Problem 117 - Chemistry: An Atoms-Focused Approach

Question

Pyrophyllite \(\left[\mathrm{Al}_{2} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\right]\) is a silicate mineral with a layered structure. The distance between the layers is \(1855 \mathrm{pm} .\) What is the smallest angle of diffraction of X-rays with \(\lambda=154 \mathrm{pm}\) from this solid?

Step-by-Step Solution

Answer

Answer: The smallest angle of diffraction is approximately 2.39°.

Step 1: Write down Bragg's Law formula

The formula for Bragg's Law is given by: nλ = 2d sinθ where n is the order of the diffraction, λ is the wavelength of the X-rays, d is the distance between the layers, and θ is the angle of diffraction.

Step 2: Input the given values into the formula

We are given: λ = 154 pm d = 1855 pm We are asked to find the smallest angle of diffraction, which refers to the first order of diffraction (n=1). Substitute these values into Bragg's Law formula: 1 (154 pm) = 2 (1855 pm) sinθ

Step 3: Solve for θ

Rearrange the formula to solve for θ: sinθ = (154 pm) / (2 * 1855 pm) Calculate the value of sinθ: sinθ ≈ 0.04156 Now, find the angle θ by taking the inverse sine (arcsin) of the value: θ = arcsin(0.04156)

Step 4: Calculate the smallest angle of diffraction

Calculate the smallest angle of diffraction: θ ≈ 2.39° The smallest angle of diffraction for X-rays with λ = 154 pm from the pyrophyllite solid is approximately 2.39°.

Key Concepts

X-ray Diffraction

X-ray diffraction is a fascinating technique that helps scientists understand the atomic structure of a material. It works by directing X-ray beams at a crystal and observing how these beams are scattered. When X-rays hit the crystal, they can reflect off the atom layers in a predictable way. This is because the wavelengths of X-rays are similar in scale to the distances between atoms in a solid.

• Bragg's Law is used to calculate the angles at which X-rays are most strongly diffracted.
• These angles help pinpoint the structure and spacing of the crystal layers.

In essence, by analyzing the pattern of X-ray diffraction, scientists can map out the arrangement of atoms in the material.

Pyrophyllite

Pyrophyllite is a unique silicate mineral that can be primarily found in metamorphic rocks. It is composed of aluminum silicate with a chemical formula of decoration_formal{Al}_{2} decor_formal{Si}_{4} decor_formal{O}_{10}(decor_formal{OH})_{2}. This mineral is soft and often has a talc-like feel, making it useful in commercial applications. Furthermore, pyrophyllite is often employed in the manufacture of ceramics, paints, and sculptures.

• It features a layered structure which makes it relatively pliable compared to other minerals.
• Because of its properties, pyrophyllite can serve as a model material for examining X-ray diffraction.

Its characteristic layered structure is critical to understanding how X-rays interact with pyrophyllite.

Layered Structure

A layered structure in minerals refers to the stacked arrangement of atoms or molecules that create a sheet-like formation. This configuration is particularly significant because it influences physical properties like slipperiness and flexibility. In layered silicates like pyrophyllite, the sheets are formed from silicon-oxygen tetrahedra. These layers can easily slide over one another, contributing to the soft texture of the mineral.

• Each layer is separated by an interlayer gap, which determines how the material interacts with X-rays.
• In the case of pyrophyllite, the layer separation is 1855 pm.

Understanding this formation helps in analyzing the diffraction patterns of X-rays, which need to pass through or reflect off these layers.

Silicate Mineral

Silicate minerals are by far the largest and most important class of minerals, making up approximately 90% of the Earth's crust. These minerals are primarily composed of silicon and oxygen, forming a structure based on the silica tetrahedron \(( decor_formal{SiO}_{4})\). The diverse arrangements of these tetrahedra lead to different categories of silicate minerals, including sheet silicates like pyrophyllite.

• Silicate minerals have varied applications, from construction materials to electronic components.
• Their structural diversity supports a wide range of physical properties, making them adaptable for many uses.

The study of X-ray diffraction of silicate minerals like pyrophyllite offers insights into their complex atomic framework and enhances our understanding of their functionality and applications.