Problem 64 - Chemistry An Atoms First Approach

Question

Bauxite, the principal ore used in the production of aluminum, has a molecular formula of \(\mathrm{Al}_{2} \mathrm{O}_{3} \cdot 2 \mathrm{H}_{2} \mathrm{O} .\) The \(\cdot \mathrm{H}_{2} \mathrm{O}\) in the formula are called waters of hydration. Each formula unit of the compound contains two water molecules. a. What is the molar mass of bauxite? b. What is the mass of aluminum in 0.58 mole of bauxite? c. How many atoms of aluminum are in 0.58 mole of bauxite? d. What is the mass of \(2.1 \times 10^{24}\) formula units of bauxite?

Step-by-Step Solution

Answer

The molar mass of bauxite is 106.00 g/mol. The mass of aluminum in 0.58 mole of bauxite is 31.72 g, and there are 6.98 × 10^23 aluminum atoms in this amount of bauxite. The mass of 2.1 × 10^24 formula units of bauxite is 369.28 g.

Step 1: 1. Determine the molar mass of each element in the formula

Using the periodic table, find the molar mass for each element in the formula: Aluminum (Al): 26.98 g/mol Oxygen (O): 16.00 g/mol Hydrogen (H): 1.01 g/mol

Step 2: 2. Calculate the molar mass of bauxite

Using the molar masses of the elements and their respective numbers in the formula, we can find the molar mass of bauxite: Molar mass of bauxite = (2 × 26.98 g/mol) + (3 × 16.00 g/mol) + (4 × 1.01 g/mol) = 53.96 g/mol + 48.00 g/mol + 4.04 g/mol = 106.00 g/mol The molar mass of bauxite is 106.00 g/mol. b. Mass of Aluminum in 0.58 mole of bauxite

Step 3: 1. Calculate the mass of 0.58 mole of bauxite

Using the calculated molar mass of bauxite: Mass of bauxite = (0.58 mol) × (106.00 g/mol) = 61.48 g

Step 4: 2. Calculate the mass of aluminum in bauxite

Bauxite has a molar mass of 106.00 g/mol, out of which aluminum has a mass of 2 × 26.98 g/mol = 53.96 g/mol. Thus, the mass of aluminum in 1 mole of bauxite is: 53.96 ÷ 106.00 So, the mass of aluminum in 0.58 mole of bauxite is: Mass of aluminum = (61.48 g) × (53.96 ÷ 106.00) = 31.72 g The mass of aluminum in 0.58 mole of bauxite is 31.72 g. c. Number of atoms of aluminum in 0.58 mole of bauxite

Step 5: 1. Calculate the number of moles of aluminum in bauxite

In one mole of bauxite, there are two moles of aluminum as in the molecular formula \(\mathrm{Al}_{2} \mathrm{O}_{3} \cdot 2 \mathrm{H}_{2} \mathrm{O}\). So, in 0.58 mole of bauxite, there are: Aluminum moles = 0.58 mole × 2 = 1.16 moles

Step 6: 2. Calculate the number of aluminum atoms

Number of atoms can be determined by using Avogadro's number (6.022 × 10^23 atoms/mol). So, the number of aluminum atoms in 1.16 moles of aluminum is: Number of atoms = (1.16 mol) × (6.022 × 10^23 atoms/mol) = 6.98 × 10^23 atoms There are 6.98 × 10^23 aluminum atoms in 0.58 mole of bauxite. d. Mass of 2.1 × 10^24 formula units of bauxite

Step 7: 1. Calculate the number of moles of bauxite

Using Avogadro's number to convert formula units to moles: 2.1 × 10^24 formula units × (1 mol / 6.022 × 10^23 formula units) = 3.48 moles

Step 8: 2. Calculate the mass of bauxite

Using the molar mass of bauxite (106.00 g/mol) to find the mass: Mass of bauxite = (3.48 mol) × (106.00 g/mol) = 369.28 g The mass of 2.1 × 10^24 formula units of bauxite is 369.28 g.

Key Concepts

Bauxite

Bauxite is the primary ore used for the production of aluminum. It is mainly composed of hydrated aluminum oxides. The chemical formula for this mineral is presented as \(\text{Al}_{2}\text{O}_{3} \cdot 2\text{H}_{2}\text{O}\), indicating that it consists of aluminum oxide molecules with water molecules bonded to them.
This formula reveals that for every formula unit of bauxite, there are two aluminum atoms, three oxygen atoms, and four hydrogen atoms unified with water molecules. The significance of extracting aluminum from bauxite lies in its abundance and the ease of its extraction compared to other minerals.
Bauxite appears typically in a reddish-brown color due to the presence of iron oxides, which may also affect the ore's properties and merit scrutiny when processing it for aluminum extraction.

Waters of Hydration

The term "waters of hydration" is used to describe the water molecules that are part of the structure of a compound. In the chemical formula of bauxite \(\text{Al}_{2}\text{O}_{3} \cdot 2\text{H}_{2}\text{O}\), these are the \(\text{H}_{2}\text{O}\) units. 
These water molecules are integrated within the crystalline structure and contribute to the properties and mass of the compound. Waters of hydration can be essential in determining the total mass of the compound. In the case of bauxite, each formula unit includes two water molecules which total to four hydrogen atoms when calculated with molecular weights.
This detail is crucial as it affects the calculation of the molar mass of bauxite, impacting subsequent computations involving the extraction and processing of aluminum. Understanding waters of hydration helps chemists predict how a compound might act or react, particularly with the loss or gain of these water molecules.

Aluminum Atoms

In bauxite, aluminum atoms hold significant value due to their role in producing aluminum metal. The formula \(\text{Al}_{2}\text{O}_{3} \cdot 2\text{H}_{2}\text{O}\) signifies that each formula unit of bauxite consists of two aluminum atoms.
To calculate how many aluminum atoms are present in any given amount of bauxite, you need to consider how many formula units or moles you have. If you calculate or are given the number of moles of bauxite, simply multiply that by two, as each mole contains two moles of aluminum atoms.
For example, in one mole of bauxite, there are two moles of aluminum atoms. Using techniques in stoichiometry, further calculations can be performed to determine quantities such as mass or even the resultant number of aluminum atoms, advancing the understanding or capability of any process involving aluminum extraction.

Avogadro's Number

Avogadro's Number is fundamental in chemistry for converting between atoms/molecules and moles. It is defined as approximately \(6.022 \times 10^{23}\) particles per mole. This principle underpins the conversion from microscopic to a more tangible macroscopic scale frequently used in laboratory settings.
When solving problems involving the mass or number of formula units, such as with bauxite, Avogadro's number enables the calculation of precise numbers of atoms or molecules. For instance, if you need to find how many atoms of aluminum exist in 0.58 moles of bauxite, you first find the moles of aluminum (which is double the moles of bauxite due to the formula) and then multiply by Avogadro's number.

• This tool provides a bridge between the statistical mechanics and chemistry's practical applications, allowing scientists to measure out quantities accurately for reactions and analyses.

Understanding Avogadro's number is critical in any work involving stoichiometry, as well as in industrial applications where substance quantification is necessary.