the topic entails info about Boltzmann Entropy and Probability and methods for Avogadro determination......

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BOLTZMANNENTROPY: PROBABILITYAND
INFORMATION
We have presentedfirstanaxiomaticderivationof Boltzmannentropyonthe basisof twoaxiomsconsistentwithtwo
basicpropertiesof thermodynamicentropy.We have thenstudiedthe relationshipbetweenBoltzmannentropyand
informationalongwithitsphysical significance.PACS:89.70, + C
Keywords:Boltzmannentropy,thermodynamicprobability,axiomaticderivation,information,statistical equilibrium.
1. INTRODUCTION
The concept of entropywasfirstintroducedinthermodynamicsbyClausiusthrough the secondlaw of thermodynamics.
It was Boltzmannwhofirstprovidedthe statistical analogue of thermodynamicentropylinkingthe conceptof entropy
withmoleculardisorderorchaos [1].The conceptof entropywaslaterdevelopedwiththe adventof stati stical
mechanicsandinformationtheory[3].Boltzmannentropyisabuildingblocktothe foundationof statistical mechanics
and the literature onBoltzmannentropyisvast.Inspiteof this,there isscope forfurtherextensionof the existing
methodsand principles. Inthe presentpaperwe have providedfirstanaxiomaticderivationof Boltzmannentropyon
the basisof twoaxiomsconsistentwiththe additivityandthe increasinglaw of thermodynamicentropy. We have then
introducedthe probabilitydistributionof amacrostate throughthe conceptof priorprobabilities.Thishasledtoan
interestingrelationconnectingnegentropyandinformation.The physical significance of thisrelationinthe
characterizationof statistical equilibriumhasbeeninvestigated.
2. THERMODYNAMIC PROBABILITY AND BOLTZMANN ENTROPY
Boltzmannentropyisdefinedby[1]
S = k ln W … (2.1)
where k isthe thermodynamicunitof the measurementof the entropyand isthe Boltzmannconstant,W calledthe
thermodynamicprobabilityorstatistical weightisthe total numberof microscopicstatesorcomplexionscompatible
withthe macroscopicstate of the system.FollowingCarnop[4],we shall,however,call Was the degree of disorderor
simplydisorderof the system.We canthusrestate Boltzmannprincipleas
Entropy= k ln(Disorder) … (2.2)
The entropySbeinga monotonicincreasingfunctionof disorder,canitself be consideredasameasure of disorderof
the system.Boltzmannentropythusprovidesameasure of the disorderof the distributionof the statesoverpermissible
microstates[4].Letus derive the expression(2.1) of entropyaxiomaticallyonthe
basisof two fundamental propertiesof thermodynamicentropy.We make the followingaxioms:
Axiom (i) :
The entropyS(W) of systemisa positive increasingfunctionof the disorderW,
that is,
S (W) ≤ S (W + 1) … (2.3)
This ismotivatedbythe increasinglawof entropyinthermodynamics.

