Topic 3 - Thermodynamics

3.3 The Ideal Gas

Gas Pressure

Pressure is defined as the force per unit area.

Pressure is measured in Pascals (Pa)

Gas pressure arises because of collisions between particles (causing the force) and the wall of the container (the area over which they act)

An Ideal Gas

Physicists made a set of observations of gases from which 4 assumptions are made to define the ideal gas.A pure gas contains identical molecules in continuous random motion. (no one particle is more special than another)

All collisions are elastic (energy is conserved)

The volume of the particles is negligible compared to the volume of the container. (it is compressible)

There are no forces on the molecules except during collisions (the particles are very far apart)

An Ideal Gas

Consider a box of dimensions x, y & z as shown

A single ideal gas particle mass m is moving in the box with speed u parallel to the x direction.

x

z

y

u

An Ideal Gas

The molecule collides with the blue wall as shown.

Its initial momentum is mu, and its final momentum in -mu

Its change in momentum is therefore 2mu

x

z

y

u

An Ideal Gas

The molecule travels a distance 2x between collisions with the blue wall.

The time between collisions is therefore:2x/u

x

z

y

u

An Ideal Gas

The force exerted on the blue wall is the rate of change of momentum.

x

z

y

u

An Ideal Gas

The pressure on the blue wall is therefore:

Where V is the volume

x

z

y

u

An Ideal Gas

In a real gas there are N molecules moving randomly. On average only 1/3 of these move in the x direction.

The molecules are not all moving with speed u but have an average (mean square) speed

x

z

y

u

Molecular Speed

For one mole of gas, the equation becomes:

This could be written as:

Where m is the average kinetic energy

Molecular Speed

From other macroscopic experiments it can be shown that:

These two equations for ideal gases must equate.

Therefore:

Molecular Speed

The ratio of the two constants (R over NA) is known as the Boltzmann constant kk=1.38 x 10-23 JK-1

That is Kinetic Energy is proportional to absolute temperature

Summary of Ideal Gases

For a real gas, the ideal gas rules can be used to give approximate answers.

An increase in volume will cause a longer time between collisions, so the collisions will be less frequent, so the pressure will decrease.

An increase in temperature, will cause a higher KE, so the time between collisions will increase and the force with which they strike the container will increase. The pressure will therefore increase.

An increase in volume at constant pressure will cause the particles to slow down, therefore causing a decrease in temperature.