Derivation of the Kinetic Theory of Gases Equation (CIE A Level Physics)

• The equation for the kinetic theory of ideal gases contains the mean square speed of the particles:

• 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
• 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

Derivation of the equation for the pressure exerted by a gas

     1. Find the change in momentum as a single molecule hits a wall perpendicularly

• One assumption of the kinetic theory is that molecules rebound elastically
• This means there is no kinetic energy lost in the collision
• If they rebound in the opposite direction to their initial velocity, their final velocity is -c
• The change in momentum is therefore:

     2. Calculate the number of collisions per second by the molecule on a wall

• The time between collisions of the molecule travelling to one wall and back is calculated by travelling a distance of 2l with speed c:

Time between collisions = 

• Note: c is not taken as the speed of light in this scenario

     3. Find the change in momentum per second

• The force the molecule exerts on one wall is found using Newton’s second law of motion:

Force = rate of change of momentum = 

• The change in momentum is +2mc since the force on the molecule from the wall is in the opposite direction to its change in momentum

     4. Calculate the total pressure from N molecules

• The area of one wall is l²
• The pressure, p is defined using the force and area:

• This is the pressure exerted on the container wall by one molecule
• To account for the large number of N molecules, the pressure can now be written as:

• Each molecule has a different velocity and they all contribute to the pressure
• The mean squared speed of c² is written with left and right-angled brackets <c²>
• The pressure is now defined as:

     5. Consider the effect of the molecule moving in 3D space

• The pressure equation still assumes all the molecules are travelling in the same direction and colliding with the same pair of opposite faces of the cube
• In reality, all molecules will be moving in three dimensions equally
• Splitting the velocity into its components c_x, c_y and c_z to denote the amount in the x, y and z directions, c² can be defined using pythagoras’ theorem in 3D:

• Since there is nothing special about any particular direction, it can be determined that:

• Therefore, <c_x²> can be defined as:

     6. Re-write the pressure equation

• The box is a cube and all the sides are of length l
  • This means l³ is equal to the volume of the cube, V

• Substituting the new values for <c²> and l³ back into the pressure equation obtains the final equation:

• Where:
  • p = pressure (Pa)
  • V = volume (m³)
  • N = number of molecules 
  • m = mass of one molecule (kg)
  • <c²> = mean square speed of the molecules (m s^–1)

• This can also be written using the density ρ of the gas:

• Rearranging the pressure equation for p and substituting the density ρ: