15.2.1 Kinetic Theory of Gases

• Gases consist of atoms or molecules randomly moving around at high speeds
• The kinetic theory of gases models the thermodynamic behaviour of gases by linking the microscopic properties of particles (mass and speed) to macroscopic properties of particles (pressure and volume)

• The theory is based on a set of the following assumptions:
  • Molecules of gas behave as identical, hard, perfectly elastic spheres
  • The volume of the molecules is negligible compared to the volume of the container
  • The time of a collision is negligible compared to the time between collisions
  • There are no forces of attraction or repulsion between the molecules
  • The molecules are in continuous random motion

• The number of molecules of gas in a container is very large, therefore the average behaviour (eg. speed) is usually considered

• The pressure of an ideal gas equation includes the mean square speed of the particles:

<c²>

• Where
  • c = average speed of the gas particles
  • <c²> has the units m² s^-2

• Since particles travel in all directions in 3D space and velocity is a vector, some particles will have a negative direction and others a positive direction
• When there are a large number of particles, the total positive and negative velocity values will cancel out, giving a net zero value overall
• In order to find the pressure of the gas, the velocities must be squared
  • This is a more useful method, since a negative or positive number squared is always positive

• To calculate the average speed of the particles in a gas, take the square root of the mean square speed:

• c_r.m.s is known as the root-mean-square speed and still has the units of m s^-1
• The mean square speed is not the same as the mean speed

Step 1:            Write out the equation for the pressure of an ideal gas with density

Step 2:            Rearrange for mean square speed

Step 3:            Substitute in values

Step 4:            To find the r.m.s value, take the square root of the mean square speed