Question

A 1.42-g sample of helium and an unknown mass of ${\mathrm{O}}_2$ are mixed in a flask at room temperature. The partial pressure of the helium is 42.5 torr, and that of the oxygen is 158 torr. What is the mass of the oxygen?

Short answer

To find the mass of oxygen, first, calculate the moles of helium using the Ideal Gas Law: $n_{He}=\frac{P_{He}V}{RT}$. Then, find the moles of oxygen using the ratio of partial pressures: $\frac{n_{He}}{n_{O_2}}=\frac{P_{He}}{P_{O_2}}$. Finally, calculate the mass of oxygen: $m_{O_2}=n_{O_2}\times M_{O_2}$ where $M_{O_2}$ is the molar mass of oxygen (32 g/mol).

Step-by-step solution

Calculate the moles of Helium

The Ideal Gas Law can be expressed as: $PV=nRT$ Where: - P is the pressure (in atm), - V is the volume (in L), - n is the moles of gas, - R is the gas constant (0.08206 L atm/(mol K)), - T is the temperature (in K). We know the pressure, mass, and molar mass of helium. First, we need to convert the given pressure from torr to atm. Then, we can calculate the moles by rearranging the Ideal Gas Law formula. 1 atm = 760 torr $P_{He}(atm)=\frac{P_{He}(torr)}{760}=\frac{42.5}{760}$ Now, rearrange the Ideal Gas Law to get moles (n): $n_{He}=\frac{PV}{RT}$

Calculate the moles of Oxygen

We are given the partial pressures of helium and oxygen. Then the total pressure in the flask can be calculated as: $P_{total}=P_{He}+P_{O_2}$ Convert $P_{O_2}$ to atm: $P_{O_2}(atm)=\frac{P_{O_2}(torr)}{760}=\frac{158}{760}$ Now, use the Ideal Gas Law to calculate the number of moles of oxygen: $n_{O_2}=\frac{P_{O_2}V}{RT}$ Since helium and oxygen are in the same flask, the volume V and temperature T are the same for both gases. Therefore, the ratio of the moles of helium to the moles of oxygen can be calculated as: $\frac{n_{He}}{n_{O_2}}=\frac{P_{He}}{P_{O_2}}$ From which we can find the moles of oxygen: $n_{O_2}=n_{He}\times \frac{P_{O_2}}{P_{He}}$

Find the mass of Oxygen

Now that we have the moles of oxygen, we can find the mass of the oxygen gas using the molar mass. The molar mass of ${\mathrm{O}}_2$ is 32 g/mol. Then the mass of oxygen (m) can be calculated as: $m_{O_2}=n_{O_2}\times M_{O_2}$ Where: - $n_{O_2}$ is the moles of oxygen, - $M_{O_2}$ is the molar mass of oxygen (32 g/mol). Substitute the values and calculate the mass of the oxygen.

Key concepts

Partial Pressure

Partial pressure is an important concept in the study of gas mixtures. It refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume by itself. This concept is essential when dealing with gas mixtures, as it helps us understand how each component contributes to the total pressure in a container.
For instance, in our problem, the partial pressures of helium and oxygen were given as 42.5 torr and 158 torr, respectively. These values indicate the individual pressures solely attributed to each type of gas within the flask.
In chemical calculations, especially when using the Ideal Gas Law, it's crucial to convert these pressures from units like torr to atmospheres (atm) since the gas constant (R) is typically expressed in atm. This standardization allows seamless integration of parts of the Ideal Gas Law for calculations.

Mole Calculation

A fundamental part of solving gas law problems involves mole calculations. The number of moles of a gas directly correlates to its pressure, volume, and temperature, as represented by the Ideal Gas Law: $$PV=nRT$$where:

• $P$ is the pressure.
• $V$ is the volume.
• $n$ is the number of moles.
• $R$ is the gas constant (0.08206 L atm/(mol K)).
• $T$ is the temperature.

When the pressure is given in torr, we convert it to atm to use this formula. By rearranging the formula to solve for moles ($n$), we can determine the quantity of each gas in a mixture.
In the example exercise, once we know the pressure and convert to atm, we can calculate the moles of helium. This step is crucial as the mole ratio between helium and oxygen later allows calculation of the moles of oxygen, using their partial pressures.

Gas Mixtures

Understanding gas mixtures is vital, as they often comprise varied gases combined in one container. Each gas retains its properties and contributes to the behavior of the mixture. In calculations, we assume gases mix ideally, meaning each behaves as if it were alone within the available volume.
In this exercise, helium and oxygen are mixed within the same flask. We use their partial pressures to understand their individual contributions to the total pressure. Hence, knowing the volume and temperature are constant for both gases, we can relate them using their partial pressures.
By calculating the moles of each gas, we can further deduce other unknown characteristics, such as the mass of oxygen in the problem. The Ideal Gas Law assists in connecting these properties (pressure, volume, temperature, and moles) to figure out such real-life scenarios in chemistry.