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Ch18 The Micro/Macro Connection

講者： 許永昌 老師

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ContentsMolecular Speeds and CollisionsPressureTemperatureThermal energy and Specific heatThermal interaction and HeatIrreversible Processes & the 2nd Law of

thermodynamics.

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The aim of this chapterUnderstandMicroscopic

CollisionAverage translational

kinetic energyMicroscopic energiesMolecular basis

Energy transfer

Probability

MacroscopicPressureTemperature

Thermal energyIdeal-gas lawSpecific HeatHeat and Thermal

equilibriumEntropy

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Molecule Speeds ( 請預讀 P542)

Changing the (1)TEMPERATURE or changing to a (2) DIFFERENT GAS changes the most likely speed, but it does not change the shape of the distribution.

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Boltzmann distribution statesProbability

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*

1 1where and .

2B

P v dv e v dv

v mvk T

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Mean Free Path ( 請預讀 P543)

The average speed of N2 at 20oC is about 500 m/s

There must be some collisions happened.Reference:

http://en.wikipedia.org/wiki/Diffusion

~10m

~0.1 s (of course not)

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Mean Free Path (continue)

More careful calculation (all molecules move)

2

1

4 2 /N V r

2

area

assume that all other molecules are fixed 2

/ / 2

1

4 /

coll cyl

coll

N N V V N V r L

L

N N V r

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Stop to ThinkWhat would happen in the room if the

molecules of the gas were not moving?

What would happen in an isolated room if the molecular collisions were not perfectly elastic?

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Pressure in a Gas ( 請預讀 P544~P545)

Objects: A wall whose normal is x direction. Molecules in the left hand side of this wall.

Condition: perfectly elastic.

= + + …

wall on molecule ˆ2 xJ mv x

density probability Just 0

2 2 2

ˆ* * 2

2

where is the probability of a particle whose velocity i

ix i

ix i ix ixi i i

V v Jiavg

avgix i ix ix i x avg

i i

Av t p v v mv xN t J

Ft t

F N N NP mv p v v mv p v m v

A V V V

p v

s , and 1

1, 0and , respectively.

0, 0

ii

v p v

xx

x

velocity: # of particles collide with the wall:

i

i

vN t

,if .i ip v p v

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The Root-Mean-Square Speed ( 請預讀 P545~P546)

2rms avgv v

2 2

2 2 2

2 2 2

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assume R , R Rotation Group

i iavgi

ix i iy i iz ii i i

x y zavg avg avg

x avg

i i

v v p v

v p v v p v v p v

v v v

v

p v p v

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Stop to Think & ExerciseWhat are the definitions of

1. rms speed2. Average speed3. Average velocity

What are the benefits of the definition of rms speed?EXERCISE:

2 particles: v=3êx+êy,

3 particles: v=-2êx+2êy,

4 particles: v=-2êy.Find: (1) rms speed (2) average speed (3) average

velocity.

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HomeworkStudent workbook:

18.118.518.6

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Temperature ( 請預讀 P546~P548)

Microscopic eavg Macroscopic T. Average translational kinetic energy:

eavg(½mv2)avg.

Kinetic Theory: PV=Nm(vx

2)avg. (v2)avg=3(vx

2)avg.

Ideal-gas Law: PV=NkBT.

We get

3

2avg Bk T

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Temperature (continue)

For a gas, this thing we call TEMPERATURE

measures the average translational kinetic energy.

This concept of temperature also gives meaning to ABSOLUTE ZERO as the temperature at which eavg=0 and all molecular motion ceases.

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2avg Bk T

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HomeworkStudent workbook:

18.718.10

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Thermal Energy for monatomic Gases ( 請預讀 P549~P550)

Eth=Kmicro+Umicro.For monatomic gases:

Eth=Kmicro.Eth=Neavg=3/2NkBT=3/2nRT.

Owing to the 1st Law of thermodynamics,

We get CV=3/2R=12.5 J/mol K.

Q: How about other systems?

th V VE Q nC T

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The Equipartition Theorem ( 請預讀P550~P551)The thermal energy of a system of particles is equally divided among

all the possible energy modes. For a system of N particles at temperature T, the energy stored in each mode (each degree of freedom) is ½NkBT. It is not proved here. To prove it, you need the concepts of

Probability States Phase space Boltzmann distribution

Example: Solid (for high enough temperature) Dulong-Petit law

Detail: Solid State Physics. It has 6 degrees of freedom

Eth=6*N*½kBT C=3R=25.0 J/mol K~6.00 cal/mol K.*Solid State Physics, Ashcroft/Mermin, P463

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Specific Heat of diatomic molecules ( 請預讀 P552~P553)

2 22

2 2 2, , ,

1 1 1

2 2 2 2 2 2avg

y xzmicro CM x CM y CM z

zz yy xx

L LLMv Mv Mv

I I I

2 212 12

vibrational energyrotational kinetic energy

1 1

2 2v k r

Why? A: In quantum mechanics, 1. <Li>=nћ2. Discrete energy

levels for bounded states.

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不正規講法Microscopic Macroscopic

For a long time averageCase 1

Case 2

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2 xmv

21

2 ymv

21

2 zmv0

Teff

2

2z

zz

L

I

2

2y

zz

L

I

'21'

2 xm v

'21'

2 ym v

'21'

2 zm v T

T

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Additional Remark ( 補充 ) For two particles system

<erot,z>~mH=1.66*10-27 kg, r~3.7*10-11 m, ћ=1.05*10-34 Js, kB=1.38*10-23 J/K Teff~180K (n=1)

2 21 1 2 2

2 212

1 1

2 21 1

,2 2

tot

CM

K m v m v

Mv v

1 21 2 12 2 1where , , .

mmM m m v v v

M

2 2 2

2

1 1 1

2 2 2 2z

B effz H

L nk T

I m r

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Thermal Interactions and Heat ( 請預讀 P554~P555)

Microscopic

CollisionsThermal Equilibrium:

(e1)avg= (e2)avg. (e1)avg : average

translational energy.Energy can transfer

from 2 to 1: YES

Macroscopic

Thermal interactionThermal

Equilibrium:T1=T2.

Energy can transfer from 2 to 1: No

Th TcHeatSystem 1 System 2

Probability

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ExerciseConditions:

System 1: 4.00 mol N2 at T1=27oC.

System 2: 1.00 mol H2 at T2=327oC.

3 s.f.Find:

Thermal energiesTf =?Heat transfer=?

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HomeworkStudent Workbook:

18.1218.1418.15

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Irreversible Processes and the 2nd Law of Thermodynamics ( 請預讀 P556~P558)

Microscopic (reversible)One particle

Macroscopic (irreversible)Many particles

Go to reach Equilibrium.

Probability

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Order, Disorder and Entropy ( 請預讀 P558~P560)Scientists and engineers use a state

variable called entropy to measure the probability that a macroscopic state will occur spontaneously.

The second Law of thermodynamics: The entropy of an ISOLATED

SYSTEM never decreases. The entropy either increases, until the system reaches equilibrium, or, if the system began in equilibrium, stays the same.

Th Tc spontaneously.

(Heat)

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HomeworkStudent Workbook:

18.1718.19

Student Textbook:335165

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