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Statistical Mechanics and
Material Properties
By Kunio TAKAHASHI
Tokyo Institute of Technology, Tokyo 152-8552, JAPAN
Phone/Fax +81-3-5734-3915 [email protected]
http://www.ide.titech.ac.jp/~KT-lab/ © Only for the students in TAKAHASHI’ s class. All right reserved. Do not distribute to others. 2013
Statistical Mechanics
and Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C )
1st low : Energy conservation
2nd low : Free energy decreases
3rd low : Entropy -> 0 at 0K
Interface essential to utilize thermodynamics
Grave-stone of Boltzmann…
Statistical Mechanics
Ergodic theorem : time average = space average, (in equilibrium)
You may have learnt Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C ) Everything is
1st low : Energy conservation derived from these
2nd low : Free energy decreases principles…
3rd low : Entropy -> 0 at 0K But !
Definitions
Temperature
Free energy
Entropy
Equilibrium Could you understand them as you could do in kinetics ?
Please remember when you have heard them first.
How they were defined ?
Definitions
Temperature
C : water Empirical temp.
F : salt water
Temp. in science ( physics or kinetics ) is same as above ?
Free energy
Helmholtz
Gibbs
Free energy is same as the energy in kinetics ?
Why they need to be defined as above ?
Entropy
What is this ?
Equilibrium
What is the definition ?
TSUF
TSPVUG
You may have learnt Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C ) Everything is
1st low : Energy conservation derived from these
2nd low : Free energy decreases principles…
3rd low : Entropy -> 0 at 0K But !
Definitions
Temperature
Free energy
Entropy
Equilibrium
Understanding of them ( = statistical mechanics )
lead you to utilize thermodynamics
Material properties
Let’s define them step by step !
Temperature ( first trial )
Example : Ideal gas (1)
Assumption
Number of molecule : in volume :
Velocity of molecule :
probability :
Momentum change = Pressure on area A
Comparison with eq. of state :
Law of equi-partition of energy
N V
),,( wvuc
wfvfufF c
PAdtduAudtV
Nufmu
0
)(2
22
0
)()(2 uNmduufumNduNuufmuPV
2222 wvuc
222 wvu
22
3
1cu
22
2
1
3
2
3
1mcNcNmPV
NkTPV
kTmc2
3
2
1 2
kTmwmvmu2
1
2
1
2
1
2
1 222
Example : Ideal gas (2)
Probability function
taking log :
diff. with u :
Const. independent of c,u,v,w :
because it is probability…
Average kinetic energy (expectated value) :
using therefor
eq. of state
Maxwell distribution :
du
fd
c
u
dc
Fd
du
dc
dc
Fd lnlnln
wfvfufF c wfvfufcF lnlnlnln
du
fd
du
Fd lnln
dw
fd
wdv
fd
vdu
fd
udc
Fd
c
ln1ln1ln1ln1
2
2
1
2)(
u
euf
dueumduemuduufmumuuu 22
2
1
22
1
222
22
1
22
1)(
2
1
2
1
2
2
22
13
mm kT
2
1
kT
m
kTum
ekT
muf
2
2
2)(
kTwvum
ekT
mwvuF
222
22
3
2),,(
2222 wvuc
Example : Ideal gas (3) Maxwell distribution
Probability function
Only for Ideal gas ?
< Remember >
Empirical knowledge is used.
Eq. of state for ideal gas
Relation between Temp. and average Kinetic energy
This is the definition of Temp. !?
kTum
ekT
muf
2
2
2)(
kTwvum
ekT
mwvuF
222
22
3
2),,(
kTe energy kinetic
Let’s consider more general case !
You may have learnt Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C ) Everything is
1st low : Energy conservation derived from these
2nd low : Free energy decreases principles.
3rd low : Entropy -> 0 at 0K
Definitions
Temperature
Free energy
Entropy
equilibrium
etc…
Temperature is kinetic energy ?
( at equilibrium )
Temperature and equilibrium
(general treatment)
Canonical ensemble (1)
Assumption 1 ( a model of system )
Num. of elements with energy is .
Total num. of elements (particle, molecule, etc…) :
Total energy :
…, therefore
is determined by the state. (density of state)
States with same are indistinguishable.
Assumption 2 ( most important !! )
The state is observed so as to maximize
the way of shuffling of the density of state
inN
iienE
inie
)( ii en
)( ii en
)( ii en
Why !?
N and E are given.
Canonical ensemble (2)
the way of shuffling of the state
To maximize W
using the definition of W
using the Stirling’s formula
…,
!!!
!
