1. lecture06 ch. 6 overview
TRANSCRIPT
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Chapter 6:Basic Methods & Resultsof Statistical Mechanics
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Key Concepts In Statistical Mechanics
Idea:Macroscopic properties are a
thermal average of microscopic properties. Replace the system with a set of systems
"identical" to the first and average over all of
the systems. We call the set of systemsThe Statistical Ensemble. Identical Systemsmeans that they are all
in the same thermodynamic state. To do any calculations we have to first
Choose an Ensemble!
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The Most Common Statistical Ensembles:
1. The Micro-Canonical Ensemble:
Isolated Systems:Constant Energy E.Nothing happens! Not I nteresting!
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The Most Common Statistical Ensembles:
1. The Micro-Canonical Ensemble:
Isolated Systems:Constant Energy E.Nothing happens! Not I nteresting!
2. The Canonical Ensemble:
Systemswith a fixed number Nof moleculesIn equilibrium with a Heat Reservoir(Heat Bath).
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The Most Common Statistical Ensembles:
1. The Micro-Canonical Ensemble:
Isolated Systems:Constant Energy E.Nothing happens! Not I nteresting!
2. The Canonical Ensemble:Systemswith a fixed number Nof molecules
In equilibrium with a Heat Reservoir(Heat Bath).
3. The Grand Canonical Ensemble:Systemsin equilibrium with a Heat Bath
which is also a Source of Molecules.
Their chemical potential is fixed.
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All Thermodynamic PropertiesCan Be
Calculated With Any Ensemble
Choose the most convenient one for a particular problem.For Gases: PVTproperties
use
The Canonical Ensemble
For Systems which Exchange Particles:
Such asVapor-Liquid Equilibriumuse
The Grand Canonical Ensemble
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J. Willard Gibbs was the first to show that
An Ensemble Average is Equal to aThermodynamic Average:
That is, for a given property F,
The Thermodynamic Averagecan be formally expressed as:
F nFnPnFnValue of F in state (configuration) nPnProbability of the system being in state
(configuration) n.
Properties of The Canonical
& Grand Canonical Ensembles
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Canonical Ensemble Probabilities
p g eQ
n n
U
canonN
n
QNcanon Canonical Partition FunctiongnDegeneracy of state n
Q g ecanonN nn
Un
Note that most texts use the notationZfor the partition function!
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Grand Canonical Ensemble Probabilities:
p
g e
Qnn
E
grand
n
E U Nn n n
Q g egrand nn
E n
Qgrand Grand Canonical Partition Functionor
Grand Partition Functiong
nDegeneracy of state n, Chemical PotentialNote that most texts use the notation
ZGfor the Grand Partition Function!
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Partition Functions I f the volume,V, the temperatureT, & the energy
levelsEn, of a system are known, in principle
The Partition FunctionZcan be calculated.
I f the partition function Zis known, it can be used
To CalculateAll Thermodynamic Properties.
So, in this way,
Statistical Mechanicsprovides a direct l inkbetween
Microscopic Quantum Mechanics&
Classical Macroscopic Thermodynamics.
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Canonical Ensemble Partition Function Z
Starting from the fundamental postulate of equala pr ior i probabil i ties,the following are obtained:
i . ALL RESULTSof Classical Thermodynamics,plus their statistical underpinnings;
i i. A MEANS OF CALCULATINGthethermodynamic variables(E, H, F, G, S) from a
single statistical parameter, the partition function Z(or Q),which may be obtained from the energy-levelsof a quantum system.
The partition function for a quantum system in
equilibrium with a heat reservoir is defined asW
Where iis the energy of the ithstate.
Z iexp(- i/kBT)
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Partition Function for a QuantumSystem in Contact with a Heat Reservoir:
,F
i= Energy of the ith state.
The connection to the macroscopic entropyfunction S is through the microscopic parameter, which, as we already know, is the number ofmicrostates in a given macrostate.
The connection between them, as discussed inprevious chapters, is
Z iexp(- i/kBT)
S = kBln .
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Relationship of Z to Macroscopic Parameters
Summary for the Canonical
Ensemble Partition Function Z:(Derivations are in the book!)
Internal Energy: E = - (lnZ)/ = [2(lnZ)/2]= 1/(kBT), kB=Boltzmanns constantt.
Entropy: S = kB+ kBlnZ
An important, frequently used result!
