Kinetic Model of an Ideal Gas
Gas Pressure
- A gas is made of a large number of particles
- Gas particles have mass and move randomly at high speeds
- Pressure in a gas is due to the collisions of the gas particles with the walls of the container that holds the gas
- When a gas particle hits a wall of the container, it undergoes a change in momentum due to the force exerted by the wall on the particle (as stated by Newton's Second Law)
- Final momentum = –mv
- Initial momentum = mv
- Therefore, the change in momentum Δp can be written as:
Δp = final momentum – initial momentum
Δp = –mv – mv = –2mv
- According to Newton's Third Law, there is an equal and opposite force exerted by the particle on the wall (i.e. F = 2mv/Δt)
A particle hitting a wall of the container in which the gas is held experiences a force from the wall and a change in momentum. The particle exerts an equal and opposite force on the wall
- Since there is a large number of particles, their collisions with the walls of the container give rise to gas pressure, which is calculated as follows:
- Where:
- p = pressure in pascals (Pa)
- F = force in newtons (N)
- A = area in metres squared (m2)
Average Random Kinetic Energy of Gas Particles
- Particles in gases have a variety of different speeds
- The average random kinetic energy of the particles EK, which can be written as follows:
- Where:
- EK = average random kinetic energy of the particles in joules (J)
- kB = 1.38 × 10–23 J K–1 (Boltzmann's constant)
- T = absolute temperature in kelvin (K)
- kB is known as Boltzmann's constant, and it can be written as follows:
- Where:
- R = 8.31 J K–1 mol–1 (ideal gas constant)
- NA = 6.02 × 1023 mol–1 (Avogadro constant)
Internal Energy of the Gas
- Using the equation of state of ideal gases, the internal energy can be written as follows:
- Where:
- U = internal energy of the gas in joules (J)
- p = gas pressure in pascals (Pa)
- V = gas volume in metres cubed (m3)
Worked example
2 mol of gas is sealed in a container, at a temperature of 47°C.
Determine:
- The average random kinetic energy of the particles in the gas
- The internal energy of the gas
Part (a)
Step 1: Write down the temperature T of the gas in kelvin (K)
T = 47°C = 320 K
Step 2: Write down the equation linking the absolute temperature T of the gas to the average random kinetic energy EK of the gas particles
Step 3: Substitute numbers into the equation
-
- From the data booklet, kB = 1.38 × 10–23 J K–1
EK = 6.6 × 10–21 J
Part (b)
Step 1: Write down the equation linking the internal energy U of the gas to the number of moles n and the absolute temperature T
Step 2: Substitute numbers into the equation
-
- From the data booklet, R = 8.31 J K–1 mol–1
U = 8000 J = 8kJ
-
- Note that, alternatively, the internal energy can be calculated using the following equation:
U = NEK = 8000 J = 8kJ
N = nNA = 2 mol × (6.02 × 1023) mol–1 = 1.2 × 1024
EK = 6.6 × 10–21 J (calculated in Step 3)
Examiner Tip
Momentum is a Mechanics topic that should have been covered in a previous unit. The above derivation of change in momentum and resultant force should have already been studied - if you're not comfortable with it then make sure you go back to revise this!