Kinetic Theory

Boyle’s Law

$P V = n RT = k$

The ideal gas law
- P = pressure, in kPa
  - Newton/area
  - 1 atm (atmosphere) ~ 100 kPa
- V = volume, in L
- n = moles
- R = universal gas constant
  - 8.3145
- T = temperature
  - Kinetic energy
  - Kelvin
Works for all gases, as long as they “behave”
All gases deviate at low temperatures and high pressures
STP: standard temperature and pressure
- 25 ˚C, and 100 kPa
- 1 mole of gas at STP takes up 22.7 L of space
Absolute 0: temperature at 0 Kelvins where movement of all particles stops

Gases consist of particles in constant random motion, spread a long way apart
When gases collide with the walls of their container, each collision exerts a force.
This causes a pressure to be exerted on the walls of the container, i.e. a force per unit area
Measured in Pascals, 1 Pa = $1 \frac{N}{m ^{2}}$
We also measure pressure in atmospheres, where 1 atm = 101.3 kPa

The kinetic theory of gases links the macroscopic behaviour of a gas with its microscopic behaviour
In macroscopic terms, a gas is a phase of matter that has a fixed mass and whose volume is equal to the volume of its container.
- The gas exerts pressure on the walls of the container
In microscopic terms, a gas is a collection of many particles that collide with each other and with the container walls
The kinetic theory relates the pressure exerted by a gas to the motion of its particles
The kinetic theory of gases was developed from experimental evidence to explain the similar behaviours of all gases
Mathematical relationships between pressure, volume, moles and temperature have been derived for an ideal gas, that is one that obeys all the equations perfectly
There is no such thing as an ideal gas - no real gases behave exactlly as the laws describe.
- However, under normal conditions, almost all gases follow the ideal gas model very closely
Real gases deviate most from ideal gas behaviour at very low temperatures and very high pressures
Gases with small particles like H and He follow the ideal gas laws most closely
- Smaller particles means less collisions

The motion of all particles is random
1. All particles are in constant, random motion
All particles travel in straight lines
intermolecular (electrostatic) and gravitational forces are negligible
All particles of a particular gas are identical and perfect spheres
Internal energy of the gas is entirely kinetic
All collisions between particles and the walls of the container are completely elastic (no loss of kinetic energy)
Particles take up negligible volume (pressure is due to mobility)
Newton’s Laws of Motion apply
1. An object travelling at a constant velocity, or at rest, will remain at that constant velocity or at rest, unless acted upon by an external force. i.e., inertia
2. Force is equal to the product of mass and acceleration
3. Every action has an equal and opposite reaction

Gases are composed of particles in continuous rapid, random motion
Attraction and repulsion between particles in the gas is negligible
The particles of a gas are widely spaced such that the total volume of all the particles is negligible compared to the volume occupied by the gas
The average kinetic energy of the particles of a gas is proportional to its temperature, and is the same for all gases at the same temperature
Particle collisions are elastic (and collisions with the container walls)
1. i.e. over time, as particles collide, they do not lose speed or slow down, thus particles do not lose kinetic energy (i.e. cool down) due to their collisions

An important aspect of kinetic theory is that temperature is related to average kinetic energy of the particles
However there is a range of kinetic energy values for the particles at any particular temperature - some move faster, and some move slower than the average
The Maxwell-Boltzmann distribution describes the distribution of kinetic energies of particles at a given temperature
- y axis is number of particles in a sample
- x axis is energy
At a given temperature, particles of all gases have the same distribution of kinetic energies, no matter what size their particles are.
The relationship between kinetic energy, mass, and velocity is $e = \frac{1}{2} m v^{2}$
This means that particles with a greater mass with have a lower average velocity, and vice versa

The absolute or thermodynamic scale of temperature is a measure of the average kinetic energy of the particles.
Measured in degrees Kelvin (K)
When temperature is measured in Kelvin, the temperature of a substance is directly proportional to the average kinetic energy of its particles
Absolute temperature starts at absolute zero - the theoretical lowest possible temperature, at which point the particles would have no kinetic energy
The absolute scale has units of the same magnitude as the celsius scale
Converting to and from Kelvin and degrees Celsius $T (K) = T (˚ C) + 273.15$ $T (˚ C) = T (K) - 273.15$

Gases diffuse quickly because gas particles move rapidly and randomly through the large amount of empty space between particles to spread out
Gases are easily compressed due to the large amount of empty space between particles
Gases spread out to fill a container due to the negligible forces of attraction between particles
Gases exert pressure because the particles move randomly/rapidly and collide with the interior surfaces of the container
Gases have low densities due to the large amount of space between particles

If the temperature and pressure are fixed at convenient standard values, the molar volume of a gas can be determined
At standard temperature and pressure (273.15 K and 100 kPa, 1 mole of any gas occupies a volume of 22.71 L. This is the molar volume)