Question

After preparing a sample of alum, \(\mathrm{K}_{2} \mathrm{SO}_{4} \cdot \mathrm{Al}_{2}\left(\mathrm{SO}_{4}\right)_{3} \cdot 24 \mathrm{H}_{2} \mathrm{O},\) an ana- lyst determines its purity by dissolving a \(1.2931-\mathrm{g}\) sample and precipitating the aluminum as \(\mathrm{Al}(\mathrm{OH})_{3}\). After filtering, rinsing, and igniting, \(0.1357 \mathrm{~g}\) of \(\mathrm{Al}_{2} \mathrm{O}_{3}\) is obtained. What is the purity of the alum preparation?

Short answer

The purity of the alum preparation is approximately 97.65%.

Step-by-step solution

Determine the moles of Al2O3 obtained

First, we need to calculate the moles of \( \text{Al}_2\text{O}_3 \) from the given mass. The molar mass of \( \text{Al}_2\text{O}_3 \) is approximately 101.96 g/mol. Thus, the number of moles is calculated as: \[ \text{moles of Al}_2\text{O}_3 = \frac{0.1357 \text{ g}}{101.96 \text{ g/mol}} \approx 0.00133 \text{ moles} \]

Calculate moles of Al in Al2O3

Each mole of \( \text{Al}_2\text{O}_3 \) contains 2 moles of Al. Therefore, the moles of Al are: \[ \text{moles of Al} = 2 \times 0.00133 = 0.00266 \text{ moles} \]

Calculate moles of alum from moles of Al

Alum, \( \text{K}_2\text{SO}_4 \cdot \text{Al}_2(\text{SO}_4)_3 \cdot 24 \text{H}_2\text{O} \), contains 2 moles of Al per mole of alum. Hence, the moles of alum are equal to the moles of Al: \[ \text{moles of alum} = 0.00266 \]

Determine the mass of pure alum

To find the mass of pure alum, we use the moles of alum and its molar mass. The molar mass of alum is approximately 474.36 g/mol. Thus, \[ \text{mass of pure alum} = 0.00266 \times 474.36 \text{ g/mol} \approx 1.2628 \text{ g} \]

Calculate the purity of the alum sample

The purity is determined by dividing the mass of pure alum by the mass of the sample and converting it to a percentage: \[ \text{Purity} = \left(\frac{1.2628 \text{ g}}{1.2931 \text{ g}} \right) \times 100\% \approx 97.65\% \]

Key concepts

Gravimetric Analysis

Gravimetric analysis is a precise and reliable method used to determine the composition of a substance by measuring its mass. The process involves several steps, starting with the separation of the desired component from a mixture, often by precipitation. After precipitation, the compound is filtered, washed, and dried until it reaches a constant mass. In the given exercise, aluminum hydroxide, \(\mathrm{Al}(\mathrm{OH})_3\), is precipitated, then converted into aluminum oxide, \(\mathrm{Al}_2\mathrm{O}_3\), by ignition.

Key steps in gravimetric analysis include:

• Formation of a precipitate: It is crucial to select a reagent that will form a precipitate with the analyte, which should be insoluble under the analysis conditions.
• Filtering and washing: This step ensures that only the analyte is weighed, free from impurities.
• Conversion to a weighable form: Often involves drying or heating to remove water or other volatile substances.

Gravimetric analysis relies on accurate weighing and the chemical transformation of the analyte into a compound of known composition and stable state. It is particularly useful in assays where high precision and accuracy are needed, such as determining the purity of compounds.

Purity Determination

Purity determination is a critical practice in both analytical chemistry and quality control. It involves quantifying how much of a sample is composed of the target compound, free from impurities. In the exercise, the purity of alum is determined by first converting a known mass of aluminum within the sample into \(\mathrm{Al}_2\mathrm{O}_3\).

Once you have isolated the target substance, purity is calculated with the formula:

\[ \text{Purity} = \left(\frac{\text{mass of pure component}}{\text{total mass of sample}}\right) \times 100\% \]

This calculation reveals the proportion of the desired substance compared to the entire sample mass. Purity determination plays a crucial role in quality assurance processes to ensure that products meet specific standards and do not contain unwanted materials. It's vital in pharmaceuticals, food production, and materials manufacturing, where knowing the purity dictates suitability for use.

Stoichiometry

Stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It is based on the conservation of mass and the principle that matter is neither created nor destroyed. In this exercise, stoichiometry is used to link the amount of \(\mathrm{Al}_2\mathrm{O}_3\) produced back to the original alum sample.

The step-by-step calculation involves:

• Determining the moles of \(\mathrm{Al}_2\mathrm{O}_3\) using its mass and molar mass.
• Using stoichiometry to find the moles of \(\mathrm{Al}\) atoms, as each formula unit of \(\mathrm{Al}_2\mathrm{O}_3\) contains 2 moles of \(\mathrm{Al}\).
• Relating the moles of \(\mathrm{Al}\) to the moles of alum, as each formula unit of alum contains 2 moles of \(\mathrm{Al}\).

Stoichiometry ensures that chemists can predict the amounts needed or given in reactions and processes. It's fundamental for converting experimental data into meaningful chemical understanding. Knowing how to apply stoichiometric calculations helps in determining yields, limiting reagents, and optimizing reactions in laboratories and industrial processes alike.