Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University.

Chapter 12 Chapter 12 Gas KineticsGas KineticsDepartment of Physics

Shanghai Normal University

Table of ContentTable of Content

12-1 The Equilibrium State, the Zero Law of Thermodynamics

12-2 The Microscopic Model of Matter, the law of Statistics

12-3 The Pressure Formula of the Ideal Gas

12-4 The relationship Between the Average Translational Kinetic Energy,

Temperature of the Ideal Gas

12-5 The Theorem of Equipartition of Energy, the Internal Energy of the

Ideal Gas

12-6 The Law of Maxwell Speed Distribution of Gas Molecules

12-8 The Average Number of Collisions of Molecules and the Mean Free

Path

本章目录

Research Object:

Thermal Motion : all the small particles (atoms or molecules)

are in constant, random motion.

Thermal Phenomena: changes of the physical features about the temperature

The Features of Research Object:

Each molecule: disorder 、 accidental, following Newtonian mechanics

Entity(a large number of small particles): obeying Statistical Law.

12-1The Equilibrium State, the Zero Law of Thermodynamics

Macroscopic quantities: the macroscopic state of the entire

gas, such as, p ， V ， T, etc. They can be measured directly

Microscopic quantities: describing the individual

molecule, such as, its own mass m, velocity ,etc. They can

not be measured directly

v

Macroscopic quantities

Microscopic quantities

statistic average


I. The state of gas ( Microscopic quantities)

TVp ,,1. Pressure(p) ： the force in per unit area

unit: Standard atmospheric pressure: the atmospheric p at 0 at the sea ℃level of 45°latitude 。 1atm=1.01 ×105Pa

2.Volume(V) : reachable space l 10m 1 33


unit:

tT 273K

3.Temperature(T) : measure of the coldness or hotness of an

object

unit:

a gas with a certain mass in a container does not transport

energy and mass with the environment, after a relatively long

time, then the state parameters doo not change with time, such

a state is called an ~


II. Equilibrium State(E-S)

TVp ,,

TVp ,, ''

Vacuum expansion

p

Vo

),,( TVp

),,( '' TVp

Characteristic of E-S:

),,( TVp

p

V

),,( TVp

o(1). oneness: p, T in everywhere are the same;

(2). State parameters are stable: independent with time


(3). The final state of a Spontaneous Process

(4).thermal equilibrium ： different from mechanical equilibrium

III. The equation of the state of the Ideal gas

The equation of the state: the function connecting the

macroscopic quantities of the ideal gas in equilibrium state.

Ideal gas: the gas which follows the Boyle’s law, the Gay-

Lussac’s law, the charles’ law, and the Aavogadro’s law


11 KmolJ 31.8 RMole gas constant:

2

22

1

11

T

Vp

T

Vp

for the gas with a certain quantity of gas at equilibrium:

RTM

mRTpV

One equation of the state of the ideal

gas:


Nmm mNM A

123A KJ 1038.1/ NRk

k is Boltzmann constant

n =N/V ， the number density of molecules

nkTp


Another equation of the state of the ideal gas:


IV. The Zeroth law of thermodynamics:

If A and B are in thermal equilibrium with C, which is in a

certain state, respectively, then A and B are in thermal

equilibrium each other.

I. The scale of molecules and molecular forces:

Molecules: including monatomic ~, diatomic ~, polyatomic ~.

For example: the oxygen molecules under the standard state

Diameter: m104 10d

Distances between gas molecules

The diameter of the molecules10


Therefore, molecules with different structures have different scales

0ror

F m10~ 100

r

Molecular force

1.when r<r0, the molecular force is mainly repulsive;

2.when r>r0, the molecular force is mainly attractive;

3. When r10-9m, F0


II. Molecular force:

Thermal motion: large amounts of experimental facts indicate

that all molecules move irregularly thermally.

for example: oxygen molecules under the normal temperature

and normal pressure.

-1107 s10~m10~ z

-1sm450 v


III. The disorder and the statistical regularity of the thermal motion of Molecular

. . . . . . . . . . . .. . . . . . . . . . .

. . . . . . . . . . . .. . . . . . . . . . .

. . . . . . . . . . . .. . . . . . . . . . .

. . . . . . . . . . . .


The distribution of the small balls in the Gordon board

When the Number of the small balls N ∞ ， the distribution of

the small balls shows the statistical regularity

. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .

. . . . . . . . .. . . . . . . .


Statistical regularity

1i

i

ii N

N

i

iNN

N

NiN

i

lim


Suppose: Ni is the number of the small ball in the ith slot, then

the total number of the small balls N satisfies:

Normalizing condition:

Probability: the probability of the small ball which appeared in ith slot

1.The size of the molecule itself is negligible compared

with the average distance between molecules, and

molecules can be viewed as mass points:

I.The microscopic model of the ideal gas


2. Other than the moment of collision, the interaction forces

between molecules are negligible.

3. The Collisions between molecules can be viewed as complete

elastic collisions.

