Solutions to Assignment #6

Transcription

1 Chem 45/544 Fall 00 0/9/0 Solutions to Assignment #6. he mola entopy change accompanying an isothemal, quasi-static expansion of an ideal gas can be expessed S IG S, S (, ln( /. (- ( (a Conside the statistical elation S k B ln Ω in light of Eq. -. What does such a compaison imply about the way in which the numbe of states available to a single molecule must scale with? (b Fo a had-sphee gas (at low densities, the equivalent of Eq. - is b S HS ln (- b whee b πσ A / with σ the had-sphee diamete. Is the entopy change of a had-sphee gas geate than o less than that of an ideal gas fo a given expansion (i.e. fo the same values of and? Does this diffeence make sense in tems of Ω aguments? IG HS (c eithe S no S depend on the tempeatue. It is instuctive to ask whethe this tempeatue independence is a geneal phenomenon. Do so by using the Maxwell elation ( S / ( / to deive expessions fo the entopy change accompanying isothemal expansions of fluids obeying the ideal gas, van de Waals, and edlick-kwong equations of state. (You can leave these expessions in diffeential fom. Is S tempeatue independent in all of these model fluids? y to povide some physical intepetation of you findings. (a Witing Eq. - in tems of entopy athe than mola entopy, S n ln( n / n kb ln( / and compaing this expession to the change pedicted by the statistical equation, S k B ln Ω kb ln Ω, leads to the conclusion that Ω / Ω ( /, i.e. that fo fixed tempeatue (which is the same as fixed enegy fo an ideal gas Ω. Aside: his esult can be ationalized in seveal ways. Fist, we have aleady found that fo an ideal gas Q. If one ecalls that Q(,, can be thought of as a measue of the effective numbe of states available to a system at constant, it is easonable to suppose that a simila behavio would be found fo Ω(,,E at constant E. Altenatively, one might pictue space as being divided up into a set of a vey lage numbe M of lattice cells, vcellm, into which molecules (<<M can be placed such that each cell is at most singly occupied. he numbe of ways of aanging indistinguishable and independent molecules into M sites is Ω ( M, M!/!( M!. In the limit of lage M and with M>> this combinatoial facto can be shown to educe to ln Ω ( M, ln M, i.e. Ω. (b he diffeence between the two entopy changes can be witten:

2 Chem 45/544 Fall 00 0/9/0 ( S HS S IG b / n ln ln b ( b ln ( b -b/ ln -b/ Given that the volumes and the size paamete b must be geate than zeo, and that fo an expansion, >, it is easy to show that b / > b / ( / < /, b / > b /. hus, the atio HS HS b / /( b / and S > S. ( > hat the change in entopy upon expansion is geate fo a had-sphee gas than fo an ideal gas seems logical. By analogy with the ideal gas, it would seem be easonable to suppose that the degeneacy of a hadsphee gas should be elated to the available volume b as Ω ( b. hus, fo a given and thee is a geate elative change in the volume available to molecules and theefoe a geate elative change in Ω and S. ( ote, howeve, that fo any the entopy of the had-sphee gas would be expected to be less than that of the ideal gas due to the fact that some of the volume is unavailable in the fome case. (c Using the Maxwell elation povided, the thee equations of state yield ideal gas: S a van de Waals b S b edlich-kwong / A B ( + B S A + B ( / + B Fo the ideal and van de Waals equations of state S is independent of. Howeve, this behavio is a esult of the tempeatue independence of the attactive tem in these two equations. Moe ealistic equations of state, like the edlich-kwong equation, incopoate tempeatue-dependent attactive (and sometimes epulsive tems, which lead to a tempeatue dependence of S.. M&S 6-0: Calculate the values of q ev and S along the path (,, (,, (,, [lines D+E in Fig. 6.] fo one mole of gas whose equations of state ae U U ( and /( b. Compae you esult with that obtained in Example 6-. Along any evesible path du δ q + δw δq d and ds δqev /. Fo a system in which ev U U (, we also have du C d. Combining the two elations fo du, solving fo δ qev, and using mola quantities povides the elations:

3 Chem 45/544 Fall 00 0/9/0 and δ q ev C d + d C d + d b C C ds d + d d + d b which hold fo any evesible pocess with this gas. Fo the paticula path D,,,, q D ev S D ( C ( d + d path D C ( d + ( path D C ( d + ln C ( d + d b b b Fo path E,,,, ( ( C ( d + d C ( d + d b ( ( q E ev ( C ( d + d C ( d path E C ( d E C ( S d + d b path E he net changes ae theefoe: C ( d C ( d q D+ E ev ( and ln S D b b Compaing to the esults of Example 6- we find, as expected, D+ E A B+ C S S S. q D+ E ev A ev B+ C ev q q but. (a M&S 6- Deive the equation dh ds + d and show that S C ln( / ln( / fo the change of one mole of an ideal gas fom (, to (, assuming that C is independent of tempeatue. (b How would the esult change if instead of being tempeatue independent the heat capacity could be witten C C ( a + a + a ?

