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Lecture 35Lecture 35 The Kinetic Theory of GasesThe Kinetic Theory of Gases

Chapter 11Chapter 11 Monday November 26thMonday November 26th

Quick review of 3rd law

Basic assumptions behind Kinetic TheoryThe molecular speed distribution function

Molecular flux

Gas pressure and the ideal gas law

Equipartition of energy (if time)

Reading:Reading: Chapter 11 (pages 181Chapter 11 (pages 181 -- 211)211)Homework 10 will be due on Fri. Nov. 30Homework 10 will be due on Fri. Nov. 30thth

Note:Note: Exam 3 will be on Mon. Dec. 3Exam 3 will be on Mon. Dec. 3rdrd in classin classI will post practice test and formula sheetI will post practice test and formula sheet

ReviewReview -- Thu. Nov. 29th at 5pmThu. Nov. 29th at 5pm

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Statements of the third lawStatements of the third law

0 0

0 0

lim lim

lim lim

T T

T T

G H

G H

=

=

The Nernst formulation of the Third Law:

All reactions in a liquid or solid in thermal equilibriumAll reactions in a liquid or solid in thermal equilibriumtake place with no change of entropy in the neighborhoodtake place with no change of entropy in the neighborhood

of absolute zero.of absolute zero.

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Statements of the third lawStatements of the third law

0 0

0 0

lim lim

lim lim

T T

T T

G H

G H

=

=

Plancks statement of the Third Law:

The entropy of a true equilibrium state of a system atThe entropy of a true equilibrium state of a system atabsolute zero is zero.absolute zero is zero.

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TheThe TdsTds equationsequations

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Specific heat of a solidSpecific heat of a solid

Copper

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Statements of the third lawStatements of the third law

0 0

0 0

lim lim

lim lim

T T

T T

G H

G H

=

=

Another statement of the Third Law is:

It is impossible to reduce the temperature of a system toIt is impossible to reduce the temperature of a system toabsolute zero using a finite number of processes.absolute zero using a finite number of processes.

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Adiabatic coolingAdiabatic cooling

T1T2T3

B1SS

TT

dU= TdSPdV= TdSMdB = QMdB

MM

B2 > B1

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Equivalence of the 3Equivalence of the 3rdrd law statementslaw statements

T1T2T3

B1SS

TT

dU= TdSPdV= TdSMdB = QMdB

0

B2 > B1

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Kinetic TheoryKinetic Theory -- Basic assumptionsBasic assumptions

1. A macroscopic volume contains a large number ofmolecules.

d d

d

2. The separation of molecules is large compared with

molecular dimensions and with the range ofintermolecular forces.

0.0002 to

0.05 eV

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3. No forces exist between molecules except thoseassociated with collisions.

We treat these collisions like hard collisions betweenbilliard balls.

4. All collisions are elastic.

In other words, energy and momentum are conserved.Again, the analogy with collisions between billiard balls.

5. The molecules are uniformly distributed within a container.

6. The directions of the velocities are uniformly distributed.

This statement characterizes the fact that the motion ofmolecules is random.

Kinetic TheoryKinetic Theory -- Basic assumptionsBasic assumptions

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The probability density functionThe probability density function

The random motions of the molecules can becharacterized by a probability distribution function.

Since the velocity directions are uniformly distributed, we

can reduce the problem to a speed distribution functionwhich is isotropic.

( )0

1v dv

=

Let f(v)dv be the fractional number of molecules in thespeed range from v to v +dv.

A probability distribution function has to satisfy thecondition

( )0v vf v dv

= ( )2 2

0v v f v dv

= 2

rmsv v=

We can then use the distribution function to compute theaverage behavior of the molecules:

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Exam 2 statisticsExam 2 statistics

20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

Mean = 68%Median = 67%

Fre

quency

Score (%)

ss

ff((ss))

( )

100

0

100

0

( ) 68s

s s f s

s f s ds

=

= =

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( )# in cylinder ( ) cosn f v dv vdt dA=

Molecular FluxMolecular Flux

Element of area onthe wall of thecontainer where

molecules exert apressure due totheir change inmomentum upon

bouncing from thewall.

# of molecules

volume

n =

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d

v

v

v

v

Molecular FluxMolecular Flux

2 sind v d d =

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Molecular FluxMolecular Flux

End result Number of molecules striking a unit area of the container

walls per unit time is called the flux :

1

4nv =

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Gas pressure and the ideal gas lawGas pressure and the ideal gas law

Assume specular collisions*

*Bold assumption but more general calculation gives same result.

( )cos cos

2 cos

mv mv

mv

=

Change in momentum:

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Gas pressure and the ideal gas lawGas pressure and the ideal gas law

End result Pressure related to the momentum transfer to the walls,

i.e.

2Force / 1

3

dp dtP nmv

dA dA= = =

Or,

2 21 2 1

3 3 2PV Nmv N mv NkT

= = =

where,1 1

23 18314 J kilomole K , 6.02 10 kilomoleAnR RNk nR kN N

= = =

k= 1.38 1023 J.K1 (Boltzmanns constant)

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Gas pressure and the ideal gas lawGas pressure and the ideal gas law

Kinetic theory provides a natural interpretation of theabsolute temperature of a dilute gas. Namely, thetemperature is proportional to the mean kinetic energy (K)of the gas molecules.

The mean kinetic energy is independent of pressure,volume, and the molecular species, i.e. it is the same forall molecules.

2 21 2 1 2

3 3 2 3PV Nmv N mv N K NkT

= = = =