Problem 69 - Chemistry: The Central Science

Question

A quantity of \(\mathrm{N}_{2}\) gas originally held at \(4.75\) atm pressure in a 1.00-L container at \(26^{\circ} \mathrm{C}\) is transferred to a 10.0-L container at \(20^{\circ} \mathrm{C}\). A quantity of \(\mathrm{O}_{2}\) gas originally at \(5.25\) atm and \(26^{\circ} \mathrm{C}\) in a 5.00-L container is transferred to this same container. What is the total pressure in the new container?

Step-by-Step Solution

Answer

The total pressure in the new container after transferring both N₂ and O₂ gases is 2.676 atm.

Step 1: Convert temperatures to Kelvin

Since we are dealing with temperatures measured in Celsius in the problem, we first need to convert them to Kelvin because the Ideal Gas Law uses Kelvin for temperature. The conversion to Kelvin can be done using the formula K = °C + 273.15. For Nitrogen (N₂): \(T_1\) = 26°C + 273.15 = 299.15 K For Oxygen (O₂): Same temperature, so \(T_1\) = 299.15 K The new temperature for the gases in the new container is 20°C: \(T_2\) = 20°C + 273.15 = 293.15 K

Step 2: Calculate pressure for N₂ in the new container

Now we will apply the Ideal Gas Law for the N₂ gas: \(P_1V_1/T_1 = P_2V_2/T_2\) Substitute all given values and values from Step 1 for N₂: \(4.75\, \text{atm} \cdot 1.00\, \text{L} / 299.15\, \text{K} = P_2 \cdot 10.0\, \text{L} / 293.15\, \text{K}\) Solve for \(P_2\): \(P_2 = 4.75\, \text{atm} \cdot 1.00\, \text{L} \cdot 293.15\, \text{K} / (299.15\, \text{K} \cdot 10.0\, \text{L}) = 0.5155\, \text{atm}\)

Step 3: Calculate pressure for O₂ in the new container

Now we will apply the Ideal Gas Law for the O₂ gas: \(P_1V_1/T_1 = P_2V_2/T_2\) Substitute all given values and values from Step 1 for O₂: \(5.25\, \text{atm} \cdot 5.00\, \text{L} / 299.15\, \text{K} = P_2 \cdot 10.0\, \text{L} / 293.15\, \text{K}\) Solve for \(P_2\): \(P_2 = 5.25\, \text{atm} \cdot 5.00\, \text{L} \cdot 293.15\, \text{K} / (299.15\, \text{K} \cdot 10.0\, \text{L}) = 2.1605\, \text{atm}\)

Step 4: Calculate the total pressure in the new container

Since both N₂ and O₂ gases occupy the same container, their pressures are additive. To find the total pressure, we will add the pressures obtained in step 2 and step 3: Total pressure = Pressure of N₂ + Pressure of O₂ Total pressure = 0.5155 atm + 2.1605 atm = 2.676 atm The total pressure in the new container is 2.676 atm.

Key Concepts

Temperature conversion to Kelvin

When working with gas laws, it's important to use the Kelvin scale for temperature. This is because gas laws, like the Ideal Gas Law, rely on absolute temperature. The Kelvin scale starts at absolute zero, which makes mathematical calculations more straightforward.

Here's how to convert from Celsius to Kelvin:

• Identify the temperature in Celsius.
• Add 273.15 to the Celsius temperature.

By following these steps, you ensure that the calculations remain consistent and error-free. For the given exercise, you needed to convert 26°C (both for \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\)) and 20°C to the Kelvin scale:

- The initial temperatures for \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) were 26°C, which equals 299.15 K.- The final temperature in the container is 20°C, which is equal to 293.15 K.

This simple conversion is a foundational step in applying any gas law so make sure not to skip this step!

Pressure calculations

Pressure plays a critical role in gas law calculations. Understanding how to calculate changes in pressure helps us predict how gases behave under different conditions.

In the exercise, we use the Ideal Gas Law to find the new pressure of each gas in the larger container. The equation used is derived from the Ideal Gas Law, which states that for a given amount of gas, the pressure times the volume divided by the temperature should remain constant if the amount of gas doesn't change:

• Original pressure: \(P_1\)
• Original volume: \(V_1\)
• Original temperature: \(T_1\)
• New pressure: \(P_2\)
• New volume: \(V_2\)
• New temperature: \(T_2\)

The formula used is \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). This formula can be rearranged to solve for \(P_2\), the pressure we need to find:

For \(\mathrm{N}_{2}\):- The initial conditions are given as 4.75 atm pressure, in 1.00 L volume, at 299.15 K.- These become 0.5155 atm in the new conditions of 10 L and 293.15 K.For \(\mathrm{O}_{2}\):- Initial conditions were 5.25 atm pressure, in a 5.00 L volume, at 299.15 K.- The new pressure calculated is 2.1605 atm.

By following these calculations, students can gain a deep understanding of how changes in temperature and volume affect pressure.

Gas law applications

The Ideal Gas Law is a valuable tool for explaining and predicting the behavior of gases. It allows us to understand how different variables interact. In this exercise, we see an application of the Ideal Gas Law when combining gases into a single container.

The formula \(PV = nRT\) is a core part of this law. Here we rearrange it to address changes in pressure and volume when the temperature also changes. This means we can predict how the pressure will change when gases are transferred from a smaller to a larger container while adjusting for temperature changes.

Applying this knowledge:

• We calculated the pressure changes for both gases, \(\mathrm{N}_2\) and \(\mathrm{O}_2\), after transferring them to a new container.
• This setup also addresses Dalton's Law, which says that the total pressure in a gas mixture is the sum of the partial pressures of the individual gases.
• The exercise combines these pressures from \(\mathrm{N}_2\) and \(\mathrm{O}_2\) to find the total pressure.

The calculation of 2.676 atm as the total pressure demonstrates the cumulative impact of separate gases in a shared space, allowing for a clearer prediction of real-world gas behavior.