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 cx, cy?and cz?to denote the amount in the x, y and z directions, c2?can be defined using Pythagoras' theorem in 3D:

c2?= cx2?+ cy2?+ cz2

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

cx2?= cy2?= cz2

• Therefore, cx2?can be defined as:

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

c2?= c12?+ c22?+ c32?+... + cN2

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?crms
• crms?is defined as:

• Therefore

N(crms)2?= c12?+ c22?+ c32?+... + cN2

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(crms)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)
  • crms?= 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: