Thermo & Stat Mech - Spring 2006 Class 14 1 Thermodynamics and Statistical Mechanics Kinetic Theory of Gases.

Thermo & Stat Mech - Spring 2006 Class 14

1

Thermodynamics and Statistical Mechanics

Kinetic Theory of Gases


2

Mixing of Two Ideal Gases

Change of Gibbs Function

)()(

222111

22112211

ififif

fffiii

ggnggnGGG

gngnGgngnG

An expression is needed for the specific Gibbs function.


3

Specific Gibbs Function

0

0

so gas, idealan for

hTsTcg

hTchdT

dhc

TshTsPvug

P

P

P


4

Specific Entropy

P

dPR

T

dTcdP

T

v

T

dTcds

vdPdTcTds

PdvvdPPdvdTc

PdvRdTdTcPdvdTRc

PdvdTcPdvduTds

PP

P

P

PP

v

)(


5


00

00

0

0

lnln

)lnln(

lnln

hTsTTcTcPRTg

hsPRTcTTcg

hTsTcg

sPRTcsP

dPR

T

dTcds

PP

PP

P

P

P


6


)(ln

lnln

lnln

00

00

PRTg

RT

h

R

sTccRTPRTg

hTsTTcTcPRTg

PP

PP


7

Mixing of Two Ideal Gases

i

f

i

f

if

if

ifif

P

Pn

P

PnRTG

PPn

PPnRTG

ggnggnG

PRTg

2

22

1

11

222

111

222111

lnln

)]ln(ln

)ln(ln[

)()(

)(ln


8

For the Same Pressure

22

22

21

222

11

11

21

111

21

2

22

1

11

so

so

lnln

xP

PP

n

nP

nn

nPxP

xP

PP

n

nP

nn

nPxP

PPP

P

Pn

P

PnRTG

i

ff

i

ff

ii

i

f

i

f


9

For the Same Pressure

2211

2211

22

11

2211

lnln)(

lnln

lnln

lnln

xxxxnRT

GS

xxxxnRTG

xn

nx

n

nnRTG

xnxnRTG

P


10

For the Same Volume

2

1ln

2

1ln

lnln

so constant, is

lnln

21

2

22

1

11

1

2

22

1

11

nnRTG

V

Vn

V

VnRTG

VPT

P

Pn

P

PnRTG

f

i

f

i

i

f

i

f


11

For the Same Volume

2ln)(

2ln)(

2

1ln)(

2

1ln

2

1ln

21

21

21

21

RnnS

RTnnG

RTnnG

nnRTG


12

Basic Assumptions

1. A macroscopic volume contains a large number of molecules.

2. The separation of molecules is large compared to molecular dimensions.

3. No forces exist between molecules except those associated with collisions

4. The collisions are elastic.


13

Basic Assumptions

When no external forces are applied:

5. The molecules are uniformly distributed within a container.

6. The directions of the velocities of the molecules are uniformly distributed.

The fraction of molecules with speeds in the range v to v + dv is: f (v) dv


14

Molecular Speeds

f (v) is the probability density.

speed squaremean Root

speed squareMean )(

speed averageor Mean )(

1)(

2

0

22

0

0

vv

dvvfvv

dvvvfv

dvvf

rms


15

Gas Pressure


16

Gas Pressure

ddvddt

vdNmv

dt

dpdt

vdNmv

dt

vdp

mvp

mvmvp

vmpdt

pdF

dA

dF

A

FP

),,(cos2

),,(cos2

),,(

cos2

)cos(cos


17

Gas Pressure


18

Molecular Flux

dddvvvfdAn

dt

vdN

dddvvfnvdtdAvdN

V

Nn

cossin])([4

),,(

4

sin)())(cos(),,(


19

Molecular Flux


20

Molecular Flux

vnvn

dddvvvfn

dddvvvfn

ddAdt

vdN

4

1)2(

2

1)(

4

cossin)(4

cossin])([4

),,(

2

0

2

00


21

Gas Pressure

22

2

0

2

0

2

0

2

3

1)2(

3

1

4

2

sincos)(4

2

cossin])([4

),,(

),,(cos2

vmnvnm

PdAdt

dp

dddvvfvdAnm

dt

dp

dddvvvfdAn

dt

vdN

ddvddt

vdNmv

dt

dp


22

Ideal Gas Law

AA

A

N

RkNkTRT

N

NPV

N

NnnRTPV

vNmvNmvmVnPV

vmnP

222

2

2

1

3

2

3

1

3

13

1


23

Molecular Kinetic Energy

kTvm

NkTvNm

NkTPVvNmPV

2

3

2

1

2

1

3

2

2

1

3

2

2

2

2


24

Equipartition of Energy

freedom of degreeper Energy 2

12

1

2

12

3

2

12

3

2

1

2

222

2

kT

kTvm

kTvvvm

kTvm

i

zyx


25

Internal Energy

nRdT

dUCnRTNkTU

f

nRdT

dUCnRTNkTU

f

fkTNfU

V

V

2

5

2

5

2

5

5 gas Diatomic2

3

2

3

2

3

3 gas MonatomicMolecule

Freedom of Degrees

2

1


26

Heat Capacities

fnRf

nRf

C

C

nRf

nRnRf

nRCC

nRf

dT

dUC

RTnfkTNfU

V

P

VP

V

21

2

12

122

2

2

1

2

1


27

Maxwell Velocity Distribution

Consider a gas at equilibrium. The number of molecules in any range of velocity does not change. Collisions cause individual molecules to change velocity, but the distribution does not change. For every collision that changes the distribution, there must be one that changes it back.

vdvvvdvF

and between number )( :Let


28

Molecular Collisions


29

Molecular Collisions

vvAevF

vvvv

vFvFvFvF

vFvF

vFvF

vvvv

)( :Then

''

:so elastic are Collisions

)'()'()()(

)'()'(collision reverse ofy Probabilit

)()(collision asuch ofy Probabilit

''

22

21

22

21

2121

21

21

2121


30

Maxwell Distribution

0

4

0

221

23

0

2

0

2

2

2

2

43

)(

4)(

4)(

to from speeds ofNumber )(

dvevAm

NkT

dvvNvmNkT

dvevAdvvNN

dvvAedvvN

dvvvdvvN

v

v

v


31

Evaluation of Constants

2

3

4

83

3

8

3

4

23

25

2

4

252

0

44

230

22

2

2

I

I

m

kT

IdvevI

dvevI

v

v


32


dvevkT

m

N

dvvNdvvf

dvevkT

mNdvvN

kT

mNNA

kT

m

kT

mv

kT

mv

22

23

22

23

2323

2

2

24

)()(

24)(

22


33



0

0.0005

0.001

0.0015

0.002

0.0025

0 200 400 600 800 1000 1200 1400

v (m/s)

Pro

bab

ility


34

Some Molecular Speeds

ProbableMost 2

3

3)(

8)(

max

2

0

22

0

m

kTv

m

kTvv

m

kTdvvfvv

m

kTdvvvfv

rms

Thermo & Stat Mech - Spring 2006 Class 14 1 Thermodynamics and Statistical Mechanics Kinetic Theory of Gases.

Documents

thermo stat mech spring

volume slide

gas pressure slide

molecular flux slide

maxwell distribution

ideal gases slide

specific entropy slide

internal energy slide