1. lecture06 ch. 6 overview

8/13/2019 1. Lecture06 Ch. 6 Overview

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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!

3


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2. The Canonical Ensemble:

Systemswith a fixed number Nof moleculesIn equilibrium with a Heat Reservoir(Heat Bath).

4


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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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Partition Function:

Z = nexp (-En), = 1/(kT)Mean Energy:

=(ln Z)/= - (1/Z)Z/

C i


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Partition Function:


=(ln Z)/= - (1/Z)Z/Mean Squared Energy:

E2 = rprEr2/rpr = (1/Z)2Z/2.

C i l E bl


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Partition Function:



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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Partition Function:



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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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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(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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CV= k2(E)2 = (E)2/kBT2so that:

(E)2 = kBT2CV

C i l E bl


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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!

1. lecture06 ch. 6 overview

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