Question

Explain why the pressure of a gas sample is directly proportional to the moles of gas particles in the sample.

Short answer

The pressure of a gas sample is directly proportional to its mole number because, with increasing moles, there are more gas molecules to collide against the container walls, increasing the pressure. This observation is given by the ideal gas law, especially when the volume and temperature are constant.

Step-by-step solution

Understanding the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \). In this study case, temperature (T) and volume (V) are held constant. Therefore, the equation simplifies to \( P = \frac{nR}{V} \). This equation shows that the pressure of a gas (P) is directly proportional to the number of moles (n) of the gas, with the proportionality constant being the ratio of the gas constant (R) to the volume (V). Therefore, if you maintain the temperature and volume constant and increase the number of moles of gas, the pressure also increases.

Kinetic Molecular Theory Perspective

From a kinetic molecular theory perspective, gas particles are in constant, rapid, random motion. They consistently collide with the walls of their container. Every time gas particles collide with the wall, they exert a force on the wall. The collective force exerted by the gas particles per unit area is what we call pressure. Therefore, the more the gas particles (moles of gas), the more frequent the collisions, and consequently, the higher the pressure.

Analogical Representation

To fully understand, imagine a room filled with people. If more people enter the room, the number of people (analogous to gas particles) increases. The busier the room gets, the higher the pressure on the walls of the room due to the interactions. Similarly, when more gas molecules are added to a fixed-volume container (introducing more moles of gas), they will 'bump' into the walls more frequently. This increase in the frequency of collisions results in a higher pressure.

Key concepts

Pressure and Moles Proportionality

To understand why the pressure of a gas is directly proportional to the number of moles of gas, we should start by examining the Ideal Gas Law, represented by the equation: \( PV = nRT \). When the temperature (T) and the volume (V) are kept constant, the equation simplifies to \( P = \frac{nR}{V} \), indicating pressure (P) is directly related to the number of moles (n).

In simpler terms, if you have a container with a certain volume and you add more moles of gas into this space without changing the temperature, the pressure will increase. This is because the constant, R, and the volume, V, act as constants in this scenario, allowing us to see a clear proportional relationship between pressure and moles when the other variables are controlled. An improvement in understanding can come from visualizing this scenario as inflating a balloon. At first, with less air (fewer moles of gas), the balloon is floppy and easy to squeeze. As you add more air, it becomes tighter and harder to compress – the pressure inside has increased with the number of air moles.

Kinetic Molecular Theory

The Kinetic Molecular Theory explains many properties of gases, including the relationship between pressure and moles. According to this theory, gas particles are constantly in motion in a random, rapid manner. This movement results in the particles colliding with each other and with the walls of the container.

Each collision against the container wall exerts a force. The collective force generated by all the particles, taken over the surface area of the walls, defines the gas pressure. The more particles (or moles of gas) present, the more collisions will occur, thus increasing the pressure inside the container. It can be helpful to think of it as a game of pinball where the balls are the gas molecules; the more balls in play, the more they will hit the sides, symbolizing more frequent particle-wall collisions in a gas container.

Gas Pressure Explanation

Understanding gas pressure is key to grasping how gases behave. Gas pressure can be thought of as the result of numerous tiny impacts of gas particles on a surface. Consider being inside a crowded subway car; people are constantly moving and occasionally bumping into you. In this analogy, you're the container wall, and the people are the gas particles.

The frequency and intensity of these 'bumps' determine the pressure exerted on the walls. Therefore, in a gas container, when more particles are added (increasing moles), or when the temperature rises (causing particles to move more energetically), we experience a rise in pressure. This is because there are more collisions (due to more particles) or more forceful collisions (due to higher speeds at increased temperatures), applying a greater force over the container's interior surface area. Understanding this can be significantly improved by recalling times of increased personal space or feeling crowded; those situations can serve as tangible examples of how gases exert pressure.