study heat phenomena and moving law of matters

Part Two Thermodynamics

Based on the mechanics law and the statistical theory.

Chapter 6 Kinetic Theory of Gases

Chapter 7 Thermodynamics

SStudy heat phenomena in the view of energy transformation based on some experimental laws.

---Microscopic theory.

---Macroscopic theory

Chapter 6 Kinetic Theory of Gases

§§6-1 6-1 Essential Concepts of the Kinetic Theory of Gases

All matters are made up by molecules and All matters are made up by molecules and molecules are separatedmolecules are separated

1. Standpoint of the kinetic theory of gases1. Standpoint of the kinetic theory of gases

Molecules are always in random motionMolecules are always in random motion

Brownian Motion:

Liquid

random motionrandom motion

Pollen grain 花粉

d

0r

RFRF

AFAF

NFNF

f

r dd ：： Effective diameterEffective diameter

When ,mr 910 f f 0 0

There is interaction between moleculesThere is interaction between molecules

Here, net forceHere, net force==00

rroo ：： Equilibrium distance~Equilibrium distance~1010-10-10mm

RF: repulsive force, AF: attractive force

2. The characters of molecules of gases2. The characters of molecules of gases

SmallSmall ：： Diameter Diameter ~~1010-10-10 mm

GreatGreat ：： 6610102323 //mol----mol----Avogadro’s constantAvogadro’s constant

FastFast ：： Ordinary state, their average Ordinary state, their average speedspeed~~hundreds meters/shundreds meters/s

ConfusionConfusion ： ： random motion, change rapidly random motion, change rapidly

3. 3. Statistical law

Gordon BoardGordon Board experiment experiment

nail

groovegroove

It is an accident It is an accident which groove a ball which groove a ball falls downfalls down

The distribution obeys The distribution obeys certain rule for large certain rule for large mount of balls.mount of balls.

A large mount of accidents A large mount of accidents appear certain rules under some conditions.appear certain rules under some conditions.

. Probability. Probability

N

NP A

NA lim oror

N

NP A

A

ProbabilityProbability (P)(P): : under some conditions, the under some conditions, the magnitude of probabilities that an accident magnitude of probabilities that an accident appears.appears.

Let Let NN--Total numbers of the experiment--Total numbers of the experiment ，， NNAA--The numbers of about accident --The numbers of about accident AA

appearanceappearance ，， thenthen

For For all the accidentsall the accidents of the experiment of the experiment ：：

N

NP i

i 1

----normalization----normalization

The probability of any accident appearance:The probability of any accident appearance:

10 iP

N

N i

. Statistical average quantity. Statistical average quantity

n

nn

NNN

NMNMNMM

21

2211

N

NM ii

Measure a physical quantity Measure a physical quantity MM: the appearance : the appearance times of times of MM11 、、 MM22 、、 MMn n is is NN11 、、 NN22 、、 NNn n rr

espectively, espectively,

The arithmetical average quantity ofThe arithmetical average quantity of MM ：：

ii PM

(Statistical average quantity)(Statistical average quantity)

N N : average quantity true quantity: average quantity true quantity

1. Equilibrium state1. Equilibrium state

§6-2 Equilibrium state Ideal gas law§6-2 Equilibrium state Ideal gas law

The temperature and pressure are the same at The temperature and pressure are the same at any point of the system and do not change with any point of the system and do not change with time when it is not influenced by outside.time when it is not influenced by outside.

1T

1p

2T

2pA B adiabatic walladiabatic wall

Two systems attain their Two systems attain their equilibrium state respectivelyequilibrium state respectively

A B

The wall conducts heat andThe wall conducts heat and permits molecules passing throughpermits molecules passing through

A B

Two systems attain Two systems attain their thermal their thermal equilibriumequilibrium

Two system Two system interactinteract

TT 、、 P P same same everywhereeverywhere

p

T

p

T

If two systems are thermal equilibrium with the If two systems are thermal equilibrium with the third system respectively, then the two systems third system respectively, then the two systems are thermal equilibrium, too.are thermal equilibrium, too.

--zero law of thermodynamics--zero law of thermodynamics

C

A B A B

C

An equilibrium state of a thermodynamic system can be described by its state parameters: Pressure p, volume V and temperature T

2. Thermal equilibrium process2. Thermal equilibrium process

The process is carried The process is carried out very slowly.out very slowly.

