Question

Bauxite, the principal ore used in the production of aluminum, has a molecular formula of \(\mathrm{Al}_{2} \mathrm{O}_{3}=2 \mathrm{H}_{2} \mathrm{O}\). a. What is the molar mass of bauxite? b. What is the mass of aluminum in \(0.58\) mol bauxite? c. How many atoms of aluminum are in \(0.58 \mathrm{~mol}\) bauxite? d. What is the mass of \(2.1 \times 10^{24}\) formula units of bauxite?

Short answer

a. The molar mass of bauxite is \(106.00 \frac{g}{mol}\). b. The mass of aluminum in \(0.58\) mol bauxite is \(31.24g\). c. There are \(6.985 \times 10^{23}\) aluminum atoms in \(0.58 \mathrm{~mol}\) bauxite. d. The mass of \(2.1 \times 10^{24}\) formula units of bauxite is \(369.41g\).

Step-by-step solution

Determine the molecular formula

The given molecular formula for bauxite is \(\mathrm{Al}_{2} \mathrm{O}_{3}\cdot 2 \mathrm{H}_{2} \mathrm{O}\).

List the molar masses of relevant elements

Aluminum (Al): \(26.98 \frac{g}{mol}\) Oxygen (O): \(16.00 \frac{g}{mol}\) Hydrogen (H): \(1.01 \frac{g}{mol}\)

Calculate the molar mass of bauxite

Using the molecular formula, we can find the molar mass of bauxite: Molar mass = (2 \(\times\) 26.98) + (3 \(\times\) 16.00) + (2 \(\times\) 2 \(\times\) 1.01) = \(53.96 + 48.00 + 4.04 = 106.00 \frac{g}{mol}\) #b. Determine the mass of aluminum in \(0.58\) mol bauxite#

Calculate the mass of aluminum in bauxite

Since there are 2 moles of aluminum per mole of the bauxite: Aluminum mass = (0.58 mol bauxite) \(\times\) (2 moles Al/mole bauxite) \(\times\) \(26.98\frac{g}{mol}\) Aluminum mass = \(31.24g\) #c. Determine the number of atoms of aluminum in \(0.58\) mol bauxite#

Convert moles of bauxite to moles of aluminum atoms

Since there are 2 moles of aluminum per mole of the bauxite: Aluminum moles = (0.58 mol bauxite) \(\times\) (2 moles Al/mole bauxite) = 1.16 mol Al

Calculate the number of aluminum atoms

Using Avogadro's constant (6.022 \(\times\) \(10^{23}\) atoms/mol), the number of aluminum atoms is: Al atoms = \(1.16 \space mol \space Al \times 6.022\times 10^{23} \mathrm{atoms \cdot mol^{-1}} = 6.985 \times 10^{23} \mathrm{atoms}\) #d. Determine the mass of \(2.1 \times 10^{24}\) formula units of bauxite#

Convert formula units to moles

Using Avogadro's constant (6.022 \(\times\) \(10^{23}\) formula units/mol), the number of moles of the bauxite is: Bauxite moles = \(\frac{2.1 \times 10^{24} \, \mathrm{formula \, units}}{6.022\times 10^{23}\, \mathrm{formula \, units \cdot mol^{-1}}}=3.485\, \mathrm{mol}\)

Calculate the mass of bauxite

Now that we have the number of moles of bauxite, we can calculate the mass using the molar mass: Bauxite mass = \(3.485\, \mathrm{mol} \times 106.00\frac{g}{mol} = 369.41\,g\)

Key concepts

Molar Mass Calculation

Understanding the molar mass of a substance is crucial for many chemistry calculations. The molar mass represents the weight of exactly one mole of a substance and is expressed in grams per mole (g/mol). To calculate the molar mass of a compound, such as bauxite with the chemical formula \( \mathrm{Al}_2\mathrm{O}_3\cdot2\mathrm{H}_2\mathrm{O} \), we must sum the masses of each constituent element, multiplied by the number of times each element appears in the compound. For instance:

• Aluminum (Al), with a molar mass of 26.98 g/mol, appears twice in bauxite.
• Oxygen (O), with a molar mass of 16.00 g/mol, appears in three units within the aluminum oxide (\mathrm{Al}_2\mathrm{O}_3) and in two molecules of water, adding to six units of oxygen.
• Hydrogen (H), with a molar mass of 1.01 g/mol, appears four times, as each water molecule (\mathrm{H}_2\mathrm{O}) contains two hydrogen atoms.

Adding up these masses gives us the total molar mass of bauxite. This fundamental step allows us to convert between grams and moles, thereby paving the way to solve stoichiometry problems.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the relative quantities of reactants and products in chemical reactions. It's rooted in the conservation of mass, where the total mass of reactants equals the total mass of products. To tackle stoichiometric calculations, we start by balancing the chemical equations and following the mole ratios.
For example, if we need to find the mass of aluminum present in a certain amount of bauxite (\(0.58 \text{ mol}\) of bauxite), we use the mole ratio from bauxite's formula, where each molecule contains two aluminum atoms. We then multiply that mole number by the molar mass of aluminum to find the mass in grams. Stoichiometry also applies in determining the number of atoms or molecules in a given mole of substance. By using the mole ratios and conversion factors, we can connect moles to mass, number of particles, or volume, depending on the situation and what units we are given or are looking for.

Avogadro's Constant

Avogadro's constant (often referred to as Avogadro's number) is a fundamental scientific figure that defines the number of constituent particles, typically atoms or molecules, that are contained in one mole of substance. Its value is approximately \(6.022 \times 10^{23}\) particles per mole.
This constant becomes extremely useful in converting between the number of atoms or molecules and the amount of substance in moles. If a problem asks for the number of aluminum atoms in \(0.58 \text{ mol}\) of bauxite, we multiply the number of moles of aluminum (considering the stoichiometric relationship within bauxite) by Avogadro's constant to find the total number of atoms. Similarly, if we are given the number of formula units, like \(2.1 \times 10^{24}\) units of bauxite, we can use Avogadro's constant to convert this to moles and subsequently, with the molar mass, to grams. Mastering the use of Avogadro's constant is key for performing accurate chemical quantifications.