8 kinetic theory of gases

1
KINETIC
THEORY OF GASES
SOLO HERMELIN
http://www.solohermelin.com

2
KINETIC THEORY OF GASES
HISTORY
STATE EQUATION (BOYLE-MARIOTTE LAW(
p - PRESSURE (FORCE / SURFACE(
V - VOLUME OF GAS
m - MASS OF GAS
T - GAS TEMPERATURE
-GAS DENSITY
[ ]m3
[ ]kg
[ ]o
K
[ ]kg m/ 3
ρ
[ ]2
/ mN
ROBERT BOYLE (1660( DISCOVERED EXPERIMENTALLY THAT THE
PRODUCT OF PRESSURE AND VOLUME IS CONSTANT FOR A FIXED
MASS OF GAS AT CONSTANT TEMPERATURE
SOLO
( )mTconstVp ,=
ROBERT BOYLE (1627-1691(
New Experiments Physio-Mechanicall,
Touching the Spring of the Air, and
Its Effects (1660(

3
HISTORY
RICHARD TOWNLEY (1628-1707( AND HENRY POWER (1623-1668( FOUND
ALSO EXPERIMENTALLY THE P*V LAW IN 1660.
SOLO
STATE EQUATION (BOYLE-MARIOTTE LAW(
EDMÉ MARIOTTE (~1620-1684( INDEPENDENTLY FINDS THE
RELATIONSHIP BETWEEN PRESSURE AND VOLUME P*V LAW IN 1676
IN HIS WORK “On the Nature of Air”.
( )mTconstVp ,=
THIS IS KNOWN AS “MARIOTTE’s LAW” IN FRANCE AND “BOYLE’s LAW”
ELSEWHERE.
DANIEL BERNOULLI (1700-1782) IN THE
TREATISE “Hydrodynamica” (1738)
DERIVES THE BOYLE-MARIOTTE LAW
USING A “BILLIARD BALL” MODEL.
HE ALSO USES CONSERVATION OF
MECHANICAL ENERGY TO SHOW THAT
THE PRESSURE CHANGES
PROPORTIONALLY TO THE SQUARE OF
PARTICLE VELOCITIES AS
TEMPERATURE CHANGES.

4
HISTORY
1787-JACQUES ALEXANDRE CHARLES (1746-1823(
THAT, AT A GIVEN PRESSURE, THE CHANGE IN
VOLUME IS PROPORTIONAL WITH THE
CHANGE IN TEMPERATURE. HE NEVER
PUBLISHED HIS WORK.
SOLO
( ) TmpKV V ,=
1702-GUILLAUME AMONTONS (1663-1705( MEASURED THE
INFLUENCE OF TEMPERATURE ON THE PRESSURE OF A FIXED
VOLUME OF A NUMBER OF DIFFERENT GASES AND PREDICTED
THAT AS AIR COOLED, THE PRESSURE SHOULD BECOME ZERO AT
SOME TEMPERATURE, WHICH HE ESTIMATED TO BE -240°C.
( ) TmVKp p ,=

5
HISTORY
JOSEPH-LOUIS GAY-LUSSAC (1778-1850( FINDS IN 1802
THAT, AT A GIVEN PRESSURE, THE CHANGE IN
VOLUME IS PROPORTIONAL WITH THE
CHANGE IN TEMPERATURE. HE DEVELOPED THE
IDEA OF ABSOLUTE ZERO TEMPERATURE AND
CALCULATE ITS VALUE TO BE -273°C.
SOLO
STATE EQUATION (GAY-LUSSAC LAW - 1802(
( ) TmpKV V ,= GAY-LUSSAC LAW - 1802

6
HISTORY
SOLO
AVOGADRO’s LAW (1811(
AMEDEO AVOGADRO (1776-1856( FINDS
IN 1811
“EQUAL VOLUMES OF GAS CONTAIN
THE SAME NUMBER OF MOLECULES N,
REGARDLESS OF PRESSURE OR
TEMPERATURE”.
23
10022137.6 ⋅=AN AVOGADRO’s NUMBER

7
HISTORY
SOLO
DEFINITIONS:
ONE GRAME-MOLE OF A PURE SUBSTANCE IS DEFINED AS THE MASS
IN GRAMES, NUMERICALLY EQUAL TO THE MOLECULAR WEIGHT OF
THE SUBSTANCE.
M
m
N =
m - MASS OF GAS [ ]kg
M – MOLECULAR WEIGHT OF GAS [ ]kg
N – NUMBER OF MOLES
ρ
M
vM
Mm
V
N
V
vM ====
/
–AVERAGE MOLAL SPECIFIC VOLUME
–AVERAGE SPECIFIC VOLUME
–AVERAGE DENSITY
m
V
v =:
V
m
=:ρ
-GAS DENSITY [ ]kg m/ 3
Vd
md
=:ρ
[ ]kg m/ 3
[ ]kgm /3

8
HISTORY
ASSUME A MASS m OF GAS WITH p, V, T (PRESSURE, VOLUME,
TEMPERATURE). LET CHANGE FIRST THE VOLUME TO V’
AT THE CONSTANT TEMPERATURE T. THE PRESSURE p* WILL
BE GIVEN BY BOYLE-MARIOTTE LAW:
SOLO
STATE EQUATION (IDEAL GAS)
VpVp ='*
LET CHANGE NOW THE VOLUME FROM V’ TO V*, KEEPING THE
PRESSURE p* CONSTANT. THE NEW TEMPERATURE T* WILL BE
GIVEN BY GAY-LUSSAC LAW:
'
*
*
V
V
TT =
BY ELIMINATING V’ BETWEEN THOSE TWO EQUATIONS WE OBTAIN:
T
Vp
T
Vp
=
*
**

9
HISTORY
SOLO
STATE EQUATION (IDEAL GAS(
THE RELATION: const
T
Vp
=
WAS OBTAINED BY CLAPEYRON IN 1834.
BENOIT PAUL ÉMILE CLAPEYRON
(1799-1864(
ACCORDING TO AVOGADRO’s LAW A GRAM-MOLECULE OF ANY GAS
HAS THE SAME VOLUME FOR A GIVEN PRESSURE AND TEMPERATURE.
T
vp M
=R UNIVERSAL GAS CONSTANT
THEREFORE THE RATIO
IS CALLED THE
DEFINE ALSO:
[ ]
( )[ ])/(1097.4
)/(8314
4
Rmolsluglbft
KmolkgJ
o
o
⋅⋅⋅×=
⋅=R
[ ]
( )[ ]Rlbft
KJ
N
k
o
o
A
/10565.0
/1038.1
23
23
⋅×=
×==
−
−R
BOLTZMZNN CONSTANT

10
HISTORY
SOLO
STATE EQUATION (IDEAL GAS(
R=
T
vp M [ ]
( )[ ])/(1097.4
)/(8314
4
Rmolsluglbft
KmolkgJ
o
o
⋅⋅⋅×=
⋅=R
( ) ( )
( ) ( )Rsluglbft
KkgJairR


⋅⋅=
⋅=
/1716
/287
UNIVERSAL
GAS CONSTANT
ρ
M
vM
Mm
VV
vM ====
/N
USING
TVp RN=
WE OBTAIN
TRTTT
m
N
vp
m
V
p ===== RR
M
R η
1
vM – AVERAGE MOLAR SPECIFIC VOLUME
–AVERAGE SPECIFIC VOLUME
m
V
v =: [ ]kgm /3
–MOLECULAR WEIGHT OF GAS [ ]kg
–NUMBER OF MOLES IN V
MR /=R
SPECIFIC
GAS CONSTANT
N
M

11
HISTORY
SOLO
STATE EQUATION (IDEAL GAS)
[ ]
( )[ ])/(1097.4
)/(8314
4
Rmolsluglbft
KmolkgJ
o
o
⋅⋅⋅×=
⋅=R
( ) ( )
( ) ( )Rsluglbft
KkgJairR