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Axiom (ii) :
The entropyS(W) isassumedtobe an additive functionof the disorderW,that is,forany twostatisticallyindependent
systemswithdegreesof disorderW1and W2 respectively,the entropyof the compositesystemisgivenby
S (W1 . W2) = S (W1) + S (W2) … (2.4)
This axiomisthe statistical analogue of the thermodynamicpostulate of additivityof entropy.We have thenthe
theorem:
Theorem(1) :
If the entropyfunction S(W) satisfiesthe above twoaxioms(i) and(ii),thenS(W) isgivenby
S = k lnW … (2.5)
where k isa positive constantdependingonthe unitof measurementof entropy.
Proof :
Let us assume thatW > e. Thisisjustifiedbythe factthat the macroscopicsystemwe are interestedinconsists
of a large numberof microscopicstatesandhence correspondstoa large value of statistical weightorthermodynamic
probabilityW.Forany integern,we can findanintegerm(n) suchthat
em(n) ≤ Wn ≤ em(n) +1 … (2.6)
or
𝑚( 𝑛)
𝑛
≤ ln 𝑊 ≤
𝑚( 𝑛)+1
𝑛
consequently, lim
𝑛→∞
𝑚( 𝑛)
𝑛
= ln 𝑊
The entropyfunctionS(W) satisfiesboththe axioms(i) and(ii).Byaxiom(i) we have thenfrom(2.6)
S (em(n)) ≤ S (Wn) ≤ S (em(n) +1) … (2.8)
Againbyaxiom(ii) we have from(2.8)
m(n) S(e) ≤ nS (W) ≤ (m(n) + 1) S(e)
consequently, , lim
𝑛→∞
𝑚( 𝑛)
𝑛
=
𝑆(𝑊)
𝑆(𝑒)
……………………………(2.9)
Comparing(2.7) and (2.9),we get
S(W) = k lnW .....................…(2.10)
where S(e) =k isa positive constant,the positivityfollowsfromthe positivityof the entropy –functionpostulatedin
axiom(i).The above theoremprovidesarigorousderivationof the entropyfunctionwhichisindependentof the micro-
model of the system – classical orquantal.In mostof the bookson statistical physicsexceptafew [6]
the expressionof entropy(2.1) isdeterminedfromthe additivitypostulate of entropyonly.Thisisnotcorrect.The
above derivationbasedonthe axioms(i) and(ii)whichare variantformsof twobasicpropertiesof thermodynamic
entropyismore sound and isvalidformacroscopicsystemcontainingalarge numberof molecules[6].
3. PRIOR PROBABILITIES: ENTROPY AND INFORMATION

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The thermodynamicprobabilitywhichappearsinBoltzmannentropy(2.1) isnota probabilitybutaninteger.
How to introduce the conceptof probability?Itwas
Einstein[7] whofirstintroducedthe conceptof probabilityof amacrostate by invertingBoltzmannprinciple [8].We
shall,however,followadifferentapproach.We consideranisolatedsystemconsisting of N moleculesclassifiedinton
energystatesεi ( i = 1, 2, ….n) andletNi ( i = 1, 2, ….n) be the occupationnumberof the ithenergy-state εi.The
macroscopicstate of the systemisgivenbythe setof occupationnumbersAn= {N1,N2, … Nn}.
We are nowinterestedinthe probabilitydistributionof the macrostate Anorthe probabilitydistributionof the
setof occupationnumbersAn= {N1,N2, …, Nn}.In view of the many-body aspect of the system, the setof
occupationnumbers{N1,N2,… Nn}canbe assumedtobe a setof randomvariables.The probabilitydistributionof the
macrostate An = {N1, N2, … Nn} is thengivenby
P (An) = P (N1,N2,… Nn) = W (An) ∏ 𝑝0𝑖
𝑁𝑛
𝑖=1 …………..(3.1)
where pi0isthe priorprobabilitythatamolecule liesinthe ithenergy-state εi.The probability(3.1) isof fundamental
importance inconnectingthe disorderW(An)withthe probabilityP(An),while the formerisrelatedtothe Boltzmann
entropyandthe later toinformationasshall see.The appearance of priorprobabilities{pi0} makesthe systemcomplex.
If no state is preferable toother,itisthenreasonable toassume thatall priorprobabilitiesare equal toone anotherso
that pio= 1/n,(i = 1, 2, … n). ThisisLaplace principle of insufficientknowledge[10].AccordingtoJaynes[3] it isthe state
of maximumpriorignorance of the system.Itthatcase itis easyto see from(3.1) that
- k lnP (An) = Smax – S … (3.2)
where Smax = k N ln n isthe maximumvalue of the entropyS.The l.h.s.of (3.2) isthe information-contentof the
macrostate An.that is,the informationgainedfromthe realizationof the macrostate An[2].The r.h.sof (3.2) isthe
negentropyof the systemgivingameasure of distance of non-equilibriumstate fromthe equilibriumstate [9,12].The
relation(3.2) thenimpliesthe equivalence of informationwithnegentropyconsistentwithBrillouin’snegentropy
principle of information[9].
Let us nowinvestigate the significance of the relation(3.2) forthe statistical equilibriumof asystem.According
to BoltzmannandPlanckthe statistical equilibriumof asystemcorrespondstothe mostprobable state of the system.
The thermodynamicprobabilityW(An) isnota probability,itisaninteger.Sothe statistical equilibriumasassumedto
correspondtothe maximumof the thermodynamicprobabilityorequivalentlythe Boltzmannentropygivesrise tosome
confusion[8].However,fromthe relation(3.2) we see thatthe mostprobable state of the systemobtainedby
maximizingthe probabilityof the macrostate P(An) isequivalenttothe maximizationof the Boltzmannentropy(2.1) or
the maximizationof the thermodynamicprobabilityW(An) providedwe acceptthe hypothesisof equal aprior
probabilitypio =1/n, (i = 1, 2, …n).It isin thiswaythe confusionaboutthe non-probabilisticcharacterof
thermodynamicprobabilityW(An) andthe most-probableinterpretationof statistical equilibriumcanbe resolved.
4. CONCLUSION
The objectof the presentpaperistopresent significantlydifferentapproachtothe characterizationof
Boltzmannentropyinrelationtoboththermodynamicprobabilityandinformation.The mainresultsof the paperare as
follows:
(i) We have presentedarigorousaxiomaticderivationof Boltzmannentropyonthe basisof the axiomsof additivity
and increasinglawof entropyconsistentwithtwobasic propertiesof thermodynamicentropy[11].The methodis
superior,bothmathematicallyandphysically,tothe existingmethods(exceptafew) basedmainlyonthe propertyof
additivityof entropyonly.

