Question

A 4.0 L sample of \(\mathrm{O}_{2}\) gas has a pressure of 1.0 atm. A 2.0 L sample of \(\mathrm{N}_{2}\) gas has a pressure of 2.0 atm. If these two samples are mixed and then compressed in a 2.0 L vessel, what is the final pressure of the mixture? Assume that the temperature remains unchanged.

Short answer

The final pressure of the mixture is 4.0 atm

Step-by-step solution

Determine the number of moles of each gas

Initially, we assume an ideal behavior for gases. Therefore, we can use the Ideal Gas Law \(PV=nRT\) to determine the number of moles. It's given that R (the gas constant) and T (temperature) are constant. So, we can simplify the formula to \(PV=n\). We calculate the number of moles for O\(_2\) and N\(_2\) gases using their initial pressures and volumes: Moles of O\(_2\) = \(P_{O_2}V_{O_2} = 1.0 atm * 4.0 L = 4.0 atm*L\) and moles of N\(_2\) = \(P_{N_2}V_{N_2} = 2.0 atm * 2.0 L = 4.0 atm*L\)

Calculate the total number of moles

We determine the total moles of the mixed gases by adding the moles of O\(_2\) and the moles of N\(_2\): Total moles = \(moles_{O_2} + moles_{N_2} = 4.0 atm*L + 4.0 atm*L = 8.0 atm*L\)

Determine the final pressure

To determine the final pressure, we use the modified Gas Law since volume and temperature are constant after the gases are mixed: \(P = n/V\), where n is the total number of moles and V is the final volume: Final Pressure = \(P = n/V = 8.0 atm*L / 2.0 L = 4.0 atm\).

Key concepts

Gas Pressure

Gas pressure is an essential concept in understanding how gases behave. It is the force exerted by gas molecules when they hit the walls of their container. The more molecules there are, or the faster they move, the greater the pressure they exert.
Pressure is often measured in atmospheres (atm), a unit that helps us express how much pressure a gas is applying relative to the earth's air pressure at sea level. In the exercise, we see gas pressure in action with the separate oxygen ( O_{2} ) and nitrogen ( N_{2} ) samples, each applying their respective pressures in their own volumes.
Understanding pressure helps predict how gases will behave when combined or confined. When the two samples were mixed, the pressure changed based on their volumes and the number of moles of gas. This adjustment is key to calculating the final pressure in any gas mixture situation.

Mole Calculation

Calculating moles of gas is like counting pieces of a puzzle. Each mole represents Avogadro's number of molecules, which is approximately 6.022 x 10^{23} molecules. Using the Ideal Gas Law, we can deduce the relationship between the pressure (P), volume (V), and number of moles (n) of a gas: \(PV = nRT\), where R is the gas constant and T is the temperature.
In the exercise, with temperature and R constant, the formula simplifies to \(PV = n\). By this method, we calculate the number of moles in each gas sample by multiplying their given pressure by their volume. This calculation gives us:

• Moles of O_{2}: \(1.0 \text{ atm} \times 4.0 \text{ L} = 4.0 \text{ atm*L}\)
• Moles of N_{2}: \(2.0 \text{ atm} \times 2.0 \text{ L} = 4.0 \text{ atm*L}\)

Once we understand the mole count for each gas, it becomes simple algebra to find the total moles in a combined gas mixture.

Gas Mixture

A gas mixture is a combination of different gases that share the same space. Each gas contributes to the total pressure of the mixture based on its own pressure and volume, a concept known as partial pressure. When oxygen and nitrogen gases are combined, as in the exercise, understanding how these partial pressures contribute helps determine the mixture's behavior.
The final pressure of the mixture is found by first calculating the sum of individual gases' pressures, represented as the total number of `atmosphere-liters` (atm*L). From this, the final pressure is determined by dividing the total atm*L by the new total volume, giving a clear view of how gases interact in a shared container.
This exercise showed that mixing a 4.0 atm*L and another 4.0 atm*L diversified sample resulted, after compression in a smaller volume, in a final pressure of 4.0 atm. This gives us powerful insight into how gas mixtures behave, based on their original conditions.