Question

(a) Explain why X-rays can be used to measure atomic distances in crystals but visible light cannot. (b) Why can't \(\mathrm{CaCl}_{2}\) have the same crystal structure as \(\mathrm{NaCl} ?\)

Short answer

(a) X-rays can be used to measure atomic distances in crystals because their wavelengths are similar to the distances between atoms, allowing them to diffract and resolve the crystal structure. Visible light has wavelengths that are too large compared to atomic distances, making it unable to provide information about the crystal structure. (b) CaCl2 cannot have the same crystal structure as NaCl due to differences in the size and charge of their constituent ions, their stoichiometry, and their coordination numbers. These differences result in distinct lattice arrangements for the two compounds.

Step-by-step solution

Wavelength Comparison

The key to understanding why X-rays are suitable for measuring atomic distances in crystals lies in the difference in their wavelengths compared to visible light. X-rays have a much shorter wavelength (around \(10^{-10}\) meters) than visible light (approx. \(400-700\) nm, or \(4 \times 10^{-7}\) to \(7 \times 10^{-7}\) meters). The wavelength of visible light is too large compared to the atomic distances in crystals, making it impossible to resolve these smaller structures with visible light.

Bragg's Law

The main principle behind X-ray diffraction for measuring atomic distances is Bragg's Law: \(n\lambda = 2d \cdot \sin{\theta}\), where \(n\) is an integer, \(\lambda\) is the wavelength of the radiation, \(d\) is the spacing between crystal planes, and \(\theta\) is the angle of incidence of the radiation. Given the small atomic distances in crystals, typically around \(10^{-10}\) meters, X-rays with a comparable wavelength can be diffracted by the crystal, providing information about the crystal's structure. #a. Conclusion# So, X-rays can be used to measure atomic distances in crystals because their wavelengths are similar to the distances between atoms, making it possible to diffract them and resolve the crystal structure. In contrast, visible light has wavelengths that are too large compared to atomic distances, making it unable to give information about the crystal structure. #b. Understanding CaCl2 and NaCl Crystal Structures#

Ionic Compounds

Both CaCl2 and NaCl are ionic compounds. NaCl is formed by ionic bonds between sodium (Na) and chlorine (Cl) atoms, while CaCl2 is formed by ionic bonds between calcium (Ca) and chlorine (Cl) atoms. Since their atoms have different sizes and charges, it leads to a different arrangement of atoms in their respective crystal structures.

Different Ratios of Ions

In the NaCl structure, there is a 1:1 ratio of sodium to chlorine atoms, while in CaCl2, there is a 1:2 ratio of calcium to chlorine atoms. The different stoichiometry results in a different arrangement of ions to maintain an overall neutral charge.

Coordination Number

The coordination number, or the number of nearest neighbors to an atom in the crystal lattice, is an important characteristic of a crystal structure. In NaCl, both the sodium and chlorine ions have a coordination number of 6, forming a face-centered cubic (FCC) lattice. In CaCl2, calcium has a coordination number of 8, and chlorine has a coordination number of 4, resulting in a different lattice arrangement. #b. Conclusion# CaCl2 cannot have the same crystal structure as NaCl because of the differences in the size and charge of their constituent ions, their stoichiometry, and their coordination numbers. These differences result in distinct lattice arrangements for the two compounds.

Key concepts

Bragg's Law

Bragg's Law is a fundamental equation that allows us to decipher the crystalline structure of materials by observing how X-rays scatter off the electron clouds within a crystal. The law relates the wavelength of electromagnetic radiation to the diffraction angle and the spacing between planes in the crystal lattice. This relationship is encapsulated in the equation:
\[\begin{equation}n\theta = 2d \times \text{sin}\theta \end{equation}\]
Here, \(n\) is the order of the diffraction, an integer representing the path difference in terms of the wavelength; \(\theta\) is the diffraction angle; \(d\) is the interplanar spacing; and \(\theta\) is the angle of incidence. When X-rays are fired at a crystal, some rays are deflected by atoms and scatter. If the scattered waves are in phase, they will constructively interfere and create a detectable pattern, revealing the internal structure of the crystal.

X-ray crystallography, which utilizes Bragg's Law, has been instrumental in determining the atomic structure of myriad substances, from simple salts to complex proteins, enabling scientists to understand the material properties at a fundamental level.

Atomic Distances in Crystals

Understanding atomic distances in crystals is vital for grasping the nature of solid materials. Atomic distances, or interatomic spacings, refer to the distances between the centers of atoms within a crystal. These distances are determined by the size of the atoms and the type of bonding that occurs between them (covalent, ionic, or metallic).

Within a crystal, atoms are arranged in a regular, repeating pattern known as a lattice. The precise measurement of these atomic distances is essential for discerning the physical properties of the material, such as its density, hardness, and electrical conductivity. X-ray crystallography is one of the most accurate methods for measuring these distances. When an X-ray beam penetrates a crystal, it is diffracted by the electron clouds surrounding the atoms in the crystal lattice. The diffracted beams produce a diffraction pattern that can be analyzed to reveal the atomic distances and the three-dimensional arrangement of atoms.

In addition, the concept of atomic distances in crystals is crucial when considering material technologies, including semiconductor device fabrication and the development of new pharmaceuticals.

Coordination Number

The coordination number in crystallography is a measure of how many nearest neighbor atoms, ions, or molecules surround a central atom, ion, or molecule within a crystal structure. It is a simple count of nearest neighbors and helps in understanding how a material is put together at the atomic level.

For example, in a sodium chloride (NaCl) crystal, each sodium ion is surrounded by six chloride ions, and vice versa, resulting in a coordination number of 6. This reflects a cubic coordination environment where each ion sits at the center of a cube made by its neighboring oppositely charged ions. The coordination number is essential since it significantly impacts several properties of the material, such as its stability, density, and melting point.

The concept of the coordination number not only aids in visualizing the three-dimensional structure of a crystal but also in predicting how changes in pressure, temperature, or composition may affect the material's properties.

Ionic Compound Crystal Structure

The crystal structure of an ionic compound is determined by the arrangement of its ions in a periodic lattice that ensures electrical neutrality and maximum stability. In crystals of ionic compounds, positively charged cations and negatively charged anions are held together by electrostatic forces in a regular and repeating three-dimensional array.

For instance, sodium chloride (NaCl) forms an iconic cube-shaped face-centered cubic (FCC) lattice. The stoichiometry of an ionic compound, such as the 1:1 ratio of Na to Cl in sodium chloride, affects how these ions can be packed together most efficiently. The arrangement aims to minimize the overall potential energy by optimizing attractions between oppositely charged ions and minimizing repulsion between like-charged ions.

Different ionic compounds may adopt different lattice structures based on factors such as the relative sizes of their ions, the charge of the ions, and their coordination numbers. Thus, even though both NaCl and calcium chloride (CaCl2) contain chloride ions, the differences in their cations' charge and size lead to very different crystal lattices. These differences in structure can have profound effects on the physical properties of the ionic compounds, including their brittleness, solubility, and melting points.