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Summary for the Canonical Ensemble
Partition Function Z:
Helmholtz Free EnergyF = ETS =(kBT)lnZ
and
dF = S dTPdV,soS =(F/T)V, P =(F/V)T
Gibbs Free Energy
G = F + PV = PVkBT lnZ.Enthalpy
H = E + PV = PV (lnZ)/
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Canonical Ensemble:Heat Capacity & Other Properties
Partition Function:
Z = nexp (-En), = 1/(kT)
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Canonical Ensemble:Heat Capacity & Other Properties
Partition Function:
Z = nexp (-En), = 1/(kT)Mean Energy:
=(ln Z)/= - (1/Z)Z/
C i
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Canonical Ensemble:Heat Capacity & Other Properties
Partition Function:
Z = nexp (-En), = 1/(kT)Mean Energy:
=(ln Z)/= - (1/Z)Z/Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2.
C i l E bl
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Canonical Ensemble:Heat Capacity & Other Properties
Partition Function:
Z = nexp (-En), = 1/(kT)Mean Energy:
=(ln Z)/= - (1/Z)Z/Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2.n
th
Moment:En = rprErn/rpr = (-1)n(1/Z) nZ/n
C i l E bl
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Canonical Ensemble:Heat Capacity & Other Properties
Partition Function:
Z = nexp (-En), = 1/(kT)Mean Energy:
=(ln Z)/= - (1/Z)Z/Mean Squared Energy:
E2 = rprEr2/rpr = (1/Z)2Z/2.n
th
Moment:En = rprErn/rpr = (-1)n(1/Z) nZ/n
Mean Square Deviation:
(E)2 = E2 - ()
2
= 2lnZ/2 = - /.
C i l E bl
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Canonical Ensemble:Constant Volume Heat Capacity
CV= /T = (/)(d/dT) = - k2/
C i l E bl
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Canonical Ensemble:Constant Volume Heat Capacity
CV= /T = (/)(d/dT) = - k2/using results for the Mean Square Deviation:
(E)2 = E2 - ()2= 2lnZ/2 = - /
C i l E bl
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Canonical Ensemble:Constant Volume Heat Capacity
CV= /T = (/)(d/dT) = - k2/using results for the Mean Square Deviation:
(E)2 = E2 - ()2= 2lnZ/2 = - /CV can be re-written as:
CV= k2(E)2 = (E)2/kBT2
C i l E bl
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Canonical Ensemble:Constant Volume Heat Capacity
CV= /T = (/)(d/dT) = - k2/using results for the Mean Square Deviation:
(E)2 = E2 - ()2= 2lnZ/2 = - /CV can be re-written as:
CV= k2(E)2 = (E)2/kBT2so that:
(E)2 = kBT2CV
C i l E bl
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Canonical Ensemble:Constant Volume Heat Capacity
CV= /T = (/)(d/dT) = - k2/using results for the Mean Square Deviation:
(E)2 = E2 - ()2= 2lnZ/2 = - /CV can be re-written as:
CV= k2(E)2 = (E)2/kBT2so that:
(E)2 = kBT2CV
Note that, since (E)2 0(i) CV 0 and(ii) /T 0.
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Ensembles in Classical
Statistical Mechanics
As weve seen, classical phase space for asystem with fdegrees of freedom is fgeneralized coordinates & fgeneralizedmomenta (q
i,p
i).
The classical mechanics problem is done inthe Hamiltonian formulation with aHamiltonian energy function H(q,p).
There may also be a few constants ofmotion such as
energy, number of particles, volume, ...
Th C i l Di ib i i
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The Canonical Distribution inClassical Statistical Mechanics
The Partition Functionhas the form:
Z d3r1d3r2d3rN d3p1d3p2d3pN e(-E/kT)
A 6N Dimensional I ntegral ! This assumes that we have already solved the
classical mechanics problemfor each particle in the
system so that we know the total energy Efor the Nparticles as a function of al lpositions ri& momenta pi.
E E(r1,r2,r3,rN,p1,p2,p3,pN)
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CLASSICAL
Statistical Mechanics:
Let A any measurable, macroscopicquantity. The thermodynamic average of
A .This is what is measured. Useprobability theory to calculate :
P(E) e[-E/(kBT)]/Z (A)d3r1d3r2d3rN d3p1d3p2d3pNP(E)
Another 6N Dimensional I ntegral!