Question

The "alum" used in cooking is potassium aluminum sulfate hydrate, \(\mathrm{KAl}\left(\mathrm{SO}_{4}\right)_{2} \cdot x \mathrm{H}_{2} \mathrm{O} .\) To find the value of \(x,\) you can heat a sample of the compound to drive off all of the water and leave only \(\mathrm{KAl}\left(\mathrm{SO}_{4}\right)_{2} .\) Assume you heat \(4.74 \mathrm{g}\) of the hydrated compound and that the sample loses \(2.16 \mathrm{g}\) of water. What is the value of \(x ?\)

Short answer

The value of \(x\) is 12.

Step-by-step solution

Determine Mass of Anhydrate

First, find the mass of the anhydrous compound after the water has been driven off. This can be done by subtracting the mass of water lost from the initial mass of the hydrated compound. \[\text{Mass of anhydrate} = 4.74\, \text{g} - 2.16\, \text{g} = 2.58\, \text{g}\]

Calculate Moles of Anhydrate

Next, calculate the moles of the anhydrous compound \(\text{KAl(SO}_4)_2\). Use the molar mass provided in the periodic table:\[\text{Molar Mass of } \text{KAl(SO}_4)_2 = K(39.10) + Al(26.98) + 2\times(S(32.07) + 4\times O(16.00))\]\[= 39.10 + 26.98 + 2 \times (32.07 + 64.00) = 258.21 \, \text{g/mol}\]Now:\[\text{Moles of anhydrate} = \frac{2.58\, \text{g}}{258.21\, \text{g/mol}} = 0.00999 \, \text{mol}\]

Calculate Moles of Water Lost

Calculate the moles of water that were lost. Use the molar mass of water (\(18.02\, \text{g/mol}\)):\[\text{Moles of } \text{H}_2\text{O} = \frac{2.16\, \text{g}}{18.02\, \text{g/mol}} = 0.1199 \, \text{mol}\]

Determine Value of x

Now find the value of \(x\) by dividing the moles of water lost by the moles of the anhydrate:\[x = \frac{0.1199 \, \text{mol}}{0.00999 \, \text{mol}} = 12.00\]Therefore, the value of \(x\) is 12. This indicates there are 12 water molecules per formula unit of the compound.

Key concepts

Molar Mass Calculation

Molar mass calculation is a fundamental concept in chemistry, crucial for converting between grams and moles of a substance. To perform a molar mass calculation, you'll need to refer to the periodic table, where each element's atomic mass is listed. The molar mass is calculated by adding up the atomic masses of all the atoms in a compound's formula.
For example, with potassium aluminum sulfate \((\mathrm{KAl(SO}_4)_2)\), you would find the molar mass by summing:

• the atomic mass of potassium (K): 39.10
• the atomic mass of aluminum (Al): 26.98
• two sulfate ions \((\mathrm{SO}_4)\)\(\Rightarrow 2 \times(32.07 + (4 \times 16.00))\)

This results in a molar mass of 258.21 g/mol for \(\mathrm{KAl(SO}_4)_2 \). Understanding how to calculate molar mass is essential for converting a substance's mass in grams to moles, which is often necessary for stoichiometric calculations.

Water of Crystallization

Water of crystallization refers to water molecules that are part of a compound's crystalline structure. It is commonly seen in hydrates, where water molecules are integrated into the crystal lattice.
A classic example is found in the hydrated form of potassium aluminum sulfate, where water molecules are a key part of the compound structure. As you determine the hydrate formula, you are essentially finding out how many water molecules are associated with each formula unit of the compound.
In practice, this might involve heating the hydrate to remove the water and analyzing the change in mass to understand the role water played in the crystalline structure. By losing water upon heating, one can determine the amount of water initially present, thus indicating how many water molecules were involved in the formation of the crystalline structure, which is crucial for determining the \(x\) value in formulas such as \(\mathrm{KAl(SO}_4)_2 \cdot x \mathrm{H}_2\mathrm{O} \). This process helps understand the stoichiometry and consumption of water molecules in such compounds.

Anhydrous Compound Analysis

Analyzing an anhydrous compound involves studying a substance that has had all its water of crystallization removed. This condition allows us to focus on the "dry" part of the compound, without the influence of water content.
When dealing with hydrated crystals, such as \(\mathrm{KAl(SO}_4)_2 \cdot x \mathrm{H}_2\mathrm{O}\), the goal of heating is to drive off the water, leaving you with an anhydrous form. By measuring the weight before and after heating, you can determine the mass of the anhydrous compound.

• Start with the initial mass of the hydrate.
• Subtract the mass of water lost to find the mass of the anhydrate.

This subtraction yields the mass of the anhydrous compound, crucial for further calculations like establishing the molar ratio between the anhydrous compound and the water of crystallization. Such analyses not only aid in determining the value of \(x\) in hydrates but also provide insights into the stability and composition of the anhydrous form compared to its hydrated counterpart.