Question

A mixture of \(2.10 \times 10^{-3} \mathrm{g}\) of \(\mathrm{O}_{2}, 3.88 \times 10^{-3} \mathrm{mol}\) of \(\mathrm{N}_{2},\) and \(5.25 \times 10^{20}\) molecules of \(\mathrm{CO}\) are placed into a vessel of volume 5.25 L at \(12.5^{\circ} \mathrm{C}\) a. Calculate the total pressure in the vessel. b. Calculate the mole fractions and partial pressures of each gas.

Short answer

a. The total pressure in the vessel is 0.2136 atm. b. The mole fractions and partial pressures of each gas are: - O2: mole fraction = 0.0141, partial pressure = 0.0030 atm. - N2: mole fraction = 0.8359, partial pressure = 0.1785 atm. - CO: mole fraction = 0.1875, partial pressure = 0.0400 atm.

Step-by-step solution

Find the number of moles of each gas

First, we need to find the number of moles of O2 and CO, as we're already given the number of moles of N2. For O2: \(n = \frac{mass}{molar\,mass}\) Given, mass = \(2.10 \times 10^{-3}\,g\) and molar mass of O2 = \(32\,g/mol\) \(n_{O_2} = \frac{2.10 \times 10^{-3}}{32} = 6.563 \times 10^{-5}\,mol\) For CO: \(n = \frac{number\,of\,molecules}{Avogadro's\,number}\) Given, number of molecules = \(5.25 \times 10^{20}\) and Avogadro's number = \(6.022 \times 10^{23}\,molecules/mol\) \(n_{CO} = \frac{5.25 \times 10^{20}}{6.022 \times 10^{23}} = 8.717 \times 10^{-4}\,mol\)

Calculate the total number of moles

Add the number of moles of O2, N2, and CO to find the total number of moles in the vessel. \(n_{total} = n_{O_2} + n_{N_2} + n_{CO}\) \(n_{total} = 6.563 \times 10^{-5} + 3.88 \times 10^{-3} + 8.717 \times 10^{-4}\) \(n_{total} = 0.004645\,mol\)

Calculate the total pressure in the vessel

Use the Ideal Gas Law to find the total pressure in the vessel. \(PV = nRT\) Convert the given temperature to Kelvin: \(T\,(\mathrm{K}) = 12.5 + 273.15 = 285.65\, \mathrm{K}\) For ideal gas constant, we use the value in the appropriate unit: \(R = 0.08206\, \frac{\mathrm{L} \cdot\mathrm{atm}}{\mathrm{mol} \cdot\mathrm{K}}\) Solve for P: \(P = \frac{nRT}{V}\) \(P = \frac{0.004645\, mol \times 0.08206\, \frac{\mathrm{L} \cdot\mathrm{atm}}{\mathrm{mol}\cdot\mathrm{K}} \times 285.65\, \mathrm{K}}{5.25\, \mathrm{L}}\) \(P = 0.2136\, \mathrm{atm}\) a. The total pressure in the vessel is 0.2136 atm.

Calculate the mole fractions of each gas

Use the formula for mole fraction to find the mole fractions of O2, N2, and CO. \(\chi_{O_2} = \frac{n_{O_2}}{n_{total}} = \frac{6.563 \times 10^{-5}}{0.004645} = 0.0141\) \(\chi_{N_2} = \frac{n_{N_2}}{n_{total}} = \frac{3.88 \times 10^{-3}}{0.004645} = 0.8359\) \(\chi_{CO} = \frac{n_{CO}}{n_{total}} = \frac{8.717 \times 10^{-4}}{0.004645} = 0.1875\)

Calculate the partial pressures of each gas

Use the formula for partial pressure to find the partial pressures of O2, N2, and CO. \(P_{O_2} = \chi_{O_2} \times P_{total} = 0.0141 \times 0.2136\, \mathrm{atm} = 0.0030\, \mathrm{atm}\) \(P_{N_2} = \chi_{N_2} \times P_{total} = 0.8359 \times 0.2136\, \mathrm{atm} = 0.1785\, \mathrm{atm}\) \(P_{CO} = \chi_{CO} \times P_{total} = 0.1875 \times 0.2136\, \mathrm{atm} = 0.0400\, \mathrm{atm}\) b. The mole fractions and partial pressures of each gas are as follows: - O2: mole fraction = 0.0141, partial pressure = 0.0030 atm. - N2: mole fraction = 0.8359, partial pressure = 0.1785 atm. - CO: mole fraction = 0.1875, partial pressure = 0.0400 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a critical tool in understanding and predicting the behavior of gases under various conditions. It connects four key properties of a gas: pressure (P), volume (V), the number of moles (n), and temperature (T) using the equation:
\(PV = nRT\), where R is the ideal gas constant. This law assumes that the gas particles have no volume and no intermolecular forces, which is a good approximation for most gases at high temperatures and low pressures.
In the context of our exercise, the Ideal Gas Law allows us to calculate the total pressure in a mixed gas system. By knowing the amount and temperature of the gases, along with the volume of the container, students can easily find out the pressure exerted by the combined gases. The relation forms the core of predicting gas behavior, vital for several fields such as chemistry, physics, and engineering. Moreover, this principle is key in understanding everyday phenomena, like why a balloon inflates or why a tire pressure changes with temperature.

Mole Fraction Calculations

Mole fraction calculations are instrumental in quantifying the composition of gas mixtures. The mole fraction, represented by \(\chi\), indicates the proportion of a component in relation to the total mixture. It is defined as the number of moles of a particular substance divided by the total number of moles in the mixture.
The formula for calculating mole fraction is: \(\chi_i = \frac{n_i}{n_{\text{total}}}\), where \(\chi_i\) is the mole fraction of gas 'i', \(n_i\) is the number of moles of gas 'i', and \(n_{\text{total}}\) is the total number of moles in the mixture.
By understanding mole fractions, students can solve part b of the exercise, which asks for the mole fractions and partial pressures of each gas. Mole fractions are particularly useful because they remain constant with changes in pressure and temperature, making them a reliable way to describe the composition of gas mixtures in various conditions.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), pertains to the number of units, often atoms or molecules, in one mole of any substance. In the context of gases, Avogadro's principle states that equal volumes of gases, at the same temperature and pressure, contain the same number of particles. This is a fundamental cornerstone of modern chemistry, especially when dealing with gas mixtures and reactions.
In the exercise, Avogadro's number enables the calculation of moles of \(\mathrm{CO}\) from the given number of molecules. It bridges the gap between the microscopic scale of individual molecules and the macroscopic scale used in chemical reactions and processes. Grasping Avogadro's concept supports students in moving from theoretical computations to practical applications in scientific research and industry, ensuring a comprehensive understanding of gas behaviors at the molecular level.