Problem 37 - Chemistry and Chemical Reactivity

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 \(\mathrm{Ar}\) in a \(3.0-\mathrm{L}\) container at \(27^{\circ} \mathrm{C} ?\) What are the partial pressures of the two gases?

Step-by-Step Solution

Answer

The total pressure is 5.70 atm. Partial pressures: \(P_{\mathrm{H}_2} = 4.06\) atm, \(P_{\mathrm{Ar}} = 1.64\) atm.

Step 1: Convert Mass to Moles

First, convert the mass of each gas to moles. The molar mass of hydrogen, \(\mathrm{H}_{2}\), is approximately 2.02 g/mol, and the molar mass of argon, \(\mathrm{Ar}\), is approximately 39.95 g/mol.For \(\mathrm{H}_{2}\):\[\text{Moles of } \mathrm{H}_{2} = \frac{1.0 \text{ g}}{2.02 \text{ g/mol}} = 0.495 \text{ mol}\]For \(\mathrm{Ar}\):\[\text{Moles of } \mathrm{Ar} = \frac{8.0 \text{ g}}{39.95 \text{ g/mol}} = 0.200 \text{ mol}\]

Step 2: Use Ideal Gas Law for Total Pressure

Use the ideal gas law \(PV = nRT\) to find the total pressure. Sum the moles found in Step 1 for total moles of gas, and use \(R = 0.0821\) L·atm/(mol·K) and temperature \(T = 27^\circ C = 300.15\) K.\[P = \frac{nRT}{V}\]Total moles, \(n_t = 0.495 + 0.200 = 0.695 \text{ mol}\)\[P = \frac{0.695 \text{ mol} \times 0.0821 \text{ L·atm/(mol·K)} \times 300.15 \text{ K}}{3.0 \text{ L}} = 5.70 \text{ atm}\]

Step 3: Calculate Partial Pressures

Use Dalton's Law of Partial Pressures; the partial pressure of a gas in a mixture is the total pressure times the mole fraction of the gas.For \(\mathrm{H}_{2}\):\[P_{\mathrm{H}_2} = 5.70 \text{ atm} \times \frac{0.495}{0.695} = 4.06 \text{ atm}\]For \(\mathrm{Ar}\):\[P_{\mathrm{Ar}} = 5.70 \text{ atm} \times \frac{0.200}{0.695} = 1.64 \text{ atm}\]

Key Concepts

Partial Pressure

Partial pressure is the pressure contributed by a single type of gas in a mixture of gases. In a container, each gas behaves as if it is alone, exerting its own pressure regardless of other gases present. The concept is crucial when dealing with gas mixtures because it allows us to determine the amount of a specific gas in a container when mixed with other gases. For each gas, its partial pressure can be calculated by knowing its mole fraction and the total pressure of the gas mixture. This information allows chemists to tailor conditions for experimental procedures involving gases.

Dalton's Law of Partial Pressures

Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas. Mathematically, it can be written as: \[ P_{total} = P_{1} + P_{2} + P_{3} + ext{...} \] where each \(P \) represents the partial pressure of a different gas.In scenarios where multiple gases are contained together, this law is especially helpful. By identifying the individual partial pressures, one can derive the overall pressure in the system. This principle underscores the idea that gases in a mixture occupy the same volume but contribute differently to the total pressure based on their mole fractions.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's a periodic table property that helps chemists convert between mass and moles, making it easier to use chemistry in quantitative analyses. For example, hydrogen (\(\mathrm{H}_{2} \)) has a molar mass of approximately 2.02 g/mol, and argon (\(\mathrm{Ar} \)) possesses a molar mass of about 39.95 g/mol. Knowing these values allows us to calculate the amount of substance in moles from a given mass, as seen in the example above where masses of hydrogen and argon are converted to moles using their respective molar masses. This conversion is foundational in using the ideal gas law for further calculations.

Mole Fraction

Mole fraction is a way of expressing the ratio of moles of one component in a mixture to the total moles of all components. It provides insight into the composition of a mixture. The mole fraction, \( X \), is calculated as: \[ X_A = \frac{n_A}{n_{total}} \] where \( n_A \) is the moles of the component and \( n_{total} \) is the total moles of the mixture. In the exercise, the mole fraction is utilized to determine each gas's contribution to the total pressure using Dalton's Law of Partial Pressures. Understanding mole fractions aids in grasping why different gases have varying impacts on total pressure.