# Pressure, Temperature and RMS Speed Revision Notes

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13.7Pressure, Temperature and RMS Speed

In these revision notes for Pressure, Temperature and RMS Speed, we cover the following key points:

• What is rms speed of a gas?
• What is the relationship between pressure temperature and rms speed in a gas?
• Why are the factors affecting the rms speed of ideal gas molecules?
• What is the translational kinetic energy of gas particles?
• What is mean free path of gas molecules?
• What are the factors affecting mean free path of gas molecules?

## Pressure, Temperature and RMS Speed Revision Notes

Molecules of a gas move in every direction within the container; they hit each other and the container walls and then, they bounce back. If we consider the gas as ideal, the collisions between particles can be neglected. Thus, we consider only the elastic collisions of gas molecules with the container walls.

We call the quantity v2average as root mean square speed (vrms). We can write for pressure of an ideal gas enclosed within a fixed container:

P = n × M/3V × v2rms

This formula indicates how a macroscopic quantity such as pressure depends on a microscopic quantity such as the speed of molecules.

As for the rms speed of gas molecules we have

vrms = √3 × R × T/M

The translational kinetic energy of a gas molecule molecule is

KE = 3/2 × k × T

We obtain a very important conclusion based on the last formula:

"All ideal gas molecules, no matter what kind of gas they belong, have the same translational kinetic energy at a given temperature."

This means that when we measure the temperature of an ideal gas, we are actually measuring the average translational kinetic energy of its particles.

Molecules of a gas collide continuously with each other; however, they move linearly at constant speed during the time interval between two consecutive collisions.

We use a parameter known as "mean free path" (in symbols, λ) to describe this random motion of gas particles. It represents the average distance travelled by a gas particle between two collisions. Mean free path depends on the following factors:

1- Density or concentration of gas. In this case, we express density in terms of concentration of molecules, i.e. as number of molecules per unit volume (in short, N/V) instead of mass per unit volume, as here we are interested on microscopic parameters such as the number of molecules.

Based on this factor, we can say that higher the concentration of particles, smaller their mean free path. Therefore, the molecular mean free path is inversely proportional to gas concentration.

2- Size of molecules. Obviously, larger the gas molecules, shorter the mean free path between them. molecules are, the smaller the mean free path. Mean free path λ, should vary (inversely) as the square of the molecular diameter d because the cross section of a molecule - not its diameter - determines its effective target area.

The equation of mean free path of gas molecules is:

λ = 1/2 × π × d2 × N/V

where N is the total number of gas molecules.

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