Question

Spinel is a mineral that contains \(37.9 \% \mathrm{Al}, 17.1 \% \mathrm{Mg}\), and \(45.0 \% \mathrm{O}\), by mass, and has a density of \(3.57 \mathrm{~g} / \mathrm{cm}^{3} .\) The unit cell is cubic, with an edge length of \(809 \mathrm{pm}\). How many atoms of each type are in the unit cell?

Short answer

In the unit cell of Spinel, there are approximately 4 Aluminum atoms, 4 Magnesium atoms, and 7 Oxygen atoms.

Step-by-step solution

Find the relative moles of each element

To find the relative moles of each element, we'll convert the given percentage to moles by dividing the mass percent by the molar mass of each element. - Aluminum (Al): \( \frac{37.9}{26.9815} = 1.405 \) - Magnesium (Mg): \( \frac{17.1}{24.305} = 0.703 \) - Oxygen (O): \( \frac{45.0}{16} = 2.813 \)

Calculate the volume and mass of the unit cell

For a cubic cell, the volume is given by: V = a^3, where a is the edge length. Convert the edge length from picometers to centimeters and then calculate the volume: \(V = (809 \times 10^{-10})^3 = 5.328 \times 10^{-23} \mathrm{cm^3} \) Next, we'll use the density (ρ) formula to find the mass of the unit cell. Density = Mass/Volume => Mass = Density × Volume Mass = \(3.57 \mathrm{g/cm^3} \times 5.328 \times 10^{-23} \mathrm{cm^3} = 1.901 \times 10^{-22} \mathrm{g} \)

Find the total moles in the unit cell

We know the weight of each of the element's proportions and we can calculate their moles from the mass of the unit cell: - Aluminum (Al): \( \frac{1.901 \times 10^{-22} \times 1.405}{37.9} = 7.059 \times 10^{-24} \) - Magnesium (Mg): \( \frac{1.901 \times 10^{-22} \times 0.703}{17.1} = 7.036 \times 10^{-24} \) - Oxygen (O): \( \frac{1.901 \times 10^{-22} \times 2.813}{45.0} = 1.118 \times 10^{-23} \)

Find the number of atoms using Avogadro's number

Now we'll find the number of atoms for each element in the unit cell using Avogadro's number, 6.022 × 10^23. Since we assume the relative number of moles is the actual number of moles: - Aluminum (Al): \( \frac{7.059 \times 10^{-24} \mathrm{mol * 6.022 \times 10^{23} \mathrm{atoms}}{\mathrm{mol}} = 4.25 \approx 4 \mathrm{atoms} \) - Magnesium (Mg): \( \frac{7.036 \times 10^{-24} \mathrm{mol * 6.022 \times 10^{23} \mathrm{atoms}}{\mathrm{mol}} = 4.24 \approx 4 \mathrm{atoms} \) - Oxygen (O): \( \frac{1.118 \times 10^{-23} \mathrm{mol * 6.022 \times 10^{23} \mathrm{atoms}}{\mathrm{mol}} = 6.73 \approx 7 \mathrm{atoms} \) In the unit cell of Spinel, there are approximately 4 Aluminum atoms, 4 Magnesium atoms, and 7 Oxygen atoms.

Key concepts

Spinel structure

Spinel is a fascinating mineral with a unique crystalline arrangement known as the spinel structure. This structure is named after the mineral itself, which is made up of a combination of several elements, typically magnesium, aluminum, and oxygen. In the context of crystallography, the spinel structure falls under the category of cubic crystal systems.
The atoms in spinel form an intricate lattice where each Mg and Al atom is surrounded by various oxygen atoms, creating a dense-packed framework. This structure is not only aesthetically mesmerizing but also functionally important in various applications, from gemstones to ceramics. Understanding its crystallography allows us to determine properties such as density and the number of atoms per unit cell, essential for both academic research and practical applications.

Unit cell calculation

A unit cell is the smallest structural repeating unit in a crystal, and for spinel, it's cubic. Calculating the properties of a unit cell involves several steps. The first step is determining the volume of the unit cell, especially since crystallographers are often interested in finite dimensions on an atomic scale.
The edge length of a cube (\( a \)) for spinel is given in picometers (pm), and to find the volume (\( V \)), we convert this to centimeters and use the formula \( V = a^3 \). For spinel in this problem, \( a = 809 \) pm is converted to centimeters, and \( a^3 \) gives us the unit cell volume. Precise calculations are crucial here as they directly influence the calculated mass and density of the unit cell.

Avogadro's number

Avogadro's number is a fundamental constant in chemistry and crystallography. It represents the number of atoms, ions, or molecules in one mole of a substance, approximately \( 6.022 imes 10^{23} \). This number is crucial for relating macroscopic quantities to molecular scales.
When determining how many atoms reside in the spinel's unit cell, Avogadro's number allows us to convert the calculated number of moles into actual atom count. The moles of each element derived from their mass portions and the unit cell's total mass are crucial for finding this count. This conversion provides a deeper insight into the composition and structure of the crystal by quantifying how atoms pack into the tiny space of a unit cell.

Density and mass in crystallography

Density in crystallography is a vital property, helping in understanding the packing efficiency of a crystal. It is defined as mass per unit volume. For spinel, knowing the density helps calculate the mass of its unit cell.
With the unit cell's volume determined from its cubic dimensions, the overall mass can be deduced using the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). By rearranging this formula, we can find the mass, given the crystal's density. This measurement further facilitates calculating how many atoms of each type are present within the unit cell, bringing together density and molecular amounts to reveal the unit cell's comprehensive makeup.