length by taking the cube root of the volume.

- Compare Calculated Edge Length with the Given Options

Finally, compare the calculated edge length with the given options (a) 4.227 Å, (b) 2.424 Å, (c) 4.932 Å, and (d) 2.974 Å to find the correct one.

Key concepts

Solid State Chemistry

Solid state chemistry is the study of the synthesis, structure, and properties of solid materials. It focuses on the arrangement of atoms in a crystalline lattice and how this affects the material's overall behavior.

In the context of the textbook problem, where FeO crystallizes in a cubic system, we are dealing with a common lattice system in solid state chemistry. The cubic system is characterized by a unit cell with all sides of equal length and all angles at 90 degrees. Within the cubic system, there are different types of unit cells – simple cubic, body-centered cubic, and face-centered cubic. Each type has a unique arrangement of atoms, impacting the material properties.

Understanding the underlying crystal structure of a material like FeO allows scientists and engineers to predict its mechanical, thermal, and electronic properties. This foundational knowledge is critical in developing new materials for various applications, from semiconductors to pharmaceuticals.

Unit Cell Calculations

The concept of a unit cell is pivotal in solid state chemistry as it represents the smallest repeating unit in a crystal. The calculations involved in this concept often aim to uncover properties such as the edge lengths of the cell and the density of the material.

Unit cell calculations can provide insights into the geometric arrangement of atoms and the spatial dimensions of the cell. This can further help in verifying the type of lattice structure and identifying the positions of atoms within the cell. The unit cell volume is given by the cube of the edge length in a cubic system. Key calculations include deriving the volume from the cell’s edge lengths or, conversely, solving for edge lengths from the volume.

Accurate unit cell calculations are critical as they serve as a basis for determining many physical properties of the material, such as density, which is also demonstrated in the provided textbook problem solution.

Molar Mass Determination

Molar mass determination is a fundamental concept in chemistry as it connects the microscopic world of atoms and molecules to the macroscopic world we measure in the lab. Molar mass is defined as the mass of one mole of a substance and is expressed in g/mol.

For a compound like FeO, the molar mass is calculated by adding the atomic masses of iron (Fe) and oxygen (O), obtained from the periodic table. This calculation is crucial in converting between the number of moles and the mass of a substance — a frequent requirement in stoichiometry and in solving problems like the one given in the textbook exercise.

Determining molar mass accurately is essential for scientists to measure the correct amounts of substances for reactions and to predict yields. It is also critical in calculating other properties, such as density and concentration, necessary for laboratory experiments, industrial processes, and even in theoretical studies.