Question

The compound \(\mathrm{As}_{2} \mathrm{I}_{4}\) is synthesized by reaction of arsenic metal with arsenic triiodide. If a solid cubic block of arsenic \(\left(d=5.72 \mathrm{g} / \mathrm{cm}^{3}\right)\) that is \(3.00 \mathrm{cm}\) on edge is allowed to react with \(1.01 \times 10^{24}\) molecules of arsenic triiodide, what mass of \(\mathrm{As}_{2} \mathrm{I}_{4}\) can be prepared? If the percent yield of \(\mathrm{As}_{2} \mathrm{I}_{4}\) was \(75.6 \%\) what mass of \(\mathrm{As}_{2} \mathrm{I}_{4}\) was actually isolated?

Short answer

The theoretical mass of As₂I₄ that can be prepared is 670.20 g, and the actual mass of As₂I₄ isolated with a percent yield of 75.6% is 506.51 g.

Step-by-step solution

Calculate the mass of arsenic metal

To calculate the mass of arsenic metal, we need to use the density formula as follows: mass = density × volume. The volume of the arsenic cubic block can be calculated as side × side × side. Density = 5.72 g/cm³ Side = 3.00 cm Volume = 3.00 cm × 3.00 cm × 3.00 cm = 27 cm³ Now, we can calculate the mass of arsenic: Mass of arsenic = density × volume = 5.72 g/cm³ × 27 cm³ = 154.44 g

Calculate the moles of arsenic and arsenic triiodide

To determine the moles of arsenic and arsenic triiodide, we will use the following relationship: moles = mass / molar mass Molar mass of arsenic (As) = 74.92 g/mol Moles of arsenic = mass of arsenic / molar mass of arsenic = 154.44 g / 74.92 g/mol = 2.061 mol We are given the number of molecules of arsenic triiodide, so we need to convert it to moles. We will use Avogadro's number, which is 6.022 × 10²³ molecules/mol: Moles of arsenic triiodide = (1.01 × 10²⁴ molecules) / (6.022 × 10²³ molecules/mol) = 1.678 mol

Determine the limiting reactant

To find out the limiting reactant, we need to compare the molar ratio of arsenic to arsenic triiodide with the stoichiometric ratio. The balanced chemical equation for the reaction is: 2 As + 2 AsI₃ → As₂I₄ Stoichiometric ratio of As to AsI₃ = 1 : 1 Molar ratio of As to AsI₃ = 2.061 mol : 1.678 mol = 1 : 0.815 Since the molar ratio of As to AsI₃ is greater than the stoichiometric ratio, arsenic triiodide is the limiting reactant.

Calculate the theoretical mass of As₂I₄ that can be prepared

As AsI₃ is the limiting reactant, we can use the stoichiometry of the reaction to find the theoretical amount of As₂I₄ that can be produced: 1.678 mol AsI₃ × (1 mol As₂I₄ / 2 mol AsI₃) = 0.839 mol As₂I₄ Now we can convert the moles of As₂I₄ to mass. The molar mass of As₂I₄ = 2 × 74.92 g/mol (As) + 4 × 126.90 g/mol (I) = 799.44 g/mol Mass of As₂I₄ = 0.839 mol × 799.44 g/mol = 670.20 g (theoretical yield)

Calculate the actual mass of As₂I₄ isolated

The percent yield of As₂I₄ is given as 75.6%. We can use this information to find the actual mass of As₂I₄ that was isolated: Actual mass of As₂I₄ = (75.6 / 100) × theoretical mass of As₂I₄ = 0.756 × 670.20 g = 506.51 g So, the actual mass of As₂I₄ isolated is 506.51 g.

Key concepts

Limiting Reactant

In any chemical reaction, the limiting reactant is the substance that determines the amount of product formed. It is the reactant that is completely consumed during the reaction, preventing any further product from being generated. This concept is crucial because it directly affects the theoretical yield of the reaction.

To find the limiting reactant, you first need to know the quantities of each reactant involved. By comparing the molar ratio of the reactants in the balanced chemical equation with the actual molar quantities available, you can identify which reactant will run out first. In our example, arsenic triiodide (\(\mathrm{AsI}_3\)) is the limiting reactant, as its molar ratio is less than that calculated from arsenic (\(2.061\) mol of As and \(1.678\) mol of AsI\(_3\)).

• Determine the moles of each reactant.
• Use the balanced equation to find the stoichiometric ratio.
• Identify which reactant provides fewer moles of product formation (this is the limiting reactant).

Theoretical Yield

The theoretical yield is the maximum amount of product that can be formed in a reaction, assuming all of the limiting reactant is converted to product. It is calculated based on the limiting reactant using stoichiometry and the balanced chemical equation.

In our scenario, the theoretical yield of \(\mathrm{As}_{2}\mathrm{I}_{4}\) is calculated using the moles of limiting reactant, arsenic triiodide. The reaction consumes \(1.678\) mol of \(\mathrm{AsI}_3\), leading to the formation of \(0.839\) mol of \(\mathrm{As}_{2}\mathrm{I}_{4}\) given the stoichiometric equation.

• Identify the moles of the limiting reactant.
• Use stoichiometric calculations to determine moles of product.
• Convert moles of product to mass using molar mass to find theoretical yield.

Percent Yield

Percent yield is a comparison of the actual yield (the amount of product actually obtained from a reaction) to the theoretical yield (the maximum possible amount of product predicted by stoichiometry). It offers insight into the efficiency of a reaction:

\[\text{Percent Yield} = \left( \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \right) \times 100\]

In this example, your reaction had a percent yield of 75.6%. This means that 75.6% of the theoretical maximum amount of \(\mathrm{As}_{2}\mathrm{I}_{4}\) was actually collected, leading to an actual yield of \(506.51\) g.

• Calculate theoretical yield from the limiting reactant.
• Measure actual yield from the experiment.
• Calculate percent yield to evaluate reaction efficiency.

Density Calculation

Density calculations are foundational in chemistry for determining the mass or volume of a substance. Density is defined as mass per unit volume and is usually expressed in \(\text{g/cm}^3\) or equivalent units.

The density formula is: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In the exercise, you need to calculate the mass of solid arsenic using its density, \(5.72\, \text{g/cm}^3\), and its volume calculated from its cubic dimensions \(3.00\, \text{cm}\) on each side, where the volume of the cube is \(27\, \text{cm}^3\). This gives a mass of arsenic of \(154.44\, \text{g}\).

• Determine the volume of the object (especially for solids).
• Use the density to compute the mass needed for reaction calculations.