Chapter 6: Basic Methods & Resultsof Statistical Mechanics

Key Concepts In Statistical MechanicsIdea: 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 systems The Statistical Ensemble. Identical Systems means that they are all in the same thermodynamic state. To do any calculations we have to first Choose an Ensemble!

The Most Common Statistical Ensembles:1. The Micro-Canonical Ensemble:Isolated Systems: Constant Energy E.Nothing happens! Not Interesting! *

The Most Common Statistical Ensembles:1. The Micro-Canonical Ensemble:Isolated Systems: Constant Energy E.Nothing happens! Not Interesting! 2. The Canonical Ensemble:Systems with a fixed number N of moleculesIn equilibrium with a Heat Reservoir (Heat Bath).*

The Most Common Statistical Ensembles:1. The Micro-Canonical Ensemble:Isolated Systems: Constant Energy E.Nothing happens! Not Interesting! 2. The Canonical Ensemble:Systems with a fixed number N of moleculesIn equilibrium with a Heat Reservoir (Heat Bath).3. The Grand Canonical Ensemble:Systems in equilibrium with a Heat Bathwhich is also a Source of Molecules.Their chemical potential is fixed.

All Thermodynamic Properties Can Be Calculated With Any EnsembleChoose the most convenient one for a particular problem.For Gases: PVT propertiesuseThe Canonical Ensemble

For Systems which Exchange Particles:Such as Vapor-Liquid EquilibriumuseThe Grand Canonical Ensemble

J. Willard Gibbs was the first to show thatAn Ensemble Average is Equal to a Thermodynamic Average:That is, for a given property F,The Thermodynamic Averagecan be formally expressed as: F nFnPnFn Value of F in state (configuration) nPn Probability of the system being in state(configuration) n. Properties of The Canonical & Grand Canonical Ensembles

Canonical Ensemble ProbabilitiesQNcanon Canonical Partition Functiongn Degeneracy of state nNote that most texts use the notation Z for the partition function!

Grand Canonical Ensemble Probabilities:Qgrand Grand Canonical Partition Function orGrand Partition Functiongn Degeneracy of state n, Chemical PotentialNote that most texts use the notation ZG for the Grand Partition Function!

Partition FunctionsIf the volume, V, the temperature T, & the energy levels En, of a system are known, in principleThe Partition Function Zcan be calculated. If the partition function Z is known, it can be used To CalculateAll Thermodynamic Properties.So, in this way,Statistical Mechanicsprovides a direct link betweenMicroscopic Quantum Mechanics &Classical Macroscopic Thermodynamics.

Canonical Ensemble Partition Function ZStarting from the fundamental postulate of equal a priori probabilities, the following are obtained:ALL RESULTS of Classical Thermodynamics, plus their statistical underpinnings;A MEANS OF CALCULATING the thermodynamic variables (E, H, F, G, S ) from a single statistical parameter, the partition function Z (or Q), which may be obtained from the energy-levels of a quantum system.The partition function for a quantum system in equilibrium with a heat reservoir is defined as W

Where i is the energy of the ith state.Z i exp(- i/kBT)

*Partition Function for a Quantum System in Contact with a Heat Reservoir:,

Fi = Energy of the ith state.The connection to the macroscopic entropy function S is through the microscopic parameter , which, as we already know, is the number of microstates in a given macrostate.The connection between them, as discussed in previous chapters, isZ i exp(- i/kBT)S = kBln .

Relationship of Z to Macroscopic ParametersSummary 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 + kBlnZAn important, frequently used result!

Summary for the Canonical Ensemble Partition Function Z:Helmholtz Free Energy F = E TS = (kBT)lnZ and dF = S dT PdV, soS = (F/T)V, P = (F/V)T Gibbs Free EnergyG = F + PV = PV kBT lnZ.EnthalpyH = E + PV = PV (lnZ)/

Canonical Ensemble:Heat Capacity & Other Properties Partition Function:Z = n exp (-En), = 1/(kT)

Canonical Ensemble:Heat Capacity & Other Properties Partition Function:Z = n exp (-En), = 1/(kT)Mean Energy: = (ln Z)/ = - (1/Z)Z/

Canonical Ensemble:Heat Capacity & Other Properties Partition Function:Z = n exp (-En), = 1/(kT)Mean Energy: = (ln Z)/ = - (1/Z)Z/ Mean Squared Energy:E2 = rprEr2/rpr = (1/Z)2Z/2.

Canonical Ensemble:Heat Capacity & Other Properties Partition Function:Z = n exp (-En), = 1/(kT)Mean Energy: = (ln Z)/ = - (1/Z)Z/ Mean Squared Energy:E2 = rprEr2/rpr = (1/Z)2Z/2.nth Moment:En = rprErn/rpr = (-1)n(1/Z) nZ/n

Canonical Ensemble:Heat Capacity & Other Properties Partition Function:Z = n exp (-En), = 1/(kT)Mean Energy: = (ln Z)/ = - (1/Z)Z/ Mean Squared Energy:E2 = rprEr2/rpr = (1/Z)2Z/2.nth Moment:En = rprErn/rpr = (-1)n(1/Z) nZ/nMean Square Deviation:(E)2 = E2 - ()2 = 2lnZ/2 = - / .

Canonical Ensemble:Constant Volume Heat CapacityCV = /T = (/)(d/dT) = - k2/

Canonical Ensemble:Constant Volume Heat CapacityCV = /T = (/)(d/dT) = - k2/ using results for the Mean Square Deviation: (E)2 = E2 - ()2 = 2lnZ/2 = - /

Canonical Ensemble:Constant Volume Heat CapacityCV = /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

Canonical Ensemble:Constant Volume Heat CapacityCV = /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

Canonical Ensemble:Constant Volume Heat CapacityCV = /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 = kBT2CVNote that, since (E)2 0 (i) CV 0 and (ii) /T 0.

Ensembles in ClassicalStatistical MechanicsAs weve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi,pi).The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q,p).There may also be a few constants of motion such asenergy, number of particles, volume, ...

The Canonical Distribution inClassical Statistical MechanicsThe Partition Functionhas the form:Z d3r1d3r2d3rN d3p1d3p2d3pN e(-E/kT)A 6N Dimensional Integral!This assumes that we have already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi.E E(r1,r2,r3,rN,p1,p2,p3,pN)

CLASSICALStatistical Mechanics:Let A any measurable, macroscopic quantity. The thermodynamic average of A . This is what is measured. Use probability theory to calculate :P(E) e[-E/(kBT)]/Z (A)d3r1d3r2d3rN d3p1d3p2d3pNP(E)Another 6N Dimensional Integral!

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