Question

A tank contains a mixture of 52.5 g oxygen gas and 65.1 \(\mathrm{g}\) carbon dioxide gas at \(27^{\circ} \mathrm{C}\) . The total pressure in the tank is 9.21 atm. Calculate the partial pressures of each gas in the container.

Short answer

The partial pressures of oxygen and carbon dioxide in the container are \(4.84 \: \text{atm}\) and \(4.37 \: \text{atm}\), respectively.

Step-by-step solution

Convert mass to moles

First, we need to convert the given mass of each gas to moles using the molar mass of each gas. The molar mass of oxygen (O₂) is 32 g/mol, and the molar mass of carbon dioxide (CO₂) is 44 g/mol. For Oxygen: moles of O₂ = mass of O₂ / molar mass of O₂ moles of O₂ = 52.5 g / 32 g/mol = 1.64 mol For Carbon dioxide: moles of CO₂ = mass of CO₂ / molar mass of CO₂ moles of CO₂ = 65.1 g / 44 g/mol = 1.48 mol

Calculate total moles and mole fractions

Next, we need to find the total moles of both gases combined, and then calculate the mole fraction for each gas. Total moles = moles of O₂ + moles of CO₂ Total moles = 1.64 mol + 1.48 mol = 3.12 mol Now, we can find the mole fraction for each gas: Mole fraction of O₂ = moles of O₂ / total moles = 1.64 mol / 3.12 mol = 0.526 Mole fraction of CO₂ = moles of CO₂ / total moles = 1.48 mol / 3.12 mol = 0.474

Calculate partial pressures using Dalton's Law

Now we can use Dalton's Law of Partial Pressures, where the partial pressure of each gas equals the product of the total pressure and the mole fraction of that gas: Partial pressure of O₂ = total pressure * mole fraction of O₂ Partial pressure of O₂ = 9.21 atm * 0.526 = 4.84 atm Partial pressure of CO₂ = total pressure * mole fraction of CO₂ Partial pressure of CO₂ = 9.21 atm * 0.474 = 4.37 atm The partial pressures of oxygen and carbon dioxide in the container are 4.84 atm and 4.37 atm, respectively.

Key concepts

Partial Pressure

Partial pressure is a crucial concept in understanding gas mixtures. It refers to the pressure that each gas in a mixture would exert if it were alone in a container. In essence, each gas feels like it is the only one present and contributes individually to the overall pressure. This contribution is known as the partial pressure of the gas.

When we know the total pressure of a gas mixture, like in our exercise (9.21 atm), we can find the individual gas contributions using Dalton's Law of Partial Pressures. According to this law, the partial pressure of a gas is the product of its mole fraction and the total pressure.

• This means each gas' partial pressure is just its fraction of the number of particles multiplied by the total pressure.
• For instance, if a gas makes up half of the gas mixture by moles, its partial pressure will be half of the total pressure.

Understanding partial pressures helps in various applications, from breathing systems to predicting reactions in chemistry.

Mole Fraction

Mole fraction is an important concept when dealing with mixtures of gases. It shows what proportion or fraction of the total moles (or amount) in the mixture a particular gas makes up. This fraction helps us understand how much of the container's contents is contributed by each component gas.

To calculate mole fraction, we use the formula:

• Mole fraction of a gas = Moles of the gas / Total moles of all gases in the mixture.

In the exercise, the mole fraction for oxygen was calculated as 0.526, and for carbon dioxide, it was 0.474. These values mean oxygen makes up 52.6% of the gas mixture by moles, and carbon dioxide makes up 47.4%.

• Knowing mole fractions is crucial because they help in computing partial pressures, as seen with Dalton's Law.

Mole fractions are unitless, making them convenient for calculations and comparisons.

Gas Mixtures

Gas mixtures consist of more than one type of gas molecule within the same container. The properties of the mixture depend on the behavior and amount of its components. Each gas behaves independently and contributes to the total pressure according to its proportion in the mixture.

In terms of chemistry, gas mixtures behave according to the Ideal Gas Law. However, Dalton's Law simplifies our understanding by allowing us to calculate the contribution of each component as if they do not interact. In real-world applications, knowing the composition of a gas mixture is essential for processes such as combustion, respiration, and in industrial chemical reactions.

• Understanding gas mixtures involves knowing not just the types but the amounts of gases present.
• In our example, we worked with oxygen and carbon dioxide, both affecting the total pressure.

The ability to analyze and predict the behavior of gas mixtures is a powerful tool in science and engineering.

Moles Calculation

Calculating moles from a given mass is a foundational skill in chemistry that allows us to understand proportions in reactions and mixtures. Moles, a measure of substance amount, connect tangible mass with the microscopic world of molecules.

To calculate moles, use the formula:

• Moles = Mass of the substance / Molar mass of the substance

In our exercise, the moles of oxygen and carbon dioxide were determined using their respective molar masses (32 g/mol for O₂ and 44 g/mol for CO₂). This calculation enabled us to find out how much of each gas was present and to move forward with computing mole fractions and partial pressures.

• Moles serve as a bridge between mass and Avogadro's number, which is crucial for understanding chemical equations and reactions.

Mastering moles calculation empowers students to tackle more complex topics such as stoichiometry and gas laws.