Assignment #7 - Solutions
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- Claude Potter
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1 hem 453/544 Fall 003 /05/03 Assgmet #7 - olutos. M& #0. 0.4: 0.: Euler s theorem says that... s homogeeous the rove Euler s theorem by deretatg the equato roblem 0- wth respect to ad the settg. Apply Euler s theorem to G G to derve Eq : se Euler s theorem to prove that c c Y Y... or ay etesve quatty Y. 0.4: Apply Euler s theorem to. Do you recoge the resultg equato? 0.: Beg wth the deto o a st order homogeeous ucto provded problem 0-: Deretate both sdes o ths epresso wth respect to as structed. Deretato o the r.h.s. s easy. he result s.... Deretato o the l.h.s. s accomplshed as ollows: I have t eplctly stated what s beg held costat these partal dervatves. What s costat are all o the products j other tha the partcular beg deretated a gve term. I ths deretato seems suspect to you just thk o deg a ew set o depedet varables y ad deretatg... y y y wth respect to. Gve that ca be ay costat set t equal to uty to d: j j j Oe ca apply ths theorem to the Gbbs eergy G G whch s a homogeeous st order ucto o the mole umbers ad but ot o ad smply requrg that be held costat durg the deretato:
2 hem 453/544 Fall 003 /05/03 G G G G G r r r r µ µ 0.3: he ormal deto o a etesve quatty s oe that s a st order homogeeous ucto o all o the mole umbers whch specy the overall se o the system. he mathematcal statemet o ths deto s: c c Y Y. As wth G above Euler s theorem ca be appled to Y we cosder ad to be costats durg the deretatos wrt mole umbers: c c c Y Y Y j... r. 0.4: All o the depedet varables the relato are etesve.e. so that applcato o Euler s theorem provdes: Idetyg the dervatves ths epresso wth basc thermodyamc dervatves oe ca rewrte t the orm: µ µ ote the resemblace to the deretal relato d d d d µ. hs resemblace makes t look as oe smply tegrated d. hs smlarty holds oly because all o the atural varables o are etesve quattes. It s mportat to recoge that smlar relatoshps est or the other thermodyamc potetals lke the Gbbs eergy o 0. but oly the etesve quattes get tegrated : hus although d d d da µ oe does t have A µ but rather G A µ etc.. I opc ummary 0 the ugacty o a pure codesed phase was wrtte as: d R ep. where s the ugacty o the or phase that coests wth the codesed phase at temperature ad pressure.e.
3 hem 453/544 Fall 003 /05/03 R ep d. R 0 a se the geeral epresso or the ugacty o a sgle-compoet system IG µ µ R ep ep d. R R 0 to derve Eq... [Ht: Break up the orgal tegral Eq.. to two parts oe over pressures 0 to ad the other over pressures to ad remember that phase coestece mples equalty o ether chemcal potetals or ugactes.] hroughout hapter 0 M& employ the appromato or ts equvalet or mtures. hs appromato assumes that the ugacty coecet o the or / s uty thereby eglectg ay or o-dealtes ad that the oytg correcto ep{ d } s also uty. Eame R both o these appromatos or the case o lqud beee at a pressure o bar ad temperatures o ad 0 usg the coestece data provded M& Fg 9.3. roceed as ollows: b Estmate the ugactes o beee or at these three temperatures usg the epresso or l / o a va der Waals lud opc ummary 8. c alculate the magtude o the oytg correcto at 30. At ths temperature the desty o lqud beee s g cm - ad or the pressures relevat here lqud beee ca be cosdered compressble. d At what pressures would the magtude o the oytg correcto o part c der rom uty by % 5% ad 0%? Aga assume that beee ca be treated as a compressble lqud. a Beg by wrtg the geeral epresso or the ugacty o the codesed phase terms o a tegral over ad the break the tegral to two peces as suggested: ep R 0 R d R ep d R R 0 R d R ep d ep R R 0 R d he rst term brackets here s smply the ugacty o the codesed phase at the temperature ad pressure. I s the equlbrum or pressure o the codesed phase or the temperature the chemcal potetals o the codesed ad or phases must be equal ad so too must 3
