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

Total pressure is approximately 5.72 atm; partial pressures are 4.08 atm for hydrogen and 1.64 atm for argon.

Step 1: Convert Mass to Moles

First, we need to convert the given mass of each gas to moles. The molar mass of hydrogen, \(\text{H}_2\), is \(2.02 \text{ g/mol}\), and the molar mass of argon, \(\text{Ar}\), is \(39.95 \text{ g/mol}\). For hydrogen, the moles \(n_{\text{H}_2}\) is \(\frac{1.0}{2.02} \approx 0.495\) moles. For argon, the moles \(n_{\text{Ar}}\) is \(\frac{8.0}{39.95} \approx 0.200\) moles.

Step 2: Use the Ideal Gas Law for Each Gas

We use the Ideal Gas Law \(PV = nRT\) to find the partial pressures. The gas constant \(R\) is \(0.0821 \text{ L atm/mol K}\), and the temperature \(27^\circ \text{C}\) must be converted to Kelvin by adding 273, giving us \(T = 300 \text{ K}\).

Step 3: Calculate Partial Pressure of Hydrogen

For hydrogen, using \(n_{\text{H}_2} = 0.495\) moles, and \(V = 3.0\, \text{L}\), plug into the ideal gas law: \[P_{\text{H}_2} = \frac{n_{\text{H}_2}RT}{V} = \frac{0.495 \times 0.0821 \times 300}{3.0} \approx 4.08 \text{ atm}\]

Step 4: Calculate Partial Pressure of Argon

For argon, using \(n_{\text{Ar}} = 0.200\) moles, and \(V = 3.0\, \text{L}\), apply the ideal gas law: \[P_{\text{Ar}} = \frac{n_{\text{Ar}}RT}{V} = \frac{0.200 \times 0.0821 \times 300}{3.0} \approx 1.64 \text{ atm}\]

Step 5: Calculate Total Pressure

The total pressure in the container is the sum of the partial pressures. Thus, \[P_{\text{total}} = P_{\text{H}_2} + P_{\text{Ar}} = 4.08 + 1.64 \approx 5.72 \text{ atm}\]

Key Concepts

Partial Pressure

In any gas mixture, the partial pressure of a gas refers to the pressure that gas would exert if it alone occupied the entire volume of the container.
This concept is crucial in understanding gas behavior in a mixture and is based on Dalton's Law of Partial Pressure. According to this law, each gas in a mixture contributes to the total pressure independently of the others.
To find the partial pressure, we use the Ideal Gas Law, where the formula is \[P = \frac{nRT}{V}\]. Here, \(P\) refers to the partial pressure, \(n\) is the moles of the gas, \(R\) is the gas constant \(0.0821 \text{ L atm/mol K}\), \(T\) is the temperature in Kelvin, and \(V\) is the volume in liters.
When the partial pressures of all gases in the mixture are summed, we obtain the total pressure. In the case of our example with hydrogen and argon, the individual partial pressures were determined first, then summed to give the total pressure of the gas mixture.

Moles Calculation

Calculating moles is an essential step when dealing with gases in chemistry. Moles provide a way to quantify the amount of a substance, allowing chemists to compare substances, especially when dealing with reactions and mixtures.
To calculate the moles of a gas, you can use the formula \[n = \frac{\text{mass}}{\text{molar mass}}\], where \(n\) is the number of moles, the mass is given in grams, and the molar mass is the atomic or molecular weight of the gas in g/mol.
For instance, in our example, we calculated the moles for hydrogen and argon using their respective molar masses of \(2.02 \text{ g/mol}\) for \(\text{H}_2\) and \(39.95 \text{ g/mol}\) for \(\text{Ar}\). This conversion from mass to moles is critical because, in gas equations like the Ideal Gas Law, we deal universally in moles.

Gas Mixtures

A gas mixture is a system consisting of more than one type of gas. The behavior of gas mixtures is often predictable and dictated by known gas laws.
In a gas mixture, each component has its own partial pressure which contributes to the total pressure observed in the mixture. The Ideal Gas Law is an excellent tool for understanding such mixtures, as it applies to each gas within the mixture independently.
Gas mixtures are common in various scientific and industrial applications. The air we breathe is a natural gas mixture primarily made of nitrogen, oxygen, and small amounts of other gases.
For our problem, understanding the concept of gas mixtures allowed us to apply Dalton's Law and the Ideal Gas Law, separately evaluating hydrogen and argon and then combining them to comprehend the system as a whole.