1.Fluctuations are important because the number of particles in a system is much less than...
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1. Fluctuations are important because the number of particles in a system is much less than Avogadro’s number;
2. Importance of the surface properties. Thermodynamic quantities no longer scale with the number of atoms in a system becuase the energy associated with the surface may be a significant fraction of the total.
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Nanosystems generally contain too many atoms to be thought Nanosystems generally contain too many atoms to be thought of as simple mechanical systems, but too few to be described by of as simple mechanical systems, but too few to be described by bulk properties.bulk properties.
Equilibrium thermodynamic properties are well defined because fluctuations are negligible in large (N=1023) systems.
Microscopic view of Microscopic view of bulk propertiesbulk properties
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P(1,2)=P(1)P(2).
E(1,2)=E(1)+E(2.).
)](Eexp[Const)(P 11
)](Eexp[Const)(P 22
,
TkB
1
kB = 1.381·10-23 J·K-1 = 8.62·10-5 eV·K-1
Normalized Boltzmann distribution
The Boltzmann DistributionThe Boltzmann Distribution
Copyright (c) Stuart Lindsay 2008
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Values for kTValues for kT
• kBT at room temperature (300K) = 4.14 10-21J
• = 25 meV (Much smaller than most bonds)
• = 0.6 Kcal·mol-1
• kBT at room temperature in terms of force· distance = 4.14 pN·nm
- molecular motors produce forces ca. ten times more over nm distances
Copyright (c) Stuart Lindsay 2008
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Normalized Boltzmann distributionNormalized Boltzmann distribution
)exp(1
)(Tk
E
ZrP
B
r
r B
r
Tk
EZ exp partition functionpartition function
In case of degeneracy g(r):
)exp()(
)(Tk
E
Z
rgEP
B
rr
The partition function enumerates all the states as a function of energy, so all the equilibrium properties of a system can be derived from it.
Copyright (c) Stuart Lindsay 2008
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• Entropy is proportional to the number of ways (statistical weight, ) a given macrostate r can occur.
)(ln)( rkrS B
Copyright (c) Stuart Lindsay 2008
The degeneracy of a state leads naturally to a statistical definition of EntropyEntropy:
In terms of the probability of the rth state:
r
B rprpkS )(ln)(
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The Equipartition TheoremThe Equipartition Theorem
• Average thermal energy of e.g., a harmonic oscillator
• For a classical system (all energies allowed) replace the Boltzmann sum with an integral and calculate the product of E and p(E), e.g. for potential energy:
22
2
1
2
1xxmE
dx
Tk
x
dxTk
xx
dxxEPxx
B
B
)2
exp(
2exp
2
1
))((2
1
2
12
22
22
Copyright (c) Stuart Lindsay 2008
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• With a change of variables and a standard integral we find
• Similarly
• The thermal average of any quantity that appears in the classical Hamiltonian as a quadratic term is
Tkx B2
1
2
1 2
Tkxm B2
1
2
1 2
TkB2
1 Equipartition Equipartition theoremtheorem
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The Equipartition theorem assumes that all degrees of freedom are in equilibrium with the heat bath and are independent.
However, it takes coupling between the degrees of freedom to ‘spread’ the thermal energy out evenly and this requires a non linear response.
Ex. It takes an anharmonic potential to couple vibrational and translational degres of freedom (V-T energy transfer).
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Thermodynamics and Statistical mechanicsThermodynamics and Statistical mechanics
• Thermodynamic potentials (“Free Energies”) can be minimized to obtain the equilibrium properties of a system.
Copyright (c) Stuart Lindsay 2008
0S
TSEA
TSPVEG
0A
0G
Isolated system
Closed system (V,T)
Closed system (p,T)
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Thermodynamic potentials in terms of Thermodynamic potentials in terms of partition functionspartition functions
r B
r
Tk
EZ exp
r
B )r(pln)r(pkS
)Tk
Eexp(
Z)r(p
B
r1
T
EZkNVTS B ln),,(
TSEA
)N,V,T(ZlnTk)N,V,T(A B
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Grand Canonical ensembleGrand Canonical ensemble
Each system is enclosed in a container whose walls are both heat conducting and permeable to the passage of molecules.→ transport of matter allowed, N variable
V,T,μ V,T,μ V,T,μ
V,T,μ V,T,μ V,T,μ
V,T,μ V,T,μ V,T,μ
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N r
Nra A
aaNrNr = number of systems in the ensemble that contain N molecules
and are in the state r.The set of occupation numbers {aNr} is a distribution.
