chapter 12 gas kinetics department of physics shanghai normal university
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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
12-1The Equilibrium State, the Zero Law of Thermodynamics
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
12-1The Equilibrium State, the Zero Law of Thermodynamics
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 ~
12-1The Equilibrium State, the Zero Law of Thermodynamics
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
12-1The Equilibrium State, the Zero Law of Thermodynamics
(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
12-1The Equilibrium State, the Zero Law of Thermodynamics
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:
12-1The Equilibrium State, the Zero Law of Thermodynamics
Nmm mNM A
123A KJ 1038.1/ NRk
k is Boltzmann constant
n =N/V , the number density of molecules
nkTp
12-1The Equilibrium State, the Zero Law of Thermodynamics
Another equation of the state of the ideal gas:
12-1The Equilibrium State, the Zero Law of Thermodynamics
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
12-2 The Microscopic Model of Matter, the law of Statistics
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
12-2 The Microscopic Model of Matter, the law of Statistics
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
12-2 The Microscopic Model of Matter, the law of Statistics
III. The disorder and the statistical regularity of the thermal motion of Molecular
. . . . . . . . . . . .. . . . . . . . . . .
. . . . . . . . . . . .. . . . . . . . . . .
. . . . . . . . . . . .. . . . . . . . . . .
. . . . . . . . . . . .
12-2 The Microscopic Model of Matter, the law of Statistics
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
. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .
. . . . . . . . .. . . . . . . .
12-2 The Microscopic Model of Matter, the law of Statistics
Statistical regularity
1i
i
ii N
N
i
iNN
N
NiN
i
lim
12-2 The Microscopic Model of Matter, the law of Statistics
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
12-3 The Pressure Formula 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
12-3 The Pressure Formula of the Ideal Gas
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
12-3 The Pressure Formula of the Ideal Gas
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:
12-3 The Pressure Formula of the Ideal Gas
(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
12-3 The Pressure Formula of the Ideal Gas
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
12-3 The Pressure Formula of the Ideal Gas
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
12-3 The Pressure Formula of the Ideal Gas
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:
12-3 The 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
12-3 The Pressure Formula of the Ideal Gas
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
12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas
(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:
12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas
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
12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas
( A ) ( B )
( C ) ( D )mpV
)(RTpV
)(kTpV
)( TmpV
kT
pVnVN nkTp
12-4 The relationship Between the Average Translational Kinetic Energy, Temperature of the Ideal Gas
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
12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas
the average rotational kinetic energy:
vrti
number of degrees of freedom
12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas
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
12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas
the average energy of a molecule can be expressed as:
kTi
2
12-5 The Theorem of Equipartition of Energy, the Internal Energy of the Ideal Gas
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:
12-5 The Theorem of Equipartition of Energy, the 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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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:
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
III. The three statistical speeds:
kNRmNM AA ,
Physical significance:
M
RT41.1p v
M
RT
m
kT41.1
2p v
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
(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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
(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:
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
2H2O
opvHpv v
)(vf
o
KT 3001
1pv 2pv
KT 200 12
v
)(vf
o
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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:
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-6 The Law of Maxwell Speed Distribution of Gas Molecules
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
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
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.
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
u
The average number of collisions per second: nudZ 2π
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
Taking into consideration of the motion of all other
molecules, then we have: v2u
ndZ v2π2
The average number of collisions per second:
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
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
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
when the pressure p of the gas is given, we have:
pd
kT2π2
Solution:
12-8 The Average Number of Collisions of Molecules and the Mean Free Path
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