Problem 87 - Chemistry The Central Science In Si Units

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 0.52 times the rate of effusion of Xe 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 \( \text{As}_4\text{S}_6 \).

Step 1: Understand the Problem

We are given that arsenic(III) sulfide sublimes and its vapor effusion rate is 0.52 times that of xenon (Xe). We need to find the molecular formula of arsenic(III) sulfide in the gas phase.

Step 2: Apply 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: \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \). Here, \( r_1 \) and \( r_2 \) are the rates of effusion, and \( M_1 \) and \( M_2 \) are the respective molar masses of Xe and arsenic(III) sulfide vapor.

Step 3: Set Up the Effusion Rate Ratio

We know that the rate of effusion of arsenic(III) sulfide is 0.52 times that of Xe, so \( \frac{r_{\text{As}_x \text{S}_y}}{r_{\text{Xe}}} = 0.52 \). Let \( M_{\text{As}_x \text{S}_y} \) be the molar mass of arsenic(III) sulfide and \( M_{\text{Xe}} \), the molar mass of xenon (131.29 g/mol).

Step 4: Substitute and Solve the Equation

Using \( \frac{r_{\text{As}_x \text{S}_y}}{r_{\text{Xe}}} \), the equation from Graham's Law becomes \( 0.52 = \sqrt{\frac{M_{\text{Xe}}}{M_{\text{As}_x \text{S}_y}}} \). Squaring both sides gives \( 0.2704 = \frac{M_{\text{Xe}}}{M_{\text{As}_x \text{S}_y}} \), so \( M_{\text{As}_x \text{S}_y} = \frac{M_{\text{Xe}}}{0.2704} \).

Step 5: Calculate Molar Mass of Arsenic(III) Sulfide

Calculate the molar mass: \( M_{\text{As}_x \text{S}_y} = \frac{131.29}{0.2704} = 485.5 \text{ g/mol} \).

Step 6: Determine the Molecular Formula

The molar mass of arsenic is approximately 74.92 g/mol, and sulfur is approximately 32.06 g/mol. Assume \( x \) arsenic atoms and \( y \) sulfur atoms: \( 74.92x + 32.06y = 485.5 \). Checking combinations, an approximate match is reached with \( x = 4 \) and \( y = 6 \), which matches the total molar mass.

Key Concepts

Effusion

Effusion is the process by which gas molecules escape through a tiny opening into a vacuum or a less pressurized environment. It's a fascinating phenomenon that plays a key role in understanding how gases behave. In simple terms, effusion rates are influenced by the size of the gas molecules and their speeds. According to Graham's Law of Effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. So, lighter gases effuse faster than heavier ones.

Graham's Law can be mathematically expressed as: \[\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\]Here, \( r_1 \) and \( r_2 \) are the rates of effusion, while \( M_1 \) and \( M_2 \) represent the molar masses of the two gases being compared. Applying this concept helps us determine unknown molecular masses when we know how two gases effuse relative to each other.

Molecular formula determination

Determining the molecular formula of a compound often involves some detective work. It's like piecing together a puzzle by finding how atoms fit into a specific combination based on their mass and the data we have.In our scenario, we had to find the molecular formula of arsenic(III) sulfide in the gaseous state, knowing that its effusion rate is 0.52 times that of xenon (Xe) under similar conditions. Using Graham's Law, we were able to calculate the molar mass of the gaseous arsenic(III) sulfide.The molar mass we obtained was 485.5 g/mol. With the approximate atomic masses of arsenic (74.92 g/mol) and sulfur (32.06 g/mol), the task became to find integer values for \( x \) and \( y \), such that the equation \( 74.92x + 32.06y = 485.5 \) holds true. Through a trial-and-error approach, or simply guessing and checking logical combinations, we discovered that 4 arsenic atoms and 6 sulfur atoms meet this requirement. Thus, the molecular formula was deduced to be As₄S₆.

Arsenic(III) sulfide

Arsenic(III) sulfide, also known as orpiment, is a fascinating compound. Its chemical formula in solid form is typically As₄S₄, but it can exhibit different structures in vapor. Arsenic(III) sulfide is characterized by its striking yellow hue and has been historically used in art and manufacturing. One unique property of arsenic(III) sulfide is its ability to sublime below its melting point of 320°C. Sublimation is a phase transition where a substance goes from solid to gas without passing through a liquid phase, often seen in substances like dry ice. This compound is particularly interesting for study because of its ability to exist in multiple forms, both in consistency and molecular arrangement. Understanding how it behaves in different phases enhances our comprehension of molecular behavior and effusion dynamics.