Any equilibrium state can be represented by a dot on the p-V diagram.

p

V

a

b

),,( ccc TVpc

The system remains The system remains approximately in approximately in thermodynamic thermodynamic equilibrium at all stages.equilibrium at all stages.

orp-T diagram, T-V diagram

.ConstT

pV

If the macroscopic parameters If the macroscopic parameters pp,,VV,,T T of an gas of an gas

satisfy satisfy It is called ideal gasIt is called ideal gas

/molm104.22 33,0

molV

KT 15.2730

)(N/mPa10013.1 250 p atm1

C00

3.The ideal gas law ( state equation of ideal gas )3.The ideal gas law ( state equation of ideal gas )

Under the standard condition:Under the standard condition:

0

00

T

Vp

T

pV

molVM

T

p,0

0

0

0

,00

T

VpR mol

RTM

PV

---- ---- state equation of ideal state equation of ideal gas gas

RM

Here Here KJ/mol31.8 ----mole gas constant----mole gas constant

For For MM kgkg ideal gas ideal gas

Molecules are regarded as particles.Molecules are regarded as particles. No interaction between molecules except for the inNo interaction between molecules except for the in

stantaneous impulsive force during the collisions.stantaneous impulsive force during the collisions. The collision are perfectly elastic.The collision are perfectly elastic.The motion of an individual molecule obey Newton’s The motion of an individual molecule obey Newton’s

law.law.

§6-3 The representation of pressure §6-3 The representation of pressure for ideal gasfor ideal gas

Find the connection between the macroscopic Find the connection between the macroscopic parameters and microscopic parameters of the parameters and microscopic parameters of the gas.gas.

1.The microscopic model of an ideal gas1.The microscopic model of an ideal gas

The possibility where a molecule locates in is The possibility where a molecule locates in is the same under the equilibrium state. the same under the equilibrium state.

i.e. the density of molecules is the same everyi.e. the density of molecules is the same everywhere.where.

Molecules have equal possibilities to move in Molecules have equal possibilities to move in all directions. all directions.

Then they are equal that the average values oThen they are equal that the average values of the components of the molecules’ velocity alf the components of the molecules’ velocity along any direction. ong any direction.

2. Statistical assumption 2. Statistical assumption

Such asSuch as222

zyx vvv

1l

2l

3l

1A2A

The impulse acts on The impulse acts on AA11

as one molecule collides as one molecule collides with with AA11 ：：

xmv2

The total impulse The total impulse of one molecule acof one molecule acts on ts on AA1 1 in one secin one sec

ondond

1

2

122

l

mv

l

vmv xx

x

3.The pressure equation of ideal gas3.The pressure equation of ideal gas

y

xz

v

xvyv

zv

The total impulse of The total impulse of NN molecules act on molecules act on AA1 1 in one in one secondsecond

1

2

1

22

1

21

l

mv

l

mv

l

mv Nxxx

i

ixvl

m 2

1

)1( ttF

i

ixvl

mF 2

1

The average force of The average force of NN molecules act on molecules act on AA11

The pressure acting on The pressure acting on AA11

32ll

Fp

N

vvv

V

Nm Nxxx22

22

1

2xvnm

i

ixvlll

m 2

321

n: the number of molecules in per unit volume

The density of number of molecules

DefinitionDefinition ：： The average translational kinetic The average translational kinetic energy of the moleculesenergy of the molecules

2

2

1vmt

Then Then

2

2

1

3

2vmnp tn

3

2

222zyx vvv 2

3

1v 2

3

1vnmp

RemarksRemarks

The result is the sameThe result is the same if we consider the if we consider the molecules collide with other wall of the molecules collide with other wall of the container. container.

The result is the sameThe result is the same if we consider the if we consider the molecules collide each other while they are molecules collide each other while they are moving toward the wall. moving toward the wall.

The result is the sameThe result is the same for any shape of the for any shape of the the container.the container.

It’s statistical result. It’s statistical result. So it can be used only So it can be used only for great number of molecules.for great number of molecules.

np tp

Let Let NN----the number of molecules with mass the number of molecules with mass MM kg kg ,, NNoo-- -- the number of molecules about the number of molecules about 11 molemole ，， mm– – the mass of one moleculethe mass of one molecule

V

RTMp

T

N

R

V

N

o

nkT

§§6-4 The temperature of ideal gas6-4 The temperature of ideal gas

NmM mN 0

nkTp i.e.i.e.

