Question

The compound \(\mathrm{XYZ}_{3}\) has a molar mass of \(100.09 \mathrm{~g}\) and a percent composition (by mass) of \(40.04 \% \mathrm{X}, 12.00 \% \mathrm{Y}\), and \(47.96 \% \mathrm{Z}\). What is the formula of the compound?

Short answer

The molecular formula of the compound is \(\mathrm{XYZ}_3\).

Step-by-step solution

Determine the Mass of Each Element in 100 g of Compound

Assume a 100 g sample of the compound. Thus, the masses of the elements would be:- X: 40.04 g- Y: 12.00 g- Z: 47.96 g

Calculate the Moles of Each Element

Use the molar masses of each element to find the number of moles:- Moles of X = \( \frac{40.04}{\text{Molar mass of X}} \)- Moles of Y = \( \frac{12.00}{\text{Molar mass of Y}} \)- Moles of Z = \( \frac{47.96}{\text{Molar mass of Z}} \)

Find the Simplest Whole Number Ratio

Divide the number of moles of each element by the smallest number of moles calculated. This will provide the simplest whole number ratio of atoms.

Determine the Empirical Formula

Use the whole number ratios to determine the empirical formula of the compound. This will be in the form of \(\mathrm{X}_a\mathrm{Y}_b\mathrm{Z}_c\).

Convert to the Molecular Formula

If necessary, use the molar mass of the compound (100.09 g/mol) and the empirical formula mass to find the molecular formula. Multiply the subscripts in the empirical formula by the ratio of the compound's molar mass to the empirical formula mass.

Key concepts

Percent Composition

Understanding percent composition is crucial for determining the chemical formula of a compound. Percent composition by mass refers to the percentage of each element in a compound relative to the total mass of the compound. For example, in the compound \(\text{XYZ}_3\), if 40.04% is \'X\', 12.00% is \'Y\', and 47.96% is \'Z\', these figures represent the mass contribution of each element to the compound. This initial data helps us determine how many grams of each element are present in a 100 g sample, simplifying the following calculations.

Molar Mass

Molar mass is the mass of one mole of a given substance, typically measured in grams per mole (g/mol). In this exercise, knowing the molar mass of each element (X, Y, and Z) is essential for converting mass (grams) to moles. For instance, the molar masses allow us to calculate the number of moles of each element by dividing the mass of each element by its molar mass: \( \text{Moles of X} = \frac{40.04 \text{ g}}{\text{Molar mass of X}} \). This conversion from mass to moles is a necessary step in determining the empirical formula.

Chemical Formula Determination

Chemical formula determination involves finding the simplest whole number ratio of atoms in a compound, known as the empirical formula. To do this, we start by calculating the moles of each element. We then divide each mole value by the smallest number of moles calculated among the elements to get a simple ratio. For example, if we calculated 2 moles of X, 1 mole of Y, and 3 moles of Z, and 1 mole is the smallest calculated amount, then the ratio is 2:1:3. These ratios give us the empirical formula (e.g., \( \text{X}_2 \text{Y}_1 \text{Z}_3 \)). The molecular formula is then found by comparing the empirical formula mass with the given molar mass of the compound. If needed, this ratio is used to scale up the empirical formula to match the given molar mass.

Mole Concept

The mole concept is central to understanding chemical reactions and stoichiometry. A mole defines a specific number of entities, usually 6.022 x 10^23 atoms or molecules (Avogadro's number). In the context of the exercise, it helps in converting mass percentages into the actual number of moles of each element. This is achieved by dividing the given mass of each element by its molar mass. Converting mass values into moles using the molar mass makes it easier to derive the simplest whole number ratio of atoms in the compound, which is key to determining both the empirical and molecular formulas.