Dimensional Equations of Entropy

Transcription

1 Amlia Carolina Sparavigna 1 1 Dpartmnt of Applid Scinc and chnology, Politcnico di orino, Italy. Abstract: Entropy is a quantity which is of grat importanc in physics and chmistry. h concpt coms out of thrmodynamics, proposd by Rudolf Clausius in his analysis of Carnot cycl and linkd by Ludwig Boltzmann to th numbr of spcific ways in which a physical systm may b arrangd. Any physics classroom, in its task of larning physics, has thrfor to fac this crucial concpt. As w will show in this papr, th lcturs can b nrichd by discussing dimnsional quations linkd to th ntropy of som physical systms. Kywords: Physics Classroom, Entropy, Einstin Modl, Blackbody Radiation, Bknstin-Hawking Black Hol Entropy, Casimir Entropy, Bos-Einstin Condnsation. 1. Introduction In physics and nginring, dimnsional analysis hlps finding rlationships btwn diffrnt physical quantitis by dtrmining som quations basd on a fw fundamntal quantitis. Usually, ths fundamntal quantitis ar lngth, tim, mass, tmpratur and lctric charg and ar rprsntd by symbols L,, M, and Q, rspctivly [1]. h dimnsions of any physical quantity can b xprssd as products of ths basic quantitis, ach raisd to a rational powr []. Any physically maningful quation rquirs th sam dimnsions on its lft and right sids, a proprty known as "dimnsional homognity". Chcking this homognity is a common application of dimnsional analysis, routinly usd to vrify th plausibility of calculations. In this analysis, quations ar turnd into dimnsional quations. In this papr w discuss som dimnsional quations rlatd to th ntropy of som modls of physics systms. In thrmodynamics, ntropy (usual symbol S ) is th physical quantity linkd to th scond law of thrmodynamics, th law which is tlling that th ntropy of an isolatd systm nvr dcrass. his systm will spontanously procd towards thrmodynamic quilibrium, which is th configuration with maximum ntropy. Entropy is an xtnsiv proprty. It has th dimnsion of nrgy dividd by tmpratur, having a unit of 1 jouls pr klvin ( J K ) in th Intrnational Systm of Units. Howvr, th ntropy of a substanc can b givn also as an intnsiv proprty, ntropy pr unit 1 1 mass (SI unit: J K kg ) or ntropy pr unit amount of substanc. h chang in ntropy ( S ) of a systm was originally dfind for a thrmodynamically rvrsibl procss as S dqrv /, whr is th absolut tmpratur of th systm. his tmpratur is dividing an incrmntal rvrsibl transfr of hat into that systm ( dq rv ). his dfinition is somtims calld th macroscopic dfinition of ntropy, bcaus it is usd without rgard to any microscopic dscription of th thrmodynamic systm. h most rmarkabl proprty of ntropy is that of bing a function of stat. In thrmodynamics, a stat function is a proprty of th systm which is dpnding only on th stat of th systm, not on th mannr this systm acquird that stat. Svral stat functions xist bsids ntropy, all abl of dscribing quantitativly an quilibrium stat of a systm. For this rason, bsids th chang of ntropy, an absolut ntropy ( S rathr than ) was dfind. In this cas, an approach using statistical mchanics is prfrrd. Lt us rmmbr that th concpt of ntropy, which cam out of thrmodynamics, as proposd by Rudolf Clausius in his analysis of Carnot cycl, was linkd by Ludwig Boltzmann to th numbr of spcific ways in which a physical systm may b arrangd, in a statistical mchanics approach. S In fact, th modrn statistical mchanics was initiatd in th 1870s, by th works of Boltzmann, mainly collctd and publishd in his 1896 Lcturs on Gas hory []. his articl is publishd undr th trms of th Crativ Commons Attribution Licns.0 Author(s) rtain th copyright of this articl. Publication rights with Alkhar Publications. Publishd at: DOI: /ijSci.811; Onlin ISSN: 05-95; Print ISSN: Amlia Carolina Sparavigna (Corrspondnc) d0000@polito.it

