Question

A mixture containing \(0.765 \mathrm{~mol} \mathrm{He}(g), 0.330 \mathrm{~mol} \mathrm{Ne}(g),\) and \(0.110 \mathrm{~mol} \mathrm{Ar}(g)\) is confined in a \(10.00-\mathrm{L}\) vessel at \(25^{\circ} \mathrm{C}\). 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 1.8559 \(atm\), 0.8077 \(atm\), and 0.2677 \(atm\) respectively, and the total pressure of the gas mixture is 2.9313 \(atm\).

Step-by-step solution

Convert temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin. We do this by adding 273.15 to the Celsius temperature. \(T(K) = T(°C) + 273.15\) \(T(K) = 25 + 273.15\) \(T(K) = 298.15\)

Calculate the partial pressures

Next, we'll find the partial pressures of the three gases (Helium, Neon, and Argon) using the Ideal Gas Law: \(P_i = \frac{n_i R T}{V}\), where \(P_i\) is the partial pressure of the gas, \(n_i\) is the number of moles of the gas, R is the ideal gas constant (0.0821 \(L\cdot atm/mol\cdot K\)), T is the temperature in Kelvin, and V is the volume of the vessel. For Helium: \(P_{He} = \frac{n_{He} R T}{V}\) \(P_{He} = \frac{0.765 \mathrm{~mol} \cdot 0.0821 \frac{L \cdot atm}{ mol \cdot K} \cdot 298.15 K}{10.00 L}\) \(P_{He} = 1.8559 \, atm\) For Neon: \(P_{Ne} = \frac{n_{Ne} R T}{V}\) \(P_{Ne} = \frac{0.330 \mathrm{~mol} \cdot 0.0821 \frac{L \cdot atm}{ mol \cdot K} \cdot 298.15 K}{10.00 L}\) \(P_{Ne} = 0.8077 \, atm\) For Argon: \(P_{Ar} = \frac{n_{Ar} R T}{V}\) \(P_{Ar} = \frac{0.110 \mathrm{~mol} \cdot 0.0821 \frac{L \cdot atm}{ mol \cdot K} \cdot 298.15 K}{10.00 L}\) \(P_{Ar} = 0.2677 \, atm\)

Calculate the total pressure

Now that we have the partial pressures of each gas, we can find the total pressure of the gas mixture using Dalton's Law of partial pressures: \(P_{total} = P_{He} + P_{Ne} + P_{Ar}\) \(P_{total} = 1.8559 \, atm + 0.8077 \, atm + 0.2677 \, atm\) \(P_{total} = 2.9313 \, atm\) So, the partial pressures of Helium, Neon, and Argon in the mixture are 1.8559 \(atm\), 0.8077 \(atm\), and 0.2677 \(atm\), respectively, and the total pressure of the gas mixture is 2.9313 \(atm\).

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law goes beyond plugging values into an equation— it’s essential for predicting how gases will behave under different conditions. The law is usually stated as \( PV = nRT \), where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant, and T is the temperature in Kelvin.

When it comes to practical applications, such as calculating the partial pressure of a gas in a mixture, the Ideal Gas Law lets us find out how much pressure one single gas contributes to the overall mixture. Remember, since gases in a mixture share the same volume and temperature, the only variable affecting each gas’s partial pressure is the amount of substance, or moles, of that specific gas.

To apply the law effectively, always ensure you convert the temperature to Kelvin, and use the value for R that matches the units of pressure and volume you need, as was demonstrated in the step-by-step solution for the exercise.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a crucial concept when dealing with gas mixtures. It states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases. Each gas in a mixture creates pressure as if it were alone in the container.

Mathematically, Dalton’s Law is expressed as \( P_{total} = P_1 + P_2 + P_3 + ... + P_n \), where each \( P \) represents the partial pressure of a gas in the mixture. The ease and beauty of this law shine when you realize it allows us to break down complex systems into simple, solvable parts. Find each gas's partial pressure — as if working with multiple miniature gas containers — and then add them up for the total pressure, just like in the solution for the given exercise.

This concept is particularly useful in a variety of scientific fields such as chemistry, environmental science, and respiratory physiology, where understanding the behavior of gas mixtures is critical.

Gas mixtures

Gas mixtures, like the ones often found in our atmosphere or industrial processes, consist of different gases occupying the same space. Unlike liquid or solid mixtures, which might show heterogeneity, gases mix uniformly due to their high kinetic energy and the space between gas particles.

In a mixture, each gas follows its own set of physical behaviors and properties, yet they don't interact with each other chemically if they are non-reactive. When we calculate the properties of gas mixtures, such as density or pressure, it's key to consider each particle type independently before summarising the mixture's overall behavior. The calculation of partial pressures is just one example where we see the individual contributions of each gas amalgamating to reveal the big picture of the gas mixture's behavior, reinforcing concepts like Dalton’s Law and Ideal Gas Law in the context of gas mixtures.