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Chapter 12: Problem 93

A solid contains \(X, Y,\) and \(Z\) atoms in a cubic lattice with \(X\) atoms in the corners, \(Y\) atoms in the bodycentered positions, and \(Z\) atoms on the faces of the cell. What is the empirical formula of the compound?

Short Answer

Expert verified
The empirical formula of the compound is \(X_1Y_1Z_3\).

Step by step solution

01

Understand the atoms' positions

In the problem, it's given that there are \(X\) atoms at the corners of the lattice, \(Y\) atoms in the body-centered positions, and \(Z\) atoms on the faces of the cell in each unit cell. Each atom contributes differently to the cell. Atoms at the corners are shared by 8 adjacent cells, those at the body-centered positions are not shared and those on the faces are shared by 2 adjacent cells.
02

Calculate X's contribution to the cell

In this cubic structure, there are 8 corners, and at each corner there is 1 atom of type X. However, only \(1/8\) of each atom is actually within the cell because it is shared among 8 cells. So the contribution of X to one unit cell is \(8*\frac{1}{8}=1\)
03

Calculate Y's contribution to the cell

There is 1 atom of type Y in the body-centered position of each cell. This atom is not shared so contributes entirely to 1 cell. So the contribution of Y to one unit cell is 1.
04

Calculate Z's contribution to the cell

In this unit cell, there are 6 faces and at each face there is 1 atom of type Z. However, only \(1/2\) of each atom is actually within the cell because it is shared among two cells. So, the contribution of Z to one unit cell is \(6*\frac{1}{2}=3\)
05

Derive the empirical formula

The empirical formula for a compound is derived based on the ratio of the atoms present in the compound. Our task was to find the contribution of atoms X, Y and Z to the unit cell, which has been calculated as 1:1:3. Hence, the empirical formula of the compound is \(X_1Y_1Z_3\)

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Most popular questions from this chapter

Explain in terms of intermolecular forces why (a) \(\mathrm{NH}_{3}\) has a higher boiling point than \(\mathrm{CH}_{4}\) and \((\mathrm{b}) \mathrm{KCl}\) has a higher melting point than \(\mathrm{I}_{2}\).

Explain how water's phase diagram differs from those of most substances. What property of water causes the difference?

How is the rate of evaporation of a liquid affected by (a) temperature, (b) the surface area of liquid exposed to air, (c) intermolecular forces?

What is the relationship between intermolecular forces in a liquid and the liquid's boiling point and critical temperature? Why is the critical temperature of water greater than that of most other substances?

Argon crystallizes in the face-centered cubic arrangement at \(40 \mathrm{~K}\). Given that the atomic radius of argon is \(191 \mathrm{pm},\) calculate the density of solid argon.
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