4 hem 453/544 Fall 003 /05/03 ther ugactes.e.. Makg ths substtuto ad breakg up the remag tegral oe has: R ep d R d ep d R ep d l / R ep d R as desred. b he pressures I read rom Fg. 9.3 or temperatures o ad 0 are: 3 56 ad 83 torr respectvely. ce I was workg Mathcad ayway I also computed these pressures rom the or pressure equato provded M& problem 9.9. he results tabulated below are close to those read drectly rom the graph. Gve a sucetly smple equato o state the tegral relatg to volumetrc data ca be perormed aalytcally. sg the va der Waals e.o.s. R a b to represet a o-deal or phase opc ummary 8 provded the ugacty the orm l / Z l Z B A/ Z. he ugacty coecet γ M& or ϕ most tets whch descrbes the devato rom dealty s thus: where ϕ { Z l Z B A Z} / ep / Z / R A a /R ad B b / R. o evaluate ugacty coecets I used a Mathcad worksheet below ad obtaed the results: / /torr /bar Z ϕ ϕz he colum labeled ϕ s the result obtaed rom a eact calculato o Z usg the d ucto o Mathcad ad the colum labeled ϕz s what s obtaed makg the appromato Z. As show above ths appromato s qute accurate the preset case. 4
5 hem 453/544 Fall 003 /05/03 ote that eve at pressures above bar the error assumg that the or s deal s oly about 4%. hese estmates are typcal o the small eect that or-phase o-dealtes have at pressures o less tha a ew bar. mlar values o ϕ are oud or most gases eve polar ad hydroge bodg gases lke ad methaol. I estmated ugacty coecets o ~ or ad methaol at ther bolg pots. he oly cases whch ths o-deal or correcto s mportat at low pressures are oes that volve molecules whch assocate strogly the or phase due to specc teractos. Oe eample o such a case s acetc acd whose or s odeal eve at low pressures because o etesve dmerato. At ts bolg pot 8 acetc acd or has a ugacty coecet o 0.4. Mathcad worksheet: Fudametal ostats & overso Factors: R 8.34 J K : mol dm : 0. m bar : 0 5 a torr 33.3a apor ressure Eq. M& prob. 9-9: ap : bp : cp : dp : ep : p : c : 56.75K t : 0 : K t 73.5 v : bar ep ap bp cp dp ep 3 p.70 v c torr Fugactes coecets rom va der Waals EO: vdw costats M&able.3: : v c A : a R B : Mathcad "olve" Block to d : a 8.876dm 6 : bar mol b R Gve : : R Fd R b a b 0.974dm 3 : mol Z : R Z φ : ep Z l Z B A Z φ c For a compressble substace the oytg correcto ca be appromated: orr ep d R ep d ep R R For the codtos: 9 torr 30 ad bar ad a molar volume o lqud beee o 90.0 cm 3 /mol see below I estmate ths oytg correcto to be a actor o
6 hem 453/544 Fall 003 /05/03 d olvg the above equato or orr R l orr / the total pressures at whch the oytg correcto reaches 5 ad 0% are.9 4 ad 7 bar respectvely. he Mathcad sheet I used or these calculatos s show below. : K d : 0.868kg dm 3 MW MW :.0783kg mol L : L m 3 d mol : 9 torr : bar orr ep L : orr.003 R R orr : L l orr.0 bar bar ` bar a M& 0-: he or pressures o beee ad toluee betwee 80 ad 0 as a ucto o Kelv temperature are gve by the emprcal ormulas: l be / torr K / ad l / torr K / Assumg that beee ad toluee orm a deal soluto use these ormulas to costruct a temperaturecomposto dagram o ths system at a ambet pressure o 760 torr. b O the plot you make llustrate the use o the lever rule or determg the % o the mture that s lqud versus or whe the system cossts o 40 mole % toluee ad the temperature s 365 K. [Ht: It s ot ecessary to solve equatos or ad y. mply solve the equatos provded * * problem 0-7 or ad y as uctos o ad or a ed total pressure ad plot { } ad { y } pots.] I wll label beee as compoet ad toluee as compoet. he relatos eeded or ths problem are the: tol 760 torr tol be be y tol 760 torr F L L L L wth all pressures epressed torr. a alculatos were perormed usg a Ecel spreadsheet ad the results plotted below. b he racto o the system the lqud s F L where the legths L ad L are deed the plot. I d: F L 0.65 F