2 Entropy is thn a fundamntal quantity for any physics classroom, in its tasks of larning and taching physics. As w ar showing in this papr, lcturs on this subjct can b nrichd by a discussion of dimnsional quations rlatd to th ntropy of som modls of physical systms. W will s som xampls basd on th ntropy of th blackbody radiation and of th black hols, among othrs. Bfor xampls, lt us discuss shortly th concpts of thrmodynamic and statistical ntropis.. Entropy and Carnot cycl h physical concpt of ntropy aros from th studis of Rudolf Clausius on th Carnot cycl []. his cycl is a thortical thrmodynamic rvrsibl cycl proposd by Nicolas Léonard Sadi Carnot in 18. It is composd of an isothrmal xpansion, followd by an adiabatic xpansion. hn w hav an isothrmal comprssion followd by an adiabatic comprssion. Whn th Carnot cycl is rprsntd on a prssur volum diagram (pv diagram), th isothrmal stags ar givn by isothrm lins of th working fluid. h adiabatic stags mov btwn isothrms. h ara boundd by th cycl rprsnts th total work don during on cycl. A Carnot cycl is also rprsntd by using a tmpraturntropy diagram (S diagram). In such diagrams, th adiabatic rvrsibl stag is an isntropic stag. In a Carnot cycl, hat H Q H is absorbd at tmpratur from a rsrvoir in th rvrsibl isothrmal xpansion, and givn up as hat Q C to a rsrvoir, with th rvrsibl isothrmal comprssion, at whr H C. C hrough th fforts of Clausius and Lord Klvin (William homson, 1st Baron Klvin, ), it is now known that th maximum work that a systm can produc is th product of th Carnot fficincy and th hat absorbd from th hot rsrvoir: C W QH 1 QH (1) H From th first law of thrmodynamics, ovr th ntir cycl, w hav: W Q H Q. hrfor, w hav: Q H C his implis that thr is a function of stat which is consrvd ovr th complt Carnot cycl. Clausius calld this stat function as ntropy. C QH C () In th Carnot cycl, w hav two ntropis: QH QC SH ; SC () H C hy hav opposit signs and thrfor, from (), adding thm w hav zro. his rsults is gnralizd to gnric rvrsibl cycls as:, dq 0 () rv. Entropy and statistical mchanics h statistical dfinition was givn by Ludwig Boltzmann in th 1870s. Boltzmann showd that his ntropy was quivalnt to that coming from Carnot cycl, within a constant numbr, th Boltzmann's constant. In th Boltzmann approach, ntropy is a masur of th numbr of ways in which a systm may b arrangd. his dfinition dscribs th ntropy as bing proportional to th natural logarithm of th numbr of possibl microscopic configurations of th individual atoms and molculs of th systm (microstats) which could giv ris to th obsrvd macroscopic stat of th systm. S k p i ln p i (5) i h sum is ovr all th possibl microstats of th systm, and is th probability that th systm is in p i th i-th microstat [5]. h constant of proportionality is th Boltzmann constant. It is a constant rlating nrgy at th individual particl lvl with tmpratur. It is also th gas constant R, dividd by th Avogadro constant. k N A h Boltzmann constant is linking macroscopic and microscopic physical quantitis; for instanc, for an idal gas, th product of prssur and volum V is proportional to th product of amount of substanc n (in mols) and absolut tmpratur : pv nr, whr is th abovmntiond gas constant. Introducing th Boltzmann constant transforms th idal gas law into an altrnativ form: pv Nk, whr N is th numbr of molculs of gas. hrfor, th quation of th idal gas is givn in a microscopic formalism: pv nr Nk (6) For a gas, th intrnal nrgy coming from th first law of thrmodynamics, is linkd to th ntropy by th scond law, in th following quation: du ds p dv (7) Lt us rmmbr that, for a fixd mass of an idal gas, th intrnal nrgy is a function only of its tmpratur, whras th ntropy dpnds on tmpratur and volum. R. Dimnsions of ntropy h Boltzmann constant k has th dimnsions of nrgy dividd by tmpratur, th sam of ntropy thn. Its valu in SI units is J / K. Any dimnsionlss quantity, multiplid by this constant, bcoms an ntropy. p k Lt us us th symbol E for th dimnsion [nrgy] and start from th Clausius ntropy. h dimnsional analysis givs: Volum August 015 (08)