The essence of temperature The essence of temperature – macroscopic – macroscopic displaying of the average translational kinetic displaying of the average translational kinetic energy of the moleculesenergy of the molecules

0N

Rk

--Boltzman Constant

And And tnp 3

2 kTt 2

3

herehere2310022.6

31.8

J/K1038.1 23

Microscopic quantity

RemarkRemark

is the Statistical average quantity.Statistical average quantity.t

Temperature has definite meaning only when Temperature has definite meaning only when the system consists a great number of the system consists a great number of molecules.molecules.

It has not any sense for one molecule or a few It has not any sense for one molecule or a few

molecules.molecules.

Kinetic energy of molecule

§6-5 The equipartition theorem of energy§6-5 The equipartition theorem of energy

1. The degree of freedom1. The degree of freedom

--The independent coordinates for determini--The independent coordinates for determining the position of a moving body in spaceng the position of a moving body in space

Translational kinetic energy=

Rotational kinetic energy+

Vibratory kinetic energy+

Train : Train : the number degree of the number degree of

freedom=freedom=11

planeplane ：： NDF=NDF=33

shipship ：： NDF=NDF=22

C

　 The degree of freedom of a rigid bodyThe degree of freedom of a rigid body

x

y

zThe total degree of freedom The total degree of freedom of rigid body:of rigid body: i i ==66

3 3 rotational degree of rotational degree of freedom.freedom.

3 3 translational degrees of translational degrees of freedomfreedom ),,( zyxCan determine the position Can determine the position of mass centerof mass center C.C.

Can determine the Can determine the direction of rigid body.direction of rigid body.

The degree of freedom of a moleculeThe degree of freedom of a molecule

平动自由度 转动自由度 总计平动自由度 转动自由度 总计 MonatomicMonatomic

DiatomicDiatomic

PolyatomicPolyatomic

3 0 33 2 53 3 6

The vibration of a molecule is not considered The vibration of a molecule is not considered at the ordinary temperature.at the ordinary temperature.

2

2

1vmt 2

2

3xvm kT

2

3

222

2

1

2

1

2

1zyx vmvmvm

2. The equipartition theorem of energy2. The equipartition theorem of energy

kT2

1

--The kinetic energy of each translational --The kinetic energy of each translational

degree of freedom isdegree of freedom is kT2

1

As the kinetic energy can be transferred from As the kinetic energy can be transferred from one molecule to another, or one kind of one molecule to another, or one kind of motion to another(such as from translation to motion to another(such as from translation to rotation)rotation)

The average kinetic energy of any freedom The average kinetic energy of any freedom

degree of a molecule isdegree of a molecule is kT2

1

-- The equipartition theorem of energy-- The equipartition theorem of energy

The average kinetic energy of a The average kinetic energy of a molecule with freedom degree molecule with freedom degree ii :: kT

ik 2

3.Internal energy of ideal gas3.Internal energy of ideal gas

For ideal gas, neglect the interaction between For ideal gas, neglect the interaction between molecules, i.e. neglect the potential energies.molecules, i.e. neglect the potential energies.

Kinetic energy of moleculeKinetic energy of moleculeInternal energy of real gasInternal energy of real gasPotential energies caused by the Potential energies caused by the

interaction between moleculesinteraction between molecules

The internal energy of The internal energy of 11molmol ideal gas ideal gas ：：

kTi

NNE kmol 200 RTi

2

----EE depends on only depends on only TT

The internal energy of The internal energy of M M kgkg ideal gas ideal gas ：：

RTiM

E2

For real gas, the interaction between molecules For real gas, the interaction between molecules could not be neglected, i.e. the potential energies could not be neglected, i.e. the potential energies between molecules is not zero. between molecules is not zero.

Depend on the distances of molecules

The internal energy ofThe internal energy of realreal gasgas depends on TT and volume V of the gas.

Example:Example:

§6-6 Mean free path§6-6 Mean free path

In the ordinary temperature In the ordinary temperature T=300KT=300K, for , for OO22 of of

the gas, its average translational kinetic energy is the gas, its average translational kinetic energy is

2

2

1vmt kT

2

3

Then the root-mean-square speed of OO22 is

)/(48332 smm

kTv Excellent speeder !

Average collision rate Average collision rate ：： the average the average number of collision per unit time a molecule number of collision per unit time a molecule suffers as it moves through the gas.suffers as it moves through the gas.

Z

Mean free path ： average distance a average distance a molecule travels between one collision and the molecule travels between one collision and the next.next.

