Statistical Thermodynamics - Snvanita
Statistical ThermodynamicsK.S .Rao for B.Sc III Sem Physics
Contents1. Introduction2. Distribution of Molecular States3. Interacting Systems – Gibbs Ensemble4. Classical Statistical Mechanics
1. Introduction Mechanics : Study of position, velocity, force andenergy Classical Mechanics (Molecular Mechanics) Molecules (or molecular segments) are treated as rigid object(point, sphere, cube,.) Newton’s law of motion Quantum Mechanics Molecules are composed of electrons, nuclei, . Schrodinger’s equation Wave function
1. Introduction Methodology of Thermodynamics and Statistical Mechanics Thermodynamics study of the relationships between macroscopic properties– Statistical Mechanics (Statistical Thermodynamics) how the various macroscopic properties arise as a consequence of the microscopic nature ofthe system– Volume, pressure, compressibility, Position and momenta of individual molecules (mechanical variables)Statistical Thermodynamics (or Statistical Mechanics) is a link between microscopicproperties and bulk (macroscopic) propertiesThermodynamic VariablesPure mechanical variablesStatisticalMechanicsMethods of QMMethods of MMUA particularmicroscopic modelcan be usedrPTV
1.Introduction Equilibrium Macroscopic Properties Properties are consequence of average of individualmolecules Properties are invariant with time Time averageMechanicalProperties ofIndividualMoleculesposition, velocityenergy, .averageover molecules2averageover tiestemperature, pressureinternal energy, enthalpy,.
1. Introduction Description of States Macrostates : T, P, V, (fewer variables) Microstates : position, momentum of each particles ( 1023 variables) Fundamental methodology of statistical mechanics Probabilistic approach : statistical average Most probable value Is it reasonable ? As N approaches very large number, then fluctuations are negligible “Central Limit Theorem” (from statistics) Deviation 1/N0.5
2. Distribution of Molecular States Statistical Distribution n : number of occurrences b : a propertyif we know “distribution”we can calculate the averagevalue of the property bnib1 2 3 4 5 6
2. Distribution of Molecular States Normalized Distribution Function Probability Distribution FunctionPi (bi ) ni (bi )n (b ) i in ni (bi )i P (b ) 1iiiPi b bi Piib1 b2 b3 b4 b5 b6b F (b) F (bi ) PiiFinding probability (distribution) function isthe main task in statistical thermodynamics
2. Distribution of Molecular States Quantum theory says , Each molecules can have only discrete values of energies Evidence Black-body radiationPlanck distributionHeat capacitiesAtomic and molecular spectraWave-Particle dualityEnergyLevels
2. Distribution of Molecular States Configuration . At any instance, there may be no molecules at e0 , n1molecules at e1 , n2 molecules at e2 , {n0 , n1 , n2 } configuratione5e4e3e2e1e0{ 3,2,2,1,0,0}
2. Distribution of Molecular States Weight . Each configurations can be achieved in different ways Example1 : {3,0} configuration 1e1e0 Example2 : {2,1} configuration 3e1ee1ee1e000
2. Distribution of Molecular States Calculation of Weight . Weight (W) : number of ways that a configuration can be achieved indifferent ways General formula for the weight of {n0 , n1 , n2 } configurationN!N!W n1!n2 !n3!. ni !Example1{1,0,3,5,10,1} of 20 objectsW 9.31E8iExample 2{0,1,5,0,8,0,3,2,1} of 20 objectsW 4.19 E10
Principles of Equal a Priori Probability All distributions of energy are equally probable If E 5 and N 5 then555432104321043210All configurations have equal probability, butpossible number of way (weight) is different.