Each possible distribution must satisfy the balance equations:
Number of systems in the ensemble
N r
NrNr Ea Ε Total energy of the ensemble
N r
Nr Na N Total number of molecules
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For any possible distribution, the number of states is given by:
N r NrNr !a
!})a({W
A
The distribution that maximizes W is:
N)V(ENr eee}*a{ Nr
N rN)V(E ee
eNr
AkT
1
kT
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NTSPVEG
is the chemical potentialchemical potential
NrNr
Nr
ENexp}*a{P
A
Nr
NrENVTZ exp),,(
Gibbs DistributionGibbs Distribution
Grand Partition Grand Partition functionfunction
),,(ln VTTkG B
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Nr
NrENexp),V,T(Z
Summing over r states, it is possible to write:
N
N
N
kT
N
)T,V,N(Qe)T,V,N(Q),V,T(Z
Canonical partition function
kTe
lnkTactivityactivity
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Ideal Gas: Z for one free particle (N=1)Ideal Gas: Z for one free particle (N=1)
222222
22 zyx kkkmm
E k
L
nk
L
nk
L
nk z
zy
yx
x
2,
2,
2
zyx nnn
zyx nnnEVTZ,,
),,(exp)1,,(
2
2
3
2
28
4
dkVkdkVk
dn
For large systems:
dkTmk
kk
VVTZ
B
0
222
2 2exp
2)1,,(
VTmk
VTZ B2
3
22)1,,(
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2
3
22
TmkB
Quantum concentration (one particle per
wavelength3)
The de Broglie wavelength for a free particle is:
2
12
22
Tmkk Bz,y,x
So, one particle occupies a quantum volume of about λ3:
2
32
3 2
TmkB
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Ex. Quantum volume for a free electron at 300K
3326
3
2
2131
683
22
791097
1014410119
1011312 2
nmm.
..
.
TmkB
NN
rr VTZ
NE
NNVTZ )1,,(
!
1)exp(
!
1),,(
For N non-interacting particles:
A sphere of a radius of 2.7nm!A sphere of a radius of 2.7nm!
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Quantum statisticsQuantum statistics
N
N
N
kT
N
)T,V,N(Qe)T,V,N(Q),V,T(Z
Expliciting Q(N,V,T) (i.e. energy distribution):
}n{
n
j
E
k
i iij ee)T,V,N(Q
Ej(N,V) = energy states available to a system containing N moleculesεk= molecular quantum statesnk= number of molecules in the kth molecular state when the system energy is Ej.
k
kkj nE k
knN
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N
n
}n{
n
N }n{
nN i ii
k
i i
k
i ii ee),V,T(Z
max, max,
iii
k
i
n
n
n
n i
nn
N }n{ i
e...e1
1
2
20 0
This last passage originates from the fact that we are summing over all values of N and that nk ranges over all possible values.
i
n
n
nn
n
n
n
nn max,i
i
ii
max, max,
e......ee00 0
1
1
2
2
2211
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Fermi-Dirac statistics: nFermi-Dirac statistics: nii=0 or 1, n=0 or 1, ni,maxi,max=1=1
i
FDieZ 1
iT,VT,V
FD i
i
e
eZlnZlnkTN
1
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Bose-Einstein statistics: nBose-Einstein statistics: nii=0, 1, 2,… n=0, 1, 2,… nmaxmax==∞∞
ii n
n
BEi
i
ii eeZ1
0
1
Where we used:
j
j
j )x(x0
11
iT,VT,V
BE i
i
e
eZlnZlnkTN
1
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i
FDieZ 1
i
BEieZ
11
iFD i
i
e
eN
1
iBE i
i
e
eN
1
k
k
kkk k
k
e
enNE
1
k
kelnkTpV 1
+ = FD+ = FD
- = BE- = BE
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Classical limit (Classical limit (λ→λ→0):0):
At the classical limit (high temperatures or low density) the number of available molecular quantum states is much greater than the number of particles.
The average number of molecules in any state is very small (nk→0, λ→0).
Thermodynamically:
(T fixed)V
N 0 fixed)
V
N(T
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iMBBEorFD
ieNNlim
0
Maxwell-Boltzmann distribution
ieni
Summing over i:
ii i
ien qeNi
i
q
e
N
n ii
i i
ieE
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Quantum gassesQuantum gasses
.....3,2,1,0rnBosonsBosons
1,0rnFermionsFermionsNn
rr
...,
221121
21
............exp),,(nn
iii nnnnnnVTZ
Using the Grand partition function:
Z
nnnnnnnnPP iii
Nr
............exp...),( 221121
21
With the Gibbs distribution
We consider just an ideal gas (non-interacting particles) now subject to restrictions on how states are counted:
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Writing numerator and denominator as products:
i
ii
i Z
nnnp
exp....),(
121
Single particle distribution
Fermi-Dirac (FD) statisticsFermi-Dirac (FD) statistics: ni=1 or 0
iiZ exp1
in i
iiiiii exp
expexp)n(pnn
1
10
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1exp
1
iin
Fermi Dirac thermal Fermi Dirac thermal average occupationaverage occupation
The chemical potential at The chemical potential at T=0 is called the Fermi T=0 is called the Fermi energy.energy.
The electronic properties of most conductors are dominated by quantum statistics.
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Ex. Fermi energy of Na is 3.24 eV.
For εi=μ:
For metals is several eVs!!
2
1in
eV.)K(kT 0250300 K,.
.).n(T i 80038
0250
24350
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• Summing Zi from n = 0 to :
1exp
1
iin <i
Copyright (c) Stuart Lindsay 2008
Bose-Einstein (BE) statisticsBose-Einstein (BE) statistics: ni=0, 1, 2, 3….
As ε approaches μ in the BE distribution, the occupation number approaches infinity, i.e. bosons condense into one quantum state at very low temperatures (Bose condensation).
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Phonons are bosons with no chemical potential (μ=0), so that the occupation number goes to zero as temperature approaches zero.
μ=0