4 Chem 45/544 Fall 00 0/9/0 (c Explain how the equation you deived in (a depends on the gas being ideal i.e. what would be diffeent fo the geneal case? (a Fo a evesible pocess involving only wok, du δ q + δw ds d he definition of enthalpy is H U +, so that the diffeential of H is dh du + d( du + d + d o dh ds + d Solving this expession fo ds and conveting to mola units one has ds dh d (4- H In the case of an ideal gas H H ( and theefoe dh d Cd. Substituting this expession fo d H into Eq. 4- and using the ideal gas law to ewite / /, we have C ds d d (4- If C is independent of tempeatue. Eq. 4- is easily integated: S (, S (, C d d C o S S, S (, C ln( / ln( / (4- as desied. d ( d (b If the heat capacity wee of the fom C a0 + a + a +... the tempeatue and pessue dependences would be sepaable as above. he only diffeence in this case is that the tem would be moe slightly complicated: so that C ( d a0 ln( + a + a +... S { a0 ln( / + a( + a( +...} ln( / (c Equation 4- is exact and does not depend on the natue of the system consideed. he main simplifications in the case of an ideal gas wee aleady mentioned in connection with deiving Eq. 4- fom Eq. 4-. In the geneal case one would have to wite 4

5 Chem 45/544 Fall 00 0/9/0 dh H H d + H d Cd + d and / would not be a function of alone. he diffeential expession fo S (, would then be the moe complicated fom: C (, H ds d + d (4-4 In geneal we expect the tem multiplying d in Eq. 4-4 to be a function of both and. Fo a non-ideal gas one also expects that C C (, as indicated. Both of these changes uin the sepaability of the and vaiables that allowed Eq. 5- to be easy integated. [Aside: Using the methods of Chapte 8 one can expess the d tem in Eq. 4-4 in tems and the volumetic equation of state of the system as H Fo an ideal gas whee / the deivative ( / is only a function of. Fo othe equations of state, ( / will in geneal be a moe complicated function of both and.] 4. A 00Ω esisto of mass.00g and heat capacity C 7.55 J K - g - is placed in an ice-wate mixtue at 0 C which is contained in a dewa flask and a cuent of 00 ma is passed though the esisto fo.00 minutes. Once the system has eached equilibium: (a Compute the entopy change of the esisto. (b Compute the entopy change of the ice-wate bath. (c Compute the quantity of ice that is melted in the pocess. (d Suppose that the esisto is emoved fom the dewa and themally insulated. Compute S fo the esisto and the suoundings. he electical enegy input to the esisto is initially dissipated as heat within the esisto. he amount of enegy involved is q It I t whee I is the cuent, the voltage dop, the esistance, and t the duation of the cuent flow. umeical evaluation yeilds: I : 0. A : 00Ω t :.00 min q : I t q.08 0 J (a he esisto in contact with the ice-wate bath will maintain a tempeatue of 0 C o at least come back to this tempeatue afte some time has passed. So if one waits until equilibium is estoed, thee will be no discenable change in the state of the esisto and thus its net change in entopy must be S 0. 5

6 Chem 45/544 Fall 00 0/9/0 (b he entopy of the ice-wate bath will change by an amount S W.95 J K -. S W q, which fo 7.5 K is (c he amount of ice melted is dictated by the balance between enegy o entopy given off by the esisto and the tansition enthalpy o entopy of the melting tansition. In tems of the numbe of moles of ice melted, n Using a value of q S W. fus HW fussw H fus W ( ba, 0 C 6008 J mol- one finds that n 0.8 mol. g of ice ae melted. (d Assuming that the esisto heat capacity is independent of tempeatue, the entopy change of the esisto would be: f C S d C ln( f / i i whee i and f ae the initial and final tempeatues of the esisto. he final tempeatue of the esisto will be dictated by the fact that the electical enegy (wok added at constant pessue must be balanced by the enthalpy incease of the esisto so that: f H Cd C ( f i w i w f i + and C S C ln + Assuming an initial tempeatue of i 0 C, this yields is S.5 J K -. If the esisto is completely insulated thee will be no effect on the suoundings and thus thee will be zeo entopy change in the suoundings. w C qe ln q 5. M&S 7-7: Show fo an ideal gas that S ln +. A he entopy of a system is elated to its canonical patition function Q by: S k B ln Q ln Q + k B, i 6

