Question

A mixture of gaseous oxygen and nitrogen is stored at atmospheric pressure in a \(3.7 \ell\) iron container maintained at constant temperature. After all the oxygen has reacted with the iron walls of the container to form solid iron oxide of negligible volume, the pressure is measured at 450 torr. Determine the final volume of nitrogen and the initial and final partial pressures of nitrogen and oxygen.

Short answer

The final volume of Nitrogen is given by: V_N2_final = ((P - P_O2_initial) * 3.7) / 450. The sum of the initial partial pressures of Nitrogen and Oxygen is equal to the atmospheric pressure: P_N2_initial + P_O2_initial = P. The final partial pressure of Nitrogen, which is the only gas remaining, is 450 torr. We cannot determine the exact initial partial pressures of Nitrogen and Oxygen due to insufficient information.

Step-by-step solution

1. Write down the given data

- Iron container volume (V) = 3.7 L - Pressure (P) = atmospheric pressure - Final pressure (P') = 450 torr - Temperature (T) = constant

2. Apply Dalton's Law of partial pressures

Dalton's Law states that the total pressure of a gas mixture is the sum of the partial pressures of its individual components: P = P1_initial + P2_initial = P_N2_initial + P_O2_initial After the reaction, only the pressure of Nitrogen is left: P' = P_N2_final

3. Apply the Ideal Gas Law to calculate Nitrogen's initial and final moles

Let's denote the initial and final moles of Nitrogen as n_N2_initial and n_N2_final, and the initial moles of Oxygen as n_O2_initial. Using the Ideal Gas Law, PV = nRT, for Nitrogen initially, we can say: P_N2_initial * V = n_N2_initial * R * T (1) For Nitrogen after the reaction: P_N2_final * V_N2_final = n_N2_final * R * T (2) Here, note that the moles of Nitrogen remain the same, as it does not react with iron walls. Therefore, n_N2_initial = n_N2_final.

4. Solve for the final volume of Nitrogen

Since the moles of Nitrogen remain constant, we can set Eq. (1) and Eq. (2) equal: P_N2_initial * V = P_N2_final * V_N2_final Now we know that P_N2_initial = P - P_O2_initial, and P_N2_final = P' = 450 torr. Plugging in these values: (P - P_O2_initial) * V = 450 * V_N2_final Now, solve for V_N2_final: V_N2_final = ((P - P_O2_initial) * V) / 450

5. Calculate the initial partial pressures of Nitrogen and Oxygen

Use the fact that P = P_N2_initial + P_O2_initial to solve for the initial partial pressures of Nitrogen and Oxygen. However, we have two unknowns (P_N2_initial and P_O2_initial) and one equation. In this case, we cannot determine the exact values of the initial partial pressures, as we were not provided the information to determine them uniquely. However, knowing the initial total pressure (P) and final pressure (P') can tell us that the sum of the initial partial pressures of Nitrogen and Oxygen is equal to the atmospheric pressure.

6. Calculate the final partial pressure of Nitrogen

Since the final pressure is equal to the partial pressure of Nitrogen after the reaction, P_N2_final is equal to the 450 torr provided: P_N2_final = 450 torr

7. Summary of results

- Final volume of Nitrogen: V_N2_final = ((P - P_O2_initial) * 3.7) / 450 - Initial partial pressures of Nitrogen and Oxygen: P_N2_initial + P_O2_initial = P (atmospheric pressure) - Final partial pressure of Nitrogen: P_N2_final = 450 torr We cannot determine the exact values of the initial partial pressures of Nitrogen and Oxygen, as we lack the information required to uniquely determine them. However, we have found expressions and relationships that involve them.

Key concepts

Partial Pressure

Partial pressure is a crucial concept in understanding gas mixtures. It refers to the pressure that a single gas component would exert if it occupied the entire volume of the mixture on its own, at the same temperature. Dalton's Law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. This is essential in analyzing gas behavior and calculating related variables in chemical reactions and industrial processes. For instance, when oxygen reacts with iron in our example, we look at the partial pressure of nitrogen to understand its behavior both before and after the reaction.

In the exercise, we started by identifying the given data and then applied Dalton's Law of partial pressures to set up equations for the initial state of the gas mixture. With the disappearance of oxygen after the reaction, we can infer the final partial pressure of nitrogen is what constitutes the total pressure in the container. Understanding this helps in subsequent calculations involving gas mixtures.

Ideal Gas Law

Integral to gas calculations is the Ideal Gas Law, represented by the equation PV = nRT, where P stands for pressure, V for volume, n for the number of moles of the gas, R is the ideal gas constant, and T represents temperature. This law combines several gas laws and provides a comprehensive equation to elucidate the behavior of an ideal gas under various conditions of pressure, volume, and temperature.

In our textbook exercise, we applied the Ideal Gas Law to calculate initial and final moles of nitrogen. Since the temperature and number of moles remain constant, and the volume of the container is unchanged, we used this relationship to accurately deduce the changes in pressure experienced by nitrogen. It's important to grasp that during the application of the Ideal Gas Law, any changes to one variable necessitate changes to at least one other, maintaining the constant proportionality defined by nRT.

Gas Mixture Pressure

Gas mixture pressure deals with total pressure exerted by a mixture of different gases contained in a given volume. According to Dalton's Law, each gas in a mixture exerts pressure independently of the others. As we saw in the exercise example, when you have a reaction that reduces the number of gases in the mixture (like oxygen reacting with iron), the pressure of the remaining gas (nitrogen, in this case) becomes the total pressure of the mixture.

Understanding gas mixture pressure is fundamental when predicting how the pressure will change as a result of chemical reactions, as well as in practical applications such as respiratory gases in medicine or the design of industrial gas systems. It can be quite a juggle to consider the individual partial pressures and their collective impact on the total pressure, but comprehending this interaction is vital in all fields involving gas mixtures.

Mole Calculation

Mole calculation is a basic yet powerful tool in chemistry that allows us to quantify the amount of substance in a given sample. One mole contains Avogadro's number (\(6.022 \times 10^{23}\) entities) of particles, whether they be atoms, molecules, ions, or electrons. Mole calculations are a staple in stoichiometry, relating the quantities of reactants and products in chemical reactions.

In the context of the exercise, mole calculation enables us to bridge the gap between the macroscopic properties of a gas (such as pressure and volume) and its microscopic properties (such as the number of particles). Since the Ideal Gas Law involves n, the number of moles, it is essential to understand how to compute moles. For our example, we used the initial and final conditions of the gas to conclude that the number of moles remained constant, and subsequently applied this information to solve for the final volume of nitrogen in the container.