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 have shorter wavelengths (0.01 to 10 nm) than visible light (400 to 700 nm), which allows them to diffract when interacting with the crystal lattice of atomic distances on the order of angstroms. This enables the extraction of information about atomic positions and interatomic distances. Visible light cannot produce meaningful diffraction patterns due to its larger wavelengths. (b) \(\mathrm{CaCl}_{2}\) and \(\mathrm{NaCl}\) have different ionic ratios, ion sizes, and bonding characteristics. The different charges, sizes, and electrostatic interactions between the cations and anions in each compound result in varied arrangements of ions in their respective crystal lattices. Thus, \(\mathrm{CaCl}_{2}\) cannot have the same crystal structure as \(\mathrm{NaCl}\).

Step-by-step solution

(a) Wavelengths of X-rays and visible light

X-rays have shorter wavelengths, typically on the order of 0.01 to 10 nanometers, compared to visible light, which has wavelengths in the range of 400 to 700 nanometers. The specific wavelength needed to measure atomic distances effectively depends on the size of the atoms and the distances between them, which are on the order of angstroms (1 Å = 0.1 nm) in crystals.

(a) Interaction with the crystal lattice

Since the atomic distances in crystals are of the order of angstroms, the X-rays' short wavelengths allow for diffraction to occur when they interact with the crystal lattice. This results in constructive and destructive interference, which can be captured as a diffraction pattern. Analyzing this diffraction pattern allows for the extraction of information about atomic positions and interatomic distances within the crystal. On the other hand, visible light's larger wavelengths in comparison to the atomic distances make it unable to interact effectively with the crystal lattice, producing less meaningful diffraction patterns. Therefore, X-rays are a better choice to measure atomic distances in crystals.

(b) Differences in ionic structure

\(\mathrm{CaCl}_{2}\) and \(\mathrm{NaCl}\) have different chemical formulas, which means the ratio of cations to anions in each compound is different. In \(\mathrm{CaCl}_{2}\), for every calcium cation (Ca\(^{2+}\)), there are two chloride anions (Cl\(^-\)), while in \(\mathrm{NaCl}\), there is one sodium cation (Na\(^+\)) for every chloride anion (Cl\(^-\)). This difference in ionic ratios affects the packing and arrangement of ions in the crystal structures.

(b) Differences in ion sizes

The size of the ions also plays a significant role in determining the crystal structure. In \(\mathrm{CaCl}_{2}\), the calcium ion (Ca\(^{2+}\)) is larger than the sodium ion (Na\(^+\)) in \(\mathrm{NaCl}\). Similarly, chloride ions (Cl\(^-\)) are larger than the sodium cations (Na\(^+\)). The differences in ion sizes lead to variations in the way these ions are packed together in the crystal lattice, resulting in different crystal structures for the two compounds.

(b) Differences in bonding characteristics

Due to the different charges and ion sizes in \(\mathrm{CaCl}_{2}\) and \(\mathrm{NaCl}\), the electrostatic interactions between the cations and anions in each compound will be different. These different bonding characteristics result in varied arrangements of ions in their respective crystal lattices. Thus, \(\mathrm{CaCl}_{2}\) cannot have the same crystal structure as \(\mathrm{NaCl}\).

Key concepts

Wavelength of X-rays

The concept of wavelength is crucial in understanding why X-rays are used to explore atomic distances in crystals. X-rays possess very short wavelengths, typically between 0.01 and 10 nanometers. This is significantly smaller compared to visible light, which has a wavelength range of 400 to 700 nanometers. The size of an atom and the distance between atoms in a crystal structure are typically on the order of angstroms, where 1 angstrom is equal to 0.1 nanometers.
This short wavelength of X-rays allows them to interact with the crystal lattice, which comprises atoms at distances comparable to the wavelength of X-rays themselves.
The resulting interaction leads to diffraction, characterized by the scattering of X-ray waves by the atoms in the crystal, producing patterns that can be analyzed to reveal the atomic structure.
Visible light, on the other hand, cannot yield such detailed diffraction patterns due to its much longer wavelengths that do not match the scale of atomic spacing in crystals.

Visible Light Diffraction

Visible light diffraction occurs when light waves bend around obstacles or spread as they pass through narrow openings. This property is more pronounced when the dimensions of these obstacles or openings are comparable to the wavelength of the light.
For visible light, with wavelengths ranging between 400 and 700 nanometers, diffraction effects can be easily observed in everyday phenomena such as the rainbow patterns on a CD or through a diffraction grating.
However, when it comes to atomic structures in crystals, where the atom-to-atom distance is typically much smaller (i.e., in the range of angstroms), visible light's wavelength is too long to produce meaningful diffraction patterns.
In essence, because the distances are too small compared to the visible light wavelength, visible light can barely "sense" the individual atoms, resulting in no substantial data about the crystal structure.

Ionic Crystal Structures

Ionic crystal structures are formed from a regular, repeating pattern of ions bound together by ionic bonds. These structures are determined by several factors, including the ratio of cations to anions, the relative sizes of the ions, and their respective charges.
For example, \( ext{NaCl}\) has an equal ratio of sodium ions (Na\(^+\)) to chloride ions (Cl\(^-\)), forming a simple cubic structure known as the rock-salt structure.
In contrast, \( ext{CaCl}_2\) has a different stoichiometry - one calcium ion (Ca\(^{2+}\)) to two chloride ions. The difference in charge and ionic ratios leads to complex packing arrangements.
The sizes of the ions also influence structure; larger cations and anions require more space and result in different packing efficiencies. \( ext{CaCl}_2\) typically forms a different crystal structure than \( ext{NaCl}\) due to these size differences.
Moreover, ionic crystals' unique bonding and the potential energy minimization play a crucial role in their stability and structural formation, preventing \( ext{CaCl}_2\) from adopting the \( ext{NaCl}\) lattice arrangement.