7 Chem 45/544 Fall 00 0/9/0 Fo an ideal gas in the dilute limit (whee Boltzmann statistics apply Q can be witten in tems of the molecula patition function q as: { q(, } Q(,,.! aking the logaithm of this expession and using Stiling s appoximation, and ln Q ln q ln! ln q ln + {ln q ln + ln e} ln qe ln Q ln q, utting these pieces togethe yeilds S k B qe ln q ln + k B ln! ln, Finally, witing this expession in mola units, n A, S ns, and A k B, povides the desied esult qe ln q S ln +. A 6. (a M&S 7-9. Substitute Eq. 7. fo the molecula patition function of a monatomic ideal gas into the geneal equation fo entopy in tems of Q(,,, Eq. 7.9, and deive the equation: S C ln + ln. (b Equation 7. and theefoe the equation deived fom it in this poblem ae not exact (even fo an ideal gas. Explain the appoximations inheent in Eq. 7. and discuss unde what conditions you would expect deviations fom the equation you deived to become impotant. (a As given in Eq. 7., the molecula patition function fo a monatomic ideal is: q / πmk B (, g e h athe than stating with Eq. 7.9, we may as well stat with the esult of the pevious poblem, which elates S diectly to q: (6-7

8 Chem 45/544 Fall 00 0/9/0 qe ln q S ln +. (6- A Because we want to descibe diffeences between state points, the constant tems in ln(q will not appea in the final esults. Fo this eason it is useful to collect these tems in the beginning and use Eq. 6- to wite lnq as: / πmk B ln q ln + ln + A whee A ln g e (6- h Diffeentiating this expession wt yields: ln q (6-4 Beaking out the extaneous constant tems in Eq. 6- and inseting Eqs. 6- and 6-4 into the esulting equation povides: ecognizing that witten: ln q S ln q + + ln { ln + ln + A} + + ln ln + ln + constants C S S (, C e A e A fo a monatomic ideal gas, the diffeence in S between two state points can be ln S (, C + ln ln C ln + ln ln (b he appoximations inheent in Eq. 7. ae the following: i teatment of tanslational motions using the paticle in a box model and assuming that the spacing of tanslational levels is small elative to k B (so that the quantum sum can be eplaced by a classical integal ii assumption of the dilute limit ρ Λ << (so that Boltzmann statistics can be used iii tuncation of the sum ove electonic states to only the st tem, i.e. g ie electonic states i exp( ε / k g ie B e + g e e ε e / kb + g e e εe / kb Of these appoximations, (i is essentially exact, (ii is usually an excellent appoximation except fo the lightest atoms (mainly He at low tempeatues and high densities, and (iii will beak down if the atom has excited states that ae low enough in enegy that k B /ε is not much less than unity. By fa the most common deviations fom Eq. 7. will occu at high tempeatues whee appoximation (iii is no longe accuate. g e 8

9 Chem 45/544 Fall 00 0/9/0 (Fo example, in class we discussed seveal cases in which electonic contibutions to heat capacities became non-negligible at tempeatues geate than 000 K. 7. M&S In each case below, pedict which molecule of the pai has geate mola entopy unde the same conditions. Assume gaseous species and explain you easoning. (a CO > CO lage numbe of atoms, geate mass, geate moment of inetia, lowe vibational fequencies (b popane > cyclopopane geate confomational feedom, geate moment of inetia, lowe vibational fequencies (c n-pentane > neopentane - geate confomational feedom, geate moment of inetia, lowe vibational fequencies (a D O > H O geate moments of inetia, geate mass, lowe vibational fequencies (b ethanol > ethylene oxide - geate confomational feedom, geate moment of inetia, lowe vibational fequencies (c -aminobutane > cyclo vesion - geate confomational feedom, geate moment of inetia, lowe vibational fequencies 8. O is an unsymmetical linea molecule. (a Use the molecula constants povided in M&S able 4-4 to compute S o ( point, K. [ote: he value of σ listed in able 4-4 is incoect.] fo O(g at its boiling (b What ae the % contibutions made by the diffeent molecula degees of feedom to the total entopy at 00, 500, and 000 K? (c M&S 7-0. [Hint: A tapezoidal integation of the solid-phase data is sufficiently accuate fo the puposes of this poblem.] (d Compae you esults fom pats (a and (c and discuss the likely souce of any diffeence you obseve. (a he Mathcad woksheet used to pefom these calculations is shown on the following page. (b he contibutions of the vaious modes ae as follows: 00 K 500 K 000 K S / (J K - mol S / % tanslation % otation % vibation

