The compound \(\mathrm{X}_{2}\left(\mathrm{YZ}_{3}\right)_{3}\) has a molar mass of \(282.23 \mathrm{~g}\) and a percent composition (by mass) of \(19.12 \% \mathrm{X}, 29.86 \% \mathrm{Y}\), and \(51.02 \% \mathrm{Z}\). What is the formula of the compound?

Percent composition by mass is an important concept in chemistry. It tells us the percentage of each element in a compound by weight. It’s calculated using the formula:

\(\text{Percent Composition} = \frac{\text{mass of element in 1 mole of the compound}}{\text{molar mass of compound}} \times 100\)

Here's how you can understand this better with the compound \(\text{X}_{2}\text{(YZ}_{3}\text{)}_{3}\). Given the percent compositions: 19.12% X, 29.86% Y, and 51.02% Z, in 100 grams of the compound, these percentages translate directly to the mass of each element: 19.12 g of X, 29.86 g of Y, and 51.02 g of Z.

Molar mass is the mass of one mole of a substance (usually in g/mol). To find the molar mass of a compound, you sum up the molar masses of all atoms in its formula.

Given the compound \(\text{X}_{2}\text{(YZ}_{3}\text{)}_{3}\), the total molar mass is 282.23 g/mol.

Therefore, the problem is worked out by assuming molar masses for X, Y, and Z. We use their masses (from percent composition) and convert these to moles by the formula:

\(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\).

For example, moles of X are calculated as:

\(\frac{19.12 \text{ g}}{M_{X}}\)

And similarly for Y and Z. By substituting these back into the compound’s structure and solving, we can find the approximate molar masses of X, Y, and Z.

The mole concept is fundamental in chemistry. A mole represents 6.022 \[ \times \] 10^\text{23} particles (atoms, molecules, ions, etc.) of a substance.

In practice, the mole concept helps convert amounts measured in grams to the amount in moles, which scientists use to relate quantities in chemical reactions.

For the compound \(\text{X}_{2}\text{(YZ}_{3}\text{)}_{3}\), we need to find how many moles of each element are present. Given the percent composition, we first convert the mass into moles using the formula:

\(\text{moles} = \frac{\text{mass}}{\text{molar mass}}\)

Then, we use these mole values to define the amounts in relation to each other in the compound. For instance, with

\(\text{X}_{2}\text{(YZ}_{3}\text{)}_{3}\), knowing that 2 moles of X, 3 moles of Y, and 9 moles of Z should add up to match the molar mass (282.23 g/mol), we can adjust to fit a common constant (a). This process allows us to better understand the relationships between elements within the compound.