3 dq E E S [ k] E (8) k E [ k] [ k] [ k] E k Lt us not that dimnsional quations ar charactrizd by th squar brackts. C V 9Nk D D / 0 ( x x x 1) dx (1) In th cas w start our discussion from th statistical ntropy, w hav th prsnc of th logarithmic function too. h discussion proposd in this papr will b in th framwork of Eq.8. W hav a clar xampl of (8) in th ntropy of an idal Frmi-Dirac gas. In a Frmi-Dirac systm of particls, only on particl may occupy ach nondgnrat nrgy lvl. Lt us considr first th absolut zro. If thr ar N particls, th lowst N nrgy stats ar occupid up to th lvl E o. At low tmpraturs, w hav th ntropy givn by [6]: k S Nk (9) E o Hr w can s xplicitly on of th dimnsional quations in (8): k k S Nk [ k] (10) Eo E Of cours, w can also hav diffrnt dimnsional quations as th following: E [ S ] [ k] [ k] (11) E In (11), is a givn powr. Hr, w hav nrgis and tmpratur: in fact, w can find svral othr dimnsionlss ratios too, for instanc of lngths, as w will s in this papr. 5. Entropy of th Dby modl of solids First of all, lt us find an ntropy which contains a ratio of tmpraturs. In th Dby modl, a solid is tratd as an isotropic lastic continuum in which th vlocity of th sound is constant. For th longitudinal and transvrs wavs, Ptr Dby ( ) put a cut-off at th uppr limit of frquncy to justify th fact that th solid is considrd as an lastic continuum. h uppr angular frquncy of wavs is D. h modl is charactrizd by a tmpratur, th Dby tmpratur, which is givn by D D / k. Lt us dfin th dimnsionlss variabl x / k. h/ is th rducd Planck constant. h hat capacity of th solid is [7]: h lattic ntropy is dfind as: In gnral: S C 0 V d S Nk Nk d D (1) D In (1), d is th dimnsion of lattic. In th cas of a two-dimnsional layr d. For a wir, d 1. W can hav thrfor diffrnt powrs as in th dimnsional quation (11). (1) 6. Entropy and condnsd mattr Lt us considr othr two xampls from condnsd mattr physics: on is concrning th vibrational ntropy, th othr th ntropy of paramagnts. In th prvious sction, w hav discussd th ntropy of Dby modls. Of cours, w can hav modls considring not only th lastic wavs having a constant spd of th sound, but containing phonons and thir tru disprsions. A phonon is a collctiv xcitation in a priodic arrangmnt of atoms or molculs, such as in crystallin solids. Phonons ar obtaind from th scond quantization of th displacmnt fild of solids. An assmbly of phonons posssss an ntropy givn by : S k i ln1 i / k (15) i / k k i i / k 1 h sum is ovr all th frquncis and polarizations of th systm [8]. Einstin proposd in 1907 that a solid could b considrd an assmbly of a larg numbr of idntical oscillators. All atoms oscillat with th sam frquncy. If w us th Einstin modl of solids, and introduc th Einstin tmpratur: E k E (16) h ntropy is: E / S Nk E / 1 (17a) E / Nk ln 1 Volum August 015 (08)

4 And also: S Nk E 1 E / E / / (17b) Nk ln(1 E ) Hr w hav a simpl ratio of tmpraturs: [ S ] [ k] E (18) h vibrational ntropy of Eqs.17 appars also in th calculation of th ntropy of diatomic gass. For ths gass w hav th contribution of rotational mods too. If th molculs hav a momnt of inrtia, at low tmpraturs w hav that [9] : I k I k S Nk 1 (19) rot I k In (19), w hav, as in (11), th following dimnsional quation: S rot [ k] I k (0) E E [ k] [ k] ML E E Lt us now discuss an xampl of ntropy of matrials having a magntisation, in particular of a spin 1/ paramagnt in trms of tmpratur and applid magntic fild [6]: 1 B / k S Nk ln 1 B / k 1 (1) 1 B / k Nk ln 1 B / k 1 In (1), w hav th magntic fild and th magntic momnt. At low tmpraturs, th first trm is ngligibl and thn: B B / k S Nk k () E [ k] E Hr w hav a ratio as in Eq Black-body radiation A quit intrsting xampl of dimnsional quation is coming from th ntropy of th black-body radiation. Calculating th proprtis of radiation from a blackbody was a major challng in thortical physics of th lat nintnth cntury. h problm was solvd in 1901 by Max Planck in th approach which is known today as th Planck's law