Question

Aluminum sulfate, \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\), is a compound used in sewage treatment plants to remove pollutants by causing them to precipitate. (a) Construct a pair of conversion factors that relate moles of aluminum to moles of sulfur for this compound. (b) Construct a pair of conversion factors that relate moles of sulfur to moles of \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\). (c) How many moles of \(\mathrm{Al}\) are in a sample of this compound if the sample also contains \(0.900 \mathrm{~mol} \mathrm{~S} ?\) (d) How many moles of \(S\) are in \(1.16 \mathrm{~mol} \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) ?

Short answer

(a) 2:3 and 3:2 Al:S mole ratio. (b) 1:3 and 3:1 S:Al2(SO4)3 mole ratio. (c) 0.600 moles Al. (d) 3.48 moles S.

Step-by-step solution

Determine mole ratio of Al to S

From the formula \(\mathrm{Al}_{2}(\mathrm{SO}_{4})_{3}\) it can be seen that there are 2 moles of Al for every 3 moles of \(\mathrm{SO}_{4}\). Since each \(\mathrm{SO}_{4}\) contains 1 mole of S, the ratio is 2 moles of Al for every 3 moles of S.

Write conversion factors for Al to S

Conversion factors can be set up as follows: \[\dfrac{2 \text{ moles Al}}{3 \text{ moles S}}\] and \[\dfrac{3 \text{ moles S}}{2 \text{ moles Al}}\]. These can be used to convert moles of Al to moles of S and vice versa.

Determine mole ratio of S to Al2(SO4)3

In one mole of \(\mathrm{Al}_{2}(\mathrm{SO}_{4})_{3}\), there are 3 moles of S since there are three \(\mathrm{SO}_{4}\) groups.

Write conversion factors for S to Al2(SO4)3

Conversion factors are: \[\dfrac{1 \text{ mole Al}_{2}(\mathrm{SO}_{4})_{3}}{3 \text{ moles S}}\] and \[\dfrac{3 \text{ moles S}}{1 \text{ mole Al}_{2}(\mathrm{SO}_{4})_{3}}\].

Calculate moles of Al using moles of S

To find moles of Al from 0.900 moles of S, use the Al to S conversion factor: \[\text{Moles of Al} = 0.900 \text{ moles S} \times \dfrac{2 \text{ moles Al}}{3 \text{ moles S}} = 0.600 \text{ moles Al}\].

Calculate moles of S using moles of Al2(SO4)3

To find moles of S from 1.16 moles of \(\mathrm{Al}_{2}(\mathrm{SO}_{4})_{3}\), use the S to Al2(SO4)3 conversion factor: \[\text{Moles of S} = 1.16 \text{ moles Al}_{2}(\mathrm{SO}_{4})_{3} \times \dfrac{3 \text{ moles S}}{1 \text{ mole Al}_{2}(\mathrm{SO}_{4})_{3}} = 3.48 \text{ moles S}\].

Key concepts

Mole Ratio

Understanding the mole ratio is crucial when performing calculations in stoichiometry, as it represents the relationship between the amounts in moles of any two compounds or elements in a chemical reaction. For instance, in the compound aluminum sulfate, \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \), we can directly observe from its chemical formula that there are 2 atoms of aluminum (Al) for every 3 sulfate groups (SO₄).

We deduce that each sulfate group has one atom of sulfur (S); hence, for every 3 sulfate groups, there are 3 moles of sulfur. This leads us to a mole ratio of 2 moles of aluminum to 3 moles of sulfur in the chemical compound. To apply this ratio to our problems, we first need to identify what we're asked to find: the amount of one substance based on the amount of another. Thus, the mole ratio serves as a fundamental conversion factor in our chemical calculations.

Conversion Factors

Conversion factors act like bridges that help us convert from one unit to another. In chemistry, these factors are incredibly useful for converting between amounts of substances, especially when dealing with moles. For the reactions involving aluminum sulfate, we construct conversion factors using the mole ratio we previously determined.

A conversion factor can take the form \( \dfrac{2 \text{ moles Al}}{3 \text{ moles S}} \) or its reciprocal \( \dfrac{3 \text{ moles S}}{2 \text{ moles Al}} \) based on the direction of the conversion we need. When we need to find out the amount of aluminum based on the sulfur amount, we'd use the first conversion factor. Conversely, if we have the aluminum amount and we want to find out the sulfur amount, we'd use the second. It's all about constructing the appropriate bridge to get to the unit we desire.

Chemical Compounds

Chemical compounds are substances formed by two or more different types of elements bonded together in fixed proportions. The chemical formula of these compounds, like \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \), gives us valuable information about their composition. The subscripts tell us how many atoms of each element are present in the compound. For instance, the '2' next to Al in aluminum sulfate indicates two atoms of aluminum for every one molecule.

Knowing the composition allows us to determine the mole ratios within the compound, which can then be applied to convert between moles of different elements, as seen in this example. With these mole ratios, we can calculate the amount of one element given the amount of another within the same compound or even in different compounds if they are part of a chemical reaction.

Molar Mass Calculations

Molar mass calculations are a pivotal part of stoichiometry. The molar mass is the weight of one mole of a given substance and is measured in grams per mole (g/mol). To compute the molar mass, we sum the masses of all the atoms in a molecule based on each atom's atomic weight found on the periodic table.

Using aluminum sulfate as an example, to find its molar mass, we would multiply the atomic mass of aluminum by 2 (since there are 2 Al atoms), the atomic mass of sulfur by 3 (for the three S atoms), and the atomic mass of oxygen by 12 (because there are four oxygen atoms in each of the three sulfate groups). Adding these together yields the molar mass of one mole of \( \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \). With this information, we can then convert between grams and moles, a necessary step for many stoichiometric calculations.