Problem 59 - Chemistry Matter and Change

Question

\begin{equation} \begin{array}{l}{\text { Determine the empirical formula for a compound that contains } 35.98 \% \text { aluminum and }} \\ {64.02 \% \text { sulfur. }}\end{array} \end{equation}

Step-by-Step Solution

Answer

The empirical formula is \(\text{Al}_2\text{S}_3\).

Step 1: Convert Percentages to Grams

To simplify the calculation, assume you have 100 grams of the compound. With this assumption, you have 35.98 grams of aluminum and 64.02 grams of sulfur.

Step 2: Convert Grams to Moles

Find the number of moles of each element using the formula \( ext{moles} = \frac{ ext{grams}}{ ext{molar mass}} \). The molar mass of aluminum (\( ext{Al} \)) is 26.98 g/mol and sulfur (\( ext{S} \)) is 32.07 g/mol.For aluminum: \( \frac{35.98}{26.98} \approx 1.334 \) molesFor sulfur: \( \frac{64.02}{32.07} \approx 1.996 \) moles.

Step 3: Find Simplest Ratio

Divide the moles of each element by the smallest number of moles calculated:\[ \frac{1.334}{1.334} = 1 \] for aluminum, \[ \frac{1.996}{1.334} \approx 1.496 \] for sulfur.Since 1.496 is approximately 1.5, we'll multiply both numbers by 2 to get whole numbers.

Step 4: Multiply to get Whole Numbers

Multiply each ratio by 2 to eliminate the fraction:For aluminum: \( 1 \times 2 = 2 \)For sulfur: \( 1.5 \times 2 = 3 \)

Step 5: Write the Empirical Formula

Using these whole number ratios, write the empirical formula as \( \text{Al}_2\text{S}_3 \).

Key Concepts

Percent Composition

Percent composition is a concept that helps understand how much each element contributes to the total mass of a compound. Think of it as a way to express the concentration of each element in a compound, using percentages. In our exercise, we have:

• 35.98% for aluminum
• 64.02% for sulfur

The sum of all percentages must equal 100%. Thus, you can use these percentages to understand the relative amounts of each element present. In calculations, it often helps to convert these percentages into grams based on a 100-gram sample of the compound.
This conversion simplifies the calculations without altering the proportion of elements. Next, these gram values are utilized for mole conversion, leading us to further insights into the compound's makeup.

Mole Conversion

Moles are a fundamental concept in chemistry used to convert an element's mass into a quantifiable number of atoms or molecules. This process involves using the formula: \[ \text{moles} = \frac{\text{grams}}{\text{molar mass}} \] The molar mass of an element, measured in g/mol, is a constant representing how much one mole of that element weighs. For aluminum, the molar mass is 26.98 g/mol, and for sulfur, it is 32.07 g/mol. Let's convert the grams from our compound's percent composition into moles:

• For 35.98 grams of aluminum, the mole conversion is: \( \frac{35.98}{26.98} \approx 1.334 \text{ moles} \)
• For 64.02 grams of sulfur, the mole conversion is: \( \frac{64.02}{32.07} \approx 1.996 \text{ moles} \)

This step helps identify how many moles of each element are in the compound, paving the way for finding their simplest ratio.

Simplest Ratio

Once moles are calculated, the next step is to determine the simplest ratio of moles of the elements present in the compound. This ratio tells us how the elements combine in the most basic unit of the compound, known as the empirical formula. To find the simplest ratio:1. Identify the smallest number of moles, which in this case is for aluminum: 1.334 moles.2. Divide each element's moles by the smallest mole value: - Aluminum: \( \frac{1.334}{1.334} = 1 \) - Sulfur: \( \frac{1.996}{1.334} \approx 1.496 \)3. If you don't get whole numbers, multiply both ratios by the same factor to reach whole numbers. Here, multiplying by 2 gives: - Aluminum: 1 x 2 = 2 - Sulfur: 1.5 x 2 = 3So, the simplest ratio of aluminum to sulfur is 2:3. This ratio is crucial to write the empirical formula, which reflects the most basic whole number ratio of elements in a compound.

Chemical Compounds

Chemical compounds are substances formed from two or more elements chemically bonded together in a definite proportion. Understanding the empirical formula is essential as it shows the simplest whole number ratio of atoms in a compound. For our example, the compound contains:

• 2 atoms of aluminum
• 3 atoms of sulfur

Thus, its empirical formula is \( \text{Al}_2\text{S}_3 \). This representation gives insight into the basic structure of the compound without detailing the actual number of atoms in a molecule or the nature of bonds. Chemical compounds can exist in various forms and apply to numerous contexts such as: - Reactions and stoichiometry - Determining molecular formulasBy understanding empirical formulas, you gain a foundation that can be built upon when exploring more complex features of compounds, like molecular structures and properties.