Kinetic Theory of Gases Equation (AQA A Level Physics)

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)

  • External forces (e.g. gravity) are ignored

  • 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)

Consider the Model

• 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 c₁, 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 moving in a single direction hits one side of the wall and rebounds elastically in the opposite direction to their initial velocity, its final velocity is –c₁

• The change in momentum is therefore:

p = mc₁

Δp = final p – initial p = −mc₁ − (+mc₁) = −mc₁ − mc₁ = −2mc₁

• Where:

  • Δp = change in momentum (kg m s^-1)

  • m = mass of the one molecule (kg)

  • c₁ = speed of the one molecule in a particular direction (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 c₁ :

• Note: c is not the speed of light in this model

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 +2mc₁ 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

• For N particles travelling in the same direction as the single particle, pressure on a single face is now:

• This pressure equation still assumes that all the molecules are travelling in the same direction

• In reality, all molecules will be moving in three dimensions equally and randomly, with average velocity c:

• Where c_x, c_y and c_z are the components of the average velocity - this equation is a result of Pythagoras' theorem in 3D

• The particles are moving randomly so we can assume the magnitude of each component is equal to the other

• Each squared x component of velocity accounts for a third of the actual velocity squared, allowing the equation for pressure on one wall to be amended:

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

• Finally the pressure equation can be written as:

• 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: