Question

A substance \(\mathrm{X}_{2} \mathrm{Z}\) has the composition (by mass) of \(40.0 \% \mathrm{X}\) and \(60.0 \%\) Z. What is the composition (by mass) of the compound \(\mathrm{XZ}_{2} ?\)

Short answer

The composition (by mass) of the compound $\mathrm{XZ}_{2}$ is 25% X and 75% Z.

Step-by-step solution

Assume a 100g sample of X2Z

Let's assume we have a 100g sample of X2Z. This will make it easier for us to work with mass percentages.

Calculate the masses of X and Z in the sample

From the given mass percentages, we can calculate the masses of X and Z in the 100g sample: - Mass of X = 40.0% of 100g = \(40g\) - Mass of Z = 60.0% of 100g = \(60g\)

Determine the moles of X and Z

In order to determine the moles of X and Z in the 100g sample of X2Z, we need to use their molar masses (Mx and Mz). We can write the following equations for moles: - Moles of X = mass of X / Mx - Moles of Z = mass of Z / Mz For X2Z, we know that there are two moles of X for every mole of Z. Therefore, we can write: 2 * moles of X = moles of Z Now we can replace the moles of X and Z from the equations above: 2 * (mass of X / Mx) = mass of Z / Mz

Calculate the moles of X and Z in XZ2

To maintain the same atomic ratio as in X2Z, in XZ2, there will be one mole of X for every two moles of Z. So, we can write: moles of X in XZ2 = moles of Z in XZ2 / 2

Determine the total mass of XZ2

Now, we can determine the total mass of XZ2 using the moles of X and Z, and their molar masses (Mx and Mz): Total mass of XZ2 = (moles of X in XZ2 * Mx) + (moles of Z in XZ2 * Mz) From step 4, we can substitute moles of X in XZ2 with moles of Z in XZ2 / 2: Total mass of XZ2 = ((moles of Z in XZ2 / 2) * Mx) + (moles of Z in XZ2 * Mz)

Calculate the mass percentages of X and Z in XZ2

Now that we have the total mass of XZ2, we can calculate the mass percentages of X and Z in XZ2 using the moles of X and Z in XZ2: - Mass percentage of X = (moles of X in XZ2 * Mx) / total mass of XZ2 * 100 - Mass percentage of Z = (moles of Z in XZ2 * Mz) / total mass of XZ2 * 100 Using our substitution from step 4 for moles of X in XZ2: - Mass percentage of X = ((moles of Z in XZ2 / 2) * Mx) / total mass of XZ2 * 100 Since, moles of Z is common in the mass percentage of X and Z, if we take the ratio, \[ \\ \frac{Mass\,\,percentage\,of\,X}{Mass\,\,percentage\,of\,Z} = \frac{Mx /2}{Mz} \] Plug in the values we get from Step 3 into the equation. We get \[ \frac{Mass\, percentage \, of \,X}{Mass\, percentage \,of \,Z} = \frac{40/2}{60} \] Now we can solve for the mass percentage of X: \[ Mass\,percentage\,of\,X = \frac{20}{60}\, * \,Mass\,percentage\,of\,Z \] Now, we know that Mass percentage of X + Mass percentage of Z = 100%, \[ \frac{20}{60}\, * \,Mass\,percentage\,of\,Z + Mass\,percentage\,of\,Z = 100 \] Solve for Mass percentage of Z, \[ Mass \,percentage\,of\,Z = 75 \% \] Now calculate the mass percentage of X: \[ Mass\,percentage\,of\,X = 100\% - Mass\,percentage\,of\,Z = 100\% - 75\% = 25\% \] Thus, the composition(by mass) of the compound XZ2 is 25% X and 75% Z.

Key concepts

Mass Percentage

Understanding the mass percentage of a substance is crucial in chemistry, especially when it comes to knowing the composition of compounds. The mass percentage, also known as mass percent or weight percent, is a way of expressing the concentration of an element within a compound. It is calculated by dividing the mass of the specific component by the total mass of the compound and multiplying by 100.

For instance, if we are given a compound with a mass percentage composition of 40% element X and 60% element Z, and we are tasked to find the mass percentages in a different compound, XZ2, we start by assuming a convenient sample size, often 100 grams. This way, calculating mass percentages becomes straightforward, as each percentage point corresponds to one gram.

Once we have the individual masses of the elements, we can then use the concept of moles and molar masses to maintain ratios and find the new mass percentages for the compound XZ2, as shown in the step-by-step exercise solution. This procedure is essential since the mass percentage does not only give us the ratio by mass but also helps us to relate masses to moles, thus connecting to the mole concept.

Mole Concept

The mole concept is one of the pillars of stoichiometry in chemistry. A mole represents a specific number of particles, atoms, ions, or molecules. It is defined as the amount of substance that contains as many entities as there are atoms in 12 grams of carbon-12. This number, known as Avogadro's number, is approximately 6.022 x 10^23 entities.

The step-by-step solution applies the mole concept by converting mass percentages to moles. This is done using the molar mass of the elements in question. The mole ratio within compounds is vital for determining the composition of other compounds with the same elements but different numbers of atoms.

For instance, when moving from compound X2Z to XZ2, the mole concept is used to maintain the ratio of atoms as the structure changes. Understanding how to convert mass to moles and vice versa is critical in solving mass percentage composition problems and allows for a consistent method of tracking and manipulating chemical quantities.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a crucial component when dealing with the mole concept as it provides a bridge between the mass of a substance and the number of moles present. The molar mass is numerically equivalent to the average atomic mass of the element or compound but scaled up from atomic mass units (amu) to grams.

In the provided exercise, molar mass enables the conversion of mass percentages to moles, which then allows for the comparison of elemental composition in different compounds. This conversion is critical, especially when dealing with compounds with varying numbers of atoms, as seen in X2Z and XZ2. Using the molar mass, chemists can determine the moles of each element present using the equation:
- Moles = Mass (g) / Molar Mass (g/mol)

This equation is fundamental in quantitative chemistry and allows for the calculation of unknowns in chemical reactions and compositions. Molar mass calculations are not only essential for scientists in the lab but are an everyday tool in industries ranging from pharmaceuticals to materials science.