Kinetic Theory of Gases Equation (Edexcel A Level Physics): Revision Note

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:

$\sqrt{<c^2>}=c_{rms}$

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

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:

$\Delta p=-mc-(+mc)=-mc-mc=-2mc$

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:

$timebetweencollisions=\frac{distance}{speed}=\frac{2l}{c}$

• 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:

$Force=rateofchangeofmomentum=\frac{\Delta p}{\Delta t}=\frac{2mc}{{\displaystyle \raisebox{1ex}{$2l$}\!\left/ \!\raisebox{-1ex}{$c$}\right.}}=\frac{m{c}^2}{l}$

• 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:

$pressure,p=\frac{force}{area}=\frac{{\displaystyle \raisebox{1ex}{$m{c}^2$}\!\left/ \!\raisebox{-1ex}{$l$}\right.}}{l^2}=\frac{m{c}^2}{l^3}$

• This is the pressure exerted from one molecule

• To account for the large number of N molecules, the pressure can now be written as:

$p=\frac{Nm{c}^2}{l^3}$

• 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:

$p=\frac{Nm<c^2>}{l^3}$

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:

$c^2=c{x}^2+c{y}^2+c{z}^2$

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

$<{c_x}^2>=<{c_y}^2>=<{c_z}^2>$

• Therefore, <c_x²> can be defined as:

$<{c_x}^2>=\frac{1}{3}<c^2>$

• 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:

$pV=\frac{1}{3}Nm<c^2>$

• This is known as the Kinetic Theory of Gases equation:

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

$\rho =\frac{mass}{volume}=\frac{Nm}{V}$

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

$p=\frac{1}{3}\rho <c^2>$