Question

Was particles travel at speeds of up to 500 meters per second. Why, then, does it take so long for gas molecules to travel the length of a room?

Short answer

Gas molecules take a long time due to frequent collisions that change their direction, despite their high speeds.

Step-by-step solution

Understanding Particle Speed

Gas particles move very rapidly, with speeds as high as 500 meters per second, due to their kinetic energy at a given temperature.

Analyzing Molecular Path

Despite their high speed, gas molecules rarely travel in a straight line over long distances due to constant collisions with other molecules. This results in a zig-zag or random walk pattern.

Exploring Mean Free Path

The mean free path is the average distance a molecule travels before colliding with another molecule. In a room full of gas, this distance is often very small compared to the dimensions of the room.

Calculating Time to Cross a Room

Because the molecules frequently change direction after collisions, they may take a significantly longer time to cover the same linear distance as compared to moving in a straight line at their top speed. The time taken is influenced by the short mean free paths and the high frequency of collisions.

Key concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In the context of gas particles, this energy is what causes them to move at such high speeds. Imagine a room filled with air. The air is made up of countless tiny molecules zipping around at speeds that can reach 500 meters per second. This high speed is due to their kinetic energy, which increases with temperature.

• Warmer temperatures provide more energy to gas particles, resulting in faster motion.
• The formula for kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \] where \(m\) is mass and \(v\) is velocity.

Despite these high speeds, the path gas particles take is not straightforward. Instead, they engage in frequent collisions, affecting the time it takes for them to move across a space.

Mean Free Path

The mean free path is the average distance a gas particle travels before hitting another particle. In a room full of gas molecules, frequent collisions occur, allowing particles limited travel distance before being redirected.

• The mean free path is influenced by the density of gas particles; denser gases lead to shorter mean free paths.
• It's calculated using the formula: \[ \lambda = \frac{kT}{\sqrt{2}\pi d^2 p} \] where \(\lambda\) is the mean free path, \(k\) is Boltzmann's constant, \(T\) is temperature, \(d\) is the diameter of the gas molecule, and \(p\) is pressure.

Understanding the mean free path helps explain why gas particles, despite their high speed, take considerable time to move noticeable distances. This occurs as their journey is constantly interrupted and adjusted by collisions.

Gas Particle Collisions

Collisions between gas particles are frequent and have a significant impact on their motion. As these particles dart around, they often collide, altering their direction and speed in a manner similar to a chaotic game of bumper cars.

• These collisions are elastic, meaning no kinetic energy is lost in the interactions.
• They are a critical factor in determining the behavior of gases, influencing properties like diffusion and gas pressure.

As particles collide, their motion becomes a random zig-zag, leading to significant time delays when traveling from one point in a room to another. The high frequency of these collisions ensures that even though individual particles move quickly, the overall movement of gas through a space is relatively slow. It is this interaction and resulting random path that accounts for the time it takes gas to permeate a room.