⋅⋅=
⋅=
/1716
/287
UNIVERSAL
GAS CONSTANT
TVp RN=
WE OBTAIN
TRTTT
m
vp
m
V
p ===== RR
M
R
N
η
1
MR /=R
SPECIFIC
GAS CONSTANT
v
1
:=ρ

TkN
T
N
NTVp
k
AN
A
=
==
R
NRN

–AVERAGE DENSITY [ ]3
/ mkg
TRTR
v
p ρ==
1
ANN N=: –NUMBER OF PARTICLES IN V
–MOLECULAR WEIGHT OF GAS [ ]kg
–NUMBER OF MOLES IN VN
M
[ ] ( )[ ]RlbftKJ
N
k oo
A
/10565.0/1038.1 2323
⋅×=×== −−R

12
HISTORY
SOLO
IDEAL GAS CYCLE
SADI CARNOT
(1796-1832(
IN 1824 SADI CARNOT PUBLISHED “RÉFLECTIONS SUR LA
PUISSANCE MOTRICE DE FEU” (REFLECTIONS ON THE
MOTIVE POWER OF HEAT(.
HE PROPOSED THE FOLLOWING REVERSIBLE ENGINE
(1( PLACE A CYLINDER CONTAINING A GAS
AT PRESSURE p1 AND VOLUME V1
IN A HEAT BATH AT TEMPERATURE
TL UNTIL IT REACHES THE VOLUME
V2 > V1 AND PRESSURE p2 < p3.
2( ISOLATE THE CYLINDER
GAS PRESSURE CHANGE TO p3 < p2
AND VOLUME TO V3 > V2.
3( PLACE THE CYLINDER TO A BATH
AT THE TEMPERATURE TL
GAS PRESSURE CHANGE TO p4 > p3
AND THE VOLUME TO V4 < V3.
4( ISOLATE THE CYLINDER
GAS PRESSURE CHANGE TO p1 > p4
AND VOLUME TO V1 < V4.
0
1
4
4
3
3
2
2
1
>+++== ∫∫∫∫∫ dVpdVpdVpdVpdVpWWORK PRODUCED

13
HISTORY
SOLO
JAMES PRESCOTT JOULE
(1818-1889)
1843-1848 JAMES PRESCOTT JOULES
ESTABLISHED THE EXACT RELATION BETWEEN
HEAT AND MECHANICAL WORK.
1847 JOULES PUBLISHED “ON MATTER, LIVING
FORCE, AND HEAT” IN THE MANCHESTER
COURIER, STATING THE PRINCIPLE OF THE
CONSERVATION OF ENERGY AND GIVING THE
CONSERVATION FROM HEAT TO KINETIC ENERGY.
HERMANN LUDWIG
FERDINAND von HELMHOLTZ
(1821-1894)
1847 HERMANN von HELMHOLTZ PUBLISHES
“ON THE CONSERVATION OF ENERGY” IN WHICH
HE GIVES A MATHEMATICAL FORMULATION
OF THE PRINCIPLE OF CONSERVATION OF
KINETIC ENERGY (INDEPENDENT OF JOULE’s
PUBLICATIONS).
CONSERVATION OF ENERGY

14
HISTORY
SOLO
1850 RUDOLF CLAUSIUS GIVES THE FORMULATION OF
THE SECOND LAW OF THERMODINAMICS, FOR WHICH
IS NO MECHANISM WHOSE WHOSE ONLY FUNCTION IS
HEAT TRANSFER.
WILLIAM THOMSON
LORD KELVIN
(1824-1907(
SECOND LAW OF THERMODYNAMICS
RUDOLF CLAUSIUS
(1822-1888(
1848 WILIAM THOMSON, LORD KELVIN REDISCOVERS
THE IDEA OF ABSOLUTE ZERO OF ABOUT -273 °C.
HE ALSO DERIVES THE SECOND LAW OF
THERMODYNAMICS USING CARNOT’s IDEA.
1854 RUDOLF CLAUSIUS PROPOSES THE FUNCTION
dQ/T AS A WAY TO COMPARE HEAT FLOWS WITH HEAT
CONVERSIONS.
1857 RUDOLF CLAUSIUS PUBLISHES A PAPER ON
MATHEMATICAL KINETIC ENERGY, ESTABLISHING
HEAT AS ENERGY DISTRIBUTED STATISTICALLY
AMONG PARTICLES.
1857 RUDOLF CLAUSIUS INTRODUCES THE IDEA OF THE MEAN FREE PATH

15
HISTORY
KRÖNING (1856) AND CLAUSIUS (1857) DERIVED THE SIMPLEST MODEL
OF AN IDEAL GAS.
SOLO
KRÖNING-CLAUSIUS MODEL
ASSUMPTIONS
:
RUDOLF CLAUSIUS
)1822-1888(
-THE GAS IS ASSUMED TO BE COMPOSED
OF INDIVIDUAL PARTICLES (ATOMS AND
MOLECULES) WHOSE ACTUAL
DIMENSIONS ARE SMALL IN COMPARISON
TO THE DISTANCE BETWEEN THEM.
-THE PARTICLES ARE IN CONSTANT
MOTION AND THEREFORE HAVE KINETIC
ENERGY.
-NEITHER ATTRACTIVE NOR REPULSIVE
FORCES EXIST BETWEEN THE PARTICLES.

16
HISTORY
SOLO
MONOATOMIC GAS MODEL
–NUMBER OF MOLECULES IN THE VOLUME XYZ
HAVING VELOCITY COMPONENT xv
CONSIDER THE IMPACT OF PARTICLES ON THE
x=X WALL IN THE INTERVAL Δt.
THE NUMBER OF PARTICLES ARIVING TO THE WALL IN THE
INTERVAL Δt. IS:
xv
x
N
X
tv ∆
THE CHANGE IN THE x-MOMENTUM DUE TO ELLASTIC IMPACT OF
ONE PARTICLE OF MASS m WITH THE WALL IS:
( ) xxxx vmvmvmp 2=−−=∆
xvN
THE FORCE APPLIED TO THE x=X WALL IS GIVEN BY:
( ) ( ) xx v
x
xxv
x
x N
X
vm
vFvmN
X
tv
tVF
2
2
2 =⇒




 ∆
=∆

17
HISTORY
SOLO
THE PRESURE ON THE x=X WALL DUE TO PARTICLES
HAVING VELOCITY COMPONENTS vx IS GIVEN BY:
( ) ( )
xv
xx
x N
V
vm
YZ
vF
vp
2
2
==
THE PRESURE ON THE x=X WALL IS DUE TO THE
CONTRIBUTION OF ALL PARTICLES HAVING
VELOCITY COMPONENTS vx >0:
22
0
22
xxvxxvx v
V
Nm
vdNv
V
m
vdNv
V
m
p xx
=== ∫∫
∞
∞−
∞
WHERE:
∫
∞
∞−
= xvxx dvNv
N
v x
22 1
:
WE HAVE:
22222222
zyxzyx vvvvvvvv ++=⇒++=
SINCE THE SYSTEM IS ISOTROPIC: 2222
3
1
vvvv zyx ===

18
HISTORY
SOLO
THE PRESURE ON THE x=X WALL DUE TO THE
CONTRIBUTION OF ALL PARTICLES IS:

E
x v
m
N
V
v
V
Nm
v
V
Nm
p 





=== 222
23
2
3
OR
EVp
3
2
=
BUT WE FOUND TkNVp =
THEREFORE εNTkNE ==
2
3
BOYLE’s LAW
WHERE Tk
2
3
=ε KINETIC ENERGY OF A MONOATOMIC MOLECULE

19
KINETIC THEORY OF GASESSOLO
MEAN FREE PATH (FIRST INTRODUCED BY CLAUSIUS IN 1858(
THE MEAN DISTANCE λ WHICH THE MOLECULES TRAVELS BEFORE
SUFFERING ITS NEXT COLLISION (OR EQUIVALENTLY, THE MEAN DISTANCE
WHICH IT HAS TRAVELED AFTER PRECEDING COLLISION( IS CALLED THE
MEAN FREE PATH OF THE MOLECULE.
DEFINE
V
N
n
∆
= - NUMBER OF MOLECULES PER UNIT VOLUME
r2=σ - DIAMETER OF THE MOLECULE
v - AVERAGE VELOCITY OF THE MOLECULES
relv - AVERAGE RELATIVE VELOCITY BETWEEN MOLECULES
IN A TIME τ THE MOLECULE A WILL COLLIDE WITH: τσπτσπ vnvnz rel
22
2==
MOLECULES
AArel vvv

