Kinetic theory

Kinetic theoryKinetic theory

Collective behaviour of large Collective behaviour of large systemssystems

Why gases exert pressureWhy gases exert pressure

Gases are mostly empty spaceGases are mostly empty space Gases contain molecules which have random Gases contain molecules which have random

motionmotion The molecules have kinetic energyThe molecules have kinetic energy The molecules act independently of each other – The molecules act independently of each other –

there are no forces between themthere are no forces between them Molecules strike the walls of the container – the Molecules strike the walls of the container – the

collisions are perfectly elasticcollisions are perfectly elastic Exchange energy with the containerExchange energy with the container The energy of the molecules depends upon the The energy of the molecules depends upon the

temperaturetemperature

Large collections are very Large collections are very predictablepredictable

Fluctuations in behaviour of a small group of Fluctuations in behaviour of a small group of particles are quite noticeableparticles are quite noticeable

Fluctuations in behaviour of a large group (a Fluctuations in behaviour of a large group (a mole) of particles are negligiblemole) of particles are negligible

Large populations are statistically very Large populations are statistically very reliablereliable

Pressure and momentumPressure and momentum

Pressure = force/unit areaPressure = force/unit area Force = mass x accelerationForce = mass x acceleration Acceleration = rate of change of velocityAcceleration = rate of change of velocity Force = rate of change of momentumForce = rate of change of momentum Collisions cause momentum changeCollisions cause momentum change Momentum is conservedMomentum is conserved

Elastic collision of a particle with the Elastic collision of a particle with the wallwall

Momentum lost by particle = -2mvMomentum lost by particle = -2mv Momentum gained by wall = 2mvMomentum gained by wall = 2mv Overall momentum change = -2mv + 2 mv = Overall momentum change = -2mv + 2 mv =

00 Momentum change per unit time = 2mv/Momentum change per unit time = 2mv/ΔΔtt

momentum change no collisions 1

collision time areaP x x

Factors affecting collision rateFactors affecting collision rate

1.1. Particle velocity – the faster the particles Particle velocity – the faster the particles the more hits per secondthe more hits per second

2.2. Number – the more particles – the more Number – the more particles – the more collisionscollisions

3.3. Volume – the smaller the container, the Volume – the smaller the container, the more collisions per unit area more collisions per unit area

222 o ovN mv N

P mvV V

Making refinementsMaking refinements

We only considered one wall – but there are We only considered one wall – but there are six walls in a containersix walls in a container Multiply by 1/6Multiply by 1/6

Replace vReplace v22 by the mean square speed of the by the mean square speed of the ensemble (to account for fluctuations in ensemble (to account for fluctuations in velocity)velocity)

21

3

om v NP

V

2 221

6 3o omv N mv N

P xV V

Boyle’s LawBoyle’s Law

Rearranging the previous equation:Rearranging the previous equation:

Substituting the average kinetic energySubstituting the average kinetic energy Compare ideal gas law PV = nRT:Compare ideal gas law PV = nRT:

The average kinetic energy of one mole of The average kinetic energy of one mole of molecules can be shown to be 3RT/2molecules can be shown to be 3RT/2

22 2

3 2 3o o k

m vPV N N E

3

2kE nRT

Root mean square speedRoot mean square speed

Total kinetic energy of one moleTotal kinetic energy of one mole

But molar mass M = NBut molar mass M = Noomm

Since the energy depends only on T, vSince the energy depends only on T, vRMSRMS decreases as M increasesdecreases as M increases

21 3

2 2k oE N m v RT

M

RT

mN

RTv

oRMS

33

Speed and temperatureSpeed and temperature

Not all molecules move at the same speed Not all molecules move at the same speed or in the same directionor in the same direction

Root mean square speed is useful but far Root mean square speed is useful but far from complete description of motionfrom complete description of motion

Description of distribution of speeds must Description of distribution of speeds must meet two criteria:meet two criteria: Particles travel with an average value speedParticles travel with an average value speed All directions are equally probableAll directions are equally probable

Maxwell meet BoltzmannMaxwell meet Boltzmann

The Maxwell-Boltzmann The Maxwell-Boltzmann distribution describes the distribution describes the velocities of particles at a velocities of particles at a given temperaturegiven temperature

Area under curve = 1Area under curve = 1 Curve reaches 0 at v = 0 Curve reaches 0 at v = 0

and and ∞∞

2 / 22

3 / 2

( )

42

Bmv k T

B

F v Kv e

mK

k T

M-B and temperatureM-B and temperature

As T increases vAs T increases vRMS RMS

increasesincreases Curve moves to rightCurve moves to right Peak lowers in height Peak lowers in height

to preserve areato preserve area

Boltzmann factor: transcends Boltzmann factor: transcends chemistrychemistry

Average energy of a particleAverage energy of a particle

From the M-B distributionFrom the M-B distribution

The Boltzmann factor – significant for any and all The Boltzmann factor – significant for any and all kinds of atomic or molecular energykinds of atomic or molecular energy

