Question

A chemistry student relates the following story: I noticed \(\mathrm{my}\) tires were a bit low and went to the gas station. As I was filling the tires, I thought about the kinetic molecular theory (KMT). noticed the tires because the volume was low, and I realized that I was increasing both the pressure and volume of the tires "Hmmm," I thought, "that goes against what I learned in chemistry, where I was told pressure and volume are inversely proportional." What is the fault in the logic of the chemistry student in this situation? Explain why we think pressure and volume to be inversely related (draw pictures and use the KMT).

Short answer

The fault in the chemistry student's logic is the assumption that increasing both the pressure and volume of the tires simultaneously contradicts the inverse relationship between pressure and volume in Boyle's Law. However, Boyle's Law only applies when the temperature and amount of gas (air) remain constant. In this situation, the student is adding more gas (air) to the tires, resulting in an increase in both pressure and volume. This does not violate Boyle's Law, as the amount of gas in the tires is changing. The Kinetic Molecular Theory explains the relationship between pressure and volume through collisions between gas particles and their container walls, which vary according to the amount of gas present and the volume of the container.

Step-by-step solution

Understand the Kinetic Molecular Theory (KMT)

KMT is a model that is used to describe the behavior of gases. It assumes that gases are made up of tiny, discrete particles in constant, random motion and interact only through elastic collisions. This theory helps in explaining why and how we came to understand the relationship between a gas's pressure, volume, and temperature.

Explain Boyle's Law

Boyle's Law states that the pressure of a gas (P) is inversely proportional to its volume (V) when the temperature (T) and the amount of gas (n) remain constant. Mathematically, this relationship is represented as: \(PV = k\), where k is a constant. This is the reason we often say that pressure and volume are inversely related in gases following the kinetic molecular theory.

Recall the KMT explanation of Boyle's Law

KMT helps explain Boyle's Law through the concept of collisions between gas particles and the walls of its container. When the volume of a container is decreased, the particles have less space to move in, leading to more collisions against the walls, thus increasing the pressure. Similarly, if the volume is increased, the particles have more space to move around, resulting in fewer collisions and lower pressure.

Identify the fault in the chemistry student's logic

The fault in the student's thinking is the assumption that increasing the pressure and volume of the tires at the same time contradicts the inverse relationship between pressure and volume in Boyle's Law. However, Boyle's Law states that pressure and volume are inversely proportional only when the temperature and amount of gas remain constant. In the given situation, the student is increasing the amount of gas (air) inside the tires. This leads to an increase in pressure as well as volume, as new air particles are being added to the tires. This difference in the scenario does not violate Boyle's Law, as the amount of gas is not constant.

Reiterate the Boyle's Law and its explanation in KMT context

In summary, the student's confusion results from an incorrect understanding of Boyle's Law, which states that pressure and volume are inversely proportional only when the temperature and amount of gas remain constant. In the situation of filling the tires, the amount of gas is increasing, causing both pressure and volume to increase simultaneously. The Kinetic Molecular Theory explains this relationship through the collisions between gas particles and their container walls.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle for understanding gas behavior, specifically relating to pressure and volume. The law outlines that, within a closed system, the pressure of a gas is inversely proportional to its volume as long as the temperature and quantity of gas remain unchanged. This means that if the volume decreases, the pressure increases, and vice versa. The formula representation is \[PV = k\]where \(P\) represents pressure, \(V\) the volume, and \(k\) is a constant. This relationship is rooted in the Kinetic Molecular Theory. Understanding Boyle's Law is crucial when conceptualizing how gases behave under different conditions.

In practical terms, imagine a syringe: when the plunger is pushed down reducing the volume, the air inside becomes compressed, increasing pressure. Conversely, pulling the plunger up increases volume and decreases pressure. Simple illustrations like these can help make sense of Boyle's Law and provide a handy reference for everyday situations.

Gas Behavior

Gas behavior is a complex concept that can be explained through the Kinetic Molecular Theory. This theory proposes that gases are composed of tiny particles in perpetual motion. These particles collide with each other and the walls of their container, generating pressure. The behavior of gas particles is deeply influenced by factors like temperature, volume, and the number of particles themselves.

Key points about gas behavior include:

• Gas particles move randomly and rapidly, filling any container entirely.
• Collisions between particles are considered elastic, meaning no energy is lost.
• The overall behavior of gas is predictable when observing large numbers of particles.

These principles help us predict and understand scenarios such as breathing, tire inflation, and even weather patterns. Recognizing how these elements interact provides solid groundwork for comprehending real-world phenomena involving gases.

Pressure and Volume Relationship

The relationship between pressure and volume is pivotal in gas behavior, as dictated by Boyle's Law. The key to understanding this relationship is recognizing that pressure results from collisions between gas particles and their container walls. When volume decreases, particles have less space to move, leading to more frequent collisions, thus increasing pressure. Conversely, increasing volume results in fewer collisions and a decrease in pressure.

In the context of the chemistry student's situation with the tires, the misunderstanding arises from not accounting for the changing amount of gas. As new air is added, both volume and pressure rise, making it seem like they contradict the inverse relationship suggested by Boyle's Law. However, Boyle's Law only applies when the amount of gas and temperature remain constant.

This highlights the importance of context when applying scientific laws and models to real-world situations. Always consider all the variables involved to ensure accurate understanding and application.