Question

A mixture containing \(0.477\) mol \(\mathrm{He}(g), 0.280\) mol \(\mathrm{Ne}(g)\), and \(0.110 \mathrm{~mol} \mathrm{Ar}(g)\) is confined in a \(7.00\) -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 He, Ne, and Ar in the mixture are \(1.63 \ \mathrm{atm}\), \(0.96 \ \mathrm{atm}\), and \(0.38 \ \mathrm{atm}\), respectively, and the total pressure of the mixture is \(2.97 \ \mathrm{atm}\).

Step-by-step solution

Write down the Ideal Gas Law formula and known values.

The Ideal Gas Law states that: PV = nRT where P is the pressure, V is the volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature in Kelvin. Known values: V = 7.00 L T = 25°C = 25 + 273.15 = 298.15 K R = 0.0821 L atm / (mol K) Moles of each gas: n_He = 0.477 mol n_Ne = 0.280 mol n_Ar = 0.110 mol

Calculate the partial pressure of each gas using the Ideal Gas Law.

To find the partial pressure of each gas, we must solve the Ideal Gas Law for pressure (P) and then substitute the known values for each gas. P = nRT / V For each gas, plug in the values of n, R, T, and V to determine the partial pressure: P_He = (0.477 mol) (0.0821 L atm / (mol K)) (298.15 K) / (7.00 L) P_Ne = (0.280 mol) (0.0821 L atm / (mol K)) (298.15 K) / (7.00 L) P_Ar = (0.110 mol) (0.0821 L atm / (mol K)) (298.15 K) / (7.00 L) Calculate these partial pressures to get: P_He = 1.63 atm P_Ne = 0.96 atm P_Ar = 0.38 atm

Calculate the total pressure of the mixture.

The total pressure of the mixture is the sum of the partial pressures of each gas component: P_total = P_He + P_Ne + P_Ar P_total = 1.63 atm + 0.96 atm + 0.38 atm Calculate the total pressure: P_total = 2.97 atm So, the partial pressures of He, Ne, and Ar in the mixture are 1.63 atm, 0.96 atm, and 0.38 atm, respectively, and the total pressure of the mixture is 2.97 atm.

Key concepts

Ideal Gas Law

The Ideal Gas Law is one of the most fundamental and insightful equations in the study of gases. It serves as a crucial tool for understanding the behavior of gases under various conditions and is represented by the formula:

\[ PV = nRT \]
where:

• \(P\) stands for the pressure of the gas,
• \(V\) is the volume it occupies,
• \(n\) is the number of moles of the gas,
• \(R\) is the universal gas constant, and
• \(T\) is the temperature in Kelvin.

By rearranging this equation, we can solve for any one of the variables in terms of the others. In the case of our exercise, we're interested in finding the partial pressures of each gas in a mixture. We do this by using the mole count of each individual gas while keeping the volume and temperature constant for the mixture as a whole. With careful manipulation, we add clarity to problems involving mixed gases and their behaviors, making it easier for students to grasp complex concepts.

Partial Pressures of Gases

The concept of partial pressure is crucial when dealing with gas mixtures. It refers to the pressure each gas in a mixture would exert if it were alone in the container at the same temperature. This concept allows us to treat each gas independently, even when they're mixed together.

To calculate the partial pressure for each gas, we adjust the Ideal Gas Law as follows:

\[ P_{\text{gas}} = \frac{n_{\text{gas}}RT}{V} \]
In this formula, we use \(n_{\text{gas}}\), which is the mole count for the specific gas that we're interested in. By applying the equation separately to helium (He), neon (Ne), and argon (Ar), we can establish their individual impacts on the mixture's total pressure. Following the steps from our problem's solution, substituting the respective values for each gas, we've determined their partial pressures which lead us to a better understanding of gas interactions within a mixture.

Total Pressure of Gas Mixture

After grasping the idea of partial pressures, the total pressure of a gas mixture is simply the sum of all individual gases' partial pressures. This is expressed in Dalton's Law of Partial Pressures which asserts that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures.

We calculate the total pressure using the formula:

\[ P_{\text{total}} = P_{\text{He}} + P_{\text{Ne}} + P_{\text{Ar}} \]
Each \(P_{\text{gas}}\) represents the partial pressure of one gas in the mixture. By adding together the pressure from each gas calculated earlier, we get the total pressure exerted by the mixture inside the container. Understanding this principle guides students through the complexities of pressure dynamics in gas mixtures, which is depicted in real-world applications ranging from scuba diving to chemical engineering processes.