Question

Spinel is a mineral that contains \(37.9 \%\) aluminum, \(17.1 \%\) magnesium, and \(45.0 \%\) oxygen, by mass, and has a density of \(3.57 \mathrm{g} / \mathrm{cm}^{3} .\) The edge of the cubic unit cell measures \(809 \mathrm{pm} .\) How many of each type of ion are present in the unit cell?

Short answer

There are 8 Al ions, 4 Mg ions, and 12 O ions present in the cubic unit cell of Spinel mineral.

Step-by-step solution

Calculate the total mass of the mineral in one unit cell

Using the given density and edge length, we can calculate the volume and then the mass of the mineral in one unit cell. Volume of the unit cell = Edge Length³ Volume = \((809 \mathrm{pm})^3\) = \((809 \times 10^{-12} \mathrm{m})^3\) Note that 1 pm = 10⁻¹² m Let's convert the volume to cm³: Volume = \((809 \times 10^{-12} \mathrm{m})^3 \times (10^2 \mathrm{cm/m})^3 = (809 \times 10^{-6} \mathrm{cm})^3\) Now, using the density, we can calculate the mass: Total Mass = Density × Volume = \(3.57 \mathrm{g/cm^3} \times (809 \times 10^{-6} \mathrm{cm})^3\)

Calculate the mass of each element in the unit cell

Now that we have the total mass, we can find the mass of each element in the unit cell using the given mass percentages. Mass of Al = (Total Mass) x (37.9%) Mass of Mg = (Total Mass) x (17.1%) Mass of O = (Total Mass) x (45.0%)

Calculate the number of ions in the unit cell

Now that we have the mass of each element in the unit cell, we can calculate the number of ions of each element in the unit cell using their molar masses: Molar mass of Al = 26.98 g/mol Molar mass of Mg = 24.31 g/mol Molar mass of O = 16.00 g/mol Number of Al ions = (Mass of Al) / (Molar Mass of Al) Number of Mg ions = (Mass of Mg) / (Molar Mass of Mg) Number of O ions = (Mass of O) / (Molar Mass of O) Remember that the number of ions must be integers, so round to the nearest whole number for each element. The result will be the number of each type of ion present in the unit cell.

Key concepts

Mineral Composition

Minerals are naturally occurring, inorganic solids with a definite chemical composition and a crystalline structure. Spinel is one such mineral with a specific arrangement of atoms in a cubic unit cell. It is composed of aluminum (Al), magnesium (Mg), and oxygen (O) ions in certain proportions. The knowledge of mineral composition is essential in geology and materials science as it helps determine the properties and applications of the mineral.

The exact composition of a mineral like spinel is expressed in mass percentages for each element. In our example, spinel contains 37.9% aluminum, 17.1% magnesium, and 45.0% oxygen by mass. These percentages are crucial for calculating how many ions of each element are present in the crystal structure.

Density Calculation

Density is a fundamental property defined as the mass per unit volume of a substance. For minerals, calculating density allows geologists and chemists to identify a mineral and infer its inner structure. The density of spinel is given as 3.57 g/cm³.

To calculate the mass inside a unit cell, first, determine the volume of the cell from its edge length, considering that the unit cell for spinel is cubic. After calculating the volume in cubic centimeters (cm³), the mass of the mineral within the unit cell is found by multiplying the volume by the density. This mass corresponds to the combined mass of aluminum, magnesium, and oxygen ions in that cell, providing a starting point to further determine the quantity of each ion type.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a bridge between atomic scale measurements and laboratory-scale measurements. By calculating the molar mass of elements like aluminum, magnesium, and oxygen, molar mass allows scientists to count the number of atoms or ions in a given sample by relating the element's mass to its molar mass.

In the case of spinel, once you have the mass of Al, Mg, and O from the proportion of the total mass, you can find the number of ions present by dividing the mass of each element by its respective molar mass. This step is necessary to determine the number of ions in each unit cell, which gives insight into the mineral's crystal structure and can affect properties like electrical conductivity, hardness, and melting point.