321 nnn
NW
maxln 21
EenNnWL
i
ii
i
i
EenNnnNL
i
ii
i
i
i
i 21!!ln
xxxx ln!ln
maxlnln 21
EenNnnnnNNNL
iii
ii
iiii
Lagrange multiplier
0
ii
L
n
L
Canonical ensemble (3)
Here, N and E are constant, therefore
Using
Comparing with Maxwell distribution
: Maxwell-Boltzmann distribution
is not limited to be the kinetic energy.
kinetic energy, potential energy, etc…
011ln 21
ii
i
enn
L
maxlnln 21
EenNnnnnNNNL
i
ii
i
i
i
iii
iii een 2121 expexp
inN
kTe
kTeN
ni
ii
i
expexp
1
kTum
ekT
muf
2
2
2)(
ii
ii
i eeeN
n22
2
expexpexp
1
ie
Canonical ensemble (3)
Here, N and E are constant, therefore
Using
Comparing with Maxwell distribution
: Maxwell-Boltzmann distribution
is not limited to be the kinetic energy.
kinetic energy, potential energy, etc…
011ln 21
ii
i
enn
L
maxlnln 21
EenNnnnnNNNL
i
ii
i
i
i
iii
iii een 2121 expexp
inN
kTe
kTeN
ni
ii
i
expexp
1
kTum
ekT
muf
2
2
2)(
ii
ii
i eeeN
n22
2
expexpexp
1
ie
Temperature is energy ! ( used in physics )
in the most probable situation.
Canonical ensemble (4)
Probability of the state of energy E :
Probability of the state whose energy > E :
“Assumption 2”
The state is observed so as to maximize
the way of shuffling of the density of state
Equilibrium !
kTE exp
kTE exp
)( ii en
The most probable situation
Equilibrium is,,, the most probable state !!
You may have learnt Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C ) Everything is
1st low : Energy conservation derived from these
2nd low : Free energy decreases principles.
3rd low : Entropy -> 0 at 0K
Definitions
Temperature
Free energy
Entropy
equilibrium
etc…
Temperature is energy ! ( at equilibrium )
Equilibrium is the most probable state.
Definitions
Temperature
Empirical temp. C : water defined one or two temp.
F : salt water
T. can be shifted and scaled, so as proportional to energy !
Free energy
Helmholtz
Gibbs
Free energy is same as the energy in physics ?
Why they need to be defined as above ?
Entropy
What is this ?
Equilibrium
The most probable state ! So, we can observe always…
TSUF
TSPVUG
Free energy and entropy
Prediction of phenomena
After the phenomena
Decreased Potential Energy Not changed
Not changed Total energy Not changed
? ( Not changed ) Free Energy Decreased
Physicists ( kinetics )
as to decrease Potential Energy
mgh
h
Chemists ( chemistry )
as to decrease Free Energy
Ideal gases A B
mixed A,B A,B
m
Chemists wanted to introduce something like potential energy, to predict the phenomena.
Definition of free energy :
Energy that can be converted into a work in a process of
constant temperature and volume (Helmholtz).
constant temperature and pressure (Gibbs).
( Two of three T, P, &V determine its state. )
TSUF
TSPVUG
The something; related to the work or energy, used in thermodynamic technology,
( ex. steam engine, chemical reaction, etc… )
Chemists wanted to introduce something like potential energy, to predict the phenomena.
TSPVUG
Energy or work in kinetics What’s this ! ??
TSPVUG What’s this ! ??
Definition of free energy :
Energy that can be converted into a work in a process
A B
A,B A,B
The TS term increases in the mixing process,
even if kinetic and potential energy never changes.
The mixing process never goes back.
Increase in TS term Decrease of G
= Increase of the usable energy
= Degrade of energy
Definition of free energy :
Energy that can be converted into a work in a process
There exists a part of energy, which we can not take as a work.
It is proportional to the temperature.
S is defined
as the entropy.
This is the definition of entropy in thermodynamics.
Degrade of the energy exits…
Empirical knowledge of chemists
TSPVUG
The degrade is proportional to the temp…
What’s the entropy ! ??
What’s this ! ??
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
A work is taken out !
P, V, T -> 1/2P, 2V, T
No work is taken out !
What happened ?
The degrade is TS.
The degrade of energy exists in lower process
TSPVUG
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
A work is taken out !
P, V, T -> 1/2P, 2V, T
No work is taken out !
Change in S is ?
