Question

A tank contains a mixture of 3.0 mol of \(\mathrm{N}_{2}, 2.0 \mathrm{mol}\) of \(\mathrm{O}_{2},\) and \(1.0 \mathrm{mol}\) of \(\mathrm{CO}_{2}\) at \(25^{\circ} \mathrm{C}\) and a total pressure of 10.0 atm. Calculate the partial pressure (in torr) of each gas in the mixture.

Short answer

The partial pressures of the gases in the mixture are as follows: \(P_{N_2(torr)} = 3800\,torr\), \(P_{O_2(torr)} = 2533.33\,torr\), and \(P_{CO_2(torr)} = 1266.67\,torr\).

Step-by-step solution

Calculate the mole fraction of each gas

To find the partial pressure of each gas, we first need to calculate the mole fraction of each gas in the mixture. The mole fraction is the ratio of the moles of a particular gas to the total moles of all gases in the mixture. Mole fraction of \(\mathrm{N}_{2}:\) \[\chi_{N_2} = \frac{n_{N_2}}{n_{total}} \] Mole fraction of \(\mathrm{O}_{2}:\) \[\chi_{O_2} = \frac{n_{O_2}}{n_{total}} \] Mole fraction of \(\mathrm{CO}_{2}:\) \[\chi_{CO_2} = \frac{n_{CO_2}}{n_{total}} \]

Calculate the total moles of gas in the mixture

Add the moles of each gas to find the total moles in the mixture. \[n_{total} = n_{N_2} + n_{O_2} + n_{CO_2} \]

Calculate the mole fractions of each gas

Substitute the given values and calculate the mole fractions for each gas. \[\chi_{N_2} = \frac{3.0}{3.0+2.0+1.0} \] \[\chi_{O_2} = \frac{2.0}{3.0+2.0+1.0} \] \[\chi_{CO_2} = \frac{1.0}{3.0+2.0+1.0} \]

Calculate the partial pressure of each gas

Use Dalton's Law to calculate the partial pressure of each gas. The partial pressure of each gas is the product of the mole fraction of the gas and the total pressure. \[P_{N_2} = \chi_{N_2} \times P_{total} \] \[P_{O_2} = \chi_{O_2} \times P_{total} \] \[P_{CO_2} = \chi_{CO_2} \times P_{total} \]

Convert the partial pressure from atm to torr

The conversion factor between atm and torr is 1 atm = 760 torr. Convert the partial pressure of each gas to torr. \[P_{N_2(torr)} = P_{N_2(atm)} \times 760 \] \[P_{O_2(torr)} = P_{O_2(atm)} \times 760 \] \[P_{CO_2(torr)} = P_{CO_2(atm)} \times 760 \]

Calculate the final partial pressures in torr

Substitute the calculated mole fractions and total pressure in the equations in Steps 4 and 5, then compute the partial pressures in torr. \[P_{N_2(torr)} = (\frac{3.0}{6.0}) \times 10.0 \times 760 \] \[P_{O_2(torr)} = (\frac{2.0}{6.0}) \times 10.0 \times 760 \] \[P_{CO_2(torr)} = (\frac{1.0}{6.0}) \times 10.0 \times 760 \]

Key concepts

Dalton's Law of Partial Pressures

Understanding Dalton's Law of Partial Pressures can help students unravel the mystery of gas mixtures. This principle asserts that within a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. Each gas behaves as if it occupies the container alone.

For an illustrative example, let's consider a tank with multiple gases. If you know each gas's pressure if it were alone in the tank (its partial pressure), you can add all these pressures up to get the total pressure in the tank. Dalton's Law simplifies the analysis of gas behavior in mixtures and is pivotal in determining the pressures of individual gases in a mixture, which is central to a variety of scientific and industrial processes.

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 that component to the total number of moles of all components in the mixture.

Understanding Mole Fraction

Consider a cake recipe as an analogy. If the recipe calls for 3 cups of flour, 2 cups of sugar, and 1 cup of cocoa, we can say the 'mole fraction' of flour is 3 out of the total 6 cups (also considering sugar and cocoa as ingredients), or 0.5. This concept is essential when applying Dalton's Law because the pressure each gas contributes is directly proportional to its mole fraction in the gas mixture.

Gas Mixture

A gas mixture is a system containing two or more different gases that haven't chemically reacted with each other. Each constituent gas in a mixture retains its own chemical identity and physical properties.

Characteristics of Gas Mixtures

Understanding the behavior of gas mixtures is crucial because the air we breathe is itself a mixture of gases, primarily nitrogen and oxygen. Other applications that rely on the understanding of gas mixtures include industrial gas production, scuba diving, and the management of controlled environments such as greenhouses.

Atmospheric Pressure Units

Atmospheric pressure is a measure of the force exerted by the atmosphere on the earth's surface. It is commonly expressed in units such as atmospheres (atm), torr, or Pascals (Pa).

One atmosphere is defined as the pressure exerted by a 760 mm column of mercury at sea level and at 0 degrees Celsius, which corresponds to 101,325 Pascals or 760 torr. Understanding these units is fundamental in fields like meteorology, aviation, and various scientific research areas.