Question

What is the total pressure in atmospheres of a gas mixture that contains \(1.0 \mathrm{g}\) of \(\mathrm{H}_{2}\) and \(8.0 \mathrm{g}\) of Ar in a 3.0-L container at \(27^{\circ} \mathrm{C}\) ? What are the partial pressures of the two gases?

Short answer

Total pressure is 5.74 atm; partial pressures are 4.10 atm for H2 and 1.64 atm for Ar.

Step-by-step solution

Convert Mass to Moles for H2

First, find the number of moles of hydrogen gas. The molar mass of \(\mathrm{H}_2\) is approximately 2 g/mol. Thus, \(1.0\,\mathrm{g} / 2 \,\mathrm{g/mol} = 0.5 \,\mathrm{moles}\) of \(\mathrm{H}_2\).

Convert Mass to Moles for Ar

Next, calculate the moles of argon gas. The molar mass of argon (Ar) is approximately 40 g/mol. So, \(8.0 \,\mathrm{g} / 40 \,\mathrm{g/mol} = 0.2 \,\mathrm{moles}\) of Ar.

Determine Total Moles and Use the Ideal Gas Law

The total moles of gas are the sum of the moles of \(\mathrm{H}_2\) and Ar: \(0.5 + 0.2 = 0.7\,\mathrm{moles}\). Use the ideal gas law \(PV = nRT\) to find the total pressure. \(R\) is 0.0821 L·atm/mol·K and \(T\) must be in Kelvin, so \(27^{\circ}\mathrm{C} = 300\,\mathrm{K}\). Rearrange the formula: \(P = \frac{nRT}{V}\).

Calculate Total Pressure

Substitute into the equation: \(P = \frac{0.7 \, \mathrm{moles} \times 0.0821 \, \mathrm{L} \cdot \mathrm{atm/mol} \cdot \mathrm{K} \times 300 \,\mathrm{K}}{3.0 \, \mathrm{L}} \approx 5.74\, \mathrm{atm}\).

Calculate Partial Pressure of H2

Use the mole fraction from Dalton's Law of Partial Pressures: \(P_{\mathrm{H}_2} = \chi_{\mathrm{H}_2} \cdot P_{\text{total}}\), where \(\chi_{\mathrm{H}_2} = \frac{0.5}{0.7}\) and \(P_{\text{total}} = 5.74\,\mathrm{atm}\). So, \(P_{\mathrm{H}_2} = 0.7143 \cdot 5.74 \approx 4.10\, \mathrm{atm}\).

Calculate Partial Pressure of Ar

Similar to \(\mathrm{H}_2\), calculate for argon: \(P_{\mathrm{Ar}} = \chi_{\mathrm{Ar}} \cdot P_{\text{total}}\), where \(\chi_{\mathrm{Ar}} = \frac{0.2}{0.7}\). So, \(P_{\mathrm{Ar}} = 0.2857 \cdot 5.74 \approx 1.64\,\mathrm{atm}\).

Key concepts

Partial Pressure

Partial pressure is the individual pressure exerted by a specific gas within a mixture of gases.
Each gas contributes to the total pressure proportional to its amount in the container. The total pressure is the sum of these individual pressures. Imagine you have several separate balloons, each filled with a different gas, and you merge them all into one larger balloon.
Each gas maintains its influence inside the larger balloon, pushing against the walls with its own partial pressure. The partial pressure can be determined using Dalton's Law of Partial Pressures. In our exercise, we found the partial pressures of hydrogen (\( P_{\mathrm{H}_2} \)) and argon (\( P_{\mathrm{Ar}} \)) gases by applying their respective mole fractions to the total pressure.

Moles of Gas

Moles are a fundamental concept in chemistry and crucial for understanding the behavior of gases. A mole is a unit to measure the amount of substance.
The number of moles can be determined using the formula: - \( ext{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \)For instance, in the exercise, to find the moles of hydrogen, we divided the mass of hydrogen by its molar mass. Knowing the moles of gases is instrumental in calculations involving reactions and properties like pressure and volume. Once you determine the moles of each gas, you can effectively use them in the ideal gas law to find other properties of the gas mixture.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures explains that in a gas mixture, the total pressure is the sum of the individual pressures that each gas would exert if it were alone in the container.
This is important since it allows us to calculate how much each gas contributes to the overall pressure. When you need to find out the pressure of a single type of gas within a mixture, Dalton's Law will guide you by using the formula: - \( P_{\text{gas}} = \chi_{\text{gas}} \cdot P_{\text{total}} \)Here, \( \chi_{\text{gas}} \) is the mole fraction of the gas. Using this law is straightforward once you have the moles and total pressure. In our solution, it helped us determine the individual pressures of hydrogen and argon by applying their mole fractions.

Gas Mixture

A gas mixture consists of different gases combined in one volume. These gases, while perhaps different, each follow the laws of physics described by the ideal gas law.
Gas mixtures behave predictably, with each gas contributing to the overall properties, like pressure and volume, of the mixture. In the exercise, we worked with a gas mixture of hydrogen (\( \mathrm{H}_2 \)) and argon (\( \mathrm{Ar} \)). By understanding the properties of these two gases and their behavior under the ideal gas law, we could address questions about total and partial pressures. Gas mixtures are found in many areas of science and industry, from the air we breathe to internal combustion engines, and handling their properties correctly is key to working with them.

Mole Fraction

The mole fraction is a way of expressing the concentration of a particular component in a mixture. It is the ratio of the number of moles of one component to the total number of moles in the mixture.
The formula for the mole fraction is: - \( \chi_{\text{gas}} = \frac{n_{\text{gas}}}{n_{\text{total}}} \)Where \( n_{\text{gas}} \) is the number of moles of the gas of interest and \( n_{\text{total}} \) is the total number of moles in the mixture. Knowing the mole fraction allows us to determine the contribution of each gas to the total pressure using Dalton's Law of Partial Pressures. In the exercise, we calculated the mole fraction for both hydrogen and argon to find out their individual pressures in the gas mixture.