NkTPV
2ln
22
NkTdVV
NkTPdV
V
V
V
V
2lnNkS
The degrade of energy exists in lower process
TSPVUG
Statistical Mechanics
and Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C )
1st low : Energy conservation
2nd low : Free energy decreases
3rd low : Entropy -> 0 at 0K
Boltzmann's entropy formula :
Grave-stone of Boltzmann…
Statistical Mechanics
Ergodic theorem : time average = space average, (in equilibrium)
WkS ln
Boltzmann's entropy formula & Free energy
W : Number of state
v : small control volume
V/v : num. of the c. volume 2V/v : num. of the c. volume
Num. of the way to shuffle to distribute the atoms to the each c. volume
P, V, T -> 1/2P, 2V, T
No work is taken out !
WkS ln
NvVW )/(0 NN WvVW 2)/2( 0
00 lnWkS
2lnNkS
2ln
2ln
0
0
0
NkS
Wk
SSS
N
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
A work is taken out !
P, V, T -> 1/2P, 2V, T
No work is taken out !
NkTPV
2ln
22
NkTdVV
NkTPdV
V
V
V
V
00 lnWkS
2lnNkS
2ln
2ln
0
0
0
NkS
Wk
SSS
N
If the entropy is defined as this, it becomes consistent.
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
P, V, T -> 1/2P, 2V, T
Internal E. U0 -> U0
Free E. G0=U0+PV-TS0 -> G0-TS
Entropy S0 -> S0 +S Entropy change
must be
Energy taken out :
Definition of entropy :
2ln
22
NkTdVV
NkTPdV
V
V
V
V
2lnNkS
WkS ln
2ln2ln
ln
000
00
NkSWkSS
WkS
N
Only entropy term must be changed as
No change.
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
A work is taken out !
P, V, T -> 1/2P, 2V, T
No work is taken out !
NkTPV
2ln
22
NkTdVV
NkTPdV
V
V
V
V
We can take out the infinite work ! ???... although internal energy must be finite.
2ln2ln
ln
000
00
NkSWkSS
WkS
N
W depends on volume ! ???... Energy never depend on volume…
Check of understanding
Boltzmann's entropy formula & Free energy
Free energy of ideal gas
A work is taken out !
P, V, T -> 1/2P, 2V, T
No work is taken out !
The energy is given from outside ! so as to keep T const.
Entropy is not energy, It is a index of randomness.
Introduction of Free energy & entropy & temperature
Definition of Free Energy :
Energy that can be converted into a work.
( There exist the energy part we can not take out. )
Definition of Entropy Definition of Temperature
Related to the randomness
WkS ln
Degrade of the energy…
TSPVUG
Internal energy = Kinetic energy
we can take out from outside ( by heat transfer… )
Temperature
Empirical temp. C : water defined one or two temp.
F : salt water
T. can be shifted and scaled, so as proportional to energy !
Free energy
Helmholtz
Gibbs
Free energy is energy that can be converted into work.
If defined as above, it become consistent.
Entropy
corresponding to number of states, i.e. randomness.
Equilibrium
Most probable state ! So, we can observe always…
These are the interface between
Thermodynamics and statistical mechanics
TSUF
TSPVUG
WkS ln
TSPVUG
Statistical Mechanics
and Thermodynamics
Thermodynamics
( 0th low : A.eq.B and B.eq.C -> A.eq.C )
1st low : Energy conservation + …
2nd low : Free energy decreases
3rd low : Entropy -> 0 at 0K
Boltzmann's entropy formula :
Grave-stone of Boltzmann…
Statistical Mechanics
Ergodic theorem : time average = space average, (in equilibrium)
WkS ln
Based of statistical mechanics,
Thermodynamics can be utilized !
Activation energy and Arrhenius plots
and rate-determining step
chemical reaction
diffusion coefficient
others… (evaporation, adsorption, etc…)
Arrhenius plots is to know the mechanism of rate-determining step.
Specific heat capacity (1: ideal gasses )
Remember thermodynamics ! (How have you learnt ?)
why ? why ?
Definition Heat required to increase unit temp. - under constant volume fixed - under constant pressure
Mayer’s relation
p
pT
H
NC
1
V
VT
U
NC
1
kCC Vp TSPVUG
TNCUGQ V
TNCVPUSTVPUGQ P
TNCVPTNC PV
Specific heat capacity (2: metals)
Einstein’s model
Dulong-Petit’s Low What is the U ?
Debye model
- Ionic cores
- Electrons = negligible
Quantum statistical mechanics
kTkTkTeeU 32
13
2
13pk
kT
Q3
Phase diagram
Stability of phases
Phase rule
Phase diagram
Calculation of Phase Diagram (CALPHAD)