9.6 Kinetic Theory of Gases Equation

• 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 the following assumptions:
  • Molecules of gas behave as identical, 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 forces of attraction or repulsion between the molecules
  • The molecules are in continuous random motion

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

Root-Mean-Square Speed

• The pressure of an ideal gas equation includes the mean square speed of the particles:

<c²>

• Where
  • c = average speed of the gas particles
  • <c²> has the units m² s^-2

• Since particles travel in all directions in 3D space and velocity is a vector, some particles will have a negative direction and others a positive direction
  • When there are a large number of particles, the total positive and negative velocity values will cancel out, giving a net zero value overall
  • In order to find the pressure of the gas, the velocities must be squared meaning that all the values end up positive

• To calculate the average speed of the particles in a gas, take the square root of the mean square speed:

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

Step 1:            Write out the equation for the pressure of an ideal gas with density

Step 2:            Rearrange for mean square speed

Step 3:            Substitute in values

Step 4:            To find the r.m.s value, take the square root of the mean square speed

Step 5:            Write the answer to the correct significant figures and include units

c_rms = 790 ms⁻¹ (2 sig figs)

Deriving the Equation for Kinetic Theory

• 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

The Situation for the Derivation

• 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

Five-Step Derivation

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 they rebound in the opposite direction to their initial velocity, their final velocity is -c
  • The change in momentum is therefore:

Step 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

Step 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

Step 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:

Step 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:

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

• Therefore, <c_x²> can be defined as:

• 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:

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

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