of black-body radiation [10]. h thrmodynamics of homognous and isotropic lctromagntic radiation in a cavity with givn volum and tmpratur is analysd in [11]. In this rfrnc w find that th ntropy is: I 5 8 k V () S 15h c Bsids th Planck constant h, w hav also th spd of light c. h Planck constant is th quantum of action, introducd to dscrib th proportionality constant btwn th nrgy of a chargd atomic oscillator in th wall of th black body, and th frquncy, of its associatd lctromagntic wav. Its rlvanc is now fundamntal for quantum mchanics, dscribing th rlationship btwn nrgy and frquncy in Planck- Einstin rlation: h E h () In (), w hav th angular frquncy and th rducd Planck constant. Action has th dimnsions of [nrgy] [tim], and its SI unit is joul-scond. hrfor, th corrsponding dimnsional quation is (lt us rmmbr that in th dimnsional quation mans tim): k [ k] L h c (5) k L E [ k] [ k] E c E 8. An idal Bos gas In fact, w can writ th last quation in (5), in a diffrnt form: k L [ k] E c (6) L [ k] L In (6), w usd [ c ] L. From (6), w hav that th ntropy can b th Boltzmann constant multiplid by a ratio of a givn powr of lngths (in this spcific cas, th third powr). Lt us discuss an xampl, that of an idal Bos gas. An idal Bos gas is th quantum-mchanical vrsion of a classical idal gas which is composd of bosons. hs particls hav an intgr valu of spin and oby Bos Einstin statistics. his statistics, dvlopd by Satyndra Nath Bos for photons, was considrd by Albrt Einstin for massiv particls. In 19 [1], Einstin dducd that an idal gas of bosons can form a condnsat at a low nough tmpratur. his condnsat is known as th Bos Einstin condnsat. his condnsat is a stat of mattr in which sparat atoms or subatomic particls, coold to nar 0 K, coalsc into a singl quantum mchanical ntity, dscribd by a wav function. As discussd in [1], thr is a first-ordr transition, having a critical tmpratur C. Volum August 015 (08)

5 h ntropy of this gas is givn by: 5 v Nk g( z) ln z C S (7) 5 v Nk g(1) C In (7), v V / N, whr V is th volum of th gas and N th numbr of particls. W find also th thrmal wavlngth, which is givn by: (8) mk In (8), w find th mass m of th particl. h quantity, which is namd fugacity, is dimnsionlss: from 0 to 1. h function g( z) z z / v z 0 dxx and its valus ar ranging is givn by: g(z) ln(1 z x 5/ 1 In (9), w s clarly that th ntropy of th Bos gas has dimnsions: v L k [ k] (0) L 9. Bknstin-Hawking ntropy of black hols Can w find ntropis, whos dimnsional quations hav a diffrnt powr of lngths? h answr is positiv. W hav it in Rf.1, which givs th ntropy of a black hol in a spcific formula. ) (9) As proposd by Jacob Bknstin, if a black hol wr an objct having no ntropy, this fact would lad to a violation of th scond law of thrmodynamics [15]. In fact, whn a hot gas with ntropy ntrs a black hol, onc it crosss th vnt horizon, its ntropy would disappar. o sav th scond law, th black hol must b an objct having an ntropy, th incras of which is gratr than th ntropy carrid by th gas. his ntropy dpnds on th obsrvabl proprtis of th black hol: mass, lctric charg and angular momntum. hs thr paramtrs ntr only in a combination which rprsnts th surfac ara of th black hol, as a consqunc of th "ara thorm" [16,17]. his thorm tlls that th ara of vnt horizon of a black hol cannot dcras. It is rminiscnt of th law concrning th thrmodynamic ntropy of closd systms. As a consqunc, th black hol ntropy is proposd as a monotonic function of ara: if A stands for th surfac ara of a black hol (ara of th vnt horizon), thn th black hol ntropy is givn by: A c A S BH k k L (1) G P his ntropy is known as th Bknstin-Hawking ntropy. In (1), stands for th Planck lngth: whil G,,c L P L P G c () dnot, rspctivly, Nwton's gravity constant, rducd Planck-Dirac constant and th spd of light. From this quation, it is clar that: L [ S BH ] [ k] () L In this ntropy, th black hol is idntifid with a constant tims its surfac ara [1]. his fact was clar aftr Stphn