Problem 81 - Chemistry: The Central Science

Question

Arsenic(III) sulfide sublimes readily, even below its melting point of \(320^{\circ} \mathrm{C}\). The molecules of the vapor phase are found to effuse through a tiny hole at \(0.28\) times the rate of effusion of Ar atoms under the same conditions of temperature and pressure. What is the molecular formula of arsenic(III) sulfide in the gas phase?

Step-by-Step Solution

Answer

The molecular formula of arsenic(III) sulfide in the gas phase is \(As_{4}S_{6}\).

Step 1: Write down Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass: \[ \frac{Rate_{1}}{Rate_{2}} = \sqrt{\frac{MolarMass_{2}}{MolarMass_{1}}} \] In this case, \(Rate_{1}\) is the rate of effusion of arsenic(III) sulfide and \(Rate_{2}\) is the rate of effusion of argon.

Step 2: Insert the given information into the equation

We are given that arsenic(III) sulfide effuses at 0.28 times the rate of argon, so we can write: \[ \frac{0.28 \times Rate_{Ar}}{Rate_{Ar}} = \sqrt{\frac{MolarMass_{Ar}}{MolarMass_{As_{x}S_{y}}}} \] Here, \(MolarMass_{Ar}\) is the molar mass of argon, which is approximately 40 g/mol, and \(MolarMass_{As_{x}S_{y}}\) is the molar mass of arsenic(III) sulfide in the gas phase, which we want to find.

Step 3: Simplify and solve for the molar mass of arsenic(III) sulfide

We can simplify the equation and solve for the molar mass of arsenic(III) sulfide: \[ 0.28 = \sqrt{\frac{40}{MolarMass_{As_{x}S_{y}}}} \] Square both sides: \[ 0.28^2 = \frac{40}{MolarMass_{As_{x}S_{y}}} \] Now, solve for \(MolarMass_{As_{x}S_{y}}\): \[ MolarMass_{As_{x}S_{y}} = \frac{40}{0.28^2} \approx 510 \, g/mol \]

Step 4: Determine the molecular formula of arsenic(III) sulfide

Now that we have the molar mass of arsenic(III) sulfide, we need to find the molecular formula. The molar mass of arsenic (As) is approximately 75 g/mol, and the molar mass of sulfur (S) is approximately 32 g/mol. Let \(x\) be the number of arsenic atoms, and \(y\) be the number of sulfur atoms in the molecular formula. We have: \[ 75x + 32y = 510 \] Based on the problem, the chemical formula should be \(As_{x}S_{y}\), where \(x\) is the number of arsenic atoms and \(y\) is the number of sulfur atoms. Start by guessing the stoichiometry of the compound: For example, if there are 2 As atoms and 3 S atoms, the formula would be \(As_{2}S_{3}\), and the molar mass would be: \[ 2 \times 75 + 3 \times 32 = 246 \] So, we can try higher stoichiometry: For 4 As atoms and 6 S atoms, the formula would be \(As_{4}S_{6}\), and the molar mass would be: \[ 4 \times 75 + 6 \times 32 = 492 \] Now, try 6 As atoms and 9 S atoms: \[ 6 \times 75 + 9 \times 32 = 738 \] This result is higher than our calculated molar mass, so we can conclude that the molecular formula of arsenic(III) sulfide in the gas phase is \(As_{4}S_{6}\).

Key Concepts

Arsenic(III) Sulfide

Arsenic(III) sulfide is a compound of arsenic and sulfur that is interesting due to its ability to sublime, which means it transitions from a solid to a gaseous state without passing through a liquid phase. This property is observed even below its melting point of 320°C. The compound exists in different forms, but one of the most notable is pararealgar (As₂S₂) commonly found in nature. However, the focus here is on its phase behavior in the gaseous state.

• This compound is generally used in materials science and electronics, particularly in producing semiconductors.
• Sublimation allows it to be used in applications where a solid-to-gas transition without melting is beneficial.
• In the vapor phase, the number of atoms in its molecular formula is derived from experimental data.

The investigation into its molecular form in gas-phase can involve experiments like effusion rate comparisons, as seen in this problem. This characteristic experiment can reveal the compound's nature without explicitly isolating it in its gaseous state.

Molecular Formula Determination

Understanding how to determine the molecular formula of a compound in the gaseous state involves analyzing its composition and mass. Through Graham's Law of Effusion, we are able to determine the molar mass of arsenic(III) sulfide in its vapor phase by comparing its effusion rate to a known gas, argon. This law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
The computed molar mass from the exercise was approximately 510 g/mol. Given that we know the atomic masses of arsenic and sulfur, we start trying different ratios that best fit this computed molar mass.

Methodology:

• Considering combinations of arsenic and sulfur atoms allows you to determine the likely stoichiometry of the compound.
• By substituting values into the equation derived from Graham's Law, you can pin down the right formula.
• The correct stoichiometry is the one that results in a calculated molar mass close to the experimental molar mass.

In this case, when computations showed masses close to 510 g/mol, it suggested a formula of As₄S₆. This iterative guessing process helps in accurately identifying the molecular formula.

Molar Mass Calculation

Calculating the molar mass of a compound like arsenic(III) sulfide involves determining the total mass of one mole of the compound based on its constituent atoms. To achieve this:

• First, use Graham's Law of Effusion to determine the molar mass experimentally.
• Compare the effusion rate of the compound to a reference gas like argon to find a critical ratio.
• Solve the equation to find the molar mass by utilizing the square relationship given by the effusion law.

For arsenic(III) sulfide, given the vapor phase comparisons to argon, the computed molar mass was around 510 g/mol.

Step-by-step:

• Measure how quickly arsenic(III) sulfide effuses compared to argon, using the given 0.28 factor.
• Use this to set up the ratio \[\frac{Rate_{As_{x}S_{y}}}{Rate_{Ar}} = rac{0.28}{1}\]
• Solve for the unknown molar mass. Remember, rate inversely ties to the square root of molar mass.
• The calculations ultimately revealed a molar mass that suggested searching for a formula close to that found for As₄S₆.

Through these calculations, a robust understanding of how to relate effusion rates to molar mass opens avenues for determining complex molecular structures in the gaseous state.