Problem 77 - Chemistry: Principles and Reactions

Question

Consider three sealed tanks all at the same temperature, pressure, and volume Tank A contains \(\mathrm{SO}_{2}\) gas. Tank B contains \(\mathrm{O}_{2}\) gas. Tank C contains \(\mathrm{CH}_{4}\) gas.

Step-by-Step Solution

Answer

Question: Compare the properties of SO2, O2, and CH4 gases in three tanks A, B, and C, all at the same temperature, pressure, and volume. Calculate the total number of moles of gas in each tank, the average mass of the gas particles in each tank, and the density of each gas in the tanks. Answer: All three tanks have the same number of moles (n) since their temperature, pressure, and volume are the same. However, the average mass of gas particles and the density of the gases differ due to their different molar masses. The molar masses are 64 g/mol for SO2, 32 g/mol for O2, and 16 g/mol for CH4. The average mass of particles and density can be calculated for each gas using these molar masses and the volume of the tanks.

Step 1: Calculating the number of moles in each tank

Since all the tanks share the same temperature, pressure, and volume, we can use the ideal gas law formula to find the number of moles in the tanks: PV = nRT Here, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (8.314 J/(mol·K)), T is temperature (in Kelvin). The ideal gas equation can be rewritten as: n = PV/RT Since P, V, and T are the same for all tanks, the number of moles (n) in each tank will also be the same.

Step 2: Calculate the average mass of gas particles in each tank

The molar mass of each gas is as follows: SO2: 1 S atom + 2 O atoms = 32 + 2*16 = 64 g/mol O2: 2 O atoms = 2*16 = 32 g/mol CH4: 1 C atom + 4 H atoms = 12 + 4*1 = 16 g/mol The average mass of particles in each tank can be calculated by dividing the molar mass by Avogadro's number (6.022 x 10^23 particles/mol).

Step 3: Calculate the density of each gas in the tanks

The density of a gas can be found using the formula: Density = (Mass of gas) / Volume We can rewrite the formula in terms of the number of moles, n, using the definition of molar mass: Density = (n x Molar mass) / V Here, V is the volume of the tank shared among all tanks. Calculate the density for each gas using their respective molar masses from Step 2 and the number of moles from Step 1 (which are equal in each tank).

Step 4: Comparing the properties of the gases in the tanks

Compare the average mass of the gas particles and the density of the gases in each tank. You'll notice that although the number of moles is the same in all three tanks, the average mass of the particles, and hence the density, will differ due to each type of gas having different molar masses.

Key Concepts

Molar Mass

Molar mass is a significant concept in chemistry, as it represents the mass of one mole of a substance. This is often measured in grams per mole (g/mol). To determine this, you simply add up the atomic masses of all the atoms present in a molecule. For example, sulfur dioxide (\( \text{SO}_2 \)) has a molar mass of 64 g/mol because it consists of one sulfur atom (32 g/mol) and two oxygen atoms (each 16 g/mol). Similarly, oxygen gas (\( \text{O}_2 \)) has a molar mass of 32 g/mol, and methane (\( \text{CH}_4 \)) has one carbon atom (12 g/mol) plus four hydrogen atoms (each 1 g/mol), totaling 16 g/mol.
Understanding molar mass helps in converting between the mass of a substance and the moles of the substance, allowing for easier calculations in chemistry. This can be extremely useful when using the ideal gas law, as it links the mass of the gas with other property measurements like pressure and volume.

Gas Density

Gas density is an important property when comparing different gases. It tells us how much mass is contained in a given volume. The formula for calculating gas density is \( \text{Density} = \frac{\text{Mass of gas}}{\text{Volume}} \).
However, in the context of gases, density can also be expressed using molar mass and the ideal gas law. For a given gas, the density \( \rho \) can be calculated as \( \rho = \frac{n \times \text{Molar Mass}}{V} \), where \( n \) is the number of moles and \( V \) is the volume. Since the molar mass varies between substances, different gases will have different densities even under the same conditions of pressure and temperature.
This property is crucial when studying gases with the same number of moles at identical conditions, as it can help explain differences in behavior due to varying molecular structures. For example, in our scenario, despite having the same number of moles, each tank will have a different density due to the gases' distinct molar masses.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, approximately \( 6.022 \times 10^{23} \), and it represents the number of particles, usually atoms or molecules, in one mole of any substance. This constant is an essential bridge between the molecular scale and real-world measurements.
For calculating the average mass of gas particles in a mole, you would divide the molar mass of a gas by Avogadro's number. For instance, with \( \text{SO}_2 \), which has a molar mass of 64 g/mol, the average mass of a single molecule would be approximately \( \frac{64}{6.022 \times 10^{23}} \) grams.
Understanding Avogadro's number is vital for determining quantities in chemical reactions, allowing scientists to predict outcomes and balances in reactions. It provides insight into the molecular structure and behavior of substances on the macroscopic level.