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Chapter 10: Problem 91

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?

Short Answer

Expert verified
The molecular formula of arsenic(III) sulfide in the gas phase is As_2S_6, determined by applying Graham's Law of effusion and using the provided rate of effusion compared to Argon atoms. Using the calculated molar mass of arsenic(III) sulfide (approximately 510.2 g/mol), the molecular formula As_2S_6 yields the appropriate molar mass.

Step by step solution

01

Recall 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. Mathematically, this can be represented as: \( \frac{Rate_1}{Rate_2} = \sqrt{ \frac{Molar \, Mass_2}{Molar \, Mass_1 }} \) Where Rate_1 and Rate_2 are the rates of effusion of gases 1 and 2, and Molar Mass_1 and Molar Mass_2 are the molar masses of the gases, respectively.
02

Calculate the molar mass of Argon

We are given the rate of effusion of Argon relative to arsenic(III) sulfide, so we will need the molar mass of Argon (Ar). The molar mass of Argon is 39.95 g/mol.
03

Set up the formula with known values and solve for the molar mass of arsenic(III) sulfide

We know that the rate of effusion of arsenic(III) sulfide is 0.28 times that of the rate of effusion of Argon atoms. Using Graham's Law of effusion: \( \frac{Rate_{As_2S_3}}{Rate_{Ar}} = \sqrt{ \frac{Molar \, Mass_{Ar}}{Molar \, Mass_{As_2S_3}}} \) Plugging in the known values: \( 0.28 = \sqrt{ \frac{39.95 \, g/mol}{Molar \, Mass_{As_2S_3}}} \)
04

Solve for the molar mass of arsenic(III) sulfide

To solve for the molar mass of arsenic(III) sulfide, first square both sides of the equation: \( 0.28^2 = \frac{39.95 \, g/mol}{Molar \, Mass_{As_2S_3}} \) Next, solve for the molar mass of arsenic(III) sulfide in the gas phase: \( Molar \, Mass_{As_2S_3} = \frac{39.95 \, g/mol}{0.28^2} \) \( Molar \, Mass_{As_2S_3} ≈ 510.2 \, g/mol \)
05

Determine the molecular formula of arsenic(III) sulfide based on the molar mass

The molecular formula for arsenic(III) sulfide is As_nS_m, where n and m are integers. Using the periodic table, the molar mass of As is 74.92 g/mol, and the molar mass of S is 32.07 g/mol. We can express the total molar mass of the formula as: \( 510.2 \, g/mol ≈ 74.92n + 32.07m \) In this case, trying the simplest ratios (n = 1, m = 3) won't result in the desired molar mass; however, doubling the ratio results in: \( 510.2 \, g/mol ≈ 74.92(2) + 32.07(6) \) Thus, the molecular formula of arsenic(III) sulfide in the gas phase is As_2S_6.

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