thermo & stat mech - spring 2006 class 14 1 thermodynamics and statistical mechanics kinetic...
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Thermo & Stat Mech - Spring 2006 Class 14
1
Thermodynamics and Statistical Mechanics
Kinetic Theory of Gases
Thermo & Stat Mech - Spring 2006 Class 14
2
Mixing of Two Ideal Gases
Change of Gibbs Function
)()(
222111
22112211
ififif
fffiii
ggnggnGGG
gngnGgngnG
An expression is needed for the specific Gibbs function.
Thermo & Stat Mech - Spring 2006 Class 14
3
Specific Gibbs Function
0
0
so gas, idealan for
hTsTcg
hTchdT
dhc
TshTsPvug
P
P
P
Thermo & Stat Mech - Spring 2006 Class 14
4
Specific Entropy
P
dPR
T
dTcdP
T
v
T
dTcds
vdPdTcTds
PdvvdPPdvdTc
PdvRdTdTcPdvdTRc
PdvdTcPdvduTds
PP
P
P
PP
v
)(
Thermo & Stat Mech - Spring 2006 Class 14
5
Specific Gibbs Function
00
00
0
0
lnln
)lnln(
lnln
hTsTTcTcPRTg
hsPRTcTTcg
hTsTcg
sPRTcsP
dPR
T
dTcds
PP
PP
P
P
P
Thermo & Stat Mech - Spring 2006 Class 14
6
Specific Gibbs Function
)(ln
lnln
lnln
00
00
PRTg
RT
h
R
sTccRTPRTg
hTsTTcTcPRTg
PP
PP
Thermo & Stat Mech - Spring 2006 Class 14
7
Mixing of Two Ideal Gases
i
f
i
f
if
if
ifif
P
Pn
P
PnRTG
PPn
PPnRTG
ggnggnG
PRTg
2
22
1
11
222
111
222111
lnln
)]ln(ln
)ln(ln[
)()(
)(ln
Thermo & Stat Mech - Spring 2006 Class 14
8
For the Same Pressure
22
22
21
222
11
11
21
111
21
2
22
1
11
so
so
lnln
xP
PP
n
nP
nn
nPxP
xP
PP
n
nP
nn
nPxP
PPP
P
Pn
P
PnRTG
i
ff
i
ff
ii
i
f
i
f
Thermo & Stat Mech - Spring 2006 Class 14
9
For the Same Pressure
2211
2211
22
11
2211
lnln)(
lnln
lnln
lnln
xxxxnRT
GS
xxxxnRTG
xn
nx
n
nnRTG
xnxnRTG
P
Thermo & Stat Mech - Spring 2006 Class 14
10
For the Same Volume
2
1ln
2
1ln
lnln
so constant, is
lnln
21
2
22
1
11
1
2
22
1
11
nnRTG
V
Vn
V
VnRTG
VPT
P
Pn
P
PnRTG
f
i
f
i
i
f
i
f
Thermo & Stat Mech - Spring 2006 Class 14
11
For the Same Volume
2ln)(
2ln)(
2
1ln)(
2
1ln
2
1ln
21
21
21
21
RnnS
RTnnG
RTnnG
nnRTG
Thermo & Stat Mech - Spring 2006 Class 14
12
Basic Assumptions
1. A macroscopic volume contains a large number of molecules.
2. The separation of molecules is large compared to molecular dimensions.
3. No forces exist between molecules except those associated with collisions
4. The collisions are elastic.
Thermo & Stat Mech - Spring 2006 Class 14
13
Basic Assumptions
When no external forces are applied:
5. The molecules are uniformly distributed within a container.
6. The directions of the velocities of the molecules are uniformly distributed.
The fraction of molecules with speeds in the range v to v + dv is: f (v) dv
Thermo & Stat Mech - Spring 2006 Class 14
14
Molecular Speeds
f (v) is the probability density.