A Dominating Configuration For large number of molecules and large number of energylevels, there is a dominating configuration. The weight of the dominating configuration is much morelarger than the other configurations.WiConfigurations{ni}
Dominating ConfigurationW 1 (5!/5!)555432104321043210W 20 (5!/3!)W 5 (5!/4!)Difference in W becomes larger when N is increased !In molecular systems (N 1023) considering themost dominant configuration is enough for average
How to find most dominantconfiguration ? The Boltzmann Distribution Task : Find the dominant configuration for given N andtotal energy E Method : Find maximum value of W which satisfies,N niiE e i nii dni 0i e dniii 0
Stirling's approximation A useful formula when dealing with factorials oflarge numbers.ln N! N ln N NN!ln W ln ln N ! ln ni !n1!n2 !n3!.i N ln N N ni ln ni nii N ln N ni ln niii
Method of UndeterminedMultipliers Maximum weight , W Recall the method to find min, max of a function d ln W 0 ln W 0 dni Method of undetermined multiplier : Constraints should be multiplied by a constant andadded to the main variation equation.
Method of UndeterminedMultipliersundetermined multipliers ln W dni dni e i dnid ln W i dni ii ln W ei dni 0 i dni ln W ei 0 dni
Method of UndeterminedMultipliersln W N ln N ni ln ni ln W ni (n j ln n j ) N ln N ni nij N ln N N 1 N ln N N ln N 1 niN ni ni j (n j ln n j ) ni n j 1 ln n j n j njj ni n ln W (ln ni 1) (ln N 1) ln i niN n j ln ni 1 ni
Method of UndeterminedMultipliers lnni e e iNni ei 0NNormalization ConditionN n j Ne eje e jj1 e e jjnie e iPi N e e jjBoltzmann Distribution(Probability function forenergy distribution)
The Molecular Partition Function Boltzmann Distributionnie e ie e ipi e jN eqj Molecular Partition Functionq e e jj Degeneracies : Same energy value but different states (gjfold degenerate)q g jelevelsj e j
How to obtain the value of beta ? Assumption : 1 / kT T 0 then q 1 T infinity then q infinity The molecular partition function gives an indication of theaverage number of states that are thermally accessible to amolecule at T.
2. Interacting Systems– Gibbs Ensemble Solution to Schrodinger equation (Eigen-value problem) Wave function Allowed energy levels : Enh2 2 i2 U E i 8 mi Using the molecular partition function, we can calculateaverage values of property at given QUANTUM STATE. Quantum states are changing so rapidly that the observeddynamic properties are actually time average overquantum states.
Fluctuation with TimestatestimeAlthough we know most probable distribution of energies of individualmolecules at given N and E (previous section – molecular partitionfunction) it is almost impossible to get time average for interactingmolecules
Thermodynamic Properties Entire set of possible quantum states 1 , 1 , 1 ,. i ,.E1 , E2 , E3 ,., Ei ,. Thermodynamic internal energyU lim 1E t iii
Difficulties Fluctuations are very small Fluctuations occur too rapidly We have to use alternative, abstract approach. Ensemble average method (proposed by Gibbs)
Alternative Procedure Canonical Ensemble Proposed by J. W. Gibbs (1839-1903) Alternative procedure to obtain average Ensemble : Infinite number of mental replicaof system of interestLarge reservoir (constant T)All the ensemble members have the same (n, V, T)Energy can be exchanged butparticles cannotNumber of Systems : NN
Two PostulateFist PostulateThe long time average of a mechanical variable M isequal to the ensemble average in the limit N timeE1E2E3E4E5Second Postulate (Ergodic Hypothesis)The systems of ensemble are distributed uniformly for (n,V,T) systemSingle isolated system spend equal amount of time
Averaging Method Probability of observing particular quantum state in iPi n ii Ensemble average of a dynamic property E Ei Pii Time average and ensemble averageU lim Ei ti lim n E Pi ii
How to find Most ProbableDistribution ? Calculation of Probability in an Ensemble Weight N!N!W n1!n2 !n3!. n i !i Most probable distribution configuration with maximum weight Task : find the dominating configuration for given N and E Find maximum W which satisfies N n i 0dn iEt Ei n i 0Edn i iiiii
Canonical Partition Function Similar method (Section 2) can be used to get mostprobable distributionnie EiPi N e E jjnie Eie EiPi E jN eQjQ ej E jCanonical Partition Function
How to obtain beta ?– Another interpretationdU d ( Ei Pi ) Ei dPi Pi dEiiiidU qrev wrev TdS pdV Ei P dE P V iiii Ei dPi idV PdV wrevi1 N( ln Pi dPi ln Q dPi ) ii1 ln P dP TdS dqiiiThe only function that links heat (path integral) andstate property is TEMPERATURE. 1 / kTrev
Properties from Canonical PartitionFunction Internal EnergyU E Ei Pi i1 EiEe iQ i ( qs ) Q Ei e Eii ( qs ) N ,V ln Q 1 Q U Q N ,V N ,V
Properties from Canonical PartitionFunction Pressure( wi ) N Pi dV Fi dxSmall Adiabatic expansion of system(dEi ) N Fi dx Pi dV widx E Pi i V NP P Pi PiFidVViP 11 Ei Ei EiPe e iQ iQ i V N E ln Q i e Ei V , N Q i V NP 1 ln Q ln V dEiEi
Thermodynamic Properties fromCanonical Partition Function ln Q)V , N ln T ln Q S k ln Q ()V , N ln T U kT ( ln Q ln Q H kT ()V , N ()T , N ln V ln T A kT ln Q ln Q G kT ln Q ()T , N ln V ln Q i kT N i T ,V , Nj i
Grand Canonical Ensemble Ensemble approach for open system Useful for open systems and mixtures Walls are replaced by permeable wallsLarge reservoir (constant T )All the ensemble members have the same (V, T, i )Energy and particles can be exchangedNumber of Systems : NN
Grand Canonical Ensemble Similar approach as Canonical Ensemble We cannot use second postulate because systems are not isolated After equilibrium is reached, we place walls around ensemble and treateach members the same method used in canonical ensembleAfterequilibriumT,V, T,V,N1T,V,N3T,V,N2Each members are (T,V,N) systems Apply canonical ensemble methods for each memberT,V,N5T,V,N4
Grand Canonical Ensemble Weight and Constraint n j ( N ) !j,N W n j ( N )!Number of ensemble membersN n j (N )Number of molecules afterfixed wall has been placedj,NEt n j ( N ) E j (V , N )j,Nj,NNt n j ( N ) NMethod of undetermined multiplierwith , ,gn*j ( N ) Ne ePj ( N ) j,N E j ( N ,V ) gNn j (N )Ne n*j ( N )N e E j ( N ,V ) gN ej,Ne E j ( N ,V ) gNe
Grand Canonical Ensemble Determination of Undetermined MultipliersU E Pj ( N ) E j ( N , V ) ej,NdU E j ( N ,V )dPj ( N ) Pj ( N )dE j ( N ,V )j,NdU E j ( N ,V ) gNj,Nj,N E ( N ,V ) gN lnP(N) ln dP(N) P(N)dV V1jjjjj,Nj,NdU TdS pdV dNComparing two equation gives, ej,N E j ( N ,V ) / kTeN / kT1kTg kTGrand Canonical Partition Functione
4. Classical Statistical Mechanics The formalism of statistical mechanics relies very much at themicroscopic states. Number of states , sum over states convenient for the framework of quantum mechanics What about “Classical Sates” ? Classical states We know position and velocity of all particles in the system Comparison between Quantum Mechanics and Classical MechanicsQM ProblemH E Finding probability and discrete energy statesCM ProblemF maFinding position and momentum of individual molecules
