Derivation of the Kinetic Theory of Gases Equation (Cambridge (CIE) A Level Physics): Revision Note

Derivation of the kinetic theory of gases equation

• Molecular movement causes the pressure exerted by a gas

  • When molecules rebound from their container wall, the change in momentum gives rise to a force exerted by the particles on the wall

  • Many molecules moving in random motion exert forces on the walls, which creates an average overall pressure (since pressure is the force per unit area)

Setting up the model

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

• The molecule has mass and moves with speed parallel to one side of the box

• It collides at regular intervals with the sides 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 a total of  molecules can be deduced

Modelling a gas molecule in a container

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

Deriving the equation step-by-step

     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 a molecule rebounds in the opposite direction to its initial velocity , its final velocity will be

• 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  with speed :

Time between collisions = 

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

     3. Find the change in momentum per second

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

• The change in momentum is 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 

• The pressure  is defined as the force per unit 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 square speed  is written with left and right-angled brackets

• 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 and randomly, with different velocities :

• Where , , and are the -, -, and - components of the velocity

  • This equation is a result of Pythagoras' theorem in 3D

• Since pressure is due to the force exerted on one face of the cube, only the component of velocity perpendicular to that face (e.g. the -direction) matters

• For a gas with molecules moving randomly in all directions, the mean square speed is evenly distributed among all three directions:

• Therefore, the mean square speed in the -direction accounts for a third of the total mean square speed

• This is why only one-third of the total mean square speed contributes to the pressure on one wall

     6. Re-write the pressure equation

• The box is a cube, and all the sides are of length 

  • This means the volume of the cube is equal to 

• Substituting the new values for and  back into the pressure equation obtains the final equation:

• Where:

  •  = pressure (Pa)

  •  = volume (m³)

  •  = number of molecules 

  •  = mass of one molecule (kg)

  • = 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 and substituting the density :

Root-mean-square speed

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

• Where

  • = velocity of one gas molecule (m s^-1)

  • = average velocity of the molecules in a gas (m s^-1)

  • = average of the square of the speeds of the molecules in a gas (m² s^-2)

• Since there are a large number of molecules in a gas, travelling in all directions in 3D space, this results in a large range of values of

• The average velocity of all the molecules in a gas gives a net zero value overall

  • This is because velocity is a vector, so some molecules will have a negative direction and others a positive direction

• Squaring the velocities gives positive values only

• Averaging the values of gives , the mean square speed

• Then, taking the square root of the mean square speed gives:

•  is known as the root-mean-square speed and has units of m s^-1

  • The root-mean-square speed is not the same as the mean speed