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Chapter 11: Problem 44

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

Short Answer

Expert verified
The empirical formula is \(X_{1}Y_{3}\)

Step by step solution

01

Understand the face-centered cubic structure

In a face-centered cubic cell, there are 8 atoms at the corners of the cube and 6 atoms at the faces. However, not all of these atoms belong exclusively to this unit cell. Each corner atom is shared by 8 cells (1/8 of each atom is in the cell), and each face atom is shared by 2 cells (1/2 of each atom is in the cell).
02

Calculate the total number of each type of atom

To determine the empirical formula, we need to calculate the number of each type of atom in the unit cell. For atoms X at the corners, the total contribution is \(8 \times 1/8 = 1\) X atom. For atoms Y at the faces, the total contribution is \(6 \times 1/2 = 3\) Y atoms.
03

Determine the empirical formula

The empirical formula is the simplest positive integer ratio of atoms in a compound. Here, the ratio of X to Y is 1 to 3. Thus, the empirical formula of the solid is \(X_{1}Y_{3}\)

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Most popular questions from this chapter

Barium metal crystallizes in a body-centered cubic lattice (the Ba atoms are at the lattice points only). The unit cell edge length is \(502 \mathrm{pm},\) and the density of the metal is \(3.50 \mathrm{~g} / \mathrm{cm}^{3}\). Using this information, calculate Avogadro's number. [Hint: First calculate the volume (in \(\mathrm{cm}^{3}\) ) occupied by 1 mole of Ba atoms in the unit cells. Next calculate the volume (in \(\mathrm{cm}^{3}\) ) occupied by one Ba atom in the unit cell. Assume that \(68 \%\) of the unit cell is occupied by Ba atoms.

A \(1.20-\mathrm{g}\) sample of water is injected into an evacuated \(5.00-\mathrm{L}\) flask at \(65^{\circ} \mathrm{C} .\) What percentage of the water will be vapor when the system reaches equilibrium? Assume ideal behavior of water vapor and that the volume of liquid water is negligible. The vapor pressure of water at \(65^{\circ} \mathrm{C}\) is \(187.5 \mathrm{mmHg}\)

What is the coordination number of each sphere in (a) a simple cubic cell, (b) a body-centered cubic cell, and (c) a face- centered cubic cell? Assume the spheres are all the same.

When X rays of wavelength \(0.090 \mathrm{nm}\) are diffracted by a metallic crystal, the angle of first-order diffraction \((n=1)\) is measured to be \(15.2^{\circ} .\) What is the distance (in pm) between the layers of atoms responsible for the diffraction?

What is an amorphous solid? How does it differ from crystalline solid?
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