Newton’s Law of Motion Three formulations for Newton’s second law of motion Newtonian formulation Lagrangian formulation Hamiltonian formulationH (r N , p N ) KE(kinetic energy) PE(potenti al energy)pH (r N , p N ) i U (r1 , r2 ,., rN )i 2mi H p i ri H r i p i ri p i t mir r (rx , ry , rz ) p i Fi tFi Fijp p( p x , p y , p z )j 1j i
Classical Statistical Mechanics Instead of taking replica of systems, use abstract “phase space”composed of momentum space and position space (total 6N-space)pNt2 2t1 1Phase space p1 , p 2 , p3 ,., p N , r1 , r1 , r3 ,., rN rN
Classical Statistical Mechanics “ Classical State “ : defines a cell in the space (small volume ofmomentum and positions)" Classical State" dqx dq y dqz drx dry drz d 3 pd 3r for simplicity Ensemble AverageU lim 1 0E ( )d lim P N ( ) E ( )d PN ( )d n Fraction of Ensemble members in this range( to d )Using similar technique used forBoltzmann distributionPN ( )d exp( H / kT )d . exp( H / kT )d
Classical Statistical Mechanics Canonical Partition FunctionPhase IntegralT . exp( H / kT )d Canonical Partition FunctionQ c . exp( H / kT )d Match between Quantumand Classical Mechanicsc limT exp( E / kT )ii . exp( H / kT )d 1N!h NFFor rigorous derivation see Hill, Chap.6 (“Statistical Thermodynamics”)c
Classical Statistical Mechanics Canonical Partition Function in Classical Mechanics1Q . exp( H / kT )d NF N!h
Example ) Translational Motion forIdeal GasH (r N , p N ) KE(kinetic energy) PE(potenti al energy)pH (r N , p N ) i U (r1 , r2 ,., rN )i 2mi3NH i2pi2miNo potential energy, 3 dimensionalspace.pi21Q . exp( )dp1.dp N dr1.drN3N N!hi 2mi 1N !h 3 N3N V p)dp dr1dr2 dr3 exp( 2mi 0 1 2 mkT N ! h 2 N3N / 2VNWe will get ideal gas lawpV nRT
Semi-Classical Partition Function The energy of a molecule is distributed in different modes Vibration, Rotation (Internal : depends only on T) Translation (External : depends on T and V) Assumption 1 : Partition Function (thus energy distribution) can be separatedinto two parts (internal center of mass motion)EiCM EiintEiCMEiintQ exp( ) exp( ) exp( )kTkTkTQ QCM ( N ,V , T )Qint ( N , T )
Semi-Classical Partition Function Internal parts are density independent and most of thecomponents have the same value with ideal gases.Qint ( N , , T ) Qint ( N ,0, T ) For solids and polymeric molecules, this assumption is notvalid any more.
Semi-Classical Partition Function Assumption 2 : for T 50 K , classical approximation can beused for translational motionH CM pix2 piy2 piz22mi U (r1 , r2 ,., r3 N )pix2 piy2 piz21Q . exp( )dp 3 N . ( U / kT )dr 3 N3N N !h2mkTi 3 N ZN!Configurational Integral1/ 2 h2 2 mkT Z . ( U / kT )dr1dr2 .dr3 NQ 1Qint 3 N ZN!
Another, Different Treatment
Statistical Thermodynamics:the basics Nature is quantum-mechanicalConsequence:–– Systems have discrete quantum states.For finite “closed” systems, the number ofstates is finite (but usually very large)Hypothesis: In a closed system, everystate is equally likely to be observed.Consequence: ALL of equilibriumStatistical Mechanics andThermodynamics
Each individualmicrostate isequally probable , but there are notmany microstates thatgive these extremeresultsBasic assumptionIf the number ofparticles is large ( 10)these functions aresharply peaked
Does the basis assumption lead to somethingthat is consistent with classicalthermodynamics?E1E2 E E1Systems 1 and 2 are weakly coupledsuch that they can exchange energy.What will be E1?W E1 , E E1 W1 E1 W2 E E1 BA: each configuration is equally probable; but the number ofstates that give an energy E1 is not know.
