Question

A tank contains a mixture of \(52.5 \mathrm{g}\) 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 pressure of oxygen gas in the container is 4.85 atm, and the partial pressure of carbon dioxide gas is 4.36 atm.

Step-by-step solution

Find the moles of each gas

Use the molar masses of oxygen and carbon dioxide to convert the given mass of each gas to moles. The molar mass of oxygen (\(O_2\)) is 32 g/mol, and the molar mass of carbon dioxide (\(CO_2\)) is 44.01 g/mol. Moles of \(O_2\) = Mass / Molar mass = \(52.5 \, \text{g} / (32 \, \text{g/mol}) = 1.64 \, \text{mol}\) Moles of \(CO_2\) = Mass / Molar mass = \(65.1 \, \text{g} / (44.01 \, \text{g/mol}) = 1.48 \, \text{mol}\)

Determine the mole fraction of each gas

Calculate the total moles of gas in the container and then find the mole fraction of each gas. Total moles = Moles of \(O_2\) + Moles of \(CO_2\) = \(1.64 \, \text{mol} + 1.48 \, \text{mol} = 3.12 \, \text{mol}\) Mole fraction of \(O_2\) = Moles of \(O_2\) / Total moles = \(1.64 \, \text{mol} / 3.12 \, \text{mol} = 0.526\) Mole fraction of \(CO_2\) = Moles of \(CO_2\) / Total moles = \(1.48 \, \text{mol} / 3.12 \, \text{mol} = 0.474\)

Calculate the partial pressures of each gas

Use the mole fractions and the total pressure to find the partial pressures of each gas. Partial pressure of \(O_2\) = Mole fraction of \(O_2\) × Total pressure = \(0.526 \times 9.21 \, \text{atm} = 4.85 \, \text{atm}\) Partial pressure of \(CO_2\) = Mole fraction of \(CO_2\) × Total pressure = \(0.474 \times 9.21 \, \text{atm} = 4.36 \, \text{atm}\) The partial pressure of oxygen gas in the container is 4.85 atm, and the partial pressure of carbon dioxide gas is 4.36 atm.

Key concepts

Mole Fraction

The concept of mole fraction is fundamental in the study of gas mixtures and their behaviors. It represents the proportion of a particular component in a mixture, relative to the total number of moles of all components. To calculate the mole fraction, one divides the number of moles of the component of interest by the total number of moles present.

For instance, if we have a gas mixture containing two different gases, we can find the mole fraction of each gas by dividing the moles of that specific gas by the sum of the moles of all gases in the mixture. The mole fraction is unitless and always ranges between 0 and 1. It is extensively used to calculate properties such as partial pressures, which are crucial to understanding the behavior of gases in different conditions.

Gas Laws

Gas laws are a set of laws that describe the relationship between the pressure, volume, temperature, and amount (in moles) of a gas. These laws are critical to understanding the behavior of gases under various conditions. Three of the most important gas laws are Boyle's Law, which states that the pressure and volume of a gas are inversely proportional at constant temperature; Charles's Law, which states that the volume and temperature of a gas are directly proportional at constant pressure; and Avogadro's Law, which asserts that the volume of a gas is directly proportional to the number of moles at constant pressure and temperature.

These individual laws are special cases of the ideal gas law, represented by the equation:
\( PV = nRT \),
where P represents the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This equation is instrumental in calculating various properties of gases, including partial pressures.

Molar Mass

Molar mass is a property that links the mass of a substance to its amount in moles. It is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol). Molar mass can be calculated by summing the atomic masses of all the atoms in a molecule. For example, the molar mass of carbon dioxide, \(CO_2\), is found by adding the atomic mass of carbon with that of two oxygen atoms.

Knowing the molar mass of each gas in a mixture is essential for converting mass measurements to moles, which is a critical step in performing stoichiometric calculations and determining mole fractions. Accurate determination of molar mass allows for reliable calculations of other properties of the gas, such as density and partial pressure.

Stoichiometry

Stoichiometry is the area of chemistry that deals with the quantitative relationships between reactants and products in chemical reactions. It relies on the principle that matter is conserved in reactions, which allows for the calculation of the amounts of substances required or produced. Stoichiometry involves the use of balanced chemical equations, molar masses, mole ratios, and Avogadro's law to predict yields, determine limiting reagents, and calculate the composition of mixtures.

In the context of gases, stoichiometry is crucial for predicting the outcomes of reactions that occur under various conditions of temperature and pressure. It is also fundamental in identifying the amounts of gases involved in reactions, which requires an understanding of how gas laws interplay with stoichiometric relationships. By mastering stoichiometry, students can tackle complex problems involving gas mixtures and reactions with confidence and precision.