−= ' vvvvvvvvvv AA
v
AA
v
AArelrelrel 22
0
''' =⋅−⋅+⋅=⋅=






THE MEAN FREE PATH λ IS GIVEN BY THE MEAN PATH <v> τ TRAVELED BY THE MOLECULE
IN THE TIME τ , DIVIDED BY THE NUMBER OF MOLECULES z IT WILL COLLIDE TO.
2
2
1
σπ
τ
λ
nz
v
==

20
MEAN FREE PATH (CONTINUE(
LET FIND HOW THE MEAN FREE PATH IS RELATED TO PRESSURE AND TEMPERATURE.
TknTk
V
N
p ==USING
2
2
1
σπ
λ
n
=
WE OBTAIN p
Tk
2
2
1
σπ
λ =
GAS λ (cm) σ (cm)
HYDROGEN (H2) 1,123x10-5
2,3x10-8
NITROGEN (N2) 0,599x10-5
3,1x10-8
OXIGEN (O2) 0,647x10-5
2,9x10-8
HELIUM (He) 1,798x10-5
1,9x10-8
ARGON (Ar) 0,666x10-5
3,6x10-8
p (mm
Hg)
λ (cm)
760 7x10-6
1 5x10-3
0,01 5x10-1
10-4
5x101
10-6
5x103
λ OF AIR FOR
DIFFERENT p
MEAN FREE PATH λ AND DIAMETERS σ FOR DIFFERENT
MOLECULES

21
VISCOSITY AND TRANSPORT OF MOMENTUM FOR RAREFIATE GAS
WE WANT TO FIND THE CHANGE IN VELOCITY COMPONENT vz ACROSS THE
SURFACE S DUE TO THE VISCOSITY EFFECT.
m
Tk
vvvvv zyx
32222
=++==
- AVERRAGE MOLECULES VELOCITY
tSvn ∆∆
6
1
- NUMBER OF MOLECULES TRAVELING IN THE x DIRECTION, IN TIME Δ t,
THROUGH THE SURFACE ΔS.
V
N
n
∆
( )λ−∆∆ xvmtSvn z
6
1
- CHANGE IN z MOMENTUM COMPONENT , DUE TO MOLECULES
CROSSING ΔS IN +x DIRECTION.
( )λ+∆∆ xvmtSvn z
6
1
- CHANGE IN z MOMENTUM COMPONENT , DUE TO MOLECULES
CROSSING ΔS IN -x DIRECTION.
SINCE ONLY THE MOLECULES AT A DISTACE LESS THAN MEAN FREE PATH BETWEEN
COLLISIONS λ WILL CROSS THE SURFACE S AND CONTRIBUTE TO A CHANGE IN LINEAR
MOMENTUM, WE CAN WRITE:

22
VISCOSITY AND TRANSPORT OF MOMENTUM FOR DILUATE GAS (CONTINUE(
( ) ( )[ ]
λ
λλτ






∂
∂
∆∆−=
+−−∆∆=∆∆
∆
x
v
mtSvn
xvxvmtSvntS
z
zz
F
xz
z
2
6
1
6
1

xzτ - STRESS [N/m2
] IN z DRECTION, DUE TO
CHANGE IN MOMENTUM.
x
v
x
v
mvn zz
xz
∂
∂
−=
∂
∂
−= µλτ
3
1
FROM THIS EQUATION WE OBTAIN:
WHERE:
2
3
2
2
1
6233
1
2
οποπ
λµ
οπ
λ
mTkvm
mvn
m
kT
v
n
==
===
NEWTONIAN FLUID

23
THERMAL CONDUCTIVITY AND TRANSPORT OF ENERGY
WE WANT TO FIND THE CHANGE IN HEAT FLOW COMPONENT Qz ACROSS THE
SURFACE S DUE TO A TEMPERATURE GRADIENT
m
Tk
vvvvv zyx
32222
=++==
tSvn ∆∆
6
1
V
N
n
∆
( )λε −∆ xSvn
6
1
- CHANGE IN ENERGY , DUE TO MOLECULES CROSSING ΔS
IN +x DIRECTION.
( )λε +∆ xSvn
6
1
- CHANGE IN ENERGY , DUE TO MOLECULES CROSSING ΔS
IN -x DIRECTION.
COLLISIONS λ WILL CROSS THE SURFACE S AND CONTRIBUTE TO A CHANGE IN THE MEAN
ENERGY OF THE MOLECULE ε (x(, WE CAN WRITE:

24
( ) ( )[ ]
λ
ε
λελε






∂
∂
∆−=
+−−∆=∆
x
Svn
xxSvnSqx
2
6
1
6
1
xq - HEAT FLOW [J/m2
] IN x DIRECTION, DUE TO
TEMPERATURE GRADIENT [°K/m]
 x
T
K
x
T
T
vn
x
vnq
c
x
∂
∂
−=
∂
∂
∂
∂
−=
∂
∂
−=
ε
λ
ε
λ
3
1
3
1
WHERE:
m
Tkcvc
cvnK
m
kT
v
n
2
3
2
2
1
6
1
233
1
2
οποπ
λ
οπ
λ ==
===
T
c
∂
∂
=
∆ ε
- HEAT CAPACITY (AT CONSTANT VOLUME) PER MOLECULE
THERMAL CONDUCTIVITY AND TRANSPORT OF ENERGY (CONTINUE(
FOURIER’s LAW

25
SELF-DIFFUSION AND TRANSPORT OF MOLECULES
WE WANT TO FIND THE MASS FLOW ACROSS THE
SURFACE S DUE TO MOLECULES CONCENTRATION GRADIENT
m
Tk
vvvvv zyx
32222
=++==
tSvn ∆∆
6
1
V
N
n
∆
( ) tSvxn ∆∆− λ
6
1
- NUMBER OF MOLECULES CROSSING ΔS IN +x DIRECTION,
IN TIME Δ t
( ) tSvxn ∆∆+ λ
6
1
- NUMBER OF MOLECULES CROSSING ΔS IN -x DIRECTION,
IN TIME Δ t.
COLLISIONS λ WILL CROSS THE SURFACE S AND CONTRIBUTE TO A CHANGE IN THE
MASS, WE CAN WRITE:

26
( ) ( )[ ]
λ
λλ






∂
∂
∆∆−=
+−−∆∆=∆
x
n
tSvm
xnxntSvmM
2
6
1
6
1
M∆ - MASS FLOW [Kg/m2
] IN x DIRECTION, DUE TO
MOLECULE CONCENTRATION GRADIENT [1/m3
]
tS
x
DtS
x
vM ∆∆
∂
∂
−=∆∆
∂
∂
−=∆
ρρ
λ
3
1
WHERE:
m
Tk
nn
v
vD
m
kT
v
n
2
3
2
2
1
1
6
1
233
1
2
οποπ
λ
οπ
λ ==
===
SELF-DIFFUSION AND TRANSPORT OF MOLECULES (CONTINUE(
SINCE:
x
n
m
∂
∂
=
∆
ρ

27
MAXWELL VELOCITY DISTRIBUTION
IN 1859 MAXWELL PROPOSED THE FOLLOWING MODEL:
ASSUME THAT THE VELOCITY COMPONENTS OF N
MOLECULES, ENCLOSED IN A CUBE WITH SIDE l, ALONG EACH
OF THE THREE COORDINATE AXES ARE INDEPENDENTLY AND
IDENTICALLY DISTRIBUTED ACCORDING TO THE DENSITY f0(α(
= f0(-α(, I.E.,
JAMES CLERK
MAXWELL
(1831 – 1879(
( ) ( ) ( ) ( )
( ) ( )[ ] zyx
zyxzzyyxx
vdvdvdvvvvBA
vdvdvdvvfvvfvvfvdvf
00
000000
3
0
exp


−⋅−−=
−−−=
f (Vi( d Vi = THE PROBABILITY THAT THE i VELOCITY
COMPONENTS IS BETWEEN vi AND vi + d vi ; i=x,y,z
MAXWELL ASSUMED THAT THE DISTRIBUTION DEPENDS
ONLY ON THE MAGNITUDE OF THE VELOCITY.

