Kinetic theory

1. 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