3. The gas laws
Temperature and pressure P.14
On page 12, we saw what can happen to the temperature of a gas when we do work on it or we allow it to do work. The gas laws are different from this. In this case, we assume that there is no work done. The other assumptions are:
  • the mass of gas is fixed
  • the gas is monatomic
  • the collisions are, on average, elastic
  • the size of the particles is negligible
  • the forces between particles are negligible (except in collisions)
  • the time spent colliding is negligible compared with the time spent between collisions.
Particles and pressure
It is the movement of the particles that produces pressure. Let's see why.

Imagine the particles are held in a container. They are bouncing off each other and off the walls of the container.

Pressure from gases
Picture 3.1. When a gas is heated its particles move faster.
Every time a particle bumps into the wall, there is a small impulse - the colliding particle pushes outwards on the container for a tiny instant. Each particle collides with the walls hundreds of times every second and there are billions of particles. Therefore, there are hundreds of billions of collisions every second. This produces a steady outwards force, and therefore pressure, on the container walls.
Particles and temperature
The kinetic energy of the particles is proportional to the temperature. This means that increasing the temperature will make the particles move faster and produce a bigger pressure. There are two reasons that faster particles produce a bigger pressure:
  • they hit the walls more often
  • they hit the walls harder.

When we measure how pressure changes with temperature, we find that they are proportional – as long as we use the absolute temperature scale. This is known as the Pressure law.

The Pressure law
Pressure is proportional to absolute temperature
we can write this as:  P = constant x T
or P/T = constant
Why is P µ T
Although the pressure law is an empirical law, we can make sense of it by relating it to the average KE of particles.

If we double the speed of particles, then their KE quadruples (KE=1/2mv2)

Therefore, the temperature is 4 times bigger (T µ KE)

However, if we double the speed there will be twice as many collisions every second.

Also, each collision has twice the impulse.

So, with twice as many, doubly hard, collisions the pressure is 4 times bigger as well.

Both the temperature and the pressure are 4 times bigger, i.e. they have gone up in proportion.

Graph of pressure against temperature
Picture 3.2. Graph of results of pressure against temperature.
Measuring pressure against temperature
We can set up an experiment to measure temperature against pressure for a fixed mass of gas. In a laboratory, the temperature range is going to be from about 10  °C to 100  °C. Plotting a graph of the results should give us a straight line. However, it clearly will not go through the origin of 0  °C (this would imply that all air particles stopped moving at 0  °C, which would have severe consequences). By extrapolating the line, we can get an estimate for a value of absolute zero.

top start units pages
Question 11
In the assumptions of an ideal gas:

a) we assume that all the collisions are, on average, elastic.
i. What would happen if they were not?

Click shift/return to get a line break in your answer
ii. Is this a reasonable assumption?
b) Why do we have to assume that the particles are monatomic?

Summary                                           Close
  • an ideal gas obeys the gas laws, including . . .
  • pressure is proportional to absolute temperature