28
MAXWELL VELOCITY DISTRIBUTION (CONTINUE)
SINCE THE DEFINITION OF THE TOTAL NUMBER OF PARTICLES N
IS:
( )∫ ∫= tvrfvdrdN ,,33 
WE HAVE IN EQUILIBRIUM
( ) ( )[ ]
( ) ( ) ( ) 2
3
222
222
0
3
expexpexp
exp






=−−−=
++−==
∫∫∫
∫ ∫ ∫∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
B
AdvvBdvvBdvvBA
dvdvdvvvvBAvfvd
V
N
zzyyxx
zyxxxx
π

WHERE V IS THE VOLUME OF THE CONTAINER
∫= rdV 3
IT FOLLOWS THAT B > 0 AND
V
NB
A
2
3






=
π
LET FIND THE CONSTANTS A, B AND IN ( ) ( )[ ]2
00 exp vvBAvf

−−=0v


29
MAXWELL VELOCITY DISTRIBUTION (CONTINUE(
LET FIND THE CONSTANTS A, B AND IN ( ) ( )[ ]2
00 exp vvBAvf

−−=
THE AVERAGE VELOCITY IS GIVEN BY:
( )
( )
( ) ( )[ ]
( ) [ ] 00
3
00
3
0
3
0
3
exp
exp
vvvBvvvd
N
VA
vvvvBvvd
N
VA
vfvd
vfvvd
v





=⋅−+=
−⋅−−==
∫
∫
∫
∫
THE AVERAGE KINEMATIC ENERGY OF THE MOLECULES ε WHEN IS00

=v
( )
( )
( ) B
m
vBvvd
N
VAm
vfvd
vfvmvd
4
3
exp
2
2
1
223
0
3
0
23
=−== ∫
∫
∫


ε
WE FOUND ALSO THAT FOR A MONOATOMIC GAS Tk
2
3
=ε
Tk
mm
B
24
3
==
ε V
N
Tk
m
V
NB
A
2
3
2
3
2 





=





=
ππ
THEREFORE

30
MAXWELL VELOCITY DISTRIBUTION BECOMES
( ) 





⋅−





= vv
Tk
m
Tk
m
V
N
vf

2
exp
2
2
3
0
π
( ) ( ) ( ) ( )
( ) zyxzyx
zyxzyx
vdvdvdvvv
Tk
m
Tk
m
V
N
vdvdvdvfvfvfvdvf






++−





=
=
222
2/3
0
2
exp
2π
3
OR

31
( ) 





⋅−





= vv
Tk
m
Tk
m
V
N
vf

2
exp
2
2
3
0
π

32
BOLTZMANN EQUATION (1872(
UNLIKE MAXWELL, WHO PRESUPOSE EQUILIBRIUM,
BOLTZMANN ASSUMED THAT THE GAS WAS NOT IN
EQUILIBRIUM.
LUDWIG
BOLTZMANN
(1844 - 1906(
ASSUMPTIONS:
(1( ONLY BINARY COLLISIONS OCCUR (RAREFIATE GAS(
(2( THE DISTRIBUTION FUNCTION FOR PAIRS OF MOLLECULES IS GIVEN BY:
( )
( ) ( ) ( )tvftvftvvf ,,,, 2121
2 
=
THIS IS BOLTZMANN’s STOSSZAHLANSATZ, OR “ASSUMPTION” OF
“MOLECULAR CHAOS”, AND THIS ASSUMPTION [ORIGINALLY DUE TO
CLAUSIUS (1857(] IT WAS AND STILL IS THE MOST WIDELY DISCUSSED.
f(v,t( d3
v - NUMBER OF PARTICLES AT TIME t WITH VELOCITY IN THE
VOLUME ELEMENT d3
v AROUND .v

HE ASSUMED THAT THE GAS IS COMPOSED BY N PARTICLES,
PERFECTLY ELASTIC SPHERES, ENCLOSED IN A VOLUME V,
WITH MASS m, DIAMETER σ, AND VELOCITY DISTRIBUTION
FUNCTION f( v,t( DEFINED BY:

33
BOLTZMANN EQUATION (1872( (CONTINUE(
MOLECULE (2( WILL
COLLIDE WITH (1(
IF IT’S CENTER IS
INSIDE THE
“COLLISION CILINDER”
OF RADIUS σ AND
DIRECTION
12 vvg

−=
THE DIFFERENTIAL
OF COLLISION SURFACE
OF THE “COLLISION
CILINDER” IS GIVEN BY:
Ω==












=
ddd
dddbdb
4
sin
4
2
sin
22
cos
22
σ
φθθ
σ
θ
θσ
φ
θ
σφ

34
BOLTZMANN EQUATION (1872) (CONTINUE)
SINCE WE ASSUMED ELASTIC COLLISIONS, WE HAVE:
'' 2121 vmvmvmvm

+=+
2
2
2
1
2
2
2
1 '
2
1
'
2
1
2
1
2
1
vmvmvmvm +=+
'' 2121 vvvv

+=+
2
2
2
1
2
2
2
1 '' vvvv +=+
CONSERVATION OF LINEAR
MOMENTUM
CONSERVATION OF ENERGY
'' 2121 vvvv

+=+ ( ) ( ) ( ) ( )'''' 21212121 vvvvvvvv

+⋅+=+⋅+
21
2
2
2
121
2
2
2
1 ''2''2 vvvvvvvv

⋅++=⋅++
2
2
2
1
2
2
2
1 '' vvvv +=+
2121 '' vvvv

⋅=⋅
21
2
2
2
112
21
2
2
2
112
''2''''
2
vvvvvv
vvvvvv


⋅−+=−
⋅−+=−
1212 '' vvvv

−=−

35
BOLTZMANN EQUATION (1872( (CONTINUE(
BY DEFINITION, THE RATE OF CHANGE OF f(v,t( d3
v IS EQUAL
TO THE NET GAIN OF MOLECULES IN d3
v AS A RESULT OF
COLLISIONS, I.E.,
( ) ( ) ( ) 1
3
11
3
11
31
,,
,
vdtvnvdtvnvd
t
tvf
outin


−=
∂
∂
WHERE
( ) vdn outin
3
-THE NUMBER OF BINARY COLLISIONS AT TIME t IN WHICH
ONE OF THE FINAL (INITIAL( MOLECULES IS IN d3
v.
( ) ( )[ ] ( )[ ]∫ ∫ −Ω= 1
3
12
3
212
2
1
3
1 ,,
4
,
2
vdtvfvdtvfvvdvdtvn
V
out
 σ
( ) ( )[ ] ( )[ ]∫ ∫ −Ω= 1
3
1
'
2
3
212
2
1
3
1 ','','''
4 2
vdtvfvdtvfvvdvdvn
v
in
 σ
WE HAVE 1212 '' vvvv

−=− AND 2
3
1
3
2
3
1
3
'' vdvdvdvd =
WE OBTAIN BOLTZMANN EQUATION (1872(
( ) ( ) ( ) ( ) ( )[ ]∫ ∫ −−Ω=
∂
∂
2
2
3
212112
2
1
,,,','
4
,
V
vdtvftvftvftvfvvd
t
tvf 

σ

36
- THE NUMBER OF MOLECUES WHICH AT TIME t
HAVE POSITIONS IN THE VOLUME ELEMENT
d 3
r AROUND IN THE VELOCITY-SPACE ELEMENT
d 3
v AROUND
GENERALIZATION OF BOLTZMANN EQUATION
( ) vdrdtvrf 33
,,

