Problem 75 - Chemistry The Central Science

Question

A quantity of \(\mathrm{N}_{2}\) gas originally held at 5.25 atm pressure in a 1.00-L container at \(26^{\circ} \mathrm{C}\) is transferred to a 12.5-L container at \(20^{\circ} \mathrm{C}\). A quantity of \(\mathrm{O}_{2}\) gas originally at \(5.25 \mathrm{~atm}\) 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

In order to find the total pressure in the new container after transferring N₂ and O₂ gases from their original containers, follow these steps: 1. Convert temperatures from Celsius to Kelvin: \( T_{1(N2)} = T_{1(O2)} = 299.15 K \), \( T_{2} = 293.15 K \) 2. Calculate moles of each gas using the Ideal Gas Law. 3. Calculate the final pressure for each gas after they have been transferred to the new container. 4. Calculate the total pressure in the new container by adding the final pressures of N₂ and O₂: \( P_{total} = P_{2(N2)} + P_{2(O2)} \)

Step 1: Convert temperatures to Kelvin

For both gases, we need to convert their initial temperatures from Celsius to Kelvin: \( T_{1(N2)} = 26^{\circ}C + 273.15 K = 299.15 K \) \( T_{1(O2)} = 26^{\circ}C + 273.15 K = 299.15 K \) \( T_{2} = 20^{\circ}C + 273.15 K = 293.15 K \)

Step 2: Calculate moles of each gas

Now, we can find the number of moles for N₂ and O₂ using the Ideal Gas Law: For N₂: \( P_{1(N2)}V_{1(N2)} = n_{N2}RT_{1(N2)} \) \( n_{N2} = \frac{P_{1(N2)}V_{1(N2)}}{RT_{1(N2)}} = \frac{(5.25 \, atm)(1.00 \, L)}{(0.0821 \, L \cdot atm / mol \cdot K)(299.15 \, K)} \) For O₂: \( P_{1(O2)}V_{1(O2)} = n_{O2}RT_{1(O2)} \) \( n_{O2} = \frac{P_{1(O2)}V_{1(O2)}}{RT_{1(O2)}} = \frac{(5.25 \, atm)(5.00 \, L)}{(0.0821 \, L \cdot atm / mol \cdot K)(299.15 \, K)} \)

Step 3: Calculate final pressure for each gas

Now, we will find the final pressure for each gas after they have been transferred to the new container: For N₂: \( P_{2(N2)}V_{2} = n_{N2}RT_{2} \) \( P_{2(N2)} = \frac{n_{N2}RT_{2}}{V_{2}} \) For O₂: \( P_{2(O2)}V_{2} = n_{O2}RT_{2} \) \( P_{2(O2)} = \frac{n_{O2}RT_{2}}{V_{2}} \)

Step 4: Calculate total pressure in the new container

\ Finally, we will find the total pressure in the new container by adding the final pressures of N₂ and O₂: \( P_{total} = P_{2(N2)} + P_{2(O2)} \) By following the above steps, we will get the total pressure in the new container after transferring N₂ and O₂ gases from their original containers.

Key Concepts

Gas Pressure and Volume

Understanding gas pressure and volume is a cornerstone in studying gases and their behaviors. Pressure, often measured in atmospheres (atm), is the force exerted by the gas particles as they collide with the surfaces of their container. Volume, on the other hand, is the space that the gas occupies, usually in liters (L). When we say a gas is 'pressurized,' it means that its particles are packed into a smaller volume, leading to more frequent collisions and therefore higher pressure.

According to Boyle's Law, pressure and volume are inversely proportional, provided the number of moles and temperature remain constant. This translates into an important relationship: when the volume of a gas is decreased, its pressure increases, and vice versa, if the temperature and amount of gas stay the same. Therefore, if we transfer a gas into a larger container, under constant temperature and amount, we can expect the pressure to decrease, which is pivotal in many calculations and applications involving gases.

Moles of Gas

The moles of gas concept is fundamental to the Ideal Gas Law and stoichiometry. A mole is a unit that measures the amount of substance, usually denoted as 'n' in equations. It is particularly handy in chemistry because it links the microscopic world of atoms and molecules to the macroscopic world we can measure. One mole contains approximately Avogadro's number of particles, which is around 6.022 x 10^23 particles.

When dealing with gases, knowing the number of moles allows us to predict behavior under varying conditions of pressure, volume, and temperature. In the given exercise, the calculation of moles of gas is crucial to determine the total pressure after a change in volume and temperature. The ability to calculate moles gives us insight into the quantitative aspects of gas reactions and transformations.

Temperature Conversion

Temperature often needs to be converted between several scales. In gas law calculations, the Kelvin scale is used because it is an absolute scale with zero being the lowest possible temperature - absolute zero. Converting Celsius to Kelvin is straightforward: simply add 273.15. For instance, at room temperature of 20°C, we have 293.15 K.

A proper temperature conversion is essential because the behavior of gases is highly temperature-dependent. An incorrect temperature can lead to errors in calculating pressure, volume, or moles using the Ideal Gas Law. Temperature in Kelvin ensures that volume and pressure are directly proportional to temperature and that these relationships remain consistent in calculations.

Combined Gas Law

The combined gas law is an integration of Boyle's, Charles's, and Gay-Lussac's laws. It shows the relationship between pressure, volume, and temperature of a fixed amount of gas. The formula can be stated as: \( \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \) where P1 and P2 are the initial and final pressures, V1 and V2 are the initial and final volumes, and T1 and T2 are the initial and final temperatures.

This law is incredibly useful when predicting the final state of a gas after changes in pressure, volume, and/or temperature occur. In practical terms, if a sealed container with gas is heated, we can expect both the pressure and volume to increase if one of them is able to change. In our exercise, both gases experienced a change in volume and temperature, making the combined gas law the perfect tool to calculate the new conditions, including the total pressure in a shared container.