Problem 125 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

The unit cell contains 8 Al, 8 Mg, and 32 O atoms.

Step 1: Determine Molar Mass of Spinel

Start by calculating the molar mass of spinel. For simplicity, assume 100 g of the mineral, giving \(37.9 \, g\) of \(\text{Al}\), \(17.1 \, g\) of \(\text{Mg}\), and \(45.0 \, g\) of \(\text{O}\).- Molar Mass of \(\text{Al} = 26.98 \, g/mol\)- Molar Mass of \(\text{Mg} = 24.31 \, g/mol\)- Molar Mass of \(\text{O} = 16.00 \, g/mol\)Calculate moles of each:\[\text{Moles of Al} = \frac{37.9}{26.98} = 1.404 \, mol\]\[\text{Moles of Mg} = \frac{17.1}{24.31} = 0.704 \, mol\]\[\text{Moles of O} = \frac{45.0}{16.00} = 2.8125 \, mol\]The simplest ratio: Al:Mg:O ≈ 1:1:4.

Step 2: Calculate Unit Cell Mass

With the density and the dimensions of the unit cell, calculate the mass of the unit cell:1. Convert edge length from picometers to centimeters: \[ 809 \, \text{pm} = 809 \times 10^{-10} \, \text{cm} \]2. Calculate volume of the unit cell: \[ \text{Volume} = (809 \times 10^{-10} \, \text{cm})^3 = 5.286 \times 10^{-22} \, \text{cm}^3 \]3. Use density \(\rho = 3.57 \, \text{g/cm}^3\) to find mass: \[ \text{Mass of Unit Cell} = \rho \times \text{Volume} = 3.57 \, \text{g/cm}^3 \times 5.286 \times 10^{-22} \, \text{cm}^3 = 1.886 \times 10^{-21} \, \text{g} \]

Step 3: Determine Number of Formula Units in the Unit Cell

Use the calculated mass of the unit cell and formula units (obtained from Step 1) to find the number of atoms:1. Calculate the approximate molar mass using estimated ratio Al:Mg:O = 1:1:4: \[ \text{Molar Mass of Spinel} = 26.98 \, g/mol + 24.31 \, g/mol + 4 \times 16.00 \, g/mol = 101.29 \, g/mol \]2. Compute the number of formula units:\[\text{Number of Formula Units} = \frac{\text{Mass of Unit Cell}}{\text{Molar Mass/Avogadro's Number}}\] Number of formula units:\[= \frac{1.886 \times 10^{-21} \, \text{g}}{101.29 \, g/mol \div 6.022 \times 10^{23} \, \text{mol}^{-1}}\]equals approximately 8 formula units.

Step 4: Calculate Number of Atoms per Unit Cell

Now that you have the number of formula units (8), calculate the number of atoms per unit cell:1. For Al and Mg: - Each formula unit has 1 Al and 1 Mg: \[ \text{Atoms Al} = \text{Atoms Mg} = 8 \times 1 = 8\]2. For O: - Each formula unit has 4 O: \[ \text{Atoms O} = 8 \times 4 = 32 \]Thus, one unit cell of spinel contains 8 Al, 8 Mg, and 32 O atoms.

Key Concepts

Unit Cell

When we talk about crystal structures, the concept of a unit cell is crucial. Imagine the unit cell as the smallest repeating "block" of the crystal. It's like a building block that, when stacked together in three-dimensional space, forms the entire crystal lattice. In our example, spinel has a cubic unit cell with an edge length of 809 picometers (pm).
To visualize this, think of a cube made up of tiny atoms at each corner. This arrangement repeats to form the extended structure. Calculating the volume of the unit cell involves using the formula: \[ \text{Volume} = \text{edge length}^3 \]Here, the edge length must be converted into centimeters for all calculations involving density and mass, as density is often expressed in grams per cubic centimeter (g/cm³). For example, the conversion from pm to cm ensures consistency and accuracy in calculations.
The unit cell not only helps us understand structural characteristics but also aids in calculating the mass and number of atoms, which are fundamental in comprehending the whole composition of a mineral like spinel.

Molar Mass

Molar mass allows us to convert between grams and moles, which are central units in chemistry. For spinel, we calculate the molar mass by looking at the constituent elements: aluminum (Al), magnesium (Mg), and oxygen (O).
Molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol). Here's how you calculate it:

• Molar mass of Al: 26.98 g/mol
• Molar mass of Mg: 24.31 g/mol
• Molar mass of O: 16.00 g/mol

To find the molar mass of the compound, we use the molar ratios based on percentage masses. After determining the moles of each component, we calculate the proportionate molar masses contributing to the total mass.
For instance, we consider the given proportions and simplify the ratio of Al:Mg:O to 1:1:4. This ratio gives a formula mass of 101.29 g/mol, which helps in finding out how many formula units fit into a unit cell.

Density

Density is the mass per unit volume of a substance, typically expressed in grams per cubic centimeter (g/cm³). Understanding density helps us calculate how tightly packed atoms are within a mineral's structure.
In our scenario, spinel has a density of 3.57 g/cm³. Calculating the mass of the unit cell involves several steps:

• First, determine the unit cell's volume from its cubic structure: \[ \text{Volume} = (809 \times 10^{-10} \, \text{cm})^3 \]
• Calculate the cell's mass using the formula:\[ \text{Mass} = \text{Density} \times \text{Volume} \]
• Resulting in a unit cell mass of approximately \[ 1.886 \times 10^{-21} \, \text{g} \]

This density calculation is vital for determining the number of formula units within the unit cell and helps us understand the spatial arrangement affecting physical properties.

Atoms in Unit Cell

To find out how many atoms are in a unit cell, we need to consider the number of formula units that fit into the unit cell. For spinel, we found there are about 8 formula units per cell.
Each formula unit of spinel consists of specific numbers of Al, Mg, and O atoms based on our calculation from the molar ratio.

• 1 Al atom per formula unit
• 1 Mg atom per formula unit
• 4 O atoms per formula unit

Thus, in one unit cell, which contains 8 formula units:

• Atoms of Al: 8 x 1 = 8
• Atoms of Mg: 8 x 1 = 8
• Atoms of O: 8 x 4 = 32

This means that a single unit cell contains 8 aluminum, 8 magnesium, and 32 oxygen atoms. By understanding these numbers, we get a clearer picture of the elemental makeup of the crystal lattice.