Ch.12 - Solids and Modern Materials

Problem 109

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Chapter 12, Problem 109

For each of the intermetallic compounds shown in Figure 12.17 determine the number of each type of atom in the unit cell. Do your answers correspond to the ratios expected from the empirical formulas: Ni3Al?

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Identify the positions of the Pt and Fe atoms in the unit cell. Pt atoms are represented by blue spheres, and Fe atoms are represented by pink spheres.

Count the number of Pt atoms in the unit cell. Note that Pt atoms are located at the face centers of the cubic unit cell.

Count the number of Fe atoms in the unit cell. Note that Fe atoms are located at the corners of the cubic unit cell.

Calculate the contribution of each Pt atom to the unit cell. Each face-centered atom contributes 1/2 of an atom to the unit cell.

Calculate the contribution of each Fe atom to the unit cell. Each corner atom contributes 1/8 of an atom to the unit cell.

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Key Concepts

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

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. It defines the arrangement of atoms in three-dimensional space and is characterized by its dimensions and the types of atoms it contains. Understanding the unit cell is crucial for determining the composition and properties of crystalline materials.

Empirical Formula

An empirical formula represents the simplest whole-number ratio of atoms of each element in a compound. For example, the empirical formula Ni3Al indicates that for every three nickel (Ni) atoms, there is one aluminum (Al) atom. This concept is essential for analyzing the composition of intermetallic compounds and verifying if the observed atomic ratios in the unit cell match the expected ratios from the empirical formula.

Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each corner and the centers of all the cube faces. This arrangement allows for a high packing efficiency and is common in metals. Understanding the FCC structure is important for determining the number of atoms per unit cell and how they relate to the empirical formula of intermetallic compounds like Ni3Al.

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Face Centered Cubic Example

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The electrical conductivity of aluminum is approximately 109 times greater than that of its neighbor in the periodic table, silicon. Aluminum has a face-centered cubic structure, and silicon has the diamond structure. A classmate of yours tells you that density is the reason aluminum is a metal but silicon is not; therefore, if you were to put silicon under high pressure, it too would act like a metal. Discuss this idea with your classmates, looking up data about Al and Si as needed.