Problem 118 - Chemistry: An Atoms-Focused Approach

Question

Minnesotaite \(\left[\mathrm{Fe}_{3} \mathrm{Si}_{4} \mathrm{O}_{10}(\mathrm{OH})_{2}\right]\) is a silicate mineral with a layered structure similar to that of kaolinite. The distance between the layers in minnesotaite is \(1940 \pm 10 \mathrm{pm}\) What is the smallest angle of diffraction of X-rays with \(\lambda=154\) pm from this solid?

Step-by-Step Solution

Answer

Answer: The smallest angle of diffraction is approximately 2.27°.

Step 1: Given variables and constants

We are given the following information: - X-ray wavelength, \(\lambda = 154\,\text{pm}\) - Interlayer spacing, \(d = 1940\,\text{pm}\)

Step 2: Using Bragg's Law to find the smallest angle

We'll use Bragg's Law to find the smallest angle of diffraction (\(\theta\)): \( n\lambda = 2d\sin\theta \) Since we need the smallest angle, we'll consider the order of diffraction \(n=1\): \( 1\times154 = 2\times1940\times\sin\theta \)

Step 3: Solving for \(\theta\)

Now, we'll solve the equation for \(\theta\) by dividing both sides by \(2\times1940\) and taking the inverse sine: \( \sin\theta = \frac{1\times154}{2\times1940} \) Finally, find the angle: \( \theta = \sin^{-1}\left(\frac{154}{2\times1940} \right) \)

Step 4: Calculate the smallest angle

Now we can simply plug in the given values into the equation to find the smallest angle of diffraction: \( \theta = \sin^{-1}\left(\frac{154}{2\times1940} \right) = \sin^{-1}\left(\frac{154}{3880} \right) \) Using a calculator, we find: \( \theta \approx 2.27^{\circ} \) So, the smallest angle of diffraction of X-rays with a wavelength of \(154\,\text{pm}\) from the solid Minnesotaite is approximately \(2.27^{\circ}\).

Key Concepts

Diffraction Angle

The concept of a diffraction angle is pivotal in understanding how waves, like X-rays, behave when they encounter an object or surface. When X-rays strike a material, particularly crystalline structures, they can be scattered in various directions. The angles at which this scattering, or diffraction, occurs are known as diffraction angles.

These angles are essential for analyzing materials through techniques such as X-ray diffraction because they provide information about the internal structure of a material. In the exercise above, the smallest diffraction angle was calculated using Bragg's Law. Bragg's Law relates the wavelength of incident X-rays to the angle and spacing between crystal planes. This relationship is given by the equation:

• \[ n\lambda = 2d\sin\theta \]

Where:

• \( n \) is the order of diffraction
• \( \lambda \) is the wavelength of the X-ray
• \( d \) is the distance between atomic layers (interlayer spacing)
• \( \theta \) is the diffraction angle

By evaluating this equation, we can find the smallest angle, helping us understand the material's structural details.

X-ray Diffraction

X-ray diffraction (XRD) is a powerful analytical technique used to study the structure of crystallized materials. When X-rays are directed at a crystal, they are diffracted in many specific directions, creating a diffraction pattern. This pattern can be analyzed to reveal a crystal’s structure.

X-ray diffraction relies on the principle of interference, where waves overlap and combine. Constructive interference, where waves add up, occurs at specific angles. This is where Bragg's Law comes into play, allowing scientists to determine these angles accurately.

• XRD helps identify the atomic structure of materials
• It can determine layer spacing within crystals
• Used in various fields like mineralogy, chemistry, and material science
• Provides data crucial for understanding physical and chemical properties

By analyzing X-ray diffraction patterns, researchers can gain insights into the size, shape, and arrangement of molecules within a crystal. This is why it is a critical tool in modern research and technology.

Interlayer Spacing

Interlayer spacing refers to the distance between parallel planes of atoms or molecules in a crystalline material. This spacing is important in understanding the physical properties of a material since it affects how substances interact with the material.

In our scenario with the mineral Minnesotaite, the interlayer spacing is provided as 1940 pm. This value is crucial as it directly influences the diffraction pattern under X-ray analysis.

• The interlayer spacing contributes to the diffraction angle calculated using Bragg's Law.
• It is a defining factor in the crystal’s ability to diffract X-rays at certain angles, influencing the efficiency of the diffraction.
• It can impact other material properties such as density, porosity, and strength.
• Accurate measurements of interlayer spacing can provide deeper insights into the mineral’s composition and other features.

Understanding interlayer spacing helps determine how materials will perform under various conditions and can influence how we use these materials in technology and industry.