W E1 , E E1 W1 E1 W2 E E1 ln W E1, E E1 ln W1 E1 ln W2 E E1 ln W E1 , E E1 0 E1 N1 ,V1 Energy is conserved!dE1 -dE2 lnW1 E1 lnW2 E E1 E1 E1 N ,V N11 02 ,V2 ln W1 E1 ln W2 E E1 E1 E2 N1 ,V1 N2 ,V2 ln W E E N ,V 1 2This can be seen as anequilibrium condition
Entropy and number ofconfigurationsConjecture:S ln WAlmost right. Good features: Extensivity Third law of thermodynamics comes for free Bad feature: It assumes that entropy is dimensionless but (forunfortunate, historical reasons, it is not )
We have to live with the past, thereforeS kB ln W E With kB 1.380662 10-23 J/KIn thermodynamics, the absolute (Kelvin)temperature scale was defined such that1 S E N ,V TndE TdS-pdV idNii 1But we found (defined): ln W E E N ,V
And this gives the “statistical” definition of temperature: ln W E 1 kB T E N ,VIn short:Entropy and temperature are both related tothe fact that we can COUNT states.Basic assumption:1. leads to an equilibrium condition: equal temperatures2. leads to a maximum of entropy3. leads to the third law of thermodynamics
Number of configurationsHow large is W? For macroscopic systems, super-astronomically large. For instance, for a glass of water at room temperature:W 102 1025 Macroscopic deviations from the second law ofthermodynamics are not forbidden, but they areextremely unlikely.
Canonical ensemble1/kBTConsider a small system that can exchange heat with a big reservoirEi ln Wln W E Ei ln W E Ei EE EilnW E Ei W E Hence, the probability to find Ei:P Ei W E Ei exp Ei k BT W E E jjEi k BTjexp E j k BT P Ei exp Ei kBT Boltzmann distribution
Example: ideal gas1recallQ N ,V , T Thermo dr (3)exp U r N!NN3NHelmholtz Free energy:N1V dF3 N S dTdr p1d V 3 N N! N!NFree energy: PressureN V F F ln 3 N P V T N ! N 3Energy: N ln N ln N ln 3 N ln Pressure:V F F T E 1 T Energy:N F P V T V F 3N 3E Nk BT 2
Ensembles Micro-canonical ensemble: E,V,NCanonical ensemble: T,V,NConstant pressure ensemble: T,P,NGrand-canonical ensemble: T,V,μ
Each individualmicrostate isequally probable , but there are notmany microstates thatgive these extremeresultsBasic assumptionIf the number ofparticles is large ( 10)these functions aresharply peaked
Does the basis assumption lead to somethingthat is consistent with classicalthermodynamics?E1E2 E E1Systems 1 and 2 are weakly coupledsuch that they can exchange energy.What will be E1?W E1 , E E1 W1 E1 W2 E E1 BA: each configuration is equally probable; but the number ofstates that give an energy E1 is not know.
W E1 , E E1 W1 E1 W2 E E1 ln W E1, E E1 ln W1 E1 ln W2 E E1 ln W E1 , E E1 0 E1 N1 ,V1 Energy is conserved!dE1 -dE2 lnW1 E1 lnW2 E E1 E1 E1 N ,V N11 02 ,V2 ln W1 E1 ln W2 E E1 E1 E2 N1 ,V1 N2 ,V2 ln W E E N ,V 1 2This can be seen as anequilibrium condition
Entropy and number ofconfigurationsConjecture:S ln WAlmost right. Good features: Extensivity Third law of thermodynamics comes for free Bad feature: It assumes that entropy is dimensionless but (forunfortunate, historical reasons, it is not )
We have to live with the past, thereforeS kB ln W E With kB 1.380662 10-23 J/KIn thermodynamics, the absolute (Kelvin)temperature scale was defined such that1 S E N ,V TndE TdS-pdV idNii 1But we found (defined): ln W E E N ,V
And this gives the “statistical” definition of temperature: ln W E 1 kB T E N ,VIn short:Entropy and temperature are both related tothe fact that we can COUNT states.Basic assumption:1. leads to an equilibrium condition: equal temperatures2. leads to a maximum of entropy3. leads to the third law of thermodynamics
Number of configurationsHow large is W? For macroscopic systems, super-astronomically large. For instance, for a glass of water at room temperature:W 102 1025 Macroscopic deviations from the second law ofthermodynamics are not forbidden, but they areextremely unlikely.