Hawking discovrd that a black hol mits radiation at a wll-dfind tmpratur : c k () 8GM his tmpratur is also known as th Hawking radiation tmpratur. M is th mass. h radius of a black hol is R GM / c, which is th Schwarzschild radius. h surfac ara is thn: G M A R 16 (5) c dq c dm h ntropy is ds whr th hat incrmnt is idntifid with th nrgy quivalnt of th in-falling mass [18]. Aftr intgration w can obtain (1). Lt us continu our dimnsional analysis: A c A S BH k [ k] L G P (6) 5 L [ k] FL M FL hn: [ S BH ] [ k] ML 5 L M 1 5 L L c [ k] [ k] 5 L L c In (6), F mans [forc] and c a [spd]. Lt us show that, starting from th dimnsions of th classical Clausius ntropy, w can arriv to BH-ntropy. E is th nrgy and W th work, which hav th sam dimnsions. k k k [ k] [ k] [ k] (8) E W pv ML (7) h volum V is th product of ara A and lngth L, thn: Volum August 015 (08) 5

6 k A k A L [ k] [ k] (9) F A L GM A L h forc F has th sam dimnsion of th gravitational forc. Considring that has th sam dimnsion of an intrnal nrgy U of a prfct gas, which is dimnsionally th sam of a kintic nrgy: U A L [ k] (0) GM A L hn: ML A L [ k] GM A L (1) L A c A [ k] [ k] 1 G M A G h And ths ar th dimnsions of Bknstin-Hawking ntropy kc A/ G. Using again (5): GM L [ k] 1 L ch [ k ] c A c G M [ k] [ k] Gh Gc h 1 E [ k] E () About th amount of ntropy, in [1] an xampl is givn. A on-solar mass Schwarzschild black hol has an horizon ara of th sam ordr as th municipal ara 77 of Atlanta or Chicago. Its ntropy is about 10 k, which is about twnty ordrs of magnitud largr than th thrmodynamic ntropy of th sun [1]. 10. h Bknstin bound Bknstin bound is th uppr limit of th ntropy S that can b containd within a givn finit rgion of spac which has a finit amount of nrgy. It is also th maximum amount of information rquird to prfctly dscrib a givn physical systm down to th quantum lvl [19]. h univrsal form of th bound was originally found by Jacob Bknstin as th inquality [19]: k RE S () c whr R is th radius of a sphr that can nclos th givn systm, E is th total mass nrgy including any rst masss. Not that th xprssion for th bound dos not contain th gravitational constant G. h Bknstin-Hawking ntropy of black hols xactly saturats th bound. Lt us considr th dimnsional quation of this bound: RE LE [ k] [ k] hc 1 hl () E E [ k] [ k] 1 E E h bound is closly associatd with black hol thrmodynamics. 11. Entropy of vacuum h vacuum has ntropy too. In quantum mchanics and in quantum fild thory, th vacuum is dfind as th stat of considrd systm with th lowst possibl nrgy. his nrgy is known as th zro-point nrgy. Sinc all quantum mchanical systms undrgo fluctuations, bcaus of thir wav-lik natur, th vacuum is subjctd to fluctuations too. h fluctuations ar tmporary changs in th amount of nrgy, as givn by th Wrnr Hisnbrg's uncrtainty principl. h zro-point nrgy of quantum lctrodynamics was an important rsult in th thory of quantizd filds [0,1]. h Casimir ffct dals with th modification of this nrgy. h original analysis, proposd in [], providd th basic problm by calculating th forc btwn two conducting plats, du to th modification imposd by thir prsnc, to th possibl lctromagntic mods. For an lctromagntic fild, th vacuum may b considrd as its quilibrium stat in th limit of vanishing tmpratur: in [0], th authors hav studid th xtnsion to finit tmpraturs. Onc th Casimir fr nrgy had bn calculatd, by th standard thrmodynamic formula, th prssur on th plats can b obtaind from it [0]. In this rfrnc, th radiation is supposd confind btwn two conducting plats. h dg siz of both plats is L. h first is placd at z 0 in th XY plan. h scond plat is placd at z a paralll to th XY plan. Lt us suppos L a. h ntropy of th zro-point fluctuations of th filds is [0]: () L a S k a (5) L [ k] L Eq.(5) is givn in th limit of high tmpratur. As in (6), this ntropy is a ratio of cubic powrs of lngths. In (5), w hav: 1 ( n) (6) n m1 m In [0], it is strssd that, whil th Casimir nrgy dnsity vanishs in th high tmpratur limit, th Casimir fr nrgy dnsity dos not. hus, th rsultant forc of attraction btwn th plats is of ntropic origin [0]. Volum August 015 (08) 6

