Question

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.

Short answer

The value of \( x \) is 18.

Step-by-step solution

Determine Molar Mass of Anhydrous Compound

First, calculate the molar mass of the anhydrous aluminum sulfate, \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \). It is composed of 2 aluminum (Al) atoms, 3 sulfur (S) atoms, and 12 oxygen (O) atoms: \[ \text{Molar mass of } \mathrm{Al}_2(\mathrm{SO}_4)_3 = 2(26.98) + 3(32.07) + 12(16.00) = 342.15 \, \text{g/mol} \]

Express Mass of Compound with Water

The full formula for the hydrated compound is \( \mathrm{Al}_2(\mathrm{SO}_4)_3 \cdot x \mathrm{H}_2 \mathrm{O} \). Its molar mass is expressed as \( 342.15 + 18x \), where \( 18 \times x \) accounts for the mass of water molecules.

Set Up Equation for Aluminum Percentage

Given that the compound contains 8.10% aluminum by mass, set up an equation: \[ \frac{2(26.98)}{342.15 + 18x} = 0.0810 \]. Simplifying the numerator gives you \( 53.96 \).

Solve for x

Solve the equation for \( x \):\[ \frac{53.96}{342.15 + 18x} = 0.0810 \]Rearranging gives\[ 53.96 = 0.0810 \times (342.15 + 18x) \]\[ 53.96 = 27.71865 + 1.458x \]\[ 26.24135 = 1.458x \]\[ x \approx \frac{26.24135}{1.458} \approx 18 \]

Verify the Calculation

Verify the solution by calculating the theoretical percentage content. With \( x = 18 \), substitute back into the equation:\[ \text{Molar Mass of Hydrate} = 342.15 + 18(18) = 666.15 \, \text{g/mol} \]\[ \frac{53.96}{666.15} \times 100 \approx 8.10\% \]This confirms our calculated \( x \) value is correct.

Key concepts

Molar Mass Calculation

Molar mass is an essential concept in chemistry, especially when dealing with substances that form hydrates. In this case, calculating the molar mass of aluminum sulfate, \(\mathrm{Al}_2(\mathrm{SO}_4)_3\), is the first step. To do this, you add up the atomic masses of all the atoms in the formula.
Here’s a quick guide on how to do it:

• Aluminum (Al) has an atomic mass of 26.98 g/mol.
• Sulfur (S) has an atomic mass of 32.07 g/mol.
• Oxygen (O) has an atomic mass of 16.00 g/mol.

Putting it all together: the molar mass of \(\mathrm{Al}_2(\mathrm{SO}_4)_3 = 342.15\, \text{g/mol}\). This number is crucial when calculating the amount of water molecules in the hydrate, often represented as \(x\) in the formula \(\mathrm{Al}_2(\mathrm{SO}_4)_3\cdot x\mathrm{H}_2\mathrm{O}\). Understanding molar mass paves the way for determining percentages and constructing detailed chemical equations.

Aluminum Percentage

Understanding the aluminum percentage in a compound involves seeing how much of the whole substance's mass is made up of aluminum. In the chemical compound \(\mathrm{Al}_2(\mathrm{SO}_4)_3\cdot x\mathrm{H}_2\mathrm{O}\), aluminum's mass can help us figure out the correct number of water molecules, \(x\). The problem specifies that aluminum makes up 8.10% of the compound's mass.
By setting up the equation \[ \frac{53.96}{342.15 + 18x} = 0.0810\], you relate the known mass of aluminum, 53.96 g/mol (from the two aluminum atoms), to the total molar mass of the compound with the unknown quantity of water. By solving this equation, you can accurately determine the number of water molecules. Utilizing known percentages helps simplify what might seem like a complex chemical makeup at first glance.

Hydrate Formation

Hydrates are fascinating as they include water molecules integrated into their structures. In aluminum sulfate hydrate, \(\mathrm{Al}_2(\mathrm{SO}_4)_3\cdot x\mathrm{H}_2\mathrm{O}\), water molecules are bonded within the crystalline framework.
The chemical formula indicates not just the original compound, but also the presence of moisture. The number of water molecules, \(x\), can deeply influence the hydrate’s properties, such as mass and reactivity. The water molecules are not merely added on; they play a structural role that may affect stability and appearance.In practical applications, knowing the extent of hydration is crucial, since it can impact both storage and usage of the compound. Thus, calculating the correct \(x\) is not only a theoretical exercise but also a vital part of chemical analysis.

Chemical Formulas

Chemical formulas are the backbone of understanding compounds. They provide a visual representation of the elements involved and their ratios. For aluminum sulfate hydrate, the formula\(\mathrm{Al}_2(\mathrm{SO}_4)_3\cdot x\mathrm{H}_2\mathrm{O}\) tells a story.
It shows that each part has a precise role. In our problem, the formula includes:

• 2 Aluminum atoms, which are crucial for determining the percentage and thus the hydrate's mass composition.
• 3 Sulfate groups, which together with aluminum define the anhydrous part.
• \(x\) molecules of water, which we solve for to understand the hydrate structure.

Each element and its count are not merely symbols but stand for specific weights and contributions to the overall nature of the compound. Chemical formulas thus bridge the microscopic world of atoms and molecules with the macroscopic properties we can observe.