Average speed of molecules: v

　MoleculeMolecule : elastic ball : elastic ball ，， effective diameter is effective diameter is dd

nudZ 2

The average number of The average number of collision the moving molecule collision the moving molecule collides with other fixed collides with other fixed molecules in molecules in 11 second: second:

d2

d

1. One molecule moves with average speed 1. One molecule moves with average speed and others are at rest.and others are at rest.

u

n:molecular densityd:molecular effective diameter

2. All molecules are moving2. All molecules are moving

nvdZ 22

3. 3. Mean free path of molecule

Z

v

pd

kT22

Average collision rateAverage collision rate

nd 22

1

vu 2 u :Average relative speed

v :Average speed

[[Exa.Exa.] Calculate the mean free path and the ] Calculate the mean free path and the average collision rate of oxygen under the staverage collision rate of oxygen under the standard condition. Suppose molecular averagandard condition. Suppose molecular average speed is e speed is 426m/s426m/s, and its effective diameter i, and its effective diameter is s 2.9×102.9×10-10-10mm

Solution Solution ：： standard conditionstandard condition

K273T Pa10013.1 5p

kT

pn

nvdz 22

2731038.1

10013.123

5

325 m1069.2 19 s1028.4

z

v

91028.4

426

m1095.9 8

§6-7 The Maxwell Speed distribution§6-7 The Maxwell Speed distribution

1.The measurement of molecular speed distribution1.The measurement of molecular speed distribution

l

v

MetalMetalvaporvapor

screenscreenExperimental deviceExperimental device

v

lt

the molecules with can the molecules with can

pass through the second slit. pass through the second slit.

lv

2. The function of speed distribution2. The function of speed distribution

Ndv

dNvf )(

Let Let NN : the total number of molecules. : the total number of molecules.

dNdN : the number of molecules in speeds in: the number of molecules in speeds interval terval v v and and v+dvv+dv. .

N

dN :the ratio of the molecules that their speed distribution is v~v+dv

then Distribution function of speed

It represents the ratio of the molecules that It represents the ratio of the molecules that their speed their speed vv in the unit speed interval in the unit speed interval adjacent to adjacent to vv

Ndv

dNvf )(

Or : Or : It represents the probability of one It represents the probability of one molecule that its speedmolecule that its speed vv in the unit in the unit speed interval adjacent to speed interval adjacent to vv

3. Maxwell speed distribution function3. Maxwell speed distribution function

22

2/3 2

24)( ve

kT

mvf kT

mv

-- Maxwell speed distribution function-- Maxwell speed distribution function

Under equilibrium state, the speed distribution Under equilibrium state, the speed distribution of gas is given byof gas is given by

Maxwell speed Maxwell speed distribution curvedistribution curve

pv

)(vf

vO

0 0 ~~ ::

0

1)( dvvf ----normalizing normalizing conditioncondition

the ratio of molecules the ratio of molecules distributed withindistributed within vv1 1 ~~ vv22::

2

1

)(v

vdvvf

N

N

)(vf

vO dv 1v 2v

the ratio of molecules within v~v+dv

dvvfN

dN)(

the area of the strip whose height is f(v) and width is dv

4.Three kinds of speed of gas4.Three kinds of speed of gas

..The most probable speed The most probable speed vvpp ：：

0)(

dv

vdfLetLet

22

2/3 2

24)( ve

kT

mvf kT

mv

pvRT2

m

kT2

Correspond to the maximum value of Correspond to the maximum value of ff((vv))

We getWe get

v..The average speedThe average speed

N

vvvv N

21

N

vvvNfvvvNf 2211 )()(

1

)(i

ii vvfv

N

vNvN 2211

0

)( dvvvfvm

kT

8

RT8

)(vf

vO pv v 2v

..The root-mean-square speedThe root-mean-square speed 2v

0

22 )( dvvfvvm

kT3

m

kTv

32 RT3

73.1:60.1:41.1

:: 2 vvv p

[[Exa.Exa.]The oxygen with ]The oxygen with 00ooCC is at equilibrium state is at equilibrium state. Calculate the ratio of the molecules at speed inte. Calculate the ratio of the molecules at speed interval rval 300--310m/s300--310m/s

SolutionSolution

vvfN

N

)(

m

kTv p

2

RT2

sm377

Since the speed interval calculated is very Since the speed interval calculated is very small relative to the mean speed of molecules, small relative to the mean speed of molecules, we can usewe can use

vvekT

m

N

N kT

mv

22

2/3 2

24

)()(4

2)(23

p

v

v

p v

ve

v

v p

)377

10()