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(ii)Boltzmannentropyisindependentof probability.We have introducedthe probabilityof amacrostate throughthe
hypothesisof priorprobability{pi0}.
(iii)The probabilitydistributionof macrostate leadstothe interestingrelation(3.2) connectinginformationand
BoltzmannentropyornegentropyconsistentwithBrillouins’negentropyprinciple of information[4].There is,however,
a difference. Inourcase informationcomesthroughthe conceptof probabilityandthe l.h.sof (3.2) isin fact,a
probabilisticmeasure of informationandnotlike the one introducedbyBrillouin[9] andothers[12].
(iv) The relation(3.2) has importantphysical significance forstatistical equilibrium.Itresolvesthe confusionaboutthe
most–probable state of statistical equilibriumachievedbymaximizingthe thermodynamicprobabilityunderthe
hypothesisof equal priorprobabilities.
(v) In the case of unequal priorprobabilities{pio} the situationbecomescomplicated and interesting forboth
physical andnon-physical systems[13-15].
METHODS FOR DERTERMINATION OF
AVOGADROS NUMBER:
AMEDEO CARLO AVOGADROwasan ItalianScientistgave the conceptof AvogadroLaw.(atoms,ions,ormolecules)
The numberof particlesone mole of asubstance isalwaysconstanti.e,6.0221407574× 1023.It isshownby 𝑁𝐴.The
experimental methjodsare asfollows:
1-By Perrinmethod(Brownianmovement)
2-By X-rayDiffraction(Crystallographicmethod)
3-By Electrochemical method
4-By ElectricCharge
5-By counting∝- particles
By Perrin Method:
In 1909 J.Perrindetermined 𝑁𝐴 byBoltzmanndistributionlaw .he determinedBoltzmannconstantKfromnumberof
particlesatdifferentheightsinasuspensionbyusingrelationship 𝑁𝐴 =
𝑅
𝑘
.Perrinobtained colloidal particlesof uniform
densityanddiameterbycentrifugingemulsionof gambage (ayellowresinusedinwatercolor).He observedthatat
differentheightsno.ofparticlesare different.He usedmicroscopeof verylow depthtosee no.of particlesinverylow
depthof ` h’height.
Consideracylinderof areasA inwhichmoleculescome inequilibriumwithgravity.Pressure isPatheighthbut Pressure
become P-dPbychangingheightdh(byincreasingvolume,pressure isdecreasedi.e.-ve)
Two forcesact
Upward force =downwardforce
Fu=Fd

