Derivation of the Kinetic Theory of Gases Equation (HL IB Physics)

• Gas pressure arises due to the collisions of the gas particles with the walls of the container that holds it
  • When a gas particle collides with a wall in the container, it exerts a force on the wall
  • The random motion of a large number of molecules exerting a force on the walls creates an overall pressure
  • This is because pressure = force per unit area

Derivation

• Picture a single molecule in a cube-shaped box with sides of equal length
• The molecule has a mass m and moves with speed c, parallel to one side of the box
• It collides at regular intervals with the ends of the box, exerting a force and contributing to the pressure of the gas
• By calculating the pressure this one molecule exerts on one end of the box, the total pressure produced by all the molecules can be deduced

A single molecule in a box collides with the walls and exerts a pressure

     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, so initial and final velocities are equal in magnitude
• When a gas particle of mass m, travels at an average speed v, and hits a wall of the container, it undergoes a change in momentum due to the force exerted by the wall on the particle

initial momentum =

final momentum = 

• Therefore, the change in momentum  is:

 = final momentum – initial momentum

• Force is equal to the rate of change of momentum (Newton's Second Law), so the force exerted by the wall on a molecule is:

• According to Newton's Third Law, the particle exerts an equal and opposite force on the wall
• Therefore force exerted on the wall by the particle is:

A particle hitting a wall of the container in which the gas is held experiences a force from the wall and a change in momentum. The particle exerts an equal and opposite force on the wall

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

• The time between collisions from one wall to the other, and then back again, over a distance of with speed v is:

time between collisions =

     3. Find the change in momentum per second

• The force the molecule exerts on one wall, F, from step 1 now becomes the following when substituting the time between collisions for Δt:

     4. Calculate the total pressure from N molecules

• The area of one wall is
• The pressure, P can be written as:

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

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

• The pressure equation written above still assumes all the molecules are travelling in the same x direction and colliding with the same pair of opposite faces of the cube
  • To reflect this, it can be rewritten as:

• Where v_x is the x component of the average velocity of all the particles
• In reality, all molecules will be moving in three dimensions equally
• Splitting the velocity into its components v_x, v_y and v_z to denote the amount in the x, y and z directions, v² (the square of the average speed) can be defined using pythagoras’ theorem in 3D:

• Since there is nothing special about any particular direction, it can be determined that the square of the average speed in each direction is equal:

• Therefore, by combining the two equations, v can be defined as:

     6. Re-write the pressure equation

• The box is a cube with a side length 
• Therefore, volume of the cube

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

• • m is the mass of a single particle so Nm is the total mass of all the particles
• Rearranging the pressure equation for p and substituting the density ρ:

• Where:
  • P = pressure of the gas (Pa)
  • ρ = density of the gas (kg m^–3)
  • v = mean square speed (m s^–1)

• This is known as the Kinetic theory of gases equation