Problem 93 - Chemistry The Science in Context

Question

A 150.0 mL flask contains \(0.391 \mathrm{g}\) of a volatile oxide of sulfur. The pressure in the flask is \(750 \mathrm{mm} \mathrm{Hg}\), and the temperature is \(22^{\circ} \mathrm{C} .\) Is the gas \(\mathrm{SO}_{2}\) or \(\mathrm{SO}_{3} ?\)

Step-by-Step Solution

Answer

Answer: To identify the gas as either SO₂ or SO₃, follow the steps to calculate its molar mass using the Ideal Gas Law. After calculating the molar mass, compare it to the molar masses of SO₂ (approximately 64 g/mol) and SO₃ (approximately 80 g/mol). The gas is more likely to be the one with a closer match to the calculated molar mass.

Step 1: Convert the given pressure to atmospheres

Since the pressure is given in mm Hg, we need to convert it to atmospheres (atm) for use in the Ideal Gas Law equation. The conversion factor is 1 atm = 760 mm Hg: $$ P_{\mathrm{atm}} = \frac{750\,\mathrm{mm}\,\mathrm{Hg}}{760\,\mathrm{mm}\,\mathrm{Hg}/\mathrm{atm}}$$

Step 2: Convert the given volume to liters

The volume should be in liters for the Ideal Gas Law equation. The conversion factor is 1 L = 1000 mL: $$ V_{\mathrm{L}} = \frac{150.0\,\mathrm{mL}}{1000\,\mathrm{mL}/\mathrm{L}}$$

Step 3: Convert the given temperature to Kelvins

To use the Ideal Gas Law, the temperature must be in Kelvins. Add 273.15 to the given temperature in Celsius to convert to Kelvin: $$ T_{\mathrm{K}} = 22^{\circ} \mathrm{C} + 273.15 $$

Step 4: Apply the Ideal Gas Law

Use the Ideal Gas Law equation, which relates pressure, volume, temperature, and the number of moles of a gas: $$ PV = nRT $$ Where P is pressure in atm, V is volume in liters, n is number of moles, R is the universal gas constant (\(0.0821\, \mathrm{L} \cdot\mathrm{atm} / \mathrm{mol} \cdot \mathrm{K}\)), and T is temperature in Kelvins. We can solve for n: $$ n = \frac{PV}{RT} $$ Plug in the values obtained in Steps 1, 2, and 3 for P, V, and T.

Step 5: Calculate the molar mass of the gas

To find the molar mass, divide the mass of the gas in grams (0.391 g) by the number of moles calculated in Step 4: $$ Molar\, Mass = \frac{0.391\,\mathrm{g}}{n}$$

Step 6: Identify the gas

Compare the calculated molar mass to the molar masses of \(\mathrm{SO}_{2}\) and \(\mathrm{SO}_{3}\). The molar mass of \(\mathrm{SO}_{2}\) is approximately 64 g/mol, while the molar mass of \(\mathrm{SO}_{3}\) is approximately 80 g/mol. Determine whether the calculated molar mass is closer to that of \(\mathrm{SO}_{2}\) or \(\mathrm{SO}_{3}\), and identify the gas accordingly.

Key Concepts

Pressure Conversion

Pressure conversion is essential when dealing with the Ideal Gas Law, especially if the pressure is not given in atmospheres (atm). In this exercise, the pressure was initially provided in millimeters of mercury (mm Hg). To convert this to atmospheres, which is needed for the Ideal Gas Law, use the conversion factor:

• 1 atm = 760 mm Hg

To convert 750 mm Hg to atmospheres, divide it by 760:\[P_{\mathrm{atm}} = \frac{750\,\mathrm{mm}\,\mathrm{Hg}}{760\,\mathrm{mm}\,\mathrm{Hg}/\mathrm{atm}} \approx 0.987 \mathrm{atm}\]This conversion allows the pressure data to be used effectively within the Ideal Gas Law calculation, ensuring compatibility with the universal gas constant \( R\) when it is expressed as 0.0821 L·atm/mol·K.Having pressure in atm is crucial for accurate calculations within gas law equations.

Temperature Conversion

Temperature conversion is necessary because the Ideal Gas Law requires temperature in Kelvin for its calculations. The original temperature in this problem was provided in degrees Celsius. To convert from Celsius to Kelvin, which is the absolute temperature scale needed:

• Add 273.15 to the Celsius temperature.

Using this conversion formula:\[T_{\mathrm{K}} = 22^{\circ} \mathrm{C} + 273.15 = 295.15 \mathrm{K}\]Converting temperature to Kelvin ensures that the calculations in the Ideal Gas Law remain accurate because the Kelvin scale starts at absolute zero, aligning perfectly with molecular kinetic energy values. This allows consistent use of temperature values across various physical calculations.

Molar Mass Calculation

Calculating the molar mass of the gas helps identify what the gas actually is. In this scenario, the gas mass is given in grams and the number of moles is determined using the Ideal Gas Law. The formula for molar mass is:

• Molar Mass \( M \) = Mass of gas (g) / Number of moles \( n \)

After using the Ideal Gas Law:\[PV = nRT \quad \Rightarrow \quad n = \frac{PV}{RT}\]Substitute the values for pressure \( P \), volume \( V \), temperature \( T \), and the gas constant \( R \) to find \( n \):Once \( n \) is found, divide the given mass of 0.391 g by \( n \) to calculate the molar mass.This calculated molar mass is then compared to known values to identify the gas.

Chemical Identification

Chemical identification in this context involves using calculated molar mass to decide which chemical compound the gas is most likely to be. Once the molar mass is calculated, it is compared to known molar masses of potential compounds. In the exercise, the options were sulfur dioxide \( \mathrm{SO}_{2} \) and sulfur trioxide \( \mathrm{SO}_{3} \).

• The molar mass of \( \mathrm{SO}_{2} \) is approximately 64 g/mol.
• The molar mass of \( \mathrm{SO}_{3} \) is approximately 80 g/mol.

By comparing these molar masses to the calculated value from the exercise, you can determine which compound the gas is. If the molar mass is closer to 64 g/mol, the gas is \( \mathrm{SO}_{2} \), and if closer to 80 g/mol, the gas is \( \mathrm{SO}_{3} \).This method provides a clear approach to identifying unknown gases through physical measurements and calculations with the Ideal Gas Law.