Describes the probability that a particle will adopt Describes the probability that a particle will adopt a specific energy given the prevailing thermal a specific energy given the prevailing thermal energyenergy

1exp

expB

B

k T

k T

2

2

mv

( ) expB

Pk T

Applying the Boltzmann factorApplying the Boltzmann factor

Population of a state at a level Population of a state at a level εε above the above the ground state depends on the relative value ground state depends on the relative value of of εε and k and kBBTT

When When εε << << kkBBT, P(T, P(εε) = 1) = 1

When When εε >> >> kkBBT, P(T, P(εε) = 0) = 0

Thermodynamics, kinetics, quantum Thermodynamics, kinetics, quantum mechanicsmechanics

( ) expB

Pk T

Collisions and mean free pathCollisions and mean free path

Collisions between Collisions between molecules impede molecules impede progressprogress

Diffusion and effusion Diffusion and effusion are the result of are the result of molecular collisionsmolecular collisions

DiffusionDiffusion

The process by which gas molecules The process by which gas molecules become assimilated into the population is become assimilated into the population is diffusiondiffusion

Diffusion mixes gases completelyDiffusion mixes gases completely Gases disperse: the concentration Gases disperse: the concentration

decreases with distance from the sourcedecreases with distance from the source

Effusion and DiffusionEffusion and Diffusion

The high velocity of molecules leads to rapid The high velocity of molecules leads to rapid mixing of gases and escape from punctured mixing of gases and escape from punctured containerscontainers

Diffusion is the mixing of gases by motionDiffusion is the mixing of gases by motion Effusion is the escape of a gas from a Effusion is the escape of a gas from a

containercontainer

Graham’s LawGraham’s Law

The rate of effusion of a gas is inversely The rate of effusion of a gas is inversely proportional to the square root of its massproportional to the square root of its mass

Comparing two gasesComparing two gases

mRate

1

1

2

1

2

2

1

m

m

m

m

Rate

Rate

Living in the real worldLiving in the real world

For many gases under most conditions, the For many gases under most conditions, the ideal gas equation works wellideal gas equation works well

Two differences between the ideal and the Two differences between the ideal and the realreal Real gases occupy nonzero volumeReal gases occupy nonzero volume Molecules do interact with each other – Molecules do interact with each other –

collisions are non-elasticcollisions are non-elastic

Consequences for the ideal gas Consequences for the ideal gas equationequation

1.1. Nonzero volume means actual pressure is Nonzero volume means actual pressure is largerlarger than predicted by ideal gas equation than predicted by ideal gas equation

Positive deviationPositive deviation

2.2. Attractive forces between molecules mean Attractive forces between molecules mean pressure exerted is lower than predicted – pressure exerted is lower than predicted – or volume occupied is less than predictedor volume occupied is less than predicted

Negative deviationNegative deviation Note that the two effects actually offset Note that the two effects actually offset

each othereach other

Van der Waals equation: Van der Waals equation: tinkering with the ideal gas equationtinkering with the ideal gas equation

Deviation from ideal is more apparent at Deviation from ideal is more apparent at high P, as V decreaseshigh P, as V decreases

Adjustments to the ideal gas equation are Adjustments to the ideal gas equation are made to make quantitative account for these made to make quantitative account for these effectseffects

nRTnbVV

anP

2

2

Correction for intermolecular

interactions

Correction for molecular volume

Real v idealReal v ideal

At a fixed temperature At a fixed temperature (300 K):(300 K): PVPVobsobs < PV < PVidealideal at low P at low P

PVPVobsobs > PV > PVidealideal at high P at high P

Effects of temperature on deviationsEffects of temperature on deviations

For a given gas the For a given gas the deviations from ideal deviations from ideal vary with Tvary with T

As T increases the As T increases the negative deviations negative deviations from ideal vanishfrom ideal vanish

Explain in terms of van Explain in terms of van der Waals equationder Waals equation

nRTnbVV

anP

2

2

Interpreting real gas behaviorsInterpreting real gas behaviors

First term is correction for First term is correction for volume of moleculesvolume of molecules Tends to increase PTends to increase Prealreal

Second term is correction Second term is correction for molecular interactionsfor molecular interactions Tends to decrease PTends to decrease Prealreal

At higher temperatures, At higher temperatures, molecular interactions are molecular interactions are less significantless significant First term increases relative to First term increases relative to

second termsecond term

2

2

2

2

V

an

nbV

nRTP

nRTnbVV

anP