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A× 𝑑𝑃 =Adh𝜌g
dP=dh𝜌g
-ve signisusedbecause pressure isdecreased
- dP=dh𝜌g
dP=-dh𝜌g……………….(1)
Value of 𝜌:
Accordingto Kineticequation 𝑃𝑣 =
2
3
𝐸. 𝑘 or 𝑉 =
2𝐸.𝑘
3𝑃
.and we know density 𝜌 =
𝑀
𝑉
𝜌 =
𝑀
2𝐸.𝑘
3𝑃
=
3𝑀𝑃
2𝐸.𝑘
by puttingvalue of 𝜌ineq1
- dP=
3𝑀𝑃𝑔𝑑ℎ
2𝐸.𝑘
…………….(2)
− dP
𝑃
=
3𝑀𝑔𝑑ℎ
2𝐸.𝑘
integrate itw.r.t P whenh=0 thenpressure is 𝑃0 whenheightishthen
pressure isP.
∫ −𝑑𝑃
𝑃
𝑃0
=
3𝑀𝑔
2𝐸.𝑘
∫ 𝑑ℎ
ℎ
0
-[ln 𝑃 − ln 𝑃0]=
3𝑀𝑔
2𝐸.𝑘
[h-0]
- lnP+ ln𝑃0 =
3𝑀𝑔ℎ
2𝐸.𝑘
lnP
Po⁄ =
3𝑀𝑔ℎ
2𝐸.𝑘
but P∝ 𝑛
thus ln n
𝑛0⁄ =
3𝑀𝑔ℎ
2𝐸.𝑘
……………….(3)
𝑛0 isno. f molseculeswhenh=0
n is no.of moleculeswhenheightish
Mg = 𝑁𝐴mg (1 -
𝜌′
𝜌
)
Thus eq3 will become
ln n
𝑛0⁄ =
3𝑁 𝐴 𝑚𝑔ℎ
2𝐸.𝑘
(1 -
𝜌′
𝜌
) ….. but E.k=
3
2
𝑅𝑇
ln n
𝑛0⁄ =
𝑁 𝐴 𝑚𝑔ℎ
𝑅𝑇
(1 -
𝜌′
𝜌
) ……. Andby multiplyingitbyRT
RT ln n
𝑛0⁄ = 𝑁𝐴 𝑚𝑔ℎ(
𝜌−𝜌′
𝜌
) ………(4) but V=
4
3
𝜋𝑟3, 𝜌 =
𝑀
𝑉
,M=
4
3
𝜋𝑟3 𝜌

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By puttingvalue of min eq(4)
RT lnn
𝑛0⁄ = 𝑁𝐴
4
3
𝜋𝑟3 h𝜌g(
𝜌−𝜌′
𝜌
)
𝑅𝑇
𝑁 𝐴
lnn
𝑛0⁄ =
4
3
𝜋𝑟3 gh(𝜌 − 𝜌′)
𝑁𝐴 =
3RT lnn
𝑛0⁄
4𝜋𝑟3 𝑔ℎ(𝜌− 𝜌′)
By puttingvaluesof RT,n,r,g,h, 𝑛0, 𝜌, 𝜌′
𝑁𝐴 =6.022 × 1023.
Where n and 𝑛0can be determinedbymicroscopeof verylow depthof field.
By Crystallographic Method:
To determine Avogadronumberbycrystallographicmethodfollowingvaluesare required.
1- Volume of one mole of a crystalline solid,itcanbe determinedbyfollowingformula;
Density=
𝑚𝑎𝑠𝑠
𝑣𝑜𝑙𝑢𝑚𝑒
or volume =
𝑚𝑎𝑠𝑠
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
2-the distance betweenparticlesinthe crystalslattice.Itcanbe determinedbyX-rays.
Example : calculate 𝑁𝐴 inLiFhaving2.65g𝑐𝑚−3densityanddistance betweenionsis2.01 𝐴0.
SOLVE: Formulamass of LiF= 6.939+18.9984=25.9374
𝑔
𝑚𝑜𝑙⁄
Volume =
𝑚𝑎𝑠𝑠
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
=
25.9374
2.65
= 9.788𝑐𝑚3 burt volume = 𝑙 × 𝑙 × 𝑙 =𝑙3, 𝑙 =√ 𝑣3
= √9.7883
= 2.139𝑐𝑚3
No.of ionsalongone length
𝑙𝑒𝑛𝑔𝑡ℎ
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑜𝑛𝑠
=
2.139
2.01×10−8
=1.064× 108
Total no.of ionsalongthree lengths=(1.064× 108)3 = 1.204 × 1024 𝑖𝑜𝑛𝑠
Thus inLiF total no,of Li+ ionsis one Avogadronumberandtotal no of F- ionsisone Avogadronumber.
Therefore, no. of onlyLi+ or F- , ions=
1.204 ×1024
2
= 6.022× 1023 ions
By Electrolysis Method:
As we knowthat
1 mole of electron=6.022× 1023 e- and1 mole of Cu = 6.022× 1023 atoms
Avogadrosnumber=
𝑛𝑜.𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠
𝑛𝑜.𝑜𝑓 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠
Thus we can calculate no.of electronsandno.of molesof electronsbyelectrolysisexperiment.
Experiment : take CuSO4 in a container .dip two strip of Cu which act as Electrode .the CuSO4 is
ionized into
CuSO4 ↔ 𝐶𝑢2+ + 𝑆𝑂4−2