FOR “NONUNIFORM SYSTEMS”, THE VELOCITY DISTRIBUTION FUNCTION
DEPENDS ALSO ON POSITION:
IN THE PRESENSE OF AN EXTERNAL FORCE THE POINT
BECOMES AT TIME dt.
( )rF

( )tvr ,,









+++ dttdt
m
F
vdtvr ,,


IN THIS CASE THE CHANGE IN THE VELOCITY DISTRIBUTION FUNCTION IS
( )
( ) ( ) ( )tvrf
m
F
tvrfv
t
tvrf
td
tvrftdttd
m
F
vdtvrf
vr
td
,,,,
,,
,,,,
lim
0







∇⋅+∇⋅+
∂
∂
=








−





+++
→
- GRADIENT WITH RESPECT TO rr∇
- GRADIENT WITH RESPECT TO vv∇

37
GENERALIZATION OF BOLTZMANN EQUATION (CONTINUE(
ONLY FOR HARD SPHERES THE
COLLISION SURFACE IS A CIRCLE
OF RADIUS σ. IN A GENERAL CASE
WE MUST REPLASE THE
INTERACTION ELEMENT BY
( ) Ω= dgIdbdb θφ ,
THEREFORE THE GENERALIZATION OF BOLTZMANN EQUATION IS
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )[ ]∫ ∫ −−Ω=
∇⋅+∇⋅+
∂
∂
2
2
3
212112
11
1
,,,,,',,',,
,,,,
,,
v
vr
vdtvrftvrftvrftvrfvvdgI
tvrf
m
F
tvrfv
t
tvrf





θ
IN THE LITERATURE WE FIND THE FOLLOWING NOTATION
( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫ −−Ω=
∆
2
2
3
212112 ,,,,,',,',,,
v
vdtvrftvrftvrftvrfvvdgIffB

θ
( ) ( ) ( ) ( )ffBtvrf
m
F
tvrfv
t
tvrf
vr ,,,,,
,,
11
1
=∇⋅+∇⋅+
∂
∂ 



GENERALIZED BOLTZMANN
EQUATION

38
BOLTZMANN’s H THEOREM
THE EQUILIBRIUM DISTRIBUTION IS DEFINED BY THE TIME-INDEPENDENT
SOLUTION OF BOLTZMANN’s EQUATION.
( )vff mequilibriu

0=
CONSIDER THE SPACIALLY UNIFORM CASE, SO THAT
FROM BOLTZMANN’s EQUATION, THE EQUILIBRIUM CONDITION IS:
( ) ( ) ( ) ( ) ( )[ ] ( ) 0,''
,
2
2
3
2010201012
10
=−−Ω=
∂
∂
∫ ∫V
vdgIvfvfvfvfvvd
t
tvf
θ


WE CAN SEE THAT
IS A SUFICIENT CONDITION FOR EQUILIBRIUM.
BOLTZMANN PROVED THAT IS ALSO A NECESSARY CONDITION FOR
EQUILIBRIUM.
( ) ( ) ( ) ( )20102010 '' vfvfvfvf

=

39
( ) ( ) ( ) ( ) ( )[ ] ( )∫ ∫ −−Ω=
∂
∂
2
2
3
212112
1
,,,,','
,
v
vdgItvftvftvftvfvvd
t
tvf
θ


IF IS A SOLUTION OF BOLTZMANN’s EQUATION:( )tvf ,1

AND H (t( IS DEFINED BY:
( ) ( ) ( )∫
∆
=
V
vdtvftvftH 3
,ln,

THEN:
( ) 0≤
td
tHd

40
SUBSTITUING THE BOLTZMANN’s EQUATION, WE
OBTAIN:
PROOF OF BOLTZMANN’s H THEOREM
( ) ( ) ( ) ( ) 1222112211 ,,'',,'',,,, vvgtvfftvfftvfftvff

−=====
∆∆∆∆∆
WHERE:
( ) ( ) ( )∫
∆
=
V
vdtvftvftH 3
,ln,
 ( ) ( ) ( )[ ]∫ +
∂
∂
=
V
vdtvf
t
tvf
td
tHd 3
,ln1
, 

( ) ( ) ( ) [ ]∫ ∫ ∫ +−Ω=
1 2
112122
3
1
3
ln1'',
v v
fffffgIgdvdvd
td
tHd
θ
INTERCHANGE
12 vv

⇔
( ) ( ) ( ) [ ]∫ ∫ ∫ +−Ω=
1 2
212121
3
2
3
ln1'',
v v
fffffgIgdvdvd
td
tHd
θ
1
2
ADD (1) AND (2) AND DIVIDE BY 2
( ) ( ) ( ) [ ]∫ ∫ ∫ +−Ω=
1 2
2112122
3
1
3
ln2'',
2
1
v v
ffffffgIgdvdvd
td
tHd
θ3

41
PROOF OF BOLTZMANN’s H THEOREM (CONTINUE)
CHANGE VARIABLES
( ) ( ) ( ) [ ]∫ ∫ ∫ +−Ω=
1 2
2112122
3
1
3
''ln2'',
2
1
V V
ffffffgIgdVdVd
td
tHd
θ4
( ) ( )2121 ',', VVVV

⇔
AND REMEMBER THAT ''' 1212 gVVVVg =−=−=

AND 2
3
1
3
2
3
1
3
'' VdVdVdVd =
TAKE ONE HALF OF (3) AND (4)
( ) ( ) ( ) [ ]∫ ∫ ∫ −−Ω=
1 2
212112122
3
1
3
''lnln'',
4
1
V V
ffffffffgIgdVdVd
td
tHd
θ5
IT IS EASY TO SHOW THAT FOR ANY TWO POSITIVE NUMBERS x AND
y
( )

( )( ) 0lnlnln
0
0
≤−−=−⇒≥
≥
≤
yxxy
y
x
xyyx

THEREFORE THE INTEGRAND IN (5) IS NEVER POSITIVE, I.E.,
( ) 0≤
td
tHd AND ( ) 0≡
td
tHd ( ) ( ) [ ] 0'',
,
2
2
3
2112
1
≡−Ω=
∂
∂
∫ ∫V
VdffffgIgd
t
tVf
θ

END OF PROOF

42
MAXWELL-BOLTZMANN DISTRIBUTION
FROM BOLTZMANN’s H THEOREM WE HAVE SEEN THAT , IN ABSENSE
OF EXTERNAL FORCES, THE EQUILIBRIUM STATE IS SPECIFIED BY THE
VELOCITY DISTRIBUTION FUNCTION SATISFYING:
( ) ( ) ( ) ( )20102010 '' vfvfvfvf

= ( ) ( ) ( ) ( )20102010 lnln'ln'ln vfvfvfvf

+=+
THIS HAS THE FORM OF A CONSERVATION LAW.
SINCE FOR SPINLESS MOLECULES (MAXWELL-BOLTZMANN
ASSUMPTION( THE ONLY CONSERVED QUANTITIES ARE ENERGY AND
LINEAR MOMENTUM, IT FOLOWS THAT log f0 (v( MUST BE A LINEAR
COMBINATION OF v2
, THE THREE COMPONENTS OF , AND A
CONSTANT:
v

( ) ( )2
00 lnln vvBAvf

−−=
( ) ( )[ ]2
00 exp vvBAvf

−−=
THIS IS THE MAXWELL DISTIBUTION FOR VELOCITIES.

