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?

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, it can be expressed as: \( \frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}} \) Here, Rate1 and Rate2 are the rates of effusion for gas1 and gas2, respectively. M1 and M2 are their respective molar masses.

We are given that the rate of effusion of arsenic (III) sulfide (AS) to that of the argon (Ar) is 0.28 (or 28%). The molar mass of argon gas is 39.95 g/mol. We can rearrange the formula to find the molar mass of arsenic (III) sulfide. \( \frac{Rate_{AS}}{Rate_{Ar}} = 0.28 = \sqrt{\frac{M_{Ar}}{M_{AS}}} \)

Now, we can calculate the molar mass of arsenic (III) sulfide by squaring both sides of the equation and rearranging it: \( M_{AS} = \frac{M_{Ar}}{0.28^2} \) \( M_{AS} = \frac{39.95\,g/mol}{0.0784} \) \( M_{AS} ≈ 509.49\,g/mol \)

Arsenic has a molar mass of approximately 74.92 g/mol and sulfur has a molar mass of approximately 32.07 g/mol. Let's assume the molecular formula to be AsxSy, where x and y are the numbers of arsenic and sulfur atoms, respectively. We can express the molar mass of arsenic (III) sulfide as: \( 509.49 \approx 74.92x + 32.07y \) Since this is arsenic (III) sulfide, the oxidation state of arsenic is +3 and each sulfur atom has an oxidation state of -2. This means that for each arsenic atom, there will be 3/2 sulfur atoms in the formula. Therefore, we get: \( y = \frac{3}{2} x \) Now we can substitute this expression into the molar mass equation: \( 509.49 \approx 74.92x + 32.07\left(\frac{3}{2}x\right) \) Solving for x, we find: \( x \approx 2 \) Since y = 3/2x: \( y \approx 3 \)

Based on our calculations, the molecular formula of arsenic (III) sulfide in the gas phase is: As2S3