Collective behaviour of large systems Kinetic theory Collective behaviour of large systems

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

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

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

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

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

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)

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

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

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

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 ∞

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

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

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

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

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

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

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

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

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

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

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

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

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