43
J.D. ANDERSON JR., “Modern Compressible Flow with Historical Perspective”
McGRAW-HILL, 1982

44
McGRAW-HILL, 1982

45
McGRAW-HILL, 1982

46
McGRAW-HILL, 1982

47
McGRAW-HILL, 1982

48
McGRAW-HILL, 1982

49
MOLECULAR MODELS
BOLTZMANN STATISTICS
• DISTINGUISHABLE PARTICLES
• NO LIMIT ON THE NUMBER OF
PARTICLES PER QUANTUM STATE.
BOSE-EINSTEIN STATISTICS
• INDISTINGUISHABLE PARTICLES
FERMI-DIRAC STATISTICS
• ONE PARTICLE PER QUANTUM STATE.
LUDWIG BOLTZMANN
SATYENDRANATH N. BOSE ALBERT EINSTEIN
ENRICO FERMI PAUL A.M. DIRAC
∏ 







=
j j
N
j
N
g
Nw
j
!
!
( )
( )∏ 







−
−+
=
j jjj
jj
Ng
Ng
w
!!1
!1
( )∏ 







−
=
j jjjj
j
NNg
g
w
!!
!
∑=
j
jNN ∑=
j
jj NE 'ε
NUMBER OF MICROSTATES
FOR A GIVEN MACROSTATE

50
MOLECULAR MODELS
• DISTINGUISHABLE PARTICLES
LUDWIG BOLTZMANN
∏ 







=
j j
N
j
Boltz
N
g
Nw
j
!
!NUMBER OF MICROSTATES
NUMBER OF WAYS N DISTINGUISHABLE
PARTICLES CAN BE DIVIDED IN GROUPS WITH
N1, N2,…,Nj,…PARTICLES IS
∑=
j
jNN
∏j
jN
N
!
!
NUMBER OF WAYS Nj PARTICLES CAN BE PLASED IN THE gj STATES IS
jN
jg
A MACROSTATE IS DEFINED BY
- QUANTUM STATES g1,g2,…,gj
AT THE ENERGY LEVELS
- NUMBER OF PARTICLES
N1,N2,…Nj
IN STATES g1,g2, …,gj
j',,',' 21 εεε 

51
THE MOST PROBABLE MACROSTATE –
THE THERMODYNAMIC EQUILIBRIUM STATE
BOLTZMANN STATISTICS ∏ 







=
j j
N
j
Boltz
N
g
Nw
j
!
!
USING STIRLING FORMULA
0'' ==⇒= ∑∑ EdNdNE
j
jj
j
jj εε
( ) aaaa −≈ ln!ln
( ) ( )∑∑ /+−+/−≈−+=
j
jjjjj
STIRLING
j
jjj NNNgNNNNNgNNw lnlnln!lnln!lnln
( ) ( ) 0lnlnln =−−= ∑j
jjjjj NdNNdgNdwd
0==⇒= ∑∑ NdNdNN
j
j
j
j
TO CALCULATE THE MOST PROBABLE MACROSTATE WE MUST COMPUTE
THE DIFFERENTIAL
CONSTRAINTED BY:

52
(CONTINUE) ∏ 







=
j j
N
j
Boltz
N
g
Nw
j
!
!
0' =∑j
jj Ndεβ
( ) 0lnln =








=− ∑j
j
j
j
Nd
g
N
wd
0=∑j
jNdα
WE OBTAIN
LET ADJOIN THE CONSTRAINTS USING THE LAGRANGE MULTIPLIERS
0'
*
ln0'ln =++








⇒=








++








∑ j
j
j
j
jj
j
j
g
N
Nd
g
N
εβαεβα
βα,
TO OBTAIN
OR j
eegN jBoltzj
'
*
εβα −−
= BOLTZMANN
MOST PROBABLE MACROSTATE

53
MOLECULAR MODELS
∑=
j
jNN
NUMBER OF WAYS Nj INDISTINGUISHABLE PARTICLES CAN BE PLASED IN
THE gj STATES IS
( )
( ) !!1
!1
jj
jj
Ng
Ng
−
−+
- QUANTUM STATES g1,g2,…,gj
AT THE ENERGY LEVELS
- NUMBER OF PARTICLES
N1,N2,…Nj
SATYENDRANATH N. BOSE
(1894-1974)
ALBERT EINSTEIN
(1879-1955)
j',,',' 21 εεε 
( )
( )∏ −
−+
=−
j jj
jj
EB
Ng
Ng
w
!!1
!1

54
(CONTINUE)
USING STIRLING FORMULA ( ) aaaa −≈ ln!ln
( )[ ] ( ) ( ) ( )[ ]
( )∑
∑∑
















+++=
/+−/+−/+/−++≈−−+=
j j
j
jjjj
j
jjjjjjjjjjjj
STIRLING
j
jjjj
g
N
gNgN
NNNggggNgNgNNggNw
1ln/1ln
lnlnln!ln!ln!ln
( ) 01ln
1
1
1
1lnln
2
=








+=














+
/+
+
−
/+








+= ∑∑ j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
Nd
N
g
Nd
g
N
g
g
N
g
N
g
N
N
g
wd
THE DIFFERENTIAL
( )
( )
( )
∏∏
+
≈
−
−+
=−
j jj
jj
j jj
jj
EB
Ng
Ng
Ng
Ng
w
!!
!
!!1
!1

55
(CONTINUE)
0' =∑j
jj Ndεβ0=∑j
jNdα
WE OBTAIN
0'
*
1ln0'1ln =−−








+⇒=








−−








+∑ j
j
j
j
jj
j
j
N
g
Nd
N
g
εβαεβα
βα,
TO OBTAIN
OR
1
* '
−
=− j
ee
g
N
j
EBj εβα
BOSE-EINSTEIN
( )
( )
( )
∏∏
+
≈
−
−+
=−
j jj
jj
j jj
jj
EB
Ng
Ng
Ng
Ng
w
!!
!
!!1
!1
( ) 01lnln =








+= ∑j
j
j
j
Nd
N
g
wd

56
MOLECULAR MODELS
∑=
j
jNN
NUMBER OF WAYS Nj INDISTINGUISHABLE PARTICLES CAN BE PLASED IN
THE gj STATES IS ( ) !!
!
jjj
j
NNg
g
−
( )∏ −
=−
j jjj
j
DF
NNg
g
w
!!
!
• ONE PARTICLE PER QUANTUM STATE.
ENRICO FERMI
(1901-1954)
PAUL A.M. DIRAC
(1902-1984)
- QUANTUM STATES g1,g2,…,gj AT THE
ENERGY LEVELS
- NUMBER OF PARTICLES N1,N2,…Nj AT
THE ENERGY LEVELS
j',,',' 21 εεε 
j',,',' 21 εεε 

57
(CONTINUE)
USING STIRLING FORMULA ( ) aaaa −≈ ln!ln
( )[ ] ( ) ( ) ( )[ ]
( ) ( )[ ]∑
∑∑
−−−−=
/+−/−/+−−−/−≈−−−=
j
jjjjjjjj
j
jjjjjjjjjjjj
STIRLING
j
jjjj
NNNgNggg
NNNNgNgNggggNNggw
lnlnln
lnlnln!ln!ln!lnln
( ) ( )[ ] 0lnlnlnln =







 −
=−−−+= ∑∑ j
j
j
jj
j
jjjjjjj Nd
N
Ng
NdNNdNdNgNdwd
THE DIFFERENTIAL
( )∏ −
=−
j jjj
j
DF
NNg
g
w
!!
!

58
(CONTINUE)
0' =∑j
jj Ndεβ0=∑j
jNdα
WE OBTAIN
0'
*
*
ln0'ln =−−







 −
⇒=








−−







 −
∑ j
j
jj
j
jj
j
jj
N
Ng
Nd
N
Ng
εβαεβα
βα,
TO OBTAIN
OR
1
* '
+
=− j
ee
g
N
j
DFj εβα
FERMI-DIRAC
( ) 0lnln =







 −
= ∑j
j
j
jj
Nd
N
Ng
wd
( )∏ −
=−
j jjj
j
DF
NNg
g
w
!!
!

59
FERMI-DIRAC
STATISTICS
OR
( )∏ −
=−
j jjj
j
DF
NNg
g
w
!!
!( )
( )∏ −
−+
=−
j jj
jj
EB
Ng
Ng
w
!!1
!1
BOSE-EINSTEIN
STATISTICS
∏ 







=
j j
N
j
Boltz
N
g
Nw
j
!
!
BOLTZMANN
STATISTICS
FOR GASES AT LOW PRESSURES OR HIGH TEMPERATURE THE NUMBER
OF QUANTUM STATES gj AVAILABLE AT ANY LEVEL IS MUCH LARGER
THAN THE NUMBER OF PARTICLES IN THAT LEVEL Nj.
jj Ng >>
( )
( ) ( )( ) ( ) j
jj
N
j
Ng
jjjjj
j
jj
gNgggg
g
Ng >>
≈−+++=
−
−+
121
!1
!1

( ) ( ) ( ) j
jj
N
j
Ng
jjjj
jj
j
gNggg
Ng
g >>
≈+−−=
−
11
!
!

