Question

A sample of 3.00 \(\mathrm{g}\) of \(\mathrm{SO}_{2}(g)\) originally in a 5.00 -L vessel at \(21^{\circ} \mathrm{C}\) is transferred to a \(10.0-\mathrm{L}\) vessel at \(26^{\circ} \mathrm{C} .\) A sample of 2.35 \(\mathrm{g}\) of \(\mathrm{N}_{2}(g)\) originally in a \(2.50-\mathrm{L}\) vessel at \(20^{\circ} \mathrm{C}\) is transferred to this same 10.0 -L vessel. (a) What is the partial pressure of \(S O_{2}(g)\) in the larger container? (b) What is the partial pressure of \(N_{2}(g)\) in this vessel? (c) What is the total pressure in the vessel?

Short answer

(a) The partial pressure of SO2(g) in the larger container is 0.363 atm. (b) The partial pressure of N2(g) in the larger container is 0.492 atm. (c) The total pressure in the larger container is 0.855 atm.

Step-by-step solution

1. Find the number of moles

Since we are given the masses of SO2(g) and N2(g), we can start by calculating the number of moles using their respective molar masses. For SO2(g), Molar mass (SO2) = 32.1 g/mol (Sulfur) + 2 * 16.0 g/mol (Oxygen) = 64.1 g/mol For N2(g), Molar mass (N2) = 2 * 14.0 g/mol (Nitrogen) = 28.0 g/mol Now, we can find the number of moles (n) using the given masses: For SO2(g), n(SO2) = mass(SO2) / molar mass(SO2) = 3.00 g / 64.1 g/mol = 0.0468 mol For N2(g), n(N2) = mass(N2) / molar mass(N2) = 2.35 g / 28.0 g/mol = 0.0839 mol

2. Convert temperatures to Kelvin

To use the Ideal Gas Law, we need to convert the given temperatures in Celsius to Kelvin: Temperature (SO2) = 21°C + 273.15 = 294.15 K Temperature (N2) = 20°C + 273.15 = 293.15 K

3. Use the Ideal Gas Law to find initial pressures

Now, we can use the Ideal Gas Law to find the initial pressures before transferring the gases to the new container: For SO2(g), PV = nRT => P = nRT / V P(SO2) = (0.0468 mol)(0.0821 L atm/mol K)(294.15 K) / 5.00 L = 0.729 atm For N2(g), P(N2) = (0.0839 mol)(0.0821 L atm/mol K)(293.15 K) / 2.50 L = 0.979 atm

4. Calculate partial pressures in the larger container

Now that we have the initial pressures for both gases, we can find the partial pressures in the larger 10.0 L container using the same Ideal Gas Law: For SO2(g), P_new(SO2) = n(SO2)(R)(new temperature) / new volume P_new(SO2) = (0.0468 mol)(0.0821 L atm/mol K)(299.15 K) / 10.0 L = 0.363 atm For N₂(g), P_new(N2) = n(N2)(R)(new temperature) / new volume P_new(N2) = (0.0839 mol)(0.0821 L atm/mol K)(299.15 K) / 10.0 L = 0.492 atm

5. Calculate the total pressure in the vessel

Now that we have the partial pressures of both gases, we can calculate the total pressure in the larger container: Total Pressure = P_new(SO2) + P_new(N2) = 0.363 atm + 0.492 atm = 0.855 atm The answers are: (a) The partial pressure of SO2(g) in the larger container is 0.363 atm (b) The partial pressure of N2(g) in the larger container is 0.492 atm (c) The total pressure in the larger container is 0.855 atm.

Key concepts

Partial Pressure

Understanding partial pressure is foundational when studying the behavior of gases, particularly in mixtures like air or chemical reactions where multiple gases are present. Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. It's essential to realize that in a mixture, each gas component acts independently and contributes to the overall pressure of the system as if it were alone in the container. This concept is highlighted in Dalton's Law of Partial Pressures, which states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases.

To calculate partial pressure, one can use the Ideal Gas Law, which is expressed as PV=nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature in Kelvin. When a gas is transferred to a new container with a different volume and temperature, its partial pressure changes and can be computed with the Ideal Gas Law formula, keeping in mind the conditions of the new environment. For instance, in the exercise above, the partial pressure of each gas in the new container was separately calculated, ultimately contributing to the total pressure.

Molar Mass Calculation

Molar mass is a crucial element in chemical calculations, allowing us to convert between grams and moles of a substance. The molar mass is the weight of one mole of a substance and is usually expressed in grams per mole (g/mol). It's determined by adding up the atomic masses of all the atoms in a molecule, based on the periodic table values.

For example, sulfur dioxide (SO₂) has a molar mass calculation which involves the atomic mass of sulfur (32.1 g/mol) and oxygen (16.0 g/mol), leading to a molar mass of SO₂ being 64.1 g/mol. Knowing the molar mass allows us to relate the mass of a substance to the number of moles, which is a pivotal step in using the Ideal Gas Law for gas-related calculations. It's a fundamental concept that students need to understand and accurately apply to interpret gas properties and behaviors correctly.

Gas Temperature Conversion

Gas temperature conversion from Celsius to Kelvin is a critical step in applying the Ideal Gas Law accurately. The Ideal Gas Law necessitates the use of absolute temperature for calculations, which is why temperatures must always be in Kelvin (K). The Kelvin scale is an absolute temperature scale that starts at absolute zero, the theoretically lowest possible temperature where particles are at rest.

To convert from Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature. This ensures that all temperature values are positive on the Kelvin scale, aligning with the concept of absolute temperature. For instance, a temperature of 21°C is equivalent to 294.15 K. Having the temperature in the correct unit is a pivotal step for calculating pressures, as seen in the provided exercise, ensuring accuracy in all gas-related computations.