377

300(

4 2)377

300(

23

e

%0.2

sm300v sm10v

§6-8 The Boltzmann Distribution§6-8 The Boltzmann Distribution

1.The Maxwell distribution of velocity

vy

vx

vz

ov

dv

The area of velocity ball shell: dvv 24

dvvekT

m

dvvfN

dN

kT

mv22

2/3 2

24

)(

is the ratio of molecules that their speeds lie in this shell.

vy

vx

vz

odvy

dvx

dvz

How much is the ratio of molecules that their velocities lie in the velocity interval vx~v+dv

x , vy~v+dvy , vz~v+dvz ?

zyxkT

mv

dvdvdvekT

m

N

Nd

2

2/3 2

2

222zyx vvv

2

2

1mv

k Molecular kinetic energy

2. The distribution of molecular number with 2. The distribution of molecular number with respect to height in gravitational fieldrespect to height in gravitational field

dhhh

Thermo-motionThermo-motion ：： makes the molecular makes the molecular density tend to uniformity in space.density tend to uniformity in space.

Gravitational forceGravitational force ：： makes makes molecules tend to fall down the molecules tend to fall down the ground.ground.

p'p

ppdp ' gdh

nmgdh

Under equilibrium state , Under equilibrium state , T T should be the same should be the same everywhere.everywhere.

nkTp kTdndp

dhkT

mg

n

dnthenthen

integrationintegrationkTmghenn 0

nn0 0 :molecular density at :molecular density at hh=0 =0

nkThp )( kTmghep 0kTmghkTen 0

As As

3.Boltzmann distribution of energy3.Boltzmann distribution of energy

pEmgh -- molecular potential energy -- molecular potential energy kTE penn

0

In the range In the range x--x+dxx--x+dx ，， y--y+dyy--y+dy ，， z--z+dzz--z+dz

ndVNd dxdydzenkTE p 0

According to the Maxwell distribution of velocity,

In the range In the range vx~v+dvx , vy~v+dvy , vz~v+dvz

zyxkT dvdvdve

kT

mNNd

k

2/3

2

Under equilibrium state , the number of molUnder equilibrium state , the number of molecules whose position lie in the range ecules whose position lie in the range x--x+dxx--x+dx，， y--y+dyy--y+dy ，， z--z+dzz--z+dz, and velocities in the ran, and velocities in the range ge vx~v+dvx , vy~v+dvy , vz~v+dvz is determined by

dxdydzdvdvdvekT

mn zyx

kTEE pk )(230 )

