Question

What would be the final partial pressure of oxygen in the following experiment? A collapsed polyethylene bag of 30 liters capacity is partially blown up by the addition of 10 liters of nitrogen gas measured at \(0.965 \mathrm{~atm}\) and \(298^{\circ} \mathrm{K}\). Subsequently, enough oxygen is pumped into the bag so that, at \(298^{\circ} \mathrm{K}\) and external pressure of \(0.990 \mathrm{~atm}\), the bag contains a full 30 liters, (assume ideal behavior.)

Short answer

The final partial pressure of oxygen gas in the bag is: \(P_{O_2} = \frac{(\frac{0.990 \times 30}{0.0821 \times 298} - \frac{0.965 \times 10}{0.0821 \times 298})(0.0821)(298)}{30} = 0.323 \text{ atm}\)

Step-by-step solution

Recall that the Ideal Gas Law is given by the equation: \(PV=nRT\), where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. We can rearrange this equation to solve for n: \(n = \frac{PV}{RT}\) Here, we have the initial pressure and volume of nitrogen gas, as well as the temperature. We can use these values to calculate the number of moles of nitrogen gas added: \(n_{N_2} = \frac{P_{N_2}V_{N_2}}{RT} = \frac{0.965 \times 10}{0.0821 \times 298}\) #Step 2: Calculate the total pressure and moles of gas in the bag after adding oxygen#

After adding oxygen, the total pressure inside the bag equals the external pressure, which is given as \(0.990 \text{ atm}\). According to Dalton's Law of Partial Pressures, the total pressure inside the bag is equal to the sum of the partial pressures of nitrogen and oxygen: \(P_{total} = P_{N_2} + P_{O_2}\) Using the Ideal Gas Law again, we can also relate the total moles of gas in the bag to the final pressure, volume, and temperature: \(n_{total} = \frac{P_{total}V}{RT} = \frac{0.990 \times 30}{0.0821 \times 298}\) #Step 3: Calculate the moles of oxygen gas added#

To find the moles of oxygen gas added to the bag, we can subtract the moles of nitrogen gas from the total moles of gas in the bag: \(n_{O_2} = n_{total} - n_{N_2} = \frac{0.990 \times 30}{0.0821 \times 298} - \frac{0.965 \times 10}{0.0821 \times 298}\) #Step 4: Calculate the partial pressure of oxygen gas (final answer)#

Since we now have the moles of oxygen gas, we can use the Ideal Gas Law one last time to find the partial pressure of oxygen gas in the bag: \(P_{O_2} = \frac{n_{O_2}RT}{V} = \frac{(\frac{0.990 \times 30}{0.0821 \times 298} - \frac{0.965 \times 10}{0.0821 \times 298})(0.0821)(298)}{30}\) After calculating all values, we get our final answer: \(P_{O_2} = X \text{ atm}\) (Your final answer should be a numerical value in units of atmospheres)

Key concepts

Partial Pressure

Partial pressure is a crucial concept when dealing with gas mixtures. It refers to the pressure that a single component of a gas mixture would exert if it occupied the entire volume by itself. Imagine each gas in the mixture having its own miniature container. The pressure in that mini-container is the partial pressure. To determine the partial pressure of a gas in a mixture, you need to know the relative amount of that gas compared to the total amount of gases present. This amount is typically measured in moles, a unit that captures the quantity of atoms or molecules involved. Once you have the moles of each gas, you can calculate the partial pressures accordingly.

Dalton's Law of Partial Pressures

Dalton’s Law of Partial Pressures is a fundamental principle for understanding how different gases behave in a mixture. According to this law, the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of each individual gas.

• The formula is written as: \(P_{total} = P_{1} + P_{2} + ... + P_{n}\), where \(P_{1}\), \(P_{2}\), ..., \(P_{n}\) are the partial pressures of each gas in the mixture.
• This concept is helpful when determining the behavior of gases in combined settings, such as in atmospheric pressure measurements or in balloons.
• By knowing the total pressure and the partial pressures of other gases, you can deduce the partial pressure of any unmeasured gas.

In summary, Dalton's Law allows you to extend your understanding of pressure from one gas to a complex mixture.

Moles of Gas

The term "moles" refers to a measure of gas that helps in quantifying how many molecules are in a given volume. Moles are seen as the connecting bridge between the macroscopic and microscopic scales, providing a way to relate physical measurements (like volume or pressure) to the number of particles present.

• The relationship between moles, pressure, volume, and temperature is described by the Ideal Gas Law: \(PV = nRT\).
• In this formula, \(n\) stands for moles, capturing the amount of substance, while \(R\) is the ideal gas constant.
• When doing calculations like those in the exercise, number of moles can be found by rearranging the Ideal Gas equation to: \(n = \frac{PV}{RT}\).

Understanding moles is fundamental because it allows you to calculate how a gas will behave under varied conditions by relating it to its physical properties.

Chemical Calculations

Chemical calculations involve using established formulas and consistent units to solve problems related to chemical reactions and processes. Such calculations ensure accurate predictions of experimental conditions and outcomes.

• In the context of gases, chemical calculations frequently use the Ideal Gas Law as a basic tool for finding unknown properties like pressure or moles.
• These calculations can help predict how much of a gas is needed or produced in a reaction and assist in determining the conditions necessary for a specific reaction to occur.
• For effective chemical calculations, familiarity with converting units and rearranging equations is quite crucial.

Overall, chemical calculations are indispensable for interpreting experimental data and predicting phenomena in chemistry.