speed squaremean Root
speed squareMean )(
speed averageor Mean )(
1)(
2
0
22
0
0
vv
dvvfvv
dvvvfv
dvvf
rms
Thermo & Stat Mech - Spring 2006 Class 14
15
Gas Pressure
Thermo & Stat Mech - Spring 2006 Class 14
16
Gas Pressure
ddvddt
vdNmv
dt
dpdt
vdNmv
dt
vdp
mvp
mvmvp
vmpdt
dA
dF
A
FP
),,(cos2
),,(cos2
),,(
cos2
)cos(cos
Thermo & Stat Mech - Spring 2006 Class 14
17
Gas Pressure
Thermo & Stat Mech - Spring 2006 Class 14
18
Molecular Flux
dddvvvfdAn
dt
vdN
dddvvfnvdtdAvdN
V
Nn
cossin])([4
),,(
4
sin)())(cos(),,(
Thermo & Stat Mech - Spring 2006 Class 14
19
Molecular Flux
Thermo & Stat Mech - Spring 2006 Class 14
20
Molecular Flux
vnvn
dddvvvfn
dddvvvfn
ddAdt
vdN
4
1)2(
2
1)(
4
cossin)(4
cossin])([4
),,(
2
0
2
00
Thermo & Stat Mech - Spring 2006 Class 14
21
Gas Pressure
22
2
0
2
0
2
0
2
3
1)2(
3
1
4
2
sincos)(4
2
cossin])([4
),,(
),,(cos2
vmnvnm
PdAdt
dp
dddvvfvdAnm
dt
dp
dddvvvfdAn
dt
vdN
ddvddt
vdNmv
dt
dp
Thermo & Stat Mech - Spring 2006 Class 14
22
Ideal Gas Law
AA
A
N
RkNkTRT
N
NPV
N
NnnRTPV
vNmvNmvmVnPV
vmnP
222
2
2
1
3
2
3
1
3
13
1
Thermo & Stat Mech - Spring 2006 Class 14
23
Molecular Kinetic Energy
kTvm
NkTvNm
NkTPVvNmPV
2
3
2
1
2
1
3
2
2
1
3
2
2
2
2
Thermo & Stat Mech - Spring 2006 Class 14
24
Equipartition of Energy
freedom of degreeper Energy 2
12
1
2
12
3
2
12
3
2
1
2
222
2
kT
kTvm
kTvvvm
kTvm
i
zyx
Thermo & Stat Mech - Spring 2006 Class 14
25
Internal Energy
nRdT
dUCnRTNkTU
f
nRdT
dUCnRTNkTU
f
fkTNfU
V
V
2
5
2
5
2
5
5 gas Diatomic2
3
2
3
2
3
3 gas MonatomicMolecule
Freedom of Degrees
2
1
Thermo & Stat Mech - Spring 2006 Class 14
26
Heat Capacities
fnRf
nRf
C
C
nRf
nRnRf
nRCC
nRf
dT
dUC
RTnfkTNfU
V
P
VP
V
21
2
12
122
2
2
1
2
1
Thermo & Stat Mech - Spring 2006 Class 14
27
Maxwell Velocity Distribution
Consider a gas at equilibrium. The number of molecules in any range of velocity does not change. Collisions cause individual molecules to change velocity, but the distribution does not change. For every collision that changes the distribution, there must be one that changes it back.
vdvvvdvF
and between number )( :Let
Thermo & Stat Mech - Spring 2006 Class 14
28
Molecular Collisions
Thermo & Stat Mech - Spring 2006 Class 14
29
Molecular Collisions
vvAevF
vvvv
vFvFvFvF
vFvF
vFvF
vvvv
)( :Then
''
:so elastic are Collisions
)'()'()()(
)'()'(collision reverse ofy Probabilit
)()(collision asuch ofy Probabilit
''
22
21
22
21
2121
21
21
2121
Thermo & Stat Mech - Spring 2006 Class 14
30
Maxwell Distribution
0
4
0
221
23
0
2
0
2
2
2
2
43
)(
4)(
4)(
to from speeds ofNumber )(
dvevAm
NkT
dvvNvmNkT
dvevAdvvNN
dvvAedvvN
dvvvdvvN
v
v
v
Thermo & Stat Mech - Spring 2006 Class 14
31
Evaluation of Constants
2
3
4
83
3
8
3
4
23
25
2
4
252
0
44
230
22
2
2
I
I
m
kT
IdvevI
dvevI
v
v
Thermo & Stat Mech - Spring 2006 Class 14
32
Maxwell Distribution
dvevkT
m
N
dvvNdvvf
dvevkT
mNdvvN
kT
mNNA
kT
m
kT
mv
kT
mv
22
23
22
23
2323
2
2
24
)()(
24)(
22
Thermo & Stat Mech - Spring 2006 Class 14
33
Maxwell Distribution
Maxwell Distribution
0
0.0005
0.001
0.0015
0.002
0.0025
0 200 400 600 800 1000 1200 1400
v (m/s)
Pro
bab
ility
Thermo & Stat Mech - Spring 2006 Class 14
34
Some Molecular Speeds
ProbableMost 2
3
3)(
8)(
max
2
0
22
0
m
kTv
m
kTvv
m
kTdvvfvv
m
kTdvvvfv
rms