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Kinetic theory Collective behaviour of large systems.

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Presentation on theme: "Kinetic theory Collective behaviour of large systems."— Presentation transcript:

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 1.Particle velocity – the faster the particles the more hits per second 2.Number – the more particles – the more collisions 3.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 v 2 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 = N o m  Since the energy depends only on T, v RMS 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 v RMS 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 k B T  When ε << k B T, P(ε) = 1  When ε >> k B T, 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 1.Nonzero volume means actual pressure is larger than predicted by ideal gas equation  Positive deviation 2.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):  PV obs < PV ideal at low P  PV obs > PV ideal 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 P real  Second term is correction for molecular interactions  Tends to decrease P real  At higher temperatures, molecular interactions are less significant  First term increases relative to second term

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