Problem 85 - Chemistry: An Atoms-Focused Approach

Question

A 30.0 mL flask contains 0.078 g of a volatile oxide of sulfur. The pressure in the flask is \(750 \mathrm{mmHg}\), and the temperature is \(22^{\circ} \mathrm{C}\) a. What is the density of the gas? b. Is the gas \(\mathrm{SO}_{2}\) or \(\mathrm{SO}_{3} ?\)

Step-by-Step Solution

Answer

The density of the gas is approximately 2.60 g/L, and it is most likely SO2.

Step 1: Find the number of moles of the gas

First, convert the given pressure, volume, and temperature to the appropriate units and then use the ideal gas law, \(PV=nRT\), to find the number of moles: - Convert pressure from mmHg to atm: \(\frac{750 \mathrm{mmHg}}{760\,\mathrm{mmHg/atm}} = 0.9868\, \mathrm{atm}\) - Convert volume from mL to L: \(\frac{30.0\, \mathrm{mL}}{1000\, \mathrm{mL/L}} = 0.03\, \mathrm{L}\) - Convert temperature from Celsius to Kelvin: \(22^{\circ} \mathrm{C}+ 273.15 = 295.15\, \mathrm{K}\) Now, use the ideal gas law to solve for the number of moles, n: \[n = \frac{PV}{RT} = \frac{(0.9868\, \mathrm{atm})(0.03\,\mathrm{L})}{(0.0821\,\mathrm{L\, atm/mol\, K})(295.15\, \mathrm{K})} \approx 0.0012\, \mathrm{mol}\]

Step 2: Calculate the molar mass of the gas

To find the molar mass, divide the mass of the gas by the number of moles: \[M = \frac{0.078\, \mathrm{g}}{0.0012\, \mathrm{mol}} \approx 65\, \mathrm{g/mol}\]

Step 3: Determine the density of the gas

We can use the mass, volume and the molar mass to determine the density of the gas: \[\rho = \frac{0.078\, \mathrm{g}}{0.03\,\mathrm{L}} \approx 2.60\, \mathrm{g/L}\] The density of the gas is approximately 2.60 g/L.

Step 4: Identify the gas as \(\mathrm{SO}_{2}\) or \(\mathrm{SO}_{3}\)

Compare the molar mass calculated in step 2 to the molar masses of \(\mathrm{SO}_{2}\) and \(\mathrm{SO}_{3}\): - Molar mass of \(\mathrm{S} = 32.07\, \mathrm{g/mol}\) - Molar mass of \(\mathrm{O} = 16.00\, \mathrm{g/mol}\) - Molar mass of \(\mathrm{SO}_{2} = 32.07 + (2 \times 16.00) = 64.07\, \mathrm{g/mol}\) - Molar mass of \(\mathrm{SO}_{3} = 32.07 + (3 \times 16.00) = 80.07\, \mathrm{g/mol}\) The molar mass found in step 2 is approximately 65 g/mol, which is closer to the molar mass of \(\mathrm{SO}_{2}\). Therefore, the gas is most likely \(\mathrm{SO}_{2}.\)

Key Concepts

Gas Density

Gas density is simply the mass of gas per unit volume. It helps us to understand how much of the gas is present in a given volume. In other words, it is a measure of how compactly the molecules of a gas are packed together in a certain space.
To calculate the density using the given exercise data, we use the formula: \[\rho = \frac{\text{mass}}{\text{volume}}\] where mass is 0.078 g and the volume is 0.03 L. Plugging in these values, we find that the density is approximately 2.60 g/L.
This results in a compact representation of the amount of gas in a specific volume, which provides insight into the behavior and properties of the gas in question. Knowing the density also helps in comparing the gas with other gases, including identifying specific properties like molar mass.

Molar Mass Calculation

Molar mass is a measure of the mass of one mole of a substance, given in g/mol. It's crucial for calculating various properties of gases, including density and identity. Given the exercise, we know the mass of the gas and need to find its molar mass.
Using the ideal gas law (\[PV = nRT\]), and having calculated the number of moles (n) to be approximately 0.0012 mol, the molar mass can be found by diving mass by the number of moles:\[M = \frac{\text{mass}}{\text{moles}} = \frac{0.078\, \mathrm{g}}{0.0012\, \mathrm{mol}} \approx 65\, \mathrm{g/mol}\]
This calculation is important as it precisely identifies the gas when compared to known molar masses of sulfur oxides. The molar mass of around 65 g/mol is a critical indicator of identifying the gas type, be it SO\(_2\) or SO\(_3\).

Sulfur Oxides

Sulfur oxides are chemical compounds composed of sulfur and oxygen. The most common types are sulfur dioxide (SO\(_2\)) and sulfur trioxide (SO\(_3\)). They have different molar masses, which help in identifying them when in a gaseous state.
For instance,

• Molar mass of sulfur dioxide (SO\(_2\)) is approximately 64.07 g/mol
• Molar mass of sulfur trioxide (SO\(_3\)) is approximately 80.07 g/mol

In our exercise, the calculated molar mass was closer to 65 g/mol, which suggests that the gas is likely SO\(_2\).
This simple yet accurate comparison highlights the importance of understanding molar mass calculations and density to identify chemical compounds in gaseous form. Knowing the specific sulfur oxide can help in many applications, from environmental monitoring to industrial processes.