Canonical ensemble1/kBTConsider a small system that can exchange heat with a big reservoirEi ln Wln W E Ei ln W E Ei EE EilnW E Ei W E Hence, the probability to find Ei:P Ei W E Ei exp Ei k BT W E E jjEi k BTjexp E j k BT P Ei exp Ei kBT Boltzmann distribution
Example: ideal gas1recallQ N ,V , T Thermo dr (3)exp U r N!NN3NHelmholtz Free energy:N1V dF3 N S dTdr p1d V 3 N N! N!NFree energy: PressureN V F F ln 3 N P V T N ! N 3Energy: N ln N ln N ln 3 N ln Pressure:V F F T E 1 T Energy:N F P V T V F 3N 3E Nk BT 2
Ensembles Micro-canonical ensemble: E,V,NCanonical ensemble: T,V,NConstant pressure ensemble: T,P,NGrand-canonical ensemble: T,V,μ
1. Introduction Methodology of Thermodynamics and Statistical Mechanics Thermodynamics study of the relationships between macroscopic properties – Volume, pressure, compressibility, Statistical Mechanics (Statistical Thermodynamics) how the various macroscopic properties arise as a consequence of the microscopic nature of the system .
1.4 Second Law of Thermodynamics 1.5 Ideal Gas Readings: M.J. Moran and H.N. Shapiro, Fundamentals of Engineering Thermodynamics,3rd ed., John Wiley & Sons, Inc., or Other thermodynamics texts 1.1 Introduction 1.1.1 Thermodynamics Thermodynamics is the science devoted to the study of energy, its transformations, and its
to calculate the observables. The term statistical mechanics means the same as statistical physics. One can call it statistical thermodynamics as well. The formalism of statistical thermodynamics can be developed for both classical and quantum systems. The resulting energy distribution and calculating observables is simpler in the classical case.
Reversible and Irreversible processes First law of Thermodynamics Second law of Thermodynamics Entropy Thermodynamic Potential Third Law of Thermodynamics Phase diagram 84/120 Equivalent second law of thermodynamics W Q 1 1 for any heat engine. Heat cannot completely be converted to mechanical work. Second Law can be formulated as: (A .
An Introduction to Statistical Mechanics and Thermodynamics. This page intentionally left blank . An Introduction to Statistical Mechanics and Thermodynamics Robert H. Swendsen 1. 3 Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford.
o First Law of Thermodynamics o Second Law of Thermodynamics o Prerequisites for courses on Fluid Mechanics, Heat and Mass Transfer, Gas Dynamics, Power and Refrigeration Cycles, HVAC, Combustion, Acoustics, Statistical Thermodynamics, High level application of these topics to the
- Systems out of equilibrium - Irreversible Thermodynamics: limited! - Statistical Physics: kinetic theory, powerful, complex! Statistical Mechanics allows calculating with an excellent accuracy the properties of systems containing 1023 atoms! II) Introduction to the methods of Statistical Mechanics 1) Definitions:
R1: "Introduction to Modern Statistical Mechanics" by David Chandler (Oxford University Press). Ch. 3-8. Additional textbooks are available at the Kline Science and Engineering library include: R2: "Introduction to Statistical Thermodynamics" by T.L. Hill (Addison Wesley), R3: "Statistical Mechanics" by D. McQuarrie (Harper & Row),
– Ossa brevia (tulang pendek): tulangyang ketiga ukurannyakira-kirasama besar, contohnya ossacarpi – Ossa plana (tulang gepeng/pipih): tulangyang ukuranlebarnyaterbesar, contohnyaosparietale – Ossa irregular (tulangtak beraturan), contohnyaos sphenoidale – Ossa pneumatica (tulang beronggaudara), contohnya osmaxilla
