Problem 70 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The partial pressure of SO2 in the larger container is 0.1144 atm, the partial pressure of N2 is 0.2049 atm, and the total pressure in the vessel is 0.3193 atm.

Step 1: Calculate the number of moles for SO2 and N2 before transferring.

First, we need to find the number of moles in each gas sample before transferring. We can use the molar mass of each gas to calculate the number of moles (n). The molar mass of SO2 is 32.07 g/mol (S) + 2 * 16.00 g/mol (O) = 64.07 g/mol, and the molar mass of N2 is 2 * 14.01 g/mol (N) = 28.02 g/mol n(SO2) = m(SO2) / M(SO2) = 3.00 g / 64.07 g/mol = 0.0468 mol n(N2) = m(N2) / M(N2) = 2.35 g / 28.02 g/mol = 0.0839 mol

Step 2: Calculate the initial pressures of SO2 and N2.

Next, we use the Ideal Gas Law to find the initial pressure of each gas sample. We need to convert the given temperatures to Kelvin before proceeding: T(SO2) = 21°C + 273.15 = 294.15 K T(N2) = 20°C + 273.15 = 293.15 K P(SO2) * V(SO2) = n(SO2) * R * T(SO2) P(SO2) = n(SO2) * R * T(SO2) / V(SO2) = (0.0468 mol) * (0.0821 L atm/mol K) * (294.15 K) / 5.00 L = 0.2307 atm P(N2) * V(N2) = n(N2) * R * T(N2) P(N2) = n(N2) * R * T(N2) / V(N2) = (0.0839 mol) * (0.0821 L atm/mol K) * (293.15 K) / 2.50 L = 0.8184 atm

Step 3: Calculate the final partial pressures of SO2 and N2.

Now, we can use the given information to calculate the partial pressures of SO2 and N2 in the new container. For SO2: V' = 10.0 L T' = 26°C + 273.15 = 299.15 K P'(SO2) = n(SO2) * R * T'(SO2) / V' = (0.0468 mol) * (0.0821 L atm/mol K) * (299.15 K) / 10.0 L = 0.1144 atm For N2: P'(N2) = n(N2) * R * T'(SO2) / V' = (0.0839 mol) * (0.0821 L atm/mol K) * (299.15 K) / 10.0 L = 0.2049 atm

Step 4: Calculate the total pressure in the new container.

Finally, to find the total pressure in the container, we simply add the partial pressures of SO2 and N2: P(total) = P'(SO2) + P'(N2) = 0.1144 atm + 0.2049 atm = 0.3193 atm Therefore, the partial pressure of SO2 is 0.1144 atm, the partial pressure of N2 is 0.2049 atm, and the total pressure in the vessel is 0.3193 atm.

Key Concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial for calculating the behavior of gases under various conditions. It is represented by the equation
\( PV = nRT \), where

• \( P \) denotes the pressure of the gas,
• \( V \) is the volume it occupies,
• \( n \) is the number of moles of gas,
• \( R \) is the universal gas constant, and
• \( T \) is the temperature in Kelvin.

In the given exercise, the Ideal Gas Law helps us calculate the initial pressure of sulfur dioxide (\( SO_2 \) gas) and nitrogen (\( N_2 \) gas) in their respective containers before being transferred into a larger vessel. This law assumes that the molecules in an ideal gas do not interact and that the size of these molecules is negligible compared to the space between them. Although no real gas perfectly fits the ideal gas model, the law provides a good approximation for gases at low pressure and high temperature. The Ideal Gas Law facilitates the understanding of how gases will react when subjected to changes in temperature, volume, and pressure, which is exactly what we see when the gases are moved to the new container at a different temperature.

Molar Mass

When performing calculations that involve chemical substances, it's essential to consider molar mass. Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). This property plays a key role in converting between the mass of a substance and the amount in moles—a common step in stoichiometric calculations.

The molar mass of a molecule, such as \( SO_2 \) or \( N_2 \) as seen in our exercise, is calculated by summing up the atomic masses of the individual atoms that comprise the molecule. For instance, sulfur dioxide's molar mass is found by adding the molar mass of one sulfur atom to that of two oxygen atoms. Knowing the molar mass allows us to calculate the number of moles present in a given sample of a substance using the formula \( n = \frac{m}{M} \), where

• \( n \) represents the number of moles,
• \( m \) is the mass of the sample, and
• \( M \) is the molar mass of the substance.

It's a fundamental step which was utilized in the exercise to convert the grams of \( SO_2 \) and \( N_2 \) into moles before applying the Ideal Gas Law.

Moles of Gas

A mole is a basic unit in chemistry, representing a specific quantity—usually of atoms or molecules—that ties the macroscopic world we observe to the atomic-level phenomena we cannot see. One mole corresponds to Avogadro's number of particles, which is approximately \( 6.022 \times 10^{23} \) entities.

In the context of gases, understanding moles is vital because gases expand to fill their containers, making it difficult to quantify them by volume alone—especially since their volumes can change with temperature and pressure. The concept of moles of gas allows us to reference a specific quantity of gas particles, no matter the volume they occupy.

To calculate the moles of a gas, we often start with its mass and molar mass, as done in the provided exercise. Once we have the moles, we can utilize the Ideal Gas Law to determine other properties such as pressure and temperature. The mole concept proves to be an indispensable tool for solving problems related to gas mixtures and reactions because it gives a count of the number of entities—a critical factor in chemical interactions and reactions.