7 1. Entropic forcs Lt us conclud th papr with anothr quit attractiv problm, that of a forc which, lik th Casimir forc, has its origin in th ntropy of th corrsponding physical systm. An ntropic forc is a phnomnological forc coming for th statistical tndncy of incrasing ntropy, rathr than from a particular undrlying microscopic forc []. h first ntropic approach was proposd for th Brownian motion in Rf.. o hav a forc from ntropy, w can us th following dimnsional quation: E S S L [ F] (7) L L L Lt us dtrmin th ntropic forc for a spcific phnomnology, that of th lasticity of polymrs. Polymrs can b modlld as frly jointd chains with on fixd nd and on fr nd [5]. h lngth of a rigid sgmnt of th chain is b ; n is th numbr of sgmnts of lngth b. r is th distanc btwn th fixd and fr nds, and is th contour lngth, qual n b L c to. Whn th polymr chain oscillats, distanc r changs ovr tim. h probability of finding th chain nds a distanc r apart is givn by th following Gaussian distribution : nb r nb f r dr (8) In (8), /. By using ntropy and th Hlmholtz fr nrgy, w can obtain a forc which is lik that of th Hook s law. Lt us considr th ntropy [5]: S k ln f (9) h ntropy is linkd to th Hlmholtz fr nrgy: r A S k (50) Lcb And thn [5]: da k F r (51) dr Lcb his is th law corrsponding to th dimnsional quation prviously proposd in Eq. 7. f. Lavnda, B.H. (009). A Nw Prspctiv on hrmodynamics, Springr. ISBN Frigg, R.; Wrndl, C. (010). Entropy - A Guid for th Prplxd. In Probabilitis in Physics; Bisbart, C. & Hartmann, S., Editors, Oxford Univrsity Prss. ISBN Dugdal, J.S. (1996). Entropy And Its Physical Maning, CRC Prss. ISBN Srivastava, J.P. (011). Elmnts of Solid Stat Physics, PHI Larning Pvt. Ltd. ISBN Fultz, B. (010). Vibrational hrmodynamics of Matrials. Progrss in Matrials Scinc 55(), 7-5. DOI /j.pmatsci Landau, L.D.; Lifshitz, E.M. (1980), Statistical Physics, Elsvir Ltd. ISBN Planck, M. (1901). On th Distribution Law of Enrgy in th Normal Spctrum, Annaln dr Physik IV(), DOI /andp Klly, R.E. (1981). hrmodynamics of Blackbody Radiation, Am. J. Phys. 9(8), 1981, DOI / Bos-Einstin Condnsat (BEC) 015. Encyclopadia Britannica Onlin. Rtrivd August 19, 015, from 1. Krson Huang (196). Statistical Mchanics, Wily. ISBN Bknstin, J.D. (008). Bknstin-Hawking Entropy. Scholarpdia, (10), 775. DOI 10.9/scholarpdia Vv. Aa. (015), Holographic Principl, Wikipdia. 16. Hawking, S.W. (1971). Gravitational Radiation from Colliding Black Hols, Phys. Rv. Lttrs 6, DOI /PhysRvLtt Misnr, C.W.; horn, K.S.; Whlr, J.A. (197). Gravitation, Frman. ISBN Bradford, R.A.W. (01). Entropy in th Univrs, Rtrivd August 19, 015, from Bknstin, J.D. (1981). Univrsal Uppr Bound on th Entropy-to-Enrgy Ratio for Boundd Systms, Phys. Rv. D (), 87-98, DOI /PhysRvD Rvzn, M.; Ophr, R.; Ophr, M.; Mann, A. (1997). Casimir s Entropy, Journal of Physics A: Mathmatical and Gnral 0, DOI: /005-70/0// Itzykson, C.; Zubr, J.B. (006). Quantum Fild hory, Dovr Books on Physics. ISBN Casimir, H.B.G. (198). On th Attraction btwn wo Prfctly Conducting Plats, Proc. K. Nd. Akad. Wt. B51, Vv. Aa. (015). Entropic Forc, Wikipdia.. Numann, R.M. (1977). h Entropy of a Singl Gaussian Macromolcul in a Nonintracting Solvnt. h Journal of Chmical Physics 66 (), DOI / Vv. Aa. (015). Rubbr Elasticity, Wikipdia. his xampl on th ntropic forc hlps us to conclud th papr strssing th importanc of dimnsional analysis too. In fact, a gussd dimnsional quation can suggst a nw approach to solv a spcific problm. Rfrncs 1. Langhaar, H.L. (1980). Dimnsional Analysis and hory of Modls, Wily. ISBN Worstll, J. (01). Dimnsional Analysis: Practical Guids in Chmical Enginring, Buttrworth- Hinmann. ISBN Ebling, W.; Sokolov, I.M. (005). Statistical hrmodynamics and Stochastic hory of Nonquilibrium Systms, World Scintific Publishing. ISBN Volum August 015 (08) 7