10 Chem 45/544 Fall 00 0/9/0 ote that the vibational contibutions to the entopy ae much smalle than to the heat capacity at a given tempeatue. ote also that even at 000 K they povide less than 0% of the total entopy. Fundamental Constants & Convesion Factos: k B : J K A : mol h : Js : k B A amu : kg mol ba : 0 5 a dm : 0.m efeence conditions: 0 : 84.67K 0 : ba 0 A Molecula Constants: M : ( amu Θ ot : 0.60K g e : σ : Θ v : 00K Θ v : 850K Θ v : 840K g v : g v : g v : Entopy Calculations: π M k B e.5 S tans ( : ln S tans 0 h.5 kb 0 e S ot ( : ln S ot 0 σθ ot ( : g S v, Θ, g ln e Θ Θ + e ( S v (, Θ v, g v Θ S vib ( : S v, Θ v, g v + + S v, Θ v, g v ( S el ( : ln g e S el ( 0 0 S ( : S tans ( + S ot ( + S vib ( + S el ( ( S ( ( ( S vib ( 0 S ( JK mol (c A plot of the heat capacity data is shown below. he contibutions to the entopy fom vaious egions between 0 K and oization ae summaized on the following page. he method used fo detemining the entopy contibution fom vaious egions was the following. (umbes efe to the tabulated esults shown below. egion #: Fo the egion between 0K and 5.7 K whee no data is available, I used a Debye ( fit to the lowest expeimental point available, i.e.: C ( a with a C ( min / min. Upon integation this appoximation povides a contibution to the entopy of C /. egion #: o integate the heat capacity in the egion whee solid-phase data ae available, I simply calculated values of C / and integated them numeically using a tapeziodal appoximation. (his fom of integation is most convenient since the data ae unevenly spaced. I did this using the Excel sheet shown ( min 0

11 Chem 45/544 Fall 00 0/9/0 at the ight above. Altenatively, one could have instead fit the C ( data in the solid egion. Doing so, I obtained the fit: C /( J K mol ( / K ( / K +. 0 ( / K which can be integated numeically to yield a value of 69. J K - mol - little diffeent (0.% fom the esult obtained above. egion 4: Given the small tempeatue ange of the liquid phase (<K, I used the aveage value of the expeimental C points to pefom the liquid-phase integation. Heat Capacity C /(J K - mol Heat Capacity of O Solid egion Debye Fit Liquid Melting ansition empeatue /K /K C C/ apezoidal Integation dx sum Sum: 69. # S Contibution /(J K - mol - C / integation K 0.97 C / integation 5.7 f ansition entopy f H / f C / integation f v.04 5 ansition entopy v H / v 89.5 otal: 96.5 (d he final esult fo the entopy of the gas at the oization point, 96.5 J K - mol -, is significantly smalle than the calculated esult (0.9 J K - mol - smalle by 6.4 J K - mol It is unlikely that a diffeence this lage could be due to expeimental eo o, as we will show on the next poblem set, to gas non-idealities. (ote that the compaison povided by ock s able 4-4 at 98 K is S o ( cal 5. J K - mol - vesus S o ( stat 0.0 J K - mol -, which diffe by a simila amount, 4.9 J K -

