1. Collective behaviour of large systems
Kinetic theory
Collective behaviour of large systems

2. Why gases exert pressure
Gases are mostly empty space
Gases contain molecules which have random motion
The molecules have kinetic energy
The molecules act independently of each other – there are no forces between them
Molecules strike the walls of the container – the collisions are perfectly elastic
Exchange energy with the container
The energy of the molecules depends upon the temperature

3. Large collections are very predictable
Fluctuations in behaviour of a small group of particles are quite noticeable
Fluctuations in behaviour of a large group (a mole) of particles are negligible
Large populations are statistically very reliable

4. Pressure and momentum Pressure = force/unit area
Force = mass x acceleration
Acceleration = rate of change of velocity
Force = rate of change of momentum
Collisions cause momentum change
Momentum is conserved

5. Elastic collision of a particle with the wall
Momentum lost by particle = -2mv
Momentum gained by wall = 2mv
Overall momentum change = -2mv + 2 mv = 0
Momentum change per unit time = 2mv/Δt

6. Factors affecting collision rate
Particle velocity – the faster the particles the more hits per second
Number – the more particles – the more collisions
Volume – the smaller the container, the more collisions per unit area

7. Making refinements
We only considered one wall – but there are six walls in a container
Multiply by 1/6
Replace v2 by the mean square speed of the ensemble (to account for fluctuations in velocity)

8. Boyle’s Law Rearranging the previous equation:
Substituting the average kinetic energy
Compare ideal gas law PV = nRT:
The average kinetic energy of one mole of molecules can be shown to be 3RT/2

9. Root mean square speed Total kinetic energy of one mole
But molar mass M = Nom
Since the energy depends only on T, vRMS decreases as M increases

10. Speed and temperature
Not all molecules move at the same speed or in the same direction
Root mean square speed is useful but far from complete description of motion
Description of distribution of speeds must meet two criteria:
Particles travel with an average value speed
All directions are equally probable

11. Maxwell meet Boltzmann
The Maxwell-Boltzmann distribution describes the velocities of particles at a given temperature
Area under curve = 1
Curve reaches 0 at v = 0 and ∞

12. M-B and temperature As T increases vRMS increases Curve moves to right
Peak lowers in height to preserve area

13. Boltzmann factor: transcends chemistry
Average energy of a particle
From the M-B distribution
The Boltzmann factor – significant for any and all kinds of atomic or molecular energy
Describes the probability that a particle will adopt a specific energy given the prevailing thermal energy

14. Applying the Boltzmann factor
Population of a state at a level ε above the ground state depends on the relative value of ε and kBT
When ε << kBT, P(ε) = 1
When ε >> kBT, P(ε) = 0
Thermodynamics, kinetics, quantum mechanics

15. Collisions and mean free path
Collisions between molecules impede progress
Diffusion and effusion are the result of molecular collisions

16. Diffusion
The process by which gas molecules become assimilated into the population is diffusion
Diffusion mixes gases completely
Gases disperse: the concentration decreases with distance from the source

17. Effusion and Diffusion
The high velocity of molecules leads to rapid mixing of gases and escape from punctured containers
Diffusion is the mixing of gases by motion
Effusion is the escape of a gas from a container

18. Graham’s Law
The rate of effusion of a gas is inversely proportional to the square root of its mass
Comparing two gases

19. Living in the real world
For many gases under most conditions, the ideal gas equation works well
Two differences between the ideal and the real
Real gases occupy nonzero volume
Molecules do interact with each other – collisions are non-elastic

20. Consequences for the ideal gas equation
Nonzero volume means actual pressure is larger than predicted by ideal gas equation
Positive deviation
Attractive forces between molecules mean pressure exerted is lower than predicted – or volume occupied is less than predicted
Negative deviation
Note that the two effects actually offset each other

21. Van der Waals equation: tinkering with the ideal gas equation
Deviation from ideal is more apparent at high P, as V decreases
Adjustments to the ideal gas equation are made to make quantitative account for these effects
Correction for intermolecular interactions
Correction for molecular volume

22. Real v ideal At a fixed temperature (300 K):
PVobs < PVideal at low P
PVobs > PVideal at high P

23. Effects of temperature on deviations
For a given gas the deviations from ideal vary with T
As T increases the negative deviations from ideal vanish
Explain in terms of van der Waals equation

24. Interpreting real gas behaviors
First term is correction for volume of molecules
Tends to increase Preal
Second term is correction for molecular interactions
Tends to decrease Preal
At higher temperatures, molecular interactions are less significant
First term increases relative to second term