7 hem 453/544 Fall 003 /05/03 a b b d Beee oluee /K be tol y / emperature /K or L L lqud Mole Fracto oluee y 4. At 60 the or pressure o methyl acetate s.6 bar ad the or pressure o methaol s bar. her ecess Gbbs eergy o mg ca be descrbed by the Margules equato: G e / R.06. a Estmate the ugactes o the two compoets the mture ad the mture s or pressure as uctos o composto at 60 ad make a plot o your results. learly state ay appromatos you make your aalyss. b What s the Hery s law costat or each compoet? a he ugactes o the two compoets the lqud mture ca be wrtte terms o ther actvty coecets by: L r L* r L* r a γ I have t eplctly dsplayed the depedece o these quattes o ad but t s to be uderstood that e these equatos apply to a specc ad as does the epresso or G. he actvty coecets are related to the Margules costat A.06 by: r r γ ep ad γ ep A A sg as the composto varable the ugactes soluto are: L L* L L* ep A ad ep A I oe gores the oytg correcto ad or o-dealty the lqud-state ugactes ca be equated to the partal pressures o or-phase compoets as: L* L* ad the total or pressure above the soluto ca be appromated by: 7
8 hem 453/544 Fall 003 /05/03 * * γ γ I dd the calculatos the Ecel spreadsheet show below ad used the results to make the plot at the rght. MeOAc ; MeOH γ γ /bar /bar /bar Fugacty or ressure / bar Ac Fg Mole Fracto Methaol b he Hery s law costats or compoet dlute whch I ll deote H ad or dlute H are deed by the tal slopes: OH H lm L* ep A ep A 0 ad H lm ep A 0 Here H OHAc.44 bar methaol solute ad H AcOH 3.5 bar methyl acetate solute. 5. Aalyss o the parttog o a speces betwee two mmscble lquds phases volves the same logc as the parttog o a soluto compoet betwee med lqud ad or phases. a how that the dstrbuto coecet deed as the rato o the cocetratos o the speces the two phases s determed by: I r II γ K II r I γ b arbo tetrachlorde ad are almost completely mmscble meag the -phase coestece o l 4 /H O mtures s such that there s lttle carbo-tetrachlorde the -rch phase ad vce-versa or ay composto o the mture. Whe eough Br s added to a -phase l 4 /H O system to make the cocetrato o Br carbo tetrachlorde.0 mol/l the Br cocetrato s oud to be H O l mol/l. What s the dstrbuto coecet ths case ad what s the rato γ / γ? Br Br 8
9 hem 453/544 Fall 003 /05/03 c Wthout usg other sources o data other tha your bra would you guess or the values o the H O l4 dvdual actvty coecets γ ad γ? Epla your reasog. Br Br H O d Estmate the actvty coecets Br l4 γ ad γ Br usg the regular soluto model. a he requremet or two phases I ad II to be equlbrum s that the chemcal potetals or the ugactes o each compoet be the same the two phases: I r I II r II he ugactes are related to actvty coecets by: I r I o r I o I r I a γ where o s the ugacty o some pre-deed stadard state or eample the pure speces whatever orm s most stable at the speced ad bar. he choce o stadard state s rrelevat here as log as the actvtes the two phases employ ths same state or reerece. he rato o cocetratos o speces the two phases s the: I o r II II γ γ K II r I I γ γ o b he data provded s sucet to calculate the mole ractos o Br the two solutos eactly. However the mole ractos should be well appromated by makg the assumpto that the ecess volume o mg s ero. e / s at most a ew percet or all lqud mtures. hs assumpto leads to the results show o the Mathcad worksheet show o the ollowg page. Here l 4 W l4 " " ad " BW " Br : B Br L: 0 3 m 3 MW W : 8.0g mol MW B : 58.83g mol d W :.00g cm 3 d B : 3.9g cm 3 MW : 53.84g mol d :.595g cm 3 l4 oluto: B : mol MW B : L B d B d : MW 9.84mol B B : B B 0.09 Water oluto: B : 0.039mol MW B W : L B d B W d W W : MW W W 55.46mol B BW : B W BW o the dstrbuto coecet s: K γ l4 Br Br l4 Br γ Br
10 hem 453/544 Fall 003 /05/03 I you smply gored the volume o Br a relatvely accurate approach the results would have bee l ad K48.6. Br c Based o the act that the termolecular teractos Br l are smlar to those l 4 l the l4 teractos both lquds are domated by dsperso teractos I would guess that γ should be close to uty. Lqud o the other had s domated by hydroge-bodg teractos. ce the teractos are very deret rom those Br l I would atcpate that γ >>. o as a rst guess I l4 would say γ γ 56. Br Br d he regular soluto model would predct: lγ l4 Br ϕ Br l4 δ δ Br ad lγ δ δ R Br l4 Appromatg the ors as deal gases Br Br ϕ Br wth δ R l4 5 Mathcad sheet below to d: γ. 0 ad γ 3 0. Br Br H Br R ad usg the data suppled I worked From ths eercse we see that the regular soluto equatos predct a value o γ l4 Br / γ Br 5 ~ 0 whch s several orders o magtude larger tha the actual value. hs dscrepacy s ot surprsg. Regular soluto theory s oly epected to