12 Chem 45/544 Fall 00 0/9/0 mol -. We must conclude theefoe that thee is esidual entopy in the cystal at 0 K i.e. that the d Law assumption doesn t apply to O. It seems plausible to ascibe this esidual entopy to end-fo-end disode of the nea symmetic linea molecule O. his disodeing mechanism would be expected to poduce a value of S esid ln 5.8 J K - mol , close to the discepancy obseved. 9. (Exta Cedit In 884 F.. outon made an expeimental obsevation that has come to be known as outon s ule. he ule states that the entopy of oization of liquids at thei boiling points ( ba o is appoximately constant, vs ( b / (a Find expeimental data fo 0 o moe substances with which to test this ule. Compile values of b and v S o ( b. (hose compounds listed in M&S poblem 7-0 ae off limits. (bdiscuss possible explanations fo this ule and the limits of its applicability. (a I ve compiled data (fom J. A. iddick, W. B. Bunge, and. K. Sakano, Oganic Solvents (Wiley, ew Yok, 986 plus the data in poblem 7-0 of M&S in the following table. I ve gouped compounds into two categoies compounds that might be unsuitable fo this type of coelation due to the pesence of significant hydogen bonding in the liquid (o o states o due to thei quantal natue (He, H, and the emainde. I ve also odeed them accoding to boiling point and plotted both sets of data at the ight. Substance b/k S/ Substance b/k S/ n-c8h A pyole 40.6 O n-c0h CH4 8.8 I Xe DMSO 46. HS 0.6 n-ch CH8 9.8 GaI 6. Cl 9 0. Hg 60. CHOCH a n-c4h CdCl. C(CH AgB ethylene oxide Ag 46.4 acetaldehyde Al 79.7 diethyl ethe B 47.5 n-c5h "Undesiables" (H-bond o Ψ CS 9 0. He 4.4 acetone H SiCl4 0.4 H 40.7 B 0.9 methanol 8.6 tetahydofuan 9.4 ethanol n-c6h n-popanol CCl wate 7. C6H fomic acid cyclohexane acetic acid acetonitile popionic acid n-c7h n-hexanol 40.8 apoization Entopy S / Summay of S/ Data ln( /K All Othes H-bonding apoization empeatue b / K

13 Chem 45/544 Fall 00 0/9/0 Conside fist the behavio of the suitable data set (blue cicles. In contast to outon s ule, fom which one would expect the to be a constant, these data appea to vay with b in a systematic S way. o a easonable st appoximation the deviation fom outon s ule seems linealy elated to ln b. he substances fo which one expects significant hydogen bonding deviate consideably fom the coelation established by the othe data. Ove a elatively naow ange of b, these substances exhibit values of S( b / both geate than (wate, H, alcohols and less than (caboxylic acids the outon value of ~0.5. (b Coesponding states agument: Fist conside how one might explain the egulaity displayed by the suitable substances. One appoach would be to invoke coesponding states ideas to explain the appoximate constancy of S( b /. One agument is the following. Suppose that instead of making compaisons at the same fixed pessue of ba (o atm these compaisons wee to be made at some fixed educed pessue / c. hen one would be compaing values of: S ( H ( (- whee is the tempeatue at which the o pessue of a given substance equals the pescibed educed pessue. o the extent that coesponding states behavio applies to liquid-o coexistence, this tempeatue must coespond to the same educed tempeatue ( fo all substances. So c ( whee ( is a constant. We will see in the next chapte that, at least to the extent that the o can be teated as ideal, H ( should also be the same fo all substances when consideed in suitably educed units. Since H epesents an enegy, the simplest way to tansfom it into a dimensionless educed quantity is to divide by c. hat is the atio H ( / c should be the same fo all substances. Symbolizing the univesal values of ( by f ( and H ( / c by g ( one can wite: S ( H ( g ( a constant fo fixed f ( (If you don t like this macoscopic agument, you can also invoke the idea that both and H ( should be scale with the well-depth paamete ε, say in a Lennad-Jones wold. But outon s ule does not compae the oization pocess at the same educed pessue, so ae such aguments useful? Yes, fo the following eason. he citical pessues of the compounds consideed hee ae on the ode of 50±0 ba. Using the Clausius-Clapyon equation (Ch. 9 one can elate H ( b to H as: ( (- H ( b b H ( ln( ln( / ba c (- Fo c 50±0 ba the last (and only vaiable tem in this expession is expected to vay by ± 0.8, which is compaable to the scatte of the data about outon s value.