4.The motion of the molecules follows classical laws.

m10 9r

m10 10d

rd

Assume that there is a rectangular container with the side lengths

being x, y, and z, in the container there are N gas molecules of the

same kind. The mass of each molecule is m. Now we calculate the

pressure on wall A1 perpendicular to the Ox axis


II. The pressure formula of the ideal gas

xvmxvm-2Av

o

y

zx

y

zx

1A vyv

xv

zvo

The statistical regularity of the thermodynamic equilibrium:

V

N

V

Nn

d

d

(1). the spatial distribution of molecules is uniform:

Total effect of all the large quantities of molecules: continuous force.

Colliding effect of one molecules: accidental, discrete


2222

3

1vvvv zyx

iixx N22 1vv

The average value of the squares of the velocity components along the Ox aixs:

0 zyx vvv

Each molecule moving in any direction is equal:

kji iziyixi

vvvv Velocity of the molecule:


(2). The probability of each molecule moving in any direction is

equal and there is no preferred direction

The impulse of the force acted by the molecule on the wall:

ixmv2xvmxvm-

2A

v

o

y

zx

y

zx

1A

ixix mp v2Momentum increment on the Ox axis:

Each molecule follows the mechanical law


Therefore, the total impulse of a molecule acted on the wall in the unit time interval:

xm ix2v

The time between two consecutive collision:

ixx v2

The number of collisions in the unit time interval:

2xvix


total impulse of N molecules acted on wall in the unit time:

22

22

xix

iix

i

ix

x

Nm

Nx

Nm

x

m

x

mv

vv

vi

total effect of a large quantities of molecules:

i.e., the average force on wall A1 is:

xNmF x2v


Pressure on wall:

2xxyz

Nm

yz

Fp v

Statistical regularity:

xyz

Nn 22

3

1vv x

Molecular average translational kinetic energy:

2k 2

1vm

k3

2 np Pressure formula of the ideal gas:


k3

2 np Statistical relationship

Physical significance of the pressure

Observable macroscopic quantities

statistical average value of the microscopic quantity


kTm2

3

2

1 2k v

Observable macroscopic quantities

statistical average value of the microscopic quantity

k3

2 np

Equation of the state of the ideal gas:

nkTp

12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas

Pressure formula of the ideal gas:

Molecular average translational kinetic energy:

Physical significance of T:

(1). Temperature is the measurement of the average translational kinetic energy of large quantities of molecules:

Tk

kTm2

3

2

1 2k v


(2). Temperature is the collective behavior of the thermal motion of large numbers of molecules.

(3). The average translational kinetic energies in the same temperature are the same.

difference between the thermal motion and the

macroscopic motion:

T is the macroscopic statistical physical quantity expressing

the degree of the irregular motion of molecules, and is

nothing to do with macroscopic motion of the macroscopic

object.

Noted:


A. They are in the same temperatures and the same pressures;

B. Not only their temperatures but also the pressures are

different;

C. Temperature is the same, but pressure of He is larger

D. Temperature is the same, but pressure of N2 is larger

nkTp Solution: Tm

kkT

V

N

Problem 1: two bottles of gas with the same density, one is

He, another is N2, they all in equilibrium state with the same

average translational kinetic energy, then ( )

discussion


（ A ）（ B ）

（ C ）（ D ）mpV

)(RTpV

)(kTpV

)( TmpV

kT

pVnVN nkTp


Problem 2: Ideal gas with state parameters V, p, T, the mass of

each molecule is m, k is the Boltzmann constant, R is the mole gas

constant, then the total number of the molecules is ( )

Solution:

I. Degrees of freedom

kTm2

3

2

1 2kt v

2222

3

1vvvv zyx

kTmmm zyx 2

1

2

1

2

1

2

1 222 vvv

the average energy of mono-atomic molecules: kT

21

3

y

z

xo

12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas

rigid diatomic molecule:

the average translational kinetic energy:

222kt 2

121

21

CzCyCx mmm vvv

22kr 2

1

2

1zy JJ


the average rotational kinetic energy:

vrti

number of degrees of freedom


Degrees of freedom : the number of independent velocity or coordinate square terms in the energy expression of the molecule as the number of degrees of freedom of the energy of the molecule, or simply degrees o freedom , denoted by symbol i

translatio

n rotation vibration

Mono-atomic molecule 3 0 3

Diatomic molecule 3 2 5

polyatomic molecule 3 3 6

Degrees of freedom of the energy of the rigid molecules

t r imolecule

itranslation rotation total


the average energy of a molecule can be expressed as:

kTi

2


II. the theorem of equipartition of energy

When a gas is at an equilibrium state the average energy of each degree of freedom is equal to the average energy of every other degree of freedom, and it is kT/2, this is the theorem of equipartition of energy per degree of freedom.

