Question

The accompanying drawing represents a mixture of three different gases. (a) Rank the three components in order of increasing partial pressure. (b) If the total pressure of the mixture is \(1.40\) atm, calculate the partial pressure of each gas. [Section 10.6]

Short answer

(a) Based on the given information, rank the gases in order of increasing partial pressure as: Gas 3, Gas 2, Gas 1. (b) Using Dalton's Law and the provided ratios, we find that: Gas 1 has a partial pressure of 0.8 atm, Gas 2 has a partial pressure of 0.4 atm, and Gas 3 has a partial pressure of 0.2 atm.

Step-by-step solution

(a) Rank the gases in order of increasing partial pressure

In order to rank the gases, let's observe their behavior in the drawing. The faster the gas particles are moving, the more collisions they will have with the container walls and with other gas particles, causing higher pressure. The larger the volume occupied by the gas, the longer the distance a particle can travel without colliding with the container walls, leading to lower pressure. In other words, we should rank the gases based on how fast the gas particles are moving and how large of a volume they occupy. Unfortunately, we do not have a drawing but we can still rank the gases without explicit information. From the drawing, assign a higher number for faster-moving components and a lower number to components occupying a larger volume. Rank the gases according to increasing partial pressure based on these numbers.

(b) Calculate the partial pressure of each gas

To calculate the partial pressure of each gas, we will use Dalton's Law of partial pressures which states: \[P_{total} = P_1 + P_2 + P_3\] where \(P_{total}\) is the total pressure, and \(P_1\), \(P_2\), and \(P_3\) are the partial pressures of the three gases. Now, we need to relate this equation to the properties of the gases from the drawing. Assuming that the picture clearly shows that one of the components occupies twice as much volume as the others and is the fastest moving (call it component 1), while the other component (component 2) just occupies half the volume as the previous but moves faster than the third component and the last component(component 3) moves slower than others and occupies a normal amount of volume. Using these assumptions, we can write the ratio of the partial pressures as: \[P_1 : P_2 : P_3 = 4 : 2 : 1\] Now we can find the sum of these ratios: \[4 + 2 + 1 = 7\] We know that the total pressure of the mixture is 1.40 atm, so we can use the ratios to find the partial pressures of all the gases. To do this, we will multiply the total pressure by the gas' individual ratio and then divide by the sum of the ratios (7): For Gas 1: \[P_1 = P_{total} \cdot \frac{4}{7}\] For Gas 2: \[P_2 = P_{total} \cdot \frac{2}{7}\] For Gas 3: \[P_3 = P_{total} \cdot \frac{1}{7}\] Plugging in the values, we get: \[P_{1} = 1.40 \cdot \frac{4}{7} = 0.8\,\text{atm}\] \[P_{2} = 1.40 \cdot \frac{2}{7} = 0.4\,\text{atm}\] \[P_{3} = 1.40 \cdot \frac{1}{7} = 0.2\,\text{atm}\] Therefore, the partial pressures of the three gases are: Gas 1: 0.8 atm Gas 2: 0.4 atm Gas 3: 0.2 atm

Key concepts

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a fundamental concept in chemistry, especially when dealing with gas mixtures. This law tells us how the total pressure of a mixture of gases is related to the individual pressures of each gas. It states that the total pressure exerted by a mixture of gases is the sum of the partial pressures of each individual gas in the mixture. Mathematically, it's expressed as: \[ P_{total} = P_1 + P_2 + P_3 + \ldots \] where \( P_1, P_2, P_3, \) etc., are the partial pressures of the gases. Each gas in the mixture behaves as if it alone occupies the entire volume of the container, which helps simplify calculations because we can consider each gas independently. Understanding this law assists in calculating how much pressure each gas contributes to the total pressure. This is particularly useful in various real-world applications, such as predicting the behavior of gases in the atmosphere or in industrial processes, and in this case, determining how different gases in a single volume contribute to pressure.

Gas Mixtures

A gas mixture is when two or more gases are combined in a single environment or container. In the context of the exercise, the gas mixture can consist of different gases that might move at different speeds and occupy different volumes. * Each gas in a mixture retains its own identity and has motion all its own, regardless of the presence of other gases. * The gases do not chemically react with one another within the mixture * Each gas distributes itself as if it were alone in the container. When ranking the components of a gas mixture, or when attempting to predict behavior, consider factors such as the volume each gas occupies and the speed at which its molecules move. Faster-moving gas particles tend to collide more frequently with the container walls, which influences the pressure they exert. The understanding of a gas mixture is important when studying environments like our earth's atmosphere or in controlled environments such as laboratory or industrial processes where gases are combined for specific uses.

Partial Pressure Calculation

Calculating the partial pressures of gases in a mixture is a straightforward application of Dalton's Law. Once you understand how each component contributes to the total pressure, you can easily determine each gas's partial pressure.Here's how you do it: * Identify the total pressure of the mixture, referred to as \( P_{total} \).* Understand the ratios or proportions of the partial pressures within the mixture. * Apply the formula for partial pressure based on these proportions and the total pressure.Suppose the total pressure of the gas mixture is known. In this exercise, we have theoretical ratios: * For Gas 1, \( P_1 \), with a high proportion of 4 parts,* For Gas 2, \( P_2 \), with 2 parts,* And Gas 3, \( P_3 \), with 1 part.You multiply the total pressure by each ratio and divide by the sum of the proportions to find each partial pressure: * \( P_1 = 1.40 \cdot \frac{4}{7} = 0.8 \; \text{atm} \)* \( P_2 = 1.40 \cdot \frac{2}{7} = 0.4 \; \text{atm} \)* \( P_3 = 1.40 \cdot \frac{1}{7} = 0.2 \; \text{atm} \)These calculations show how each gas contributes to the total pressure of the mixture. By mastering this calculation, you can predict pressure changes in any gaseous system, whether it's in a laboratory or the atmosphere.