2(

),( vrdN

一一 .. 真实气体的等温线真实气体的等温线

压强计压强计

18691869 年，安德鲁斯年，安德鲁斯画出了画出了 COCO22 在不同在不同温度下的等温线温度下的等温线

§§6-9 6-9 真实气体真实气体

------ 在在 TT 、、 PP 变 化变 化的更大范围内实的更大范围内实际气体的性质际气体的性质

CO2 等温压缩实验

压强计

汽汽 液液

液

汽液共存

汽

p

vO

CO2 等温压缩实验

压强计

汽汽 液液

液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

汽

CO2 等温压缩实验液

汽液共存

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

压强计

汽汽 液液

CO2 等温压缩实验液

汽液共存

汽

p

vO

液液

汽汽汽液共存汽液共存饱和蒸汽饱和蒸汽

O V

p

气气

C 临界点临界点

二二 .. 范德瓦耳斯方程范德瓦耳斯方程

11molmol 理气：理气： RTpVm

模型：模型：分子分子 ------ 有吸引力的刚性小球有吸引力的刚性小球

mV --- 可被压缩的、 即分子可自由活动的体积

分子线度不可忽略， 需修正mV

分子引力不可忽略， p 需修正

实际气体：

r

rA B

3

3

4d

d

1.1. 分子体积修正分子体积修正

AA 占据空间占据空间

对对 1mol1mol 气体，分气体，分子的接近是相互的子的接近是相互的

30 3

4

2

1dNb

3

23

44

dN o

bb------ 范德瓦耳斯修正系数范德瓦耳斯修正系数

所有分子占据的总体积：所有分子占据的总体积：

RTbVp m )(

r

rA B

d

分子自由活动的体积：分子自由活动的体积：VVm m - b- b

2.2. 分子间引力修正分子间引力修正

A: A: 作用球内其它分子作用球内其它分子对对 AA 的作用相互抵消的作用相互抵消

BB

AA

分子力分子力作用半径作用半径

R

分子作用球分子作用球

B: B: 球内分子对球内分子对 BB 有有引力作用引力作用

使分子对器壁的压使分子对器壁的压强减小强减小 ppii ：：-------- ppii 内压强内压强

im

pbV

RTp

RTbVpp mi ))((

ip

作用球内的分子数作用球内的分子数器壁附近单位面积被吸引的分子数器壁附近单位面积被吸引的分子数

2npi 2/1 mV 2m

iV

ap

aa --- --- 范德瓦耳斯修正系数范德瓦耳斯修正系数

1mol1mol 气体的范德瓦耳斯方程：气体的范德瓦耳斯方程：

RTbVV

ap m

m

))(( 2

-------- 范德瓦耳斯方程范德瓦耳斯方程

对对 MMkgkg 、摩尔质量、摩尔质量的气体：的气体：

mVM

V

即即 VM

Vm

RTM

bM

VV

aMp

))((

22

2

-------- 范德瓦耳斯方程范德瓦耳斯方程

三三 .. 范德瓦耳斯等温线范德瓦耳斯等温线

O v

p 真实气体真实气体等温线等温线

O v

p 范德瓦耳范德瓦耳斯等温线斯等温线

临界线以下汽态和液态临界线以下汽态和液态段基本一致，汽液共存区段基本一致，汽液共存区差异明显差异明显

临界等温线以上两者相临界等温线以上两者相似似

都有一条临界等温线，都有一条临界等温线，线上拐点处的切线和横线上拐点处的切线和横轴平行轴平行

与真实气体实验等温线比较：与真实气体实验等温线比较：

O v

p 范德瓦耳范德瓦耳斯等温线斯等温线

[[ 例例 ]] 由由 1mol1mol 气体的范德瓦尔斯方程气体的范德瓦尔斯方程 ((pp++

aa//VVmm22)()(VVmm--bb)=)=RTRT ，证明气体在临界点温，证明气体在临界点温

度度 TTcc 、压强、压强 ppcc 及摩尔体积及摩尔体积 VVcc 分别为 分别为 TTcc==

88aa/27/27bRbR ， ， ppcc==aa/27/27bb22, , VVcc=3=3bb

证：证： 2mm V

a

bV

RTp

TT 不变时：不变时： 32

2

)()(

mmT

m V

a

bV

RT

dV

dp

432

2 6

)(

2)(

mmT

m V

a

bV

RT

dV

pd

临界点临界点 CC 切线为水平线，且为拐点切线为水平线，且为拐点

32

2

)()(

CC

CT

m V

a

bV

RT

dV

dpC

0

432

2 6

)(

2)(

CC

CT

m V

a

bV

RT

dV

pdC

0

解得解得 bVC 3bR

aTC 27

8

代入范德瓦尔斯方程可得代入范德瓦尔斯方程可得

227b

apC

输运过程输运过程 (( 迁移现象迁移现象 ):): 气体状态由不平衡趋向气体状态由不平衡趋向

于平衡的现象于平衡的现象

§§6-10 6-10 气体内的输运过程气体内的输运过程

一一 .. 内摩擦现象内摩擦现象 (( 粘滞现象粘滞现象 ))

1.1. 宏观现象及规律宏观现象及规律

气层流速不同气层流速不同而发而发生的现象生的现象

x

yAu

Bu

f

f

SB

A

实验表明：实验表明： Sdy

duf

气体分子气体分子动量动量定向定向迁移迁移

------ 粘滞系数，正负号表示内摩擦力成对粘滞系数，正负号表示内摩擦力成对出现出现

x

y

S

Au

Bu

f

f

B

A

)(mu

2.2. 微观本质微观本质

y

x

S

A B

BA TT

AT BT

实验表明：实验表明： Sdx

dT

t

Q

------ 热导率或导热系数，负号表热量从高热导率或导热系数，负号表热量从高温处传向低温处温处传向低温处

二二 .. 热传导热传导1.1. 宏观现象及规律宏观现象及规律气体内部气体内部温度不同温度不同

而发生的现象而发生的现象

y

x

S

A B

BA TT

AT BT

三三 .. 扩散扩散

1.1. 宏观现象及规律宏观现象及规律 气体分子气体分子数密度数密度不不

均匀而发生的现象均匀而发生的现象x

A B

S

BA m

2.2. 微观本质微观本质气体分子热运动气体分子热运动

动能动能的迁移的迁移 Q

气体分子气体分子质量质量的迁移的迁移

实验表明实验表明 Sdx

dD

t

M

DD------ 扩散系数，负号表从密度较大处向密扩散系数，负号表从密度较大处向密度较低处扩散度较低处扩散

x

A B

S

BA m

2.2. 微观本质微观本质