The aluminum sulfate hydrate \(\left[\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \cdot x \mathrm{H}_{2} \mathrm{O}\right]\)contains 8.10 percent \(\mathrm{Al}\) by mass. Calculate \(x-\) that is, the number of water molecules associated with each \(\mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) unit.

Understanding molar mass is essential for various chemical calculations. The molar mass, expressed in grams per mole (g/mol), is the weight of one mole of a substance. To calculate the molar mass of a compound, such as aluminum sulfate hydrate, one must sum the molar masses of all the atoms in the formula.

As demonstrated in the exercise solution, the molar mass of aluminum sulfate \(\mathrm{Al}_{2}(\mathrm{SO}_{4})_{3}\) is calculated by adding the molar masses of 2 aluminum atoms, 3 sulfur atoms, and 12 oxygen atoms. The molar masses of these elements—Aluminum (26.98 g/mol), Sulfur (32.06 g/mol), and Oxygen (16.00 g/mol)—are constants found on the periodic table.

The calculated molar mass is vital in subsequent steps to determine the percentage composition and the number of hydrate water molecules in the compound.

The percent composition by mass indicates the proportion of each element in a compound, relative to the total compound mass. To find the percent composition by mass of aluminum in the aluminum sulfate hydrate, the exercise solution applies the formula: \[%\ composition = \left( \frac{\text{mass of element}}{\text{total mass of compound}} \right) \times 100\].

In this case, the mass of aluminum is obtained by multiplying the molar mass of aluminum sulfate without the water by the given mass percentage of aluminum (8.10%). This provides the actual mass of aluminum in the sample. Subtracting this mass from the total mass gives the mass of water, as only aluminum and water are considered.

Understanding the percent composition helps students to calculate empirical formulas and to deduce the number of moles of each element within a sample, which are fundamental skills in chemistry.

Hydrates are crystalline compounds that contain water molecules associated with their ions. The number of water molecules (x) in an aluminum sulfate hydrate \(\mathrm{Al}_{2}(\mathrm{SO}_{4})_{3} \cdot x \mathrm{H}_{2} \mathrm{O}\) is determined by dividing the mass of water (found in the previous step) by the molar mass of water (18.01 g/mol) to find the number of moles. Since one mole of water molecules corresponds to the molar mass of water, the result provides the mole ratio of water molecules to the mole of aluminum sulfate in the sample.

The exercise solution illustrates that, once the amount of water (in grams) in the hydrate is known and divided by the molar mass of water, the result is the number of moles of water per mole of aluminum sulfate. If necessary, this number can then be rounded to the nearest whole number for the final hydrate formula, taking into account significant figures and the purity of the sample.