Problem 103 - Chemistry The Science in Context

Question

What pressure is exerted by a gas mixture containing \(2.00 \mathrm{g}\) of \(\mathrm{H}_{2}\) and \(7.00 \mathrm{g}\) of \(\mathrm{N}_{2}\) at \(273^{\circ} \mathrm{C}\) in a \(10.0 \mathrm{L}\) container? What is the contribution of \(\mathrm{N}_{2}\) to the total pressure?

Step-by-Step Solution

Answer

Based on the given information, calculate the total pressure exerted by a gas mixture of Hydrogen (H2) and Nitrogen (N2) and the individual pressure contribution of Nitrogen (N2). Mass of Hydrogen (H2): 2.00 g Mass of Nitrogen (N2): 7.00 g Temperature: 273 °C Volume of the container: 10.0 L Provide the answer in atmospheres (atm).

Step 1: Calculate moles of given gases

First, we need to determine the moles of each gas present in the gas mixture. We are given: Mass of H2 = 2.00 g Molar mass of H2 = 2 g/mol Moles of H2 = (Mass of H2) / (Molar mass of H2) n(H2) = 2.00 g / 2 g/mol = 1.00 mol Now, for N2: Mass of N2 = 7.00 g Molar mass of N2 = 28 g/mol Moles of N2 = (Mass of N2) / (Molar mass of N2) n(N2) = 7.00 g / 28 g/mol = 0.25 mol

Step 2: Calculate the Individual pressure of H2 and N2 using the ideal gas law

Now, we have to find the pressure exerted by each gas using the ideal gas law: PV = nRT R (Gas constant) = 0.0821 L atm/(mol K) Given: Volume (V) = 10.0 L Temperature = 273 °C + 273.15 = 546.15 K First, we will find out the pressure exerted by H2. P(H2) V = n(H2) R T P(H2) = n(H2) R T / V P(H2) = (1.00 mol)(0.0821 L atm/mol K)(546.15 K) / (10.0 L) P(H2) ≈ 4.49 atm Now, the pressure exerted by N2. P(N2) V = n(N2) R T P(N2) = n(N2) R T / V P(N2) = (0.25 mol)(0.0821 L atm/mol K)(546.15 K) / (10.0 L) P(N2) ≈ 1.12 atm

Step 3: Calculate the total pressure

The total pressure exerted by the gas mixture is the sum of the pressures of H2 and N2. P(total) = P(H2) + P(N2) P(total) = 4.49 atm + 1.12 atm P(total) ≈ 5.61 atm So, the total pressure exerted by the gas mixture is approximately 5.61 atm. The contribution of N2 to the total pressure is 1.12 atm.

Key Concepts

Gas Pressure Calculations

When dealing with gas pressure, particularly in a mixture, we rely on the ideal gas law: \( PV = nRT \). This formula helps us determine the pressure that a gas or a mixture of gases exerts within a container. In this equation:

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

The key to solving gas pressure problems lies in understanding that pressure comes from the collisions of gas particles within the container walls. By knowing the moles of each gas and the conditions (like temperature and volume), we can calculate the pressure each gas exerts. For instance, in a container with different gases, each gas contributes individually to the total pressure. These contributions are summed up to determine the overall pressure of the gas mixture.

Mole Calculation

The concept of moles is fundamental when we talk about gas reactions and calculations. The mole is a unit that measures the amount of a substance. To find the number of moles of a gas, we use the formula:

• \( \, \text{Moles} = \frac{\text{Mass of the substance}}{\text{Molar mass of the substance}} \, \)

Molar mass is specific to each gas and denotes the mass of one mole of that substance. For example, the molar mass of hydrogen \(\mathrm{H}_2\) is 2 g/mol, while for nitrogen \(\mathrm{N}_2\), it is 28 g/mol. By calculating moles, we can determine the number of particles in a gas sample. This is essential in pressure calculations because the pressure exerted by a gas depends heavily on the number of particles present. More particles generally mean more collisions and, thus, higher pressure.

Gas Mixtures

Gas mixtures occur when different gases share a common container, contributing to the overall behavior of the gas system. A key property of gas mixtures is that each gas behaves independently, following the ideal gas law, and contributes to the total pressure through its partial pressure.

• Partial Pressure: This is the pressure exerted by a single gas in a mixture. It can be calculated using the ideal gas law for each gas separately.
• Total Pressure: The sum of all partial pressures of gases in the mixture. In our example, hydrogen and nitrogen each contribute separately to the total pressure, which is determined by summing their individual pressures.

Understanding the behavior of gas mixtures is crucial in various scientific fields. It helps in predicting how these mixtures will react under different conditions. By knowing each component's contribution, one can easily calculate the necessary values for industrial applications or scientific research.