be o quattatve value whe dealg wth two lquds havg smlar ad opolar teractos. Although ths restrcto s ullled the case o Br / l 4 t s clearly ot the case or Br /. evertheless the theory does predct the correct qualtatve behavor γ Br >> dcatg that Br should be spargly soluble. H B : J mol H : J mol H W : J mol : 98.5K mb MW B : d B m MW : d mw MW W : d W H B R δ B : mb H R δ : m H W R δ W : γ ep mb : δ B δ R L mw δ B a γ W W ep mb : δ B δ W R L δ W δ a 0 6 a 6.89 γ.00 γ W sg the data o mtures o ethaol ad provded M& questo 0-50: a ompute the actvtes actvty coecets o both compoets ad the Gbbs ecess ad plot them as uctos o. 0
11 hem 453/544 Fall 003 /05/03 b Ft the actvty coecet data to the va Laar equatos to determe the costats α ad β. Also predct G e / R rom your t ad compare t to the epermetal data. c Ft G e / R versus to Redlch-Kster epresso usg as may terms as are eeded or good t. Also compute the dvdual actvty coecets rom ths t ad compare them to the epermetal data. d ommet o the relatve qualty o the results obtaed wth these two emprcal ttg methods. e I the va Laar equatos are terpreted terms o regular soluto theory oe ds α R δ δ R ad β δ δ Accordg to these epressos the rato o the α ad β costats should be equal to the rato o the molar volumes o the compoets. Is ths bore out the preset case? a Assumg deal ors ad eglectg the oytg correcto the soluto-phase actvtes ad actvty coecets ca be estmated rom the observed or pressures va: * a / / ad he ecess Gbbs eergy s gve by: G e γ a / { lγ γ } µ R. e e µ l I computed these varous quattes Ecel ad the results are tabulated ad plotted Fgs. 6- ad 6-3 o the ollowg page. ote that ths table ad throughout the aalyss I ve labeled compoet # ad compoet # ethaol. /torr /torr a a γ γ G e α β R
12 hem 453/544 Fall 003 /05/03.0 Fg. 6- Fg. 6- Actvty a Actvty oecet γ a a γ 3 γ Mole Fracto Ethaol Gbbs Ecess G e /R Mole Fracto Ethaol b he values o α ad β computed usg the epressos α lγ lγ lγ lγ ad β l γ lγ ad the epermetal actvty coecet data are lsted the last two colums the table o the prevous page. ote that the values o α ad β are ot costat whch relects the act that the va Laar equatos do ot t these data eactly. ome average sort o average values o α ad β must be chose order to provde a overall represetato o the data. I cosderg best values or these costats oe should ote that the ature o the above relatos s such that large ucertates values o α are epected whe s ear ero γ ad large ucertates β whe s ear ero γ. For ths reaso I dd ot use the etremes o the data set or determg α ad β. I eamed the results obtaed by averagg the dvdual values o α ad β over two rages o the data: # rage α β β/α he actvty coecets ad Gbbs ecesses calculated usg both sets are show Fgs. 6-3 ad 6-4. he sold curves are the results obtaed usg parameter set # ad the dashed les wth set #.
13 hem 453/544 Fall 003 /05/03 Actvty oecet γ Actvty oecet γ Fg. 6-3 Redlch-Kster Fts va Laar Fts Mole Fracto Ethaol Gbbs Ecess G e /R Gbbs Ecess G e /R Fg RK Fts v Laar Fts Mole Fracto Ethaol c I used gmalot to t the Gbbs ecess data drectly to the ucto: { A B...} G e / R he results o ths t are dsplayed Fg he best t parameters were A.8 B I oud o mprovemet the t upo addg a urther term to the epaso. A alteratve method o aalyg the data would have bee to t G e / R to a le a b. he RK parameters would have bee related to ths lear t by: A a b / ad B b /. Actvty coecets are obtaed rom the RK t usg the epressos: γ ep{ A 3B B } ad γ ep{ A 3B B } A comparso o the observed ad calculated actvty coecets s provded Fg d Frst t should be oted that ether ucto reproduces the data perectly over the etre composto rage. O these two -parameter equatos the Redlch-Kster ucto appears to provde superor ts to the preset set o data. e he molar volumes o ethaol ad are 58.7 cm 3 mol - EtOH ad 8.cm 3 mol - H O. Accordg to regular soluto theory the rato β / α EtOH / H O. he rato o the estmates o α ad β yeld a rato o.7 whereas the volume rato s 3.. he regular soluto model predcts the rght drecto or the derece betwee α ad β but s slghtly o o the magtude. 3 3
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