Question

What is the partial pressure of each gas in a mixture which contains 40 g. \(\mathrm{He}, 56\) g. \(\mathrm{N}_{2}\), and \(16 \mathrm{~g} . \mathrm{O}_{2}\), if the total pressure of the mixture is 5 atmospheres.

Short answer

The partial pressures of Helium, Nitrogen, and Oxygen in the mixture are 4 atm, 0.8 atm, and 0.2 atm, respectively.

Step-by-step solution

Calculate moles of each gas

To find the moles of each gas, use the molar mass of each gas. The molar masses of Helium (He), Nitrogen (\(N_2\)), and Oxygen (\(O_2\)) are approximately 4 g/mol, 28 g/mol, and 32 g/mol, respectively. For Helium: Moles of He = \(\frac{40 \,\text{g}}{4 \,\text{g/mol}}\) = 10 mol For Nitrogen: Moles of \(N_2\) = \(\frac{56 \,\text{g}}{28 \,\text{g/mol}}\) = 2 mol For Oxygen: Moles of \(O_2\) = \(\frac{16 \,\text{g}}{32 \,\text{g/mol}}\) = 0.5 mol

Calculate mole fractions for each gas

Mole fraction is the ratio of the moles of a gas to the total moles in the mixture. First, find the total moles in the mixture to calculate the mole fraction: Total moles = Moles of He + Moles of \(N_2\) + Moles of \(O_2\) = 10 + 2 + 0.5 = 12.5 mol Now, calculate the mole fractions: Mole fraction of He = \(\frac{10 \,\text{mol}}{12.5 \,\text{mol}}\) = 0.8 Mole fraction of \(N_2\) = \(\frac{2 \,\text{mol}}{12.5 \,\text{mol}}\) = 0.16 Mole fraction of \(O_2\) = \(\frac{0.5 \,\text{mol}}{12.5 \,\text{mol}}\) = 0.04

Calculate partial pressures of each gas

To find the partial pressure of each gas, multiply its mole fraction by the total pressure of the mixture (5 atmospheres in this case). Partial pressure of He = Mole fraction of He × Total pressure = 0.8 × 5 atm = 4 atm Partial pressure of \(N_2\) = Mole fraction of \(N_2\) × Total pressure = 0.16 × 5 atm = 0.8 atm Partial pressure of \(O_2\) = Mole fraction of \(O_2\) × Total pressure = 0.04 × 5 atm = 0.2 atm Thus, the partial pressures of Helium, Nitrogen, and Oxygen in the mixture are 4 atm, 0.8 atm, and 0.2 atm, respectively.

Key concepts

Mole Fraction

In chemistry, the mole fraction plays a crucial role in understanding gas mixtures. It is defined as the ratio of the number of moles of a particular component to the total number of moles in the mixture.
The formula is: Mole Fraction = \( \frac{\text{moles of component}}{\text{total moles in mixture}} \)
For example, if a mixture consists of 10 moles of Helium and the total number of moles in the mixture is 12.5, the mole fraction of Helium would be \( \frac{10}{12.5} = 0.8 \).

• This concept is dimensionless and ranges from 0 to 1.
• Mole fractions can be used to find the concentration and composition of each component in a gas mixture.
• In gas problems, knowing the mole fractions helps when applying further concepts like Dalton's Law of Partial Pressures.

Understanding mole fraction is essential, as it helps dissect the contribution of each gas in a mixture and allows for calculating properties like partial pressures.

Ideal Gas Law

The Ideal Gas Law is a valuable equation in chemistry that connects the pressure, volume, temperature, and amount of gas in moles. It is given by the equation:
\[ PV = nRT \]
Where:

• \(P\) stands for pressure,
• \(V\) signifies volume,
• \(n\) denotes the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) represents temperature in Kelvin.

This law assumes gases behave ideally. It helps to predict how changes in one of these variables affect the others. Although our exercise focuses on partial pressures and mole fractions, understanding the Ideal Gas Law is key when calculating the total pressure of a gas mixture, especially when individual gases interact in a container. Such knowledge provides a foundation for handling real-world applications and further complex gas-related problems.

Gas Mixtures

Gas mixtures contain multiple types of gases dispersed together. Unlike chemical compounds that have fixed ratios, gas mixtures can vary in proportion.
Key facts about gas mixtures include:

• Each gas in the mixture retains its original properties.
• Gases mix homogeneously, meaning the composition is the same throughout.
• Gas mixtures follow both the Ideal Gas Law and Dalton’s Law when calculating overall or individual gas pressures.

In your calculation, identifying the amount of each gas in moles is essential as it informs the mole fraction and partial pressure calculations. When dealing with gas mixtures, remember that gases don’t chemically bond but rather coexist in the same volume. This is why gas mixtures are greatly utilized in various scientific and industrial processes.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures helps us understand the behavior of gases within a mixture. It states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases.
Mathematically, it can be expressed as:
\[ P_{\text{total}} = P_1 + P_2 + P_3 + \ldots \]
in which \( P_1, P_2, P_3 \), are the partial pressures of the respective gases.
To find the partial pressure of a given gas, multiply the mole fraction of the gas by the total pressure of the mixture:

• Partial pressure of gas = Mole fraction × Total pressure.

Using Dalton's Law not only helps in understanding the contribution of each gas to the overall pressure but also offers insights into how gases behave when combined.
For students, mastering this law allows them to solve problems regarding gas mixtures easily and clearly, making it a cornerstone concept in gas chemistry.