14 Chem 45/544 Fall 00 0/9/0 Within this coesponding states viewpoint the fact that hydogen bonding (and quantal fluids do not fit outon s expectations can simply be ascibed to the fact that they do not follow the same coesponding states behavio as othe fluids. Statistical mechanical agument : A moe satisfying explanation fo the simple behavio of suitable substances can be found by consideing molecula intepetations of the entopy. Given that the only entopies we know how to calculate ae those of ideal gases, suppose we assume that the liquid to o tansition can be viewed meely as an isothemal expansion of an ideal gas between liq ( b and ( b. his (cude pictue would pedict: IG S ( b exps ( b ln{ ( b / liq ( b } (-4 If one assumes that the atio ( b / liq ( b is oughly constant fo all substances at thei boiling points, 4 and if this constant wee equal to exp( , outon s ule would esult. Is this intepetation plausible? If one looks up values fo the mola volumes of liquids at 98 K ( liq ~00 cm mol - IG 4 - and assumes ideal gas behavio (.5 0 cm mol, one estimates that / liq is on the ode of 50 at 98 K much smalle than the value needed. Howeve, it is easonable to say that the ideal gas volume that should be consideed in Eq. -4 is not the mola volume itself but athe some sot of effective fee volume δ eff in which molecules can actually move, i.e. the mola volume minus the volume excluded by the pesence of othe molecule ( -b in van de Waals language. his diffeence in pespective would not change substantially, but it would have a lage effect on the value used fo the liquid. In ode to epoduce outon s ule via such easoning one would need δ eff liq. hat is, molecules in the liquid would have to be thought of as ideal-gas-like species moving within a volume that is less than % of the total volume pe molecule. Since solids expand by something like 0% upon melting, a factional fee volume of <% is not a ealistic estimate of the actual fee volume available fo motion of molecules in a liquid, but it is at least within an ode of magnitude of expectations. If one likes this geneal appoach to ationalizing outon s ule (I do, one says that an ode of magnitude is close enough and ecognizes that otational motions ae also esticted by liquid confinement and that the feeing up of otational and othe motions in the oization pocess ae also hidden in this δ and ende it smalle than othewise expected. eff An inteesting aspect of the above pespective is that it leads quite natually to an impovement in the pedictive abilities of outon s ule. Hildeband, fist ecognized that it ( b / δ eff ( b is not eally expected to be constant, since vaiations in b among diffeent substances will cause ( b to vay substantially. He suggested that S should be constant fo the same concentation of o -- i.e. fo oization to a common efeence volume ef, athe than to the volume petaining to b. Eveett, (and late Shinoda suggested that one can easily coect fo this vaiation by noting that the vaiations in IG caused by these vaiations in o volume ae given by: S IG IG { S ( S ( }/ ln( / ln( / ln( / K + a constant (-5 b ef b ef b ef b 4

15 Chem 45/544 Fall 00 0/9/0 his ealization leads to the impoved expession known as the HE (outon-hildeband-eveett equation: S ( b ln( b / K (-6 A fit of the class data to this geneal fom yields S ( b / ln( b / K, vey close to this accepted semi-empiical ule. he Undesiables: Finally, what can be said about the cases that don t fit on this HE coelation? he hydogen bonding species ange fom having values that ae highe than the nominal S ( b / 0. 5 by +5 to, which tanslate into Ω factos of Ω exp( S / 50 to /0. In the case of the positive deviations displayed by wate, ammonia, and alcohols, one could say that hydogen bonding in the liquid state esticts feedom in the liquid state but that this estiction is lagely o entiely lifted in the o state. hus the entopy change is lage fo such species than fo molecules not having this type of association. In the case of the negative deviations fom outon s value one must conclude that the o is less disodeed elative to the liquid than expected. he caboxylic acids, which show such negative deviations ae well-known fo dimeizing in the o phase and one can easonably attibute thei deviant behavio to this dimeization*. (*One should be a little caeful hee. If using S data that efe to the actual oization pocess gas-nonidealities such as dimeization ae eflected in the values. Howeve, if one wee to look up standad state data fo the liquid and o states and subtact the two to obtain a standad value of o S a diffeent answe would be obtained. he o state in this case is the hypothetical ideal gas state which does not contain the effects of dimeization and the esult would theefoe be entiely diffeent. esumably in this case one would find a positive deviation fom outon s ule as in the othe hydogen bonding species. efeences on outon s ule and apoization Entopies:. S.. Logan, "he ationale of outon's ule and of the outon-hildeband-eveett ule," Z. atufosch. 5a, ( the only simple statistical "deivation" I've seen. L. K. ash, "outon and -H-E ule," J. Chem. Ed. 6, ( a discussion of the histoy on the centennial annivesay of outon's ule. K. Shinoda, "Entopy of apoization at the Boiling oint," J. Chem. hys. 78, 4784 (98. - a ediscovey of Eveett s contibution 4. J. Wisniak, "Fedeick homas outon: he Man, the ule, and the atio," Chemical Educato 6, 55-6 (00. - a popula account (not teibly savvy when it comes to chemisty 5. I. C. Sanchez,. M. uskett, and. J. in 't eld, "Configuational opeties and Coesponding States in Simple Fluids and Wate," J. hys. Chem. B 0, ( a moden discussion of the eal theoetical basis of liquid entopies 5