6.6.2 Kinetic Theory of Gases Equation

Assumptions in Kinetic Theory

• 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 a gas behave as identical (or all have the same mass)
  • Molecules of gas are 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 intermolecular forces between the molecules (except during impact)
  • The molecules move in continuous random motion
  • Newton's laws apply
  • There are a very large number of molecules

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

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)
• Take 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 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 N molecules can be deduced

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

1. Determine 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 the particle hits one side of the wall and rebounds elastically in the opposite direction to their initial velocity, their final velocity is –c
• The change in momentum is therefore:

p = mc

Δp = final p – initial p = −mc − (+mc) = −mc − mc = −2mc

• Where:
  • Δp = change in momentum (kg m s^-1)
  • m = mass of the molecule (kg)
  • c = speed of the molecule (m s^-1)

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

• The time between collisions of the molecule travelling to the opposite facing 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. Calculate the force exerted by the molecule on the wall

• 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 for one molecule

• The area of one wall is L²
• The pressure is defined as the force per unit area:

• This is the pressure exerted from one molecule in a particular direction

5. Consider the effect of N molecules moving randomly in 3D space

• The pressure equation still assumes that all the molecules are travelling in the same direction and collide with the same pair of opposite faces of the cube
• In reality, all molecules will be moving in three dimensions equally and randomly
• By 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 deduced that:

c_x² = c_y² = c_z²

• Therefore, c_x² can be defined as:

• Where c² is the sum of the squared speeds of all the molecules

c² = c₁² + c₂² + c₃² +... + c_N²

6. Consider the speed of the molecules as an average speed

• Each molecule has a different speed and they all contribute to the pressure
• Therefore, the square root of the average of the square velocities is taken as the speed instead
• This is called the root-mean-square speed or c_rms
• c_rms is defined as:

• Therefore

N(c_rms)² = c₁² + c₂² + c₃² +... + c_N²

7. Consider the volume of the box

• 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 N(c_rms)² and L³ back into the pressure equation obtains the equation:
  

• This is the pressure parallel to the x (or y or z axis)

• Multiplying both sides by the volume V gives the final Kinetic Theory of Gases Equation:

• Where:
  • p = pressure (Pa)
  • V = volume (m³)
  • N = number of molecules
  • m = mass of one molecule of gas (kg)
  • c_rms = root mean square speed of the molecules (m s^-1)

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

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