15.2.2 Derivation of the Kinetic Theory of Gases Equation

• When molecules rebound from a wall in a container, the change in momentum gives rise to a force exerted by the particle on the wall
• Many molecules moving in random motion exert forces on the walls which create an average overall pressure, since pressure is the force per unit area

 

• Picture a single molecule in a cube-shaped box with sides of equal length l
• 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

5 Step Derivation

     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:

Δp = −mc − (+mc) = −mc − mc = −2mc

     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:

• 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:

• 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 is defined using the force and area:

• This is the pressure exerted from 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:

c² = c_x² + c_y² + c_z²

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

<c_x²> = <c_y²> = <c_z²>

• 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:

• This is known as the Kinetic Theory of Gases equation

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

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