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Chapter 10: Problem 75

What is the formula for the compound that crystallizes with a cubic closest packed array of sulfur ions, and that contains zinc ions in \(\frac{1}{8}\) of the tetrahedral holes and aluminum ions in \(\frac{1}{2}\) of the octahedral holes?

Short Answer

Expert verified
The formula for the compound that crystallizes with a cubic closest packed array of sulfur ions, and that contains zinc ions in $\frac{1}{8}$ of the tetrahedral holes and aluminum ions in $\frac{1}{2}$ of the octahedral holes is ZnAl₂S₄.

Step by step solution

01

Determine the number of sulfur ions in the unit cell

In a cubic closest packed array, the sulfur ions will occupy the lattice points. In a cubic unit cell, there is 1 lattice point at each corner (8 corners) and 1 lattice point at the center of each face (6 faces). Each corner contributes 1/8 of an ion to the unit cell and each face contributes 1/2 of an ion to the unit cell. So, the total number of sulfur ions in the unit cell is: \[ N_\text{S} = 8\left(\frac{1}{8}\right) + 6\left(\frac{1}{2}\right) = 1+3 = 4 \]
02

Determine the number of tetrahedral and octahedral holes in the unit cell

In a ccp structure, the number of tetrahedral holes is twice the number of lattice points, and the number of octahedral holes is equal to the number of lattice points. Since the total number of sulfur ions in the unit cell is 4 (from step 1), there are 8 tetrahedral holes and 4 octahedral holes in the unit cell.
03

Determine the number of zinc ions in the unit cell

Zinc ions occupy 1/8 of the tetrahedral holes. Since there are 8 tetrahedral holes in the unit cell, the number of zinc ions in the unit cell can be calculated as: \[ N_\text{Zn} = \frac{1}{8}(8) = 1 \]
04

Determine the number of aluminum ions in the unit cell

Aluminum ions occupy 1/2 of the octahedral holes. Since there are 4 octahedral holes in the unit cell, the number of aluminum ions in the unit cell can be calculated as: \[ N_\text{Al} = \frac{1}{2}(4) = 2 \]
05

Write the formula of the compound

Now we can write the formula of the compound using the number of ions of each type in the unit cell. The formula is: \[ \text{Zn}_1\text{Al}_2\text{S}_4 \] Since all the coefficients are in the simplest whole-number ratio, the final formula is: \[ \text{ZnAl}_2\text{S}_4 \] The formula for the compound is ZnAl₂S₄.

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Most popular questions from this chapter

General Zod has sold Lex Luthor what Zod claims to be a new copper-colored form of kryptonite, the only substance that can harm Superman. Lex, not believing in honor among thieves, decided to carry out some tests on the supposed kryptonite. From previous tests, Lex knew that kryptonite is a metal having a specific heat capacity of 0.082 $\mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C}\( and a density of 9.2 \)\mathrm{g} / \mathrm{cm}^{3}$ Lex Luthor's first experiment was an attempt to find the specific heat capacity of kryptonite. He dropped a \(10 \mathrm{g} \pm 3 \mathrm{g}\) sample of the metal into a boiling water bath at a temperature of $100.0^{\circ} \mathrm{C} \pm 0.2^{\circ} \mathrm{C}$ . He waited until the metal had reached the bath temperature and then quickly transferred it to \(100 \mathrm{g} \pm\) 3 \(\mathrm{g}\) of water that was contained in a calorimeter at an initial temperature of \(25.0^{\circ} \mathrm{C} \pm 0.2^{\circ} \mathrm{C}\) . The final temperature of the metal and water was \(25.2^{\circ} \mathrm{C} .\) Based on these results, is it possible to distinguish between copper and kryptonite? Explain. When Lex found that his results from the first experiment were inconclusive, he decided to determine the density of the sample. He managed to steal a better balance and determined the mass of another portion of the purported kryptonite to be \(4 \mathrm{g} \pm 1 \mathrm{g} .\) He dropped this sample into water contained in a \(25-\mathrm{mL}\) graduated cylinder and found that it displaced a volume of \(0.42 \mathrm{mL} \pm 0.02 \mathrm{mL}\) . Is the metal copper or kryptonite? Explain. Lex was finally forced to determine the crystal structure of the metal General Zod had given him. He found that the cubic unit cell contained four atoms and had an edge length of \(600 . \mathrm{pm}\) . Explain how this information enabled Lex to identify the metal as copper or kryptonite. Will Lex be going after Superman with the kryptonite or seeking revenge on General Zod? What improvements could he have made in his experimental techniques to avoid performing the crystal structure determination?

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