Question

A mixture containing \(0.765 \mathrm{~mol} \mathrm{He}(\mathrm{g}), 0.330 \mathrm{~mol} \mathrm{Ne}(\mathrm{g})\), and \(0.110 \mathrm{~mol} A r(g)\) is confined in a \(10.00-\mathrm{L}\) vessel at \(25^{\circ} \mathrm{C}\). (a) Calculate the partial pressure of each of the gases in the mixture. (b) Calculate the total pressure of the mixture.

Short answer

The partial pressures of helium, neon, and argon in the mixture are \(1.885\,\mathrm{atm}\), \(0.811\,\mathrm{atm}\), and \(0.270\,\mathrm{atm}\), respectively. The total pressure of the mixture is \(2.966\,\mathrm{atm}\).

Step-by-step solution

Collect given information

From the exercise, we are given the following information: - Moles of He: 0.765 mol - Moles of Ne: 0.330 mol - Moles of Ar: 0.110 mol - Volume of vessel: 10.00 L - Temperature: 25 °C First, convert the temperature to Kelvin: \(T = 25 + 273.15 = 298.15\,\mathrm{K}\).

Calculate partial pressures using ideal gas law

The ideal gas law is given by the equation \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. We'll use the ideal gas law to calculate the partial pressure of each gas by substituting the values for each gas: For helium, \(P_{\mathrm{He}} = \dfrac{n_{\mathrm{He}}RT}{V} = \dfrac{0.765\,\mathrm{mol} \cdot 0.0821\,\dfrac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}} \cdot 298.15\,\mathrm{K}}{10.00\,\mathrm{L}} = 1.885 \, \mathrm{atm} \) For neon, \(P_{\mathrm{Ne}} = \dfrac{n_{\mathrm{Ne}}RT}{V} = \dfrac{0.330\,\mathrm{mol} \cdot 0.0821\,\dfrac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}} \cdot 298.15\,\mathrm{K}}{10.00\,\mathrm{L}} = 0.811\, \mathrm{atm}\) For argon, \(P_{\mathrm{Ar}} = \dfrac{n_{\mathrm{Ar}}RT}{V} = \dfrac{0.110\,\mathrm{mol} \cdot 0.0821\,\dfrac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}} \cdot 298.15\,\mathrm{K}}{10.00\,\mathrm{L}} = 0.270\, \mathrm{atm}\) These are the partial pressures of each of the gases in the mixture.

Calculate the total pressure using Dalton's Law

According to Dalton's Law of partial pressures, the total pressure of a mixture of gases is the sum of the partial pressures of its individual gases: Total pressure \(P_{total} = P_{He} + P_{Ne} + P_{Ar}\) Substitute the values calculated in step 2: Total pressure \(P_{total} = 1.885\,\mathrm{atm} + 0.811\,\mathrm{atm} + 0.270\,\mathrm{atm} = 2.966\,\mathrm{atm}\) So, the partial pressures of helium, neon, and argon are 1.885 atm, 0.811 atm, and 0.270 atm, respectively, and the total pressure of the mixture is 2.966 atm.

Key concepts

Understanding the Ideal Gas Law

The ideal gas law is a fundamental principle used to describe the behavior of gases under various conditions. It is expressed as the formula \(PV = nRT\), where:\\(P\) represents the pressure of the gas,\(V\) indicates its volume,\(n\) is the number of moles of the gas,\(R\) is the ideal gas constant, approximately \(0.0821 \dfrac{L\cdot atm}{mol\cdot K}\),and \(T\) is the temperature measured in Kelvin.The law assumes that gases are composed of particles that move randomly and do not interact with one another, fitting most real gases under normal conditions.

When solving problems involving the ideal gas law, one key step is making sure to have all units consistent, notably temperature in Kelvin. To calculate partial pressures in a gas mixture, you can rearrange the ideal gas law to\(P = \dfrac{nRT}{V}\), which allows you to compute the pressure contributed by each individual gas in the mixture by knowing its amount (moles), the temperature, and the volume of the container. Understanding the ideal gas law is crucial as it lays the foundation for exploring more complex gas laws and mixtures.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is an essential concept when studying gaseous mixtures. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases.

A partial pressure of a gas is the pressure that gas would exert if it alone occupied the entire volume of the mixture at the same temperature. To find the partial pressure of each gas, we use the ideal gas law for that gas, accounting for its specific moles, volume, and temperature.

For example, in a mixture, helium (He), neon (Ne), and argon (Ar) all contribute to the total pressure. After calculating each gas's partial pressure, simply add them together to find the total pressure:
\(P_{total} = P_{He} + P_{Ne} + P_{Ar}\). This is useful when working with gas mixtures in sealed containers or predicting how gases will behave when combined. It's a principle widely applied in fields ranging from meteorology to respiratory science.

The Composition of Gaseous Mixtures

Gaseous mixtures, such as the one presented in our exercise, are common in both natural and industrial processes. The atmosphere itself is a large gaseous mixture. A gas mixture has distinct partial pressures contributed by each gas constituting the mixture, and understanding this is key in various applications, including environmental engineering and materials science.

For a gas within a mixture, it behaves independently of the other gases, but the total effect on the mixture is cumulative. This independent behavior allows the use of the ideal gas law to calculate the pressure it would exert alone, known as its partial pressure.

Each gas's ability to exert pressure, its molar amount, and the role of temperature and volume as described by the ideal gas law, inform the processes of diffusion, effusion, and reaction rates. As such, they are not only important for theoretical calculations but also for practical applications, from designing safe chemical storage to understanding respiratory physiology.