Ch.12 - Solids and Modern Materials

Problem 58

Chapter 12, Problem 58

The unit cell of a compound containing Co and O has a unit cell shown in the diagram. The Co atoms are on the corners, and the O atoms are completely within the unit cell. a. What is the empirical formula of this compound? b. What is the oxidation state of the metal?

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Determine the number of Co atoms in the unit cell. Since Co atoms are located at the corners of the unit cell, and each corner atom is shared by eight adjacent unit cells, each Co atom contributes 1/8 to the unit cell. Calculate the total contribution of Co atoms.

Determine the number of O atoms in the unit cell. Since O atoms are completely within the unit cell, each O atom contributes fully to the unit cell. Count the total number of O atoms.

Calculate the ratio of Co to O atoms to find the empirical formula. Use the contributions from the previous steps to determine the simplest whole number ratio of Co to O.

To find the oxidation state of Co, use the fact that the compound is neutral overall. Assume the oxidation state of O is -2, and set up an equation where the sum of the oxidation states of Co and O equals zero.

Solve the equation for the oxidation state of Co, using the empirical formula and the known oxidation state of O.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Cell Structure

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In this case, the Co atoms are located at the corners of the unit cell, contributing 1/8 of an atom per corner to the overall count, while the O atoms are fully contained within the cell, contributing fully to the empirical formula.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of elements in a compound. To determine the empirical formula from the unit cell, one must calculate the total number of Co and O atoms present in the unit cell and express their ratio in the simplest form.

Oxidation State

The oxidation state of an element in a compound indicates the degree of oxidation or reduction of that element, reflecting its electron loss or gain. In this question, determining the oxidation state of cobalt (Co) involves analyzing its bonding with oxygen (O) and the overall charge balance in the compound.

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