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Chapter 12: Problem 50

A face-centered cubic cell contains \(8 \mathrm{X}\) atoms at the corners of the cell and \(6 \mathrm{Y}\) atoms at the faces. What is the empirical formula of the solid?

Short Answer

Expert verified
The empirical formula of the solid is \(XY_{3}\).

Step by step solution

01

Understand the Structure of FCC

In a face-centered cubic (fcc) structure, there are 8 corner atoms and 6 face atoms. In such a structure, each corner atom is shared by eight adjacent cubes and each face atom is shared by two adjacent cubes.
02

Calculate the Contribution of X Atoms

Given that there are 8 corner atoms of type X, and each corner atom is shared by eight adjacent cubes, the contribution of X atoms to one unit cell is \(8*(1/8)\) = 1 atom.
03

Calculate the Contribution of Y Atoms

Given that there are 6 face atoms of type Y, and each face atom is shared by two adjacent cubes, the contribution of Y atoms to one unit cell is \(6*(1/2)\) = 3 atoms.
04

Determine the Empirical Formula

The empirical formula is decided by the ratio of the number of X atoms to the number of Y atoms in the unit cell. Since the unit cell contains 1 X atom and 3 Y atoms, the empirical formula of the solid is \(X_{1}Y_{3}\), or simply \(XY_{3}\).

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Most popular questions from this chapter

Which member of each of these pairs of substances would you expect to have a higher boiling point: (a) \(\mathrm{O}_{2}\) or \(\mathrm{N}_{2},\) (b) \(\mathrm{SO}_{2}\) or \(\mathrm{CO}_{2},\) (c) HF or HI?

List the types of intermolecular forces that exist in each of these species: (a) benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\), (b) \(\mathrm{CH}_{3} \mathrm{Cl},\) (c) \(\mathrm{PF}_{3},\) (d) \(\mathrm{NaCl}\), (e) \(\mathrm{CS}_{2}\).

The compounds \(\mathrm{Br}_{2}\) and \(\mathrm{ICl}\) have the same number of electrons, yet \(\mathrm{Br}_{2}\) melts at \(-7.2^{\circ} \mathrm{C}\), whereas \(\mathrm{ICl}\) melts at \(27.2^{\circ} \mathrm{C}\). Explain.

Crystalline silicon has a cubic structure. The unit cell edge length is \(543 \mathrm{pm}\). The density of the solid is \(2.33 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate the number of \(\mathrm{Si}\) atoms in one unit cell.

Describe the geometries of these cubic cells: simple cubic cell, body-centered cubic cell, face-centered cubic cell. Which of these cells would give the highest density for the same type of atoms?
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