∏ 







=≈≈
>>
−
>>
−
j j
N
jBoltz
Ng
DF
Ng
EB
N
g
N
w
ww
jjjjj
!!
AND j
jjjj
eegNNN jBoltzj
Ng
DFj
Ng
EBj
'
***
εβα −−
>>
−
>>
− =≈≈

60
∏ 







=≈≈
>>
−
>>
−
j j
N
jBoltz
Ng
DF
Ng
EB
N
g
N
w
ww
jjjjj
!!
j
jjjj
eegNNN jBoltzj
Ng
DFj
Ng
EBj
'
***
εβα −−
>>
−
>>
− =≈≈
DIVIDING THE VALUE OF w FOR BOLTZMANN STATISTICS, WHICH
ASSUMED DISTINGUISHABLE PARTICLES, BY N! HAS THE EFFECT OF
DISCOUNTING THE DISTINGUISHABILITY OF THE N PARTICLES.

61
WE HAVE: ∑∑
−−
==
j
j
j
j
j
egeNN
'
*
εβα
DEFINE: ∑
−
=
j
j
j
egZ
'
:
εβ
PARTITION FUNCTION
USING THE DEFINITION WE HAVE:
ZeN α−
=
jj
eg
Z
N
eegN jjj
''
*
εβεβα −−−
==
Vj
jj
j
jj
Z
Z
N
eg
Z
N
NE j






∂
∂
−=== ∑∑
−
β
εε
εβ '
'*'
∑
−
−=





∂
∂
j
jj
V
j
eg
Z '
'
εβ
ε
β

62
THE CLASSICAL MONOATOMIC GAS
CONSIDER THE GAS TO BE IDEAL AND MONOATOMIC, GROUND
ELECTRONIC LEVEL ONLY, AND DESCRIBED BY THE BOLTZMANN
STATISTICS. IN THIS CASE THE GAS HAS ONLY KINETIC ENERGY.
( )222
2
1
' zyxVVV vvvmzyx
++=ε
THE TRANSLATIONAL ENERGY LEVELS ARE:
THE PARTITION FUNCTION IS:
( )
∑
++−
=
zyx
zyx
zyx
vvvall
vvvm
vvv egZ
,,
2
222β
- THE NUMBER OF QUANTUM STATES BETWEENzyx vvvg zzyyxxzyx
vvvvvvvvv ∆+∆+∆+÷ ,,,,
WE HAVE ( ) h
vXm
vmx
vXm
g x
PRINCIPLE
HEISENBERG
x
x
vx
∆
=
∆∆
∆
=
min
GENERALIZING zyx
zyx
vvvvvv vvv
h
Vm
h
vZm
h
vYm
h
vXm
gggg zyxzyx
∆∆∆=
∆∆∆
== 3
3

63
THE CLASSICAL MONOATOMIC GAS (CONTINUE)
LET COMPUTE
( )
2/3
2
3
3
3
3
2
3
3
2
3
3
2
22222






=






=








== ∫∫ ∫ ∫
∞
−∞=
−++−
h
m
V
mh
Vm
dve
h
Vm
dvdvdve
h
Vm
Z
x
x
x y z
zyx
v
x
vm
V V V
zyx
vvvm
β
π
β
πββ
FROM WHICH WE GET
β
β
π
β
Z
h
m
V
Z
V
2
32
2
3 2/5
2/3
2
−=





−=





∂
∂ −
AND
βββ
NZ
Z
NZ
Z
N
E
V
2
3
2
3
=





−−=





∂
∂
−=
COMPARING WITH
WE OBTAIN
TkNE
2
3
=
Tk
1
=β

64
MAXWELL-BOLTZMANN VELOCITY DISTRIBUTION
THE MOST PROBABLE MACROSTATE IS
WHITH
j
eg
Z
N
N jj
'
*
εβ−
=
zyxvvv dvdvdv
h
Vm
g zyx 3
3
=
THEREFORE WE RECOVERED THE MAXWELL-BOLTZMANN
VELOCITY DISTRIBUTION
( ) ( )
( )
( ) zyxzyx
zyxzyx
vvvvvv
dvdvdvvvv
kT
m
Tk
m
hTkmV
dVdVdVVVV
kT
m
hVm
Z
eg
N
Nd zvyvxv
zyxzyx






++−





=




++−
==
−
222
2/3
2
22233'
2
exp
2
/2
2
exp/
π
π
εβ
IN POLAR COORDINATES
vdddvvdvdvd
vv
vv
vv
xxx
z
y
x
φθθ
θ
φθ
φθ
sin
cos
sinsin
cossin
2
=





=
=
=
LUDWIG
BOLTZMANN
JAMES CLERK
MAXWELL

65
MAXWELL-BOLTZMANN VELOCITY DISTRIBUTION (CONTINUE)
THEREFORE
dvdd
kT
vm
v
Tk
m
N
Nd v
φθθ
π
φθ
sin
2
exp
2
2
2
2/3






−





=
USING THIS EQUATION WE CAN COMPUTE
MAXWELL-BOLTZMANN
VELOCITY DISTRIBUTION
dv
kT
vm
v
Tk
m
dddv
kT
vm
v
Tk
m
N
Nd
N
Nd vv






−





=






−





== ∫ ∫∫∫ = =
2
exp
2
4
sin
2
exp
2
2
2
2/3
2
0 0
2
2
2/3
π
π
φθθ
π
π
φ
π
θφ θ
φθ
THE MOST PROBABLE VELOCITY IS OBTAINED BY
02
2
exp
2
4
22
2
2/3
=





−





−





=





kT
vm
kT
vm
v
Tk
m
vdN
Nd
vd
d v
π
π
FROM WHICH
Tk
vm
m
Tk
v
PM
PROBABLEMOST =⇒





=
2
2
22/1

66
MAXWELL-BOLTZMANN VELOCITY DISTRIBUTION (CONTINUE)
WE CAN ALSO COMPUTE
PM
v
v
m
Tk
m
Tk
Tk
m
dv
kT
vm
v
Tk
m
dv
dvN
Nd
vv
1284.1
8
2
2
4
2
exp
2
4
2/122/3
0
2
3
2/3
0
=





=




















=






−





=





= ∫∫
∞∞
ππ
π
π
π
AND
( )2
2/52/3
0
2
4
2/3
0
22
2248.1
32
8
3
2
4
2
exp
2
4
PM
v
v
m
Tk
m
Tk
Tk
m
dv
kT
vm
v
Tk
m
dv
dvN
Nd
vv
==




















=






−





=





= ∫∫
∞∞
π
π
π
π
π
FROM WHICH
N
E
Tk
vm
===
2
3
2
2
ε
MONOATOMIC MOLECULE
KINETIC ENERGY

67
CONSIDER A SYSTEM IN EQUILIBRIUM OF VOLUME V AND TEMPERATURE T
EVALUATION OF THERMODYNAMIC PROPERTIES IN TERMS OF
THE PARTITION FUNCTION
( )TVfegZ
j
kT
j
j
,
/
== ∑
−ε
THE MOST PROBABLE MACROSTATE IS
kT
jj
j
eg
Z
N
N
/
*
ε−
=
THE INTERNAL ENERGY E, IS
∑∑
−
==
j
kT
j
j
jj
j
eg
Z
N
NE
/
*
ε
ε
THE SET OF AVAILABLE QUANTUM STATES εj IS FIXED BY THE VOLUME V.
fixedgconstTV jj ε,, ⇒=
WE HAVE
Vj
kT
jj
j
kT
jj
V
T
Z
Tkegeg
TkT
Z jj






