Question

Spinel is a mineral that contains \(37.9 \%\) 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 \(8.09 \AA\). How many atoms of each type are in the unit cell?

Short answer

The Spinel unit cell contains 2 Al atoms, 1 Mg atom, and 3 O atoms.

Step-by-step solution

Find the mass of the unit cell

To find the mass of the unit cell, we first need to calculate the volume of the unit cell and then multiply it by the density of Spinel. Volume of the unit cell = edge length^3 Spinel edge length = 8.09 Å = 8.09 * 10^{-10} m Volume = (8.09 * 10^{-10} m)^3 = 5.307 * 10^{-28} m^3 Next, we have to convert the volume from m^3 to cm^3 by multiplying by 10^6: Volume = 5.307 * 10^{-28} m^3 * 10^6 = 5.307 * 10^{-22} cm^3 Now, multiplying the volume by the density: Mass = (5.307 * 10^{-22} cm^3) * (3.57 g/cm^3) = 1.895 * 10^{-21} g

Calculate the mass of each element

Using the given percentage compositions of each element in Spinel, we can find the mass of each element in the unit cell. Mass of Al = (1.895 * 10^{-21} g) * (37.9/100) = 7.184 * 10^{-22} g Mass of Mg = (1.895 * 10^{-21} g) * (17.1/100) = 3.239 * 10^{-22} g Mass of O = (1.895 * 10^{-21} g) * (45.0/100) = 8.525 * 10^{-22} g

Calculate the number of atoms of each element

Now, we divide the mass of each element by its respective molar mass to find the number of moles, and then multiply by Avogadro's number to get the number of atoms. Molar mass of Al = 26.98 g/mol Molar mass of Mg = 24.31 g/mol Molar mass of O = 16.00 g/mol Moles of Al = (7.184 * 10^{-22} g) / (26.98 g/mol) = 2.664 * 10^{-23} mol Moles of Mg = (3.239 * 10^{-22} g) / (24.31 g/mol) = 1.332 * 10^{-23} mol Moles of O = (8.525 * 10^{-22} g) / (16.00 g/mol) = 5.328 * 10^{-23} mol Avogadro's number = 6.022 * 10^{23} atoms/mol Number of Al atoms = (2.664 * 10^{-23} mol) * (6.022 * 10^{23} atoms/mol) ≈ 1.6 atoms Number of Mg atoms = (1.332 * 10^{-23} mol) * (6.022 * 10^{23} atoms/mol) ≈ 0.8 atoms Number of O atoms = (5.328 * 10^{-23} mol) * (6.022 * 10^{23} atoms/mol) ≈ 3.2 atoms Since there can't be fractional atoms, we need to round these values to the nearest whole number: Number of Al atoms in the unit cell: 2 Number of Mg atoms in the unit cell: 1 Number of O atoms in the unit cell: 3

Key concepts

Mineral Density Calculation

Understanding the density of minerals is crucial in various fields like material science, geology, and mineralogy. Density is defined as mass per unit volume and is a key property that can help identify minerals and predict their behavior in different environmental conditions.

To calculate the density of a mineral, first determine the volume of the mineral's unit cell. In crystallography, a unit cell is the smallest divisible unit of a crystal that retains the geometric and physical characteristic of the crystal. For cubic crystals, like Spinel, this involves cubing the edge length of the unit cell. Then, by knowing the density of the mineral, we can calculate the mass of one unit cell by multiplying its volume by the mineral's density. This step lays the groundwork for determining the composition of the mineral at the atomic level.

A key point to remember is that real-world objects can't have a fractional number of atoms. Therefore, when calculating the number of atoms in a unit cell from the mass, it is essential to round to the nearest whole number to reflect a physically possible scenario.

Atomic Mass Calculation

The atomic mass plays a pivotal role in understanding the composition of substances in the field of chemistry. It allows the calculation of moles from a given mass and is crucial for stoichiometry and analytical chemistry.

To find the number of atoms of each element within the unit cell, we must convert the mass of each element (obtained from the density calculation) to moles using the atomic mass. The atomic mass of an element, often found on the periodic table, is the average mass of atoms of an element measured in atomic mass units (u) or grams per mole (g/mol).

Once the number of moles is determined, we multiply by Avogadro's number (\(6.022 \times 10^{23}\text{atoms/mol}\)) to find the number of atoms. Avogadro's number is a constant that represents the number of constituent particles, typically atoms or molecules, in one mole of a given substance.

Crystallography

Crystallography is the study of the arrangement of atoms within solids and plays an instrumental role in understanding materials' properties. Every crystal type has a unique 3D geometrical structure that repeats periodically in space. These structures are described by unit cells, which can be regarded as the building blocks of crystals.

In our example, the unit cell of Spinel is cubic, which means its cell geometry has equal edge lengths and 90-degree angles between them. The edge length is a critical parameter as it helps to calculate the volume of the unit cell, impacting the determination of its mass and the atoms within it.

Crystallography combines elements of mathematics, physics, and chemistry. It requires precise measurements and calculations to understand the complex arrangements of atoms. One must account for the concept of symmetry, lattice parameters, and the positions of atoms within the unit cell to comprehensively describe a crystal's structure.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It's a fundamental concept that involves the calculation of masses, volumes, and number of moles in chemical processes.

In the context of our problem, stoichiometry enables us to figure out the exact number of aluminum, magnesium, and oxygen atoms within the Spinel unit cell by analyzing the mass percentages of these elements and applying stoichiometric calculations. By knowing the atomic masses of these elements, we convert the mass of each element present in the unit cell to moles, and then to atoms, using Avogadro's number.

To improve comprehension, it is critical to understand the law of conservation of mass, mole concept, and how to use the molar mass as a conversion factor in stoichiometry. This helps to logically deduce the amount of substance and make necessary conversions or approximations, such as rounding number of atoms to whole numbers, ensuring our calculations yield physically meaningful results.