Problem 73 - Chemistry: The Central Science

Question

A quantity of \(\mathrm{N}_{2}\) gas originally held at \(5.25 \mathrm{~atm}\) pressure in a \(1.00\) - \(\mathrm{L}\) container at \(26^{\circ} \mathrm{C}\) is transferred to a \(12.5\) - \(\mathrm{L}\) 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 \(2.53\,\mathrm{atm}\).

Step 1: Write down given information and the ideal gas law

Write down the information given in the problem and the ideal gas law, which is \(PV = nRT\). We have the initial conditions for N₂ gas (V₁(N₂), T₁(N₂), and P₁(N₂)) and O₂ gas (V₁(O₂), T₁(O₂), and P₁(O₂)). We also have the final volume (V₂) and final temperature (T₂) for both gases after they have been transferred to the new container.

Step 2: Convert temperature to Kelvin

Convert the given temperatures from degrees Celsius to Kelvin by adding 273.15 to the given values. This is important because the ideal gas law uses Kelvin as the unit for temperature. T₁(N₂) = \(26 + 273.15 = 299.15\,\mathrm{K}\) T₂ = \(20 + 273.15 = 293.15\,\mathrm{K}\)

Step 3: Calculate moles of N₂ and O₂ using the ideal gas law

Use the ideal gas law to calculate the number of moles for N₂ and O₂. We can rearrange the equation to find moles (n): n = \(\frac{PV}{RT}\) n(N₂) = \(\frac{P₁(N₂) V₁(N₂)}{R T₁(N₂)} = \frac{5.25\,\mathrm{atm} \times 1.00\,\mathrm{L}}{0.0821\,\mathrm{L\,atm/mol\,K} \times 299.15\,\mathrm{K}} = 0.217\,\mathrm{mol}\) n(O₂) = \(\frac{P₁(O₂) V₁(O₂)}{R T₁(O₂)} = \frac{5.25\,\mathrm{atm} \times 5.00\,\mathrm{L}}{0.0821\,\mathrm{L\,atm/mol\,K} \times 299.15\,\mathrm{K}} = 1.086\,\mathrm{mol}\)

Step 4: Calculate total moles, nTotal

Sum the moles of the two gases to get the total number of moles: nTotal = n(N₂) + n(O₂) = 0.217 + 1.086 = 1.303\,\mathrm{mol}

Step 5: Calculate total pressure in the new container using the ideal gas law

We have the total moles (nTotal), final volume (V₂), and final temperature (T₂), so we can plug these values into the ideal gas law and solve for the new pressure (P₂): \(P₂V₂ = n_\text{Total} R T₂\) Rearrange the equation to find the new pressure (P₂): \(P₂ = \frac{n_\text{Total} R T₂}{V₂} = \frac{1.303\,\mathrm{mol} \times 0.0821\,\mathrm{L\,atm/mol\,K} \times 293.15\,\mathrm{K}}{12.5\,\mathrm{L}} = 2.53\,\mathrm{atm}\) The total pressure in the new container is 2.53 atm.

Key Concepts

Understanding Gas Pressure

Gas pressure is a fundamental concept in the study of gases. It is the force that the gas exerts on the walls of its container, and it is a result of gas particles colliding with the container's surfaces. The more frequent and forceful these collisions, the higher the pressure. When solving problems involving gas pressure, it's important to understand that pressure can be affected by factors such as volume, temperature, and the amount of gas present.

In the given exercise, we dealt with two gases, \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\), each at an initial pressure of 5.25 atm, which later combined in a new container, leading to a change in the total pressure. The Ideal Gas Law was used to determine this change and find the final pressure in the new container. It is crucial to convert the given temperatures to Kelvin since the Ideal Gas Law requires temperature in absolute units (Kelvin) and defines zero volume at zero Kelvin.

The Process of Moles Calculation

Calculating moles of a gas is crucial when working with chemical reactions and gas laws. The mole is a unit of measurement for the amount of substance. In chemistry, it represents Avogadro's number (approximately \(6.022 \times 10^{23}\)) of molecules or atoms. When you know the pressure, volume, and temperature of a gas, you can use the Ideal Gas Law \(PV = nRT\) to calculate the number of moles (n).

In the exercise, the number of moles for \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) were calculated using the initial conditions before the gases were combined. These calculations are essential because they help us understand how much of each gas is present, which in turn determines the final gas pressure after combining them in a new volume. This mole calculation is a key step in solving many problems in chemistry and provides the link between the macroscopic properties we can measure and the microscopic events happening with individual gas particles.

Applying Gas Laws

Gas laws explain how gases behave under different conditions of temperature, pressure, and volume. The Ideal Gas Law is one of the most important equations in this field as it combines several individual gas laws, including Boyle's Law, Charles's Law, and Avogadro's Law. It is represented as \(PV = nRT\), where P is pressure, V is volume, n is the amount of substance in moles, R is the universal gas constant, and T is temperature in Kelvin.

To tackle problems involving changes in gas conditions, a clear comprehension of these laws and proper application of the Ideal Gas Law is essential. In the exercise, the total pressure in a new container after transferring \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) gases was found by first applying the Ideal Gas Law to each gas to find the number of moles, then adding these moles to find the total. Finally, the gas law was used again with the total moles, new volume, and temperature to find the total pressure. It demonstrates the utility of grasping these gas laws to predict the behavior of gases in a variety of situations.