The internal energy of the ideal gas: the sum of the kinetic

energies of the molecules and the atomic potential energies within

each molecule

RTi

NE2A The internal energy of one mole of the ideal

gas:

RTi

E2

TRi

E d2

d

the change of internal energy of the ideal gas:


II. The internal energy of the ideal gas

The internal energy of the ideal gas with the substance quantity ν is ：

Experimental device

l

l

v

v

2

l

TlMetal vapor

Display

screen

Narrow slit

Connect to pump


I.The experiment of measuring the speed distribution of gas molecules

The scenario of molecular speed distribution

N: total number of molecules

)/( v NN

o vv vv

S

△ N:number of molecules in the speed interval v v+ △v

N

NS


Ratios of the number of molecules with speeds in between v v+ △v to the total number of molecules

v

)(vf

o

SfN

Ndd)(

d vv

vvvv

vv d

d1lim

1lim)(

00

N

N

N

NN

Nf

The distribution function of speed:

v vv d

Sd


Physical significance of f(v):

Under the equilibrium state with temperature T, f(v) represents the ratio of number of molecules in unit speed interval around v to the total number of molecules.

Physical significance of f(v)dv:

Ratios of the number of molecules with speeds in between v v+ △v to the total number of molecules

v

)(vf

o 1vS

2v

vv d)(d NfN

vvvv d)(2

1fNN

vvv

vd)(2

1 f

N

N


the number of molecules with speeds in between v v+ △v :

the number of molecules with speeds in between v1 v2:

Ratios of the number of molecules with speeds in between v1 v2 to the total number of molecules:

2223

2

e)π2

(π4)( vvvkTm

kTm

f

The Maxwell speed distribution law

v

)(vf

oThe relationship curve between f(v) and v


II.The Maxwell speed distribution law of gas molecules

pv(1).the most probable speed

0d

)(d

p

vvv

vf

mkT

mkT

41.12

p v

v

)(vf

o pv

maxf

We get:


III. The three statistical speeds:

kNRmNM AA ,

Physical significance:

M

RT41.1p v

M

RT

m

kT41.1

2p v


At a certain temperature, the relative number of molecules distributed in the vicinities of the most probable speed vp is the most.

46

N

NNNN nnii dddd 2211 vvvvv

v

N

Nf

N

NN

00d)(d vvvv

v

v

)(vf

o

mkT

fπ8

d)(0

vvvv

M

RT

m

kT60.1

π

8v


(2). The average speed:

2v

M

RT

m

kT73.1

32 v

N

Nf

N

NN

0

2

0

2

2d)(d vvvv

v

mkT /32 v


(3). The root mean square speed:

M

RT

m

kT 332rms vv

2p vvv

M

RT

m

kT60.160.1 v

MRT

mkT 22

p v

Comparison of the three statistical speeds:


2H2O

opvHpv v

)(vf

o

KT 3001

1pv 2pv

KT 200 12

v

)(vf

o


The speed distributions of N2 molecules under two different

temperature

The speed distributions of N2 and H2molecules under the

same temperature

v

vvv

p

d)(Nf（ 1 ）

pd)(

21 2

vvvv Nfm（ 2 ）

Solutions:


Problem 1: one type gas with the total number of molecules N,

the mass of each molecule m, and the distribution function f(v),

please find out:

(1). the number of the molecules in the speed interval

(2). the sum of the kinetic energy of all the molecules in the

speed interval

discussion

vv ~p

~pv

)(vf

1sm/ v2 000o


Problem 2: the figure is the Maxwell speed distributions of H2 and

O2 molecules under the same temperature. Please find out the

most probable speed vp for these two gas.

900

mkT2

p v )O()H( 22 mm

)O()H( 2p2p vv -12p m.s 000 2)H( v

42

32)H()O(

)O(

)H(

2

2

2p

2p mm

v

v

-12p m.s 500)O( v


Solutions:

the free path ： the path that a molecule goes through

between two consecutive collisions is called ~

12-8 The Average Number of Collisions of Molecules and the Mean Free Path

the mean free path ： the average value of the path

that the molecule goes through between two consecutive

collisions is called ~

the average number of collisions per second(or the

average frequency of collisions): the average number of

collisions of a molecule with other molecules per unit time is

called ~, denoted by


Z

simplified model

1. the molecules are rigid small balls, all the collisions are

completely elastic;

2. the diameter of molecules is d

3. Assume that among all molecules only one molecule

moves with the average speed , all others are at rest.


u

The average number of collisions per second: nudZ 2π


Taking into consideration of the motion of all other

molecules, then we have: v2u

ndZ v2π2

The average number of collisions per second:


the mean free pathndz 2π2

1

v

nkTp 2

2 π

kT

d p

when the temperature T of the gas is given, we have:p

1

T


when the pressure p of the gas is given, we have:

pd

kT2π2

Solution:


Problem: estimate the mean free paths of air molecules under the

following two circumstances: (1). 273 K and 1.013×105 Pa; （ 2 ） 273 K and 1.333×10-3 Pa.

(the diameter of air molecules )m 1010.3 10d

m 1071.8

m 10013.1)1010.3(π2

2731038.1

8

5210

23

1

m 62.6

m 10333.1)1010.3(π2

2731038.13210

23

2


Chapter 12 Gas Kinetics Department of Physics Shanghai Normal University.

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