∂
∂
=⇒=





∂
∂
∑∑
−− 2//
2
1 εε
εε

68
( )TVfegZ
j
kT
j
j
,
/
== ∑
−ε
WE HAVE
Vj
kT
j
T
Z
TkNeg
Z
N
E j






∂
∂
== ∑
− ln2/ε
[ ] ( )[ ]RlbftKJ
N
k oo
A
/10565.0/1038.1 2323
⋅×=×== −−R
BOLTZMANN CONSTANT
WE DEFINED
23
10022137.6 ⋅=AN AVOGADRO’s NUMBER
UNIVERSAL GAS CONSTANT
[ ]
( )[ ])/(1097.4
)/(8314
4
Rmolsluglbft
KmolkgJ
o
o
⋅⋅⋅×=
⋅=R
VA T
Z
T
N
NE 





∂
∂
=
ln2R

69
( )TVfegZ
j
kT
j
j
,
/
== ∑
−ε
IF m IS THE MASS OF ONE MOLECULES, THAN THE MASS OF THE GAS IS N m.
THE INTERNAL ENERGY PER UNIT MASS e, IS:
VA T
Z
T
mNmN
E
e 





∂
∂
==
∆ ln2R
( ) ( )
( ) ( )Rsluglbft
KkgJairR


⋅⋅=
⋅=
/1716
/287
( )mNR A/R=
SPECIFIC
GAS CONSTANT
USING
WE OBTAIN
V
T
Z
TRe 





∂
∂
=
ln2
ENERGY PER UNIT MASS

70
( )TVfegZ
j
kT
j
j
,
/
== ∑
−ε
THE SPECIFIC ENTHALPY h, IS DEFINED AS:
WE OBTAIN
TR
T
Z
TRh
V
+





∂
∂
=
ln2
SPECIFIC ENTHALPY
TRevpeh +=+=
∆

71
2.BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
)2.5(CONSTITUTIVE RELATIONS )CONTINUE(
)2.5.2(STATE EQUATION (CONTINUE)
VAN DER WAALS (1873)
EQUATION
ARISE FROM THE EXISTENCE
OF INTERNAL FORCES BETWEEN
GAS MOLECULES
REAL GAS
( ) TRbV
V
a
p =−





+ 2
2
/Va
IS PROPORTIONAL TO THE
VOLUME OCCUPIED BY THE
GAS MOLECULES THEMSELVES
b
( )
070.15100
488.01400
686.0920
510.0350
587.0344
427.08.62
372.057.8
2
2
2
2
3
2
6
Hg
OH
CO
O
Air
H
He
molelbm
ft
molelbm
ftatm
baGAS






−





−
⋅
SOLO
JOHANNES DIDERIK
VAN DER WAALS
)1837-1923(

72
2.BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
)2.5(CONSTITUTIVE RELATIONS )CONTINUE(
)2.5.2(STATE EQUATION (CONTINUE)
VAN DER WAALS (1873)
EQUATION
REAL GAS
( ) TRbV
V
a
p =−





+ 2
SOLO
JOHANNES DIDERIK
VAN DER WAALS
)1837-1923(

73
REFERENCES:
1. K.J. LAIDLER & J.H. MEISER, “ Physical Chemistry”, 2nd
Ed., HOUGHTON
MIFFLIN COMPANY, 1995
2. S.E. FRIŞ & A.V. TIMOREVA, “Fizica Generalã”, (romanian translation),
Vol.1, Ch.7, “Gazele”, 1961
3. BERKELEY PHYSICS COURSE VOL. 5 – STATISTICAL PHYSICS’
CH.8, “Elementary Kinetic Theory of Transport Processes”, McGRAW-HILL,
1965
4. C.J. THOMPSON, “Mathematical Statistical Mechanics”, PRINCETON
UNIVERSITY PRESS, 1972
5. R.E. SONNTAG & G.J. van WYLEN, “Introduction to Thermodynamics:
Classical and Statistical”, WILEY INTERNATIONAL, 1971
6. J.D. ANDERSON JR., “Modern Compressible Flow with Historical Perspective”
McGRAW-HILL, 1982
SOLO

74
REFERENCES:
7. E. SEGRÈ, “ From Falling Body to Radio Waves”, W. H. FREEMAN AND
COMPANY, 1984
8. D. FLAMM, “History and Outlook of Statistical Physics”, physics/9803005,
4 March 1998
9. J. BIGGUS, “Sketching the History of Statistical Mechanics and Thermodynamics
(From about 1575 to 1980), HyperJeff Network
SOLO

January 5, 2015 75
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –2013
Stanford University
1983 – 1986 PhD AA

76
SOLO Permutation & Combinations
Permutations
Given n objects, that can be arranged in a row, how many different permutations
(new order of the objects) are possible?
To count the possible permutations , let start by moving only the first object {1}.
1
Number of permutations
2
3
n
By moving only the first object {1}, we obtained n permutations.

77
Permutations (continue -1)
Since we obtained all the possible position of the first object, we will perform the same
procedure with the second object no {2}, that will change position with all other objects,
in each of the n permutations that we obtained before .
For example from the group 1 we obtain the following new permutations
Number of new permutations
Since this is true for all permutations (n-1 new permutations for each of the first n
permutations) we obtain a total of n (n-1) permutations .
1
2
n-2
n-1

78
Permutations (continue -2)
If we will perform the same procedure with the third object {3}, that will change position
with all other objects, besides those with objects no {1} and {2} that we already obtained,
in each of the n (n-1) permutations that we obtained before , we will obtain a total
of n (n-1) (n-2) permutations.
We continue the procedure with the objects {4}, {5}, …, {n}, to obtain finally the total
number of permutations of the n objects:
n (n-1) (n-2) (n-3)… 1 = n !
The gamma function Γ is defined as: ( ) ( )∫
∞
−
−=Γ
0
1
exp dttta a
Gamma Function Γ
If a = n is an integer then:
( )  ( ) ( ) ( )
( )nn
dtttnttdtttn nn
dv
u
n
Γ=
−+−−=−=+Γ ∫∫
∞
−∞
∞
0
1
0
0
expexpexp1

( ) ( ) ( ) 1expexp1 0
0
=−−=−=Γ
∞
∞
∫ tdtt
Therefore: ( ) ( ) ( ) ( ) ( )!11211 −=−−==Γ=+Γ nnnnnnn 

79
Combinations
Given k boxes, each box having a maximum capacity (for box i the maximum
object capacity is ni ).
Given also n objects, that must be arranged in k boxes, each box must be filled to it’s
maximum capacity :
The order of the objects in the box is not important.
Example: A box with a capacity of three objects in which we arranged the objects {2}, {4}, {7)
42 7
4
2 74 27
427
42 7
4 2 7
4 27
Equivalent
3!=6 arrangements
1 outcome
nnnn k =+++ 21

80
Combinations (continue - 1)
In order to count the different combinations we start with n ! different arrangements of the
n objects.
nnnn k =+++ 21
In each of the n! arrangements the first n1 objects will go to box no. 1, the next n2
objects in box no. 2, and so on, and the last nk objects in box no. k, and since:
all the objects are in one of the boxes.

81
But since the order of the objects in the boxes is not important, to obtain the number of
different combinations, we must divide the total number of permutations n! by n1!, because
of box no.1, as seen in the example bellow, where we used n1=2.
1 2 3 nn-1
n1=2 n2 nk
Box 1 Box 2 Box k
12 3 nn-1
kn
!n
21 =n 2n
123 nn-1
123 nn-1
12nn-1
12n n-1
4
4
4
4
n-2
n-2
n-3
n-3
!1n
!1n
!1n
Same
Combination
Same
Combination
Same
Combination
!
!
1n
n
Therefore since the order of the objects in the boxes is not important, and because
the box no.1 can contain only n1 objects, the number of combination are

82
Since the order of the objects in the boxes is not important, to obtain the number of
different combinations, we must divide the total number of arrangements n! by n1!, because
of box no.1, by n2!, because of box no.2, and so on, until nk! because of box no.k, to obtain
!!!
!
21 knnn
n

Combinations

8 kinetic theory of gases

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