prof. fred remer university of north dakota kinetic theory o2o2 n2n2 n2n2 o2o2 h2oh2o n2n2 n2n2 n2n2...

Kinetic TheoryKinetic TheoryProf. Fred RemerUniversity of North Dakota

O2

N2

N2

O2 H2O

N2 N2

N2

N2

Kinetic TheoryKinetic Theory(or life as a molecule)(or life as a molecule)


ObjectiveObjective

• Be able to define temperature and Be able to define temperature and pressurepressure

• Be able to perform simple Be able to perform simple calculations using the Ideal Gas Lawcalculations using the Ideal Gas Law


ReadingReading

• Wallace and Hobbs, Wallace and Hobbs, pp. 64, 74pp. 64, 74

• Bohren and Bohren and AlbrechtAlbrecht– pp. 1-30pp. 1-30


Kinetic TheoryKinetic Theory

• Once Upon A Time There Was A Once Upon A Time There Was A MoleculeMolecule



• The molecule had no size or internal The molecule had no size or internal structure, but it was a happy structure, but it was a happy molecule. Her name was Point Mass.molecule. Her name was Point Mass.

m = massm = mass



• One day, Point Mass decided to One day, Point Mass decided to move. He only moved in one move. He only moved in one direction. He moved a a constant direction. He moved a a constant speed.speed.

x - directionx - direction velocity = vvelocity = vxx


x - directionx - direction velocity = vvelocity = vxx


• He had momentum!He had momentum!

momemtummomemtum

= mv= mvxx



• He had so much momentum, he He had so much momentum, he could not slow down when he saw could not slow down when he saw the wall!the wall!



• He hit the wall!He hit the wall!

xx

AreaArea

AA

OUCH!


Kinetic TheoryKinetic Theory• But much to his surprise, he rebounded! His collision But much to his surprise, he rebounded! His collision

was perfectly elastic! No energy was lost in the collision.was perfectly elastic! No energy was lost in the collision.



• He had the same momentum leaving He had the same momentum leaving the wall as he had before the collision, the wall as he had before the collision, but in the opposite direction.but in the opposite direction.

momentummomentum

xmv



• His change in momentum wasHis change in momentum was

Change in MomentumChange in Momentum

xxx mvmvmv 2



• The force exerted on Point Mass by The force exerted on Point Mass by the wall wasthe wall was

oror

Time

MomentuminChangeForce

dtFmv xx 2Change Momentum



• Point Mass had other friends who are Point Mass had other friends who are molecules identical to himself.molecules identical to himself.



• They all move at the same velocity vThey all move at the same velocity vxx



• The molecules do not interact The molecules do not interact between themselves ...between themselves ...


Kinetic TheoryKinetic Theory• ……but they all interacted with the wallbut they all interacted with the wall

xx



• The number of molecules (N) in a The number of molecules (N) in a given volume (V) is the number given volume (V) is the number density (n)density (n)

V = VolumeV = Volume

N = # of N = # of moleculesmolecules

V

Nn

n = number densityn = number density



• The flux of molecules headed toward The flux of molecules headed toward the wall is ...the wall is ...

1/2 moving towards at v1/2 moving towards at vxx

1/2 moving away at v1/2 moving away at vxxvvxx

vvxx

2

nv x



• The number of molecules striking the The number of molecules striking the wall (A) during a time period (t) is ...wall (A) during a time period (t) is ...

AA

vvxx

vvxx

tt

At2

nv x



• The total time integrated force on the The total time integrated force on the wall (A) is ...wall (A) is ...

AA At2

nvmv2dtF x

xx



• The time-averaged force on the wall The time-averaged force on the wall is ...is ...

AA AnmvdtFt

1F 2

xxx



• The average force per unit area is ...The average force per unit area is ...

AA A

AnmvdtF

At

1

A

F 2x

xx



• ……Which is pressure!Which is pressure!

AA2

xx nmv

A

Fp



• Lets modify one assumption. The Lets modify one assumption. The molecules are moving at different molecules are moving at different speeds.speeds.



• Let’s replace vLet’s replace vxx22 with an average. with an average.

2xvnmp



• In reality, the molecules are moving In reality, the molecules are moving in all directions (not just x).in all directions (not just x).

2z

2y

2x vvv

22z

2y

2x vvvv

22x v

3

1v



• Substitute back into the equationSubstitute back into the equation

2xvnmp

22x v

3

1v

2mvn3

1p



• This looks like Kinetic Energy!This looks like Kinetic Energy!

KE

MonatomicGas

2mvn3

1p

2mv2

1KE

2mv2

1n

3

2p



• Definition of TemperatureDefinition of Temperature

– Temperature is a measure of the Temperature is a measure of the average KE of the molecules! average KE of the molecules!

where k = Boltzmann Constant = 1.38 x 10-23 J/K

2mv2

1n

3

2kT



• Substitute temperature into pressureSubstitute temperature into pressure

2mv2

1n

3

2p 2mv

2

1

V

N

3

2p

2mv2

1n

3

2kT



• Ideal Gas LawIdeal Gas Law

kTV

Np

NkTpV

or

where ...

p = pressureV = volumeN = number of moleculesT = temperaturek = Boltzman Constant



• Monatomic MoleculesMonatomic Molecules– Energy Is a Result of Atom’s Motion Energy Is a Result of Atom’s Motion

OnlyOnly



• Polyatomic MoleculesPolyatomic Molecules– Energy Is a Result ofEnergy Is a Result of

• Atom’s MotionAtom’s Motion• Rotation, Vibration and Oscillation of Rotation, Vibration and Oscillation of

MoleculeMolecule



• Polyatomic MoleculesPolyatomic Molecules– Need to Account for Need to Account for

Other Forms of Other Forms of Molecular EnergyMolecular Energy

TotalMolecular

Energy

Kinetic EnergyDue to Motion

Kinetic EnergyDue to Rotation &

Vibration= +



• Polyatomic MoleculesPolyatomic Molecules– More Complex More Complex

Molecules Have More Molecules Have More Rotational & Vibrational Rotational & Vibrational EnergyEnergy

TotalMolecular

Energy



Vibration= +



• Polyatomic MoleculesPolyatomic Molecules– Low PressureLow Pressure

• Approximates Ideal GasApproximates Ideal Gas

– High PressureHigh Pressure• Deviates MoreDeviates More

TotalMolecular

Energy



Vibration= +



• SummarySummary– pressure is a measure of the total kinetic energy pressure is a measure of the total kinetic energy

of molecules, the force per unit area of these of molecules, the force per unit area of these moleculesmolecules

– temperature is proportional to the average temperature is proportional to the average kinetic energy of moleculeskinetic energy of molecules

– from this kinetic theory viewpoint, we can from this kinetic theory viewpoint, we can derive the perfect gas law:derive the perfect gas law:

NkTpV


Kinetic TheoryKinetic Theory• We will return to the perfect gas law from a We will return to the perfect gas law from a

macroscopic point of view and derive macroscopic point of view and derive exactly the same relationship:exactly the same relationship:

NkTpV

RTp

oror

nMmm

NkR

V

m ,,

where ...

p = pressureV = volume, n = number of molesN = number of moleculesT = temperaturek = Boltzmann constantm = mass, M = molecular weight

prof. fred remer university of north dakota kinetic theory o2o2 n2n2 n2n2 o2o2 h2oh2o n2n2 n2n2 n2n2...

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