The mineral wustite is a nonstoichiometric iron oxide with the empirical formula FexO, where x is a number slightly less than 1. Wustite can be regarded as an FeO in which some of the Fe sites are vacant. It has a density of 5.75 g>cm3, a cubic unit cell with an edge length of 431 pm, and a facecentered cubic arrangement of oxygen atoms. (d) Using X rays with a wavelength of 70.93 pm, at what angle would third-order diffraction be observed from the planes of atoms that coincide with the faces of the unit cells? Third-order diffraction means that the value of n in the Bragg equation is equal to 3.

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1. Identify the Bragg's Law equation which is used to find the angle at which diffraction occurs: $n\lambda = 2d\sin(\theta)$, where $n$ is the order of diffraction, $\lambda$ is the wavelength of the X-rays, $d$ is the spacing between the planes in the crystal, and $\theta$ is the angle of diffraction.
2. Substitute the given values into the Bragg's Law equation. Here, $n = 3$ (since it's third-order diffraction) and $\lambda = 70.93 \text{ pm}$.
3. Calculate the spacing between the planes ($d$) in the crystal lattice. For a face-centered cubic lattice, the plane spacing from the center to a face can be calculated using the formula $d = \frac{a}{\sqrt{2}}$, where $a$ is the edge length of the cubic unit cell.
4. Substitute the value of $a = 431 \text{ pm}$ into the formula to find $d$.
5. Solve the Bragg's Law equation for $\theta$ to find the angle at which third-order diffraction occurs.

Key Concepts

Bragg's Law

Bragg's Law relates the angles at which X-rays are diffracted by a crystal lattice to the wavelength of the X-rays and the distance between the crystal planes. It is expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the distance between planes, and θ is the angle of diffraction. This law is fundamental in determining the structure of crystalline materials through X-ray diffraction techniques.

Nonstoichiometric Compounds

Nonstoichiometric compounds, like wustite (FexO), do not have a fixed ratio of elements, which means the composition can vary. In wustite, the value of x is slightly less than 1, indicating that some iron sites are vacant. This variability affects the physical properties of the material, such as its density and electrical conductivity, and is important in understanding its behavior in various applications.

Cubic Unit Cell

A cubic unit cell is the simplest repeating unit in a crystal lattice, characterized by equal edge lengths and right angles between the edges. In the case of wustite, it has a face-centered cubic arrangement of oxygen atoms, which influences the packing efficiency and overall structure of the material. Understanding the unit cell dimensions and arrangement is crucial for calculating properties like density and diffraction patterns.