Question

What is the formula of the magnetic oxide of cobalt, used in recording tapes, that crystallizes with cobalt atoms occupying one-eighth of the tetrahedral holes and one half of the octahedral holes in a closest packed array of oxide ions? (a) \(\mathrm{Co}_{2} \mathrm{O}_{3}\) (b) \(\mathrm{Co}_{5} \mathrm{O}_{s}\) (c) \(\mathrm{CoO}\) (d) \(\mathrm{Co}_{3} \mathrm{O}_{4}\)

Short answer

\(\mathrm{Co}_{3}\mathrm{O}_{4}\)

Step-by-step solution

Identify the Structure

Recognize that in a closest packed array of oxide ions, each oxide ion can be a part of tetrahedral and octahedral voids.

Determine the Number of Voids

In a closest packed structure, the number of tetrahedral holes is twice the number of atoms present, and the number of octahedral holes is the same as the number of atoms.

Calculate the Number of Cobalt Ions

Since cobalt occupies one-eighth of the tetrahedral holes and one half of the octahedral holes, start by assuming there are x oxide ions. Then there are 2x tetrahedral and x octahedral voids available. Cobalt fills one-eighth of 2x, which is x/4, and one half of x, which is x/2. The total number of cobalt atoms is x/4 + x/2 = (x+2x)/4 = 3x/4.

Create Ratio and Simplify

This gives a ratio of Co to O as (3/4)x : x, which simplifies to 3:4. This can be represented as a ratio of their subscripts in the chemical formula: \(\mathrm{Co}_{3}\mathrm{O}_{4}\).

Key concepts

Chemical Formula Determination

Determining the chemical formula of a compound requires understanding the quantitative composition of the elements within the substance. For the magnetic oxide of cobalt used in recording tapes, we start by looking at the crystalline structure where the cobalt atoms occupy certain voids within a closely packed array of oxide ions. The key is to quantify the ratio of cobalt to oxygen atoms.

Given that cobalt atoms fill one-eighth of the tetrahedral holes and half of the octahedral holes in this closely packed array, we can derive the formula by first assuming the presence of a certain number of oxygen ions, typically represented as 'x'. Since the number of tetrahedral holes in closely packed structures is twice the number of atoms, and the number of octahedral holes is equal to the number of atoms, we can calculate the number of cobalt atoms filling these voids. By simplifying the resulting ratios, the formula emerges as Co₃O₄, which represents the ratio of cobalt to oxygen in the compound.

Close-Packed Structures

Close-packed structures are arrangements of atoms or ions in a crystal where they are packed together as closely as possible. There are two primary types of such structures: the face-centered cubic (fcc) and hexagonal close-packed (hcp) structures. Each oxygen ion in these structures touches several other ions, minimizing empty space, and maximizing structural stability.

In the context of the magnetic oxide of cobalt, the oxide ions form a close-packed structure providing a framework within which the smaller cobalt ions can reside. The knowledge of close packing is essential in understanding how atoms or ions fill the available voids – the tetrahedral and octahedral holes – in these structures. This closely packed arrangement influences the overall properties of the material, including its magnetic and electrical characteristics.

Tetrahedral and Octahedral Voids

Tetrahedral and octahedral voids are small spaces formed between the atoms or ions in a crystalline structure. In the face of a densely packed array, tetrahedral voids are surrounded by four atoms, forming the shape of a tetrahedron. Octahedral voids, on the other hand, have six neighboring atoms and take on an octahedral shape.

In the chemical structure of cobalt's magnetic oxide, the cobalt atoms occupy these voids in specific proportions - one-eighth of tetrahedral voids and half of the octahedral voids. Understanding these voids is crucial because it allows us to determine the formula of the compound. With double the number of tetrahedral holes to atoms and an equal number of octahedral holes to atoms, our formula determination rests upon accounting correctly for these spaces occupied by cobalt ions. This precise arrangement impacts the electronic structure of the oxide, contributing to the compound's magnetic properties, which are utilized in recording tapes.