Question

The compounds \(\mathrm{Na}_{2} \mathrm{O}, \mathrm{CdS}\), and \(\mathrm{Zr}_{4}\) all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case?

Short answer

The fraction of tetrahedral holes occupied for each compound is: Na\(_2\)O: \( \frac{1}{2} \) CdS: \( \frac{1}{8} \) Zr\(_4\): 1

Step-by-step solution

Determine the unit cell ratio of cations to anions

First, we analyze the chemical compositions of the given compounds and find the ratio of cations to anions in each case: For Na\(_2\)O: 2 Na\(^+\) ions (cations) and 1 O\(^{2-}\) ion (anion) For CdS: 1 Cd\(^{2+}\) ion (cation) and 1 S\(^{2-}\) ion (anion) For Zr\(_4\): 4 Zr ions (cations) and 1 imaginary anion

Determine the number of anions in cubic closest packed structure

In a ccp structure, anions occupy all the face centers and vertices of the cube. The total number of anions in the unit cell structure can be calculated as follows: 8 vertices x (1 anion shared by 8 unit cells) = 8 x (1/8) = 1 anion 6 face centers x (1 anion shared by 2 unit cells) = 6 x (1/2) = 3 anions Total anions in ccp structure = 1 + 3 = 4 anions

Determine the number of tetrahedral holes in ccp structure

In a ccp structure, there are 2 tetrahedral holes associated with each anion. So, the total number of tetrahedral holes can be calculated as follows: Total tetrahedral holes = 4 anions x 2 = 8 tetrahedral holes

Calculate the fraction of tetrahedral holes occupied by cations

Now, we can calculate the fraction of occupied tetrahedral holes for each compound using the ratio of cations to anions and the total number of tetrahedral holes: For Na\(_2\)O: - Cations to Anions ratio: 2 Na\(^+\) ions : 1 O\(^{2-}\) ion - Fraction of occupied tetrahedral holes = (Number of cations) / (Number of tetrahedral holes) = 2/4 = 1/2 For CdS: - Cations to Anions ratio: 1 Cd\(^{2+}\) ion : 1 S\(^{2-}\) ion - Fraction of occupied tetrahedral holes = (Number of cations) / (Number of tetrahedral holes) = 1/8 For Zr\(_4\): - Cations to Anions ratio: 4 Zr ions : 1 imaginary anion - Fraction of occupied tetrahedral holes = (Number of cations) / (Number of tetrahedral holes) = 4/4 = 1 So, the fraction of tetrahedral holes occupied for each case is: Na\(_2\)O: 1/2 CdS: 1/8 Zr\(_4\): 1

Key concepts

Tetrahedral Holes Occupancy

Understanding how cations fill the spaces in a crystal structure helps us grasp the properties of various materials. In cubic closest packed (ccp) structures, anions make up the lattice framework, forming a tightly packed arrangement. The smaller crevices among these anions are where cations can fit, and these spaces are known as tetrahedral holes.

A ccp structure has two tetrahedral holes per anion; thus, with four anions in a ccp unit cell, there are eight tetrahedral holes available. Cations slot into these holes in a ratio that depends on the stoichiometry of the compound. As seen in the textbook exercise, the fraction of filled tetrahedral holes varies: Na₂O occupies half of these holes, CdS only an eighth, and Zr₄ fills all of them. This is crucial for predicting the electronic and geometric structure of the crystal.

Cation to Anion Ratio

The cation to anion ratio is a simple concept that carries significant implications for the crystal's electrical neutrality and stoichiometry. It represents the balance of positive and negative charges within the crystal.

In the case of Na₂O, there are two sodium cations (Na⁺) for each oxide anion (O^2-), leading to a ratio of 2:1. For CdS, with each Cd²⁺ matching one S^2-, the ratio is 1:1. Zr₄ has an unusual 4:1 ratio which, in actuality, forms a more complex structure. The ratio determines how cations fill the tetrahedral holes and thus affects the compound's overall structural stability and electronic properties.

Crystal Lattice Structure

The term crystal lattice structure refers to the orderly, repeating three-dimensional arrangement of atoms within a solid. In the context of our exercise, the ccp structure is highlighted where anions comprise the lattice points.

A ccp lattice implies that each anion is surrounded by 12 nearest neighbors in a pattern that repeats itself throughout the crystal. The ccp structure is one of the densest packings possible for spherical particles, leading to a high level of structural efficiency. It's vital to comprehend this as it directly affects material properties such as density, melting point, and strength.

Stoichiometry in Solid-State Chemistry

In solid-state chemistry, stoichiometry involves the quantitative relationship between the reactants and products in a crystalline material. This relationship guides the formation of crystal structures.

Understanding stoichiometry helps explain the occupancy of tetrahedral holes in ccp structures. The stoichiometric composition of a compound determines which and how many holes are filled by cations. For instance, in our exercise, the 2:1 stoichiometry of Na₂O means half of the tetrahedral holes are occupied, aligning with the chemical formula's indication of two Na⁺ for every O^2-. Accuracy in this concept is essential for predicting the formation and characteristics of crystal solids.