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 pressures of oxygen and carbon dioxide in the tank are 4.94 atm and 4.27 atm, respectively.

Step-by-step solution

Convert the temperature to Kelvin

We are given the temperature as 27°C. In order to use the Ideal Gas Law, we need to convert this to Kelvin. This conversion is done by adding 273.15 to the Celsius temperature: \( T(K) = T(^\circ C) + 273.15 \) \( T(K) = 27 + 273.15 = 300.15 K \)

Calculate the number of moles for each gas

In order to find the number of moles for each gas, we use the given masses and their molar masses. The molar mass of oxygen (O₂) is 32.00 g/mol and the molar mass of carbon dioxide (CO₂) is 44.01 g/mol: For oxygen gas: \( n_{O_2} = \frac{m_{O_2}}{M_{O_2}} \) \( n_{O_2} = \frac{52.5 g}{32.00 \mathrm{~g/mol}} \) \( n_{O_2} = 1.6406 \mathrm{~mol} \) For carbon dioxide gas: \( n_{CO_2} = \frac{m_{CO_2}}{M_{CO_2}} \) \( n_{CO_2} = \frac{65.1 g}{44.01 \mathrm{~g/mol}} \) \( n_{CO_2} = 1.4790 \mathrm{~mol} \)

Apply Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure of a mixture of gases is the sum of their partial pressures, which gives us: \( P_{total} = P_{O_2} + P_{CO_2} \) We know the total pressure is 9.21 atm, and we need to find the partial pressures of each gas.

Calculate the partial pressures

Using the Ideal Gas Law for each gas, we have: For oxygen gas: \( P_{O_2} = \frac{n_{O_2}RT}{V} \) For carbon dioxide gas: \( P_{CO_2} = \frac{n_{CO_2}RT}{V} \) Since both gases share the same volume and temperature in the container, we can rearrange and solve for the partial pressures of each gas: \( P_{O_2} = \frac{n_{O_2}}{n_{O_2} + n_{CO_2}} \times P_{total} \) \( P_{O_2} = \frac{1.6406}{1.6406 + 1.4790} \times 9.21 ~\mathrm{atm} \) \( P_{O_2} = 4.94 ~\mathrm{atm} \) Similarly, for carbon dioxide: \( P_{CO_2} = \frac{n_{CO_2}}{n_{O_2} + n_{CO_2}} \times P_{total} \) \( P_{CO_2} = \frac{1.4790}{1.6406 + 1.4790} \times 9.21 ~\mathrm{atm} \) \( P_{CO_2} = 4.27 ~\mathrm{atm} \) Thus, the partial pressures of oxygen and carbon dioxide in the tank are 4.94 atm and 4.27 atm, respectively.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used in chemistry to relate the pressure, volume, temperature, and number of moles of a gas. This equation is written as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant, typically \( 0.0821 \) L atm/mol K
• \( T \) is the temperature in Kelvin

To solve problems involving multiple gases, like finding partial pressures, we assume that each gas behaves ideally. Using the Ideal Gas Law is crucial for calculations in various scenarios, including finding the partial pressure of gases in a mixture, as it helps isolate different components while considering their shared physical conditions such as temperature and volume. In this scenario, even though volume \( V \) is not provided, it isn't needed as the aim is to find partial pressures using mole fractions.

partial pressure

Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the individual partial pressures of each component gas. The formula used is: \[ P_{total} = P_1 + P_2 + P_3 + ext{...} \] where each \( P_i \) represents the partial pressure of a gas component.

In practical terms, partial pressure is found using the expression: \[ P_i = \left( \frac{n_i}{n_{total}} \right) P_{total} \] where \( n_i \) is the number of moles of gas \( i \), and \( n_{total} \) is the total number of moles of all gases.Applying this idea, by knowing the total pressure and mole fraction (or ratio of moles of each gas), you can determine the contribution of each gas to the total pressure, allowing calculation of the partial pressure which influences many physical and chemical processes.

molar mass

Molar mass is a critical property that helps bridge mass to amount of substance, particularly in the calculations involving the Ideal Gas Law. It is expressed as grams per mole (g/mol) and is specific to each element or compound. To find the number of moles, you use the formula: \[ n = \frac{m}{M} \] where:

• \( n \) is the number of moles
• \( m \) is the mass of the substance
• \( M \) is the molar mass

For example, oxygen \( O_2 \) has a molar mass of 32.00 g/mol, while carbon dioxide \( CO_2 \) is 44.01 g/mol. This tells us how much of each gas we have in terms of chemical substance amounts, allowing for their partial pressures to be calculated.Overall, understanding molar mass is fundamental in assessing how much of each gas is present when solving for partial pressures or other related gas properties in a chemical reaction.

temperature conversion

Temperature conversion, especially from Celsius to Kelvin, is a standard process necessary for gas law calculations. The gas law equations, including the Ideal Gas Law, require temperature measurements in Kelvin because the Kelvin scale ensures proportionality to absolute zero—the point where all molecular motion ceases.

To convert Celsius to Kelvin, the formula is simple: \[ T(K) = T(^{\circ}C) + 273.15 \]This conversion is crucial in ensuring accurate applications of the Ideal Gas Law, such as when determining the behavior of gases concerning changes in pressure, volume, or temperature. Using Kelvin allows a consistent and precise framework for analysis.