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Electronsmove towards –ve electrode .one Cuelectrode loses2e-s
Cu → 𝐶𝑢+2 + 2𝑒−
Thus massof one Cu electrode decreases(∆𝑚1). 𝐶𝑢+2ionsinsolutiongain2e-sand deposite of otherelectrode mass
of thiselectrode increases(∆𝑚2).duringreaction(∆𝑚1 ≠ ∆𝑚2) butwhenreactioncomplete then(∆𝑚1 = ∆𝑚2)
To determine no.of electronsandno.of molesof electronisdifficultbutwe cancalculate charge of electron.
In physicscharge Q = I × 𝑡 𝑠𝑒𝑐
But charge of 1 e- = 1.6022× 10−19 coulombs
Thus Q charge =𝐼(𝑒𝑚𝑝) × 𝑡 𝑠𝑒𝑐 ×
1 𝑒−
1.602×10−19 𝑐𝑜𝑙𝑜𝑢𝑚𝑏𝑠
From chemical reactioneq
No.of e-s=no.of mole of electrons
No.of e-=
𝐼 × 𝑡
1.602×10−19
byputtingno of e-s
No.of Cu atoms =
𝐼 × 𝑡
1.602×10−19 ×2𝑒−𝑓𝑜𝑟 1 𝐶𝑢 𝑎𝑡𝑜𝑚
………..(2)
Now calculate no. of molesofCu
From massof Cu we can calculate moles of Cu
No.of mole of Cu= ∆𝑚2 ×
1 𝑚𝑜𝑙𝑒 𝑜𝑓 𝐶𝑢
63.543
…………..(3)
We knowthat No.of atoms =no. of moles × 𝑁𝐴
Or Avogadrono.=
𝑛𝑜.𝑜𝑓 𝐶𝑢 𝑎𝑡𝑜𝑚𝑠
𝑛𝑜.𝑜𝑓 𝑚𝑜𝑙𝑒𝑠 𝑜𝑓 𝐶𝑢
By puttingvaluesfromeq2 and eq3
𝑁𝐴 =
𝐼 × 𝑡
1.602×10−19 ×2𝑒−𝑓𝑜𝑟 1 𝐶𝑢 𝑎𝑡𝑜𝑚
∆𝑚2
1 𝑚𝑜𝑙𝑒 𝑜𝑓 𝐶𝑢
63.543
=
𝐼 × 𝑡 ×63.543
1.60210−19 × 2× ∆𝑚2
By puttingvaluesof I,tand ∆𝑚2 we can calculate Avogadronumber.

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Composed by
Maria Aamer Sabah
Roll No: MCEC-17-09

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