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. 0.6 0.3: 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:
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.
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 30 70 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 0.8683 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
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 30 70 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 30 9 0.59 0.996 0.996 0.996 70 55 0.735 0.986 0.986 0.986 0 756.34 0.964 0.965 0.966 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
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 ~0.94-0.98 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 : 0.655375 bp : 3.949 cp :.38874 dp : 0.085593 ep : 7.9556 p : 4.8478 c : 56.75K t : 0 : K t 73.5 v : bar ep ap bp cp dp ep 3 p.70 v c torr.756 0 3 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 0.964 φ : ep Z l Z B A Z φ 0.965 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.003. 5
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. : 303.5 K d : 0.868kg dm 3 MW MW :.0783kg mol L : L 8.999 0 5 m 3 d mol : 9 torr : bar orr ep L : orr.003 R R orr : L l orr.0 bar.945.05 bar 3.83.0 ` bar 6.85 3. 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 7.55 3856.6K / ad l / torr 8.397 454.6K / 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 0.35 6
hem 453/544 Fall 003 /05/03 a b b d Beee 7.55-3856.6 353.4 oluee 8.397-454.6 383.77 30.53 /K be tol y / 353.4 760.0 74.9 0.000 0.000 80.09 356.30 834.5 306.8 0.4 0.057 83.5 359.35 94.9 34.6 0.70 0. 86.0 36.40 00.5 379.8 0.388 0.94 89.5 365.46 094.6 4.4 0.497 0.76 9.3 368.5 94.6 466.9 0.597 0.367 95.36 37.56 30.9 56.3 0.690 0.469 98.4 374.6 46.8 570.0 0.776 0.58 0.47 377.67 539.8 68.4 0.856 0.707 04.5 380.7 67. 69.6 0.930 0.846 07.57 emperature /K 390 385 380 375 370 365 360 or L L lqud 355 350 0.0 0. 0.4 0.6 0.8.0 383.77 8.4 760.0.000.000 0.6 Mole Fracto oluee y 4. At 60 the or pressure o methyl acetate s.6 bar ad the or pressure o methaol s 0.847 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
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 0.00.000.886.6 0.000.6 0.05.003.603.073 0.0.83 0.0.0.360.04 0.00.4 0.5.04.5 0.980 0.73.53 0.0.043.97 0.940 0.334.74 0.5.068.85 0.90 0.384.87 0.30.00.68 0.867 0.47.94 0.35.39.565 0.833 0.464.97 0.40.85.465 0.800 0.496.97 0.45.39.378 0.768 0.55.93 0.50.303.303 0.734 0.55.86 0.55.378.39 0.698 0.577.76 0.60.465.85 0.660 0.60.6 0.65.565.39 0.67 0.67.44 0.70.68.00 0.568 0.65.0 0.75.85.068 0.5 0.679.90 0.80.97.043 0.444 0.707.5 0.85.5.04 0.363 0.737.0 0.90.360.0 0.66 0.770.036 0.95.603.003 0.47 0.807 0.953.00.886.000 0.000 0.847 0.847 Fugacty or ressure / bar.4..0 0.8 0.6 0.4 0. Ac Fg. 6 0.0 0.0 0. 0.4 0.6 0.8.0 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 l4 0.039 mol/l. What s the dstrbuto coecet ths case ad what s the rato γ / γ? Br Br 8
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 5.98 0 4 o the dstrbuto coecet s: K γ l4 Br Br l4 Br γ Br 55.6 9
hem 453/544 Fall 003 /05/03 I you smply gored the volume o Br a relatvely accurate approach the results would have bee l4 0.088 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 : 9.45 0 3 J mol H : 30.0 0 3 J mol H W : 43.99 0 3 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 3.04 0 6 a γ W W ep mb : δ B δ W R L δ W δ 47.93 0 6 a 0 6 a 6.89 γ.00 γ W 3.35 0 5 6. 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
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 0.00 0.0 3.8.000 0.000.000 0.000 0.0 4.3 3.3 0.980 0.07.000 3.65 0.06.95.309 0.05 0.0.7 0.953 0.68.003 3.365 0.064.300.350 0.08 4.8. 0.98 0.5.009 3.33 0.099.334.35 0.0 7.7.7 0.93 0.98.04.98 0..3.355 0.0 7.0 0.3 0.85 0.456.064.8 0.5.56.400 0.30 3. 9.3 0.83 0.58.6.758 0.74.04.48 0.40 33.9 8.5 0.778 0.573.97.433 0.300 0.96.56 0.50 36.9 7.3 0.77 0.63.454.45 0.97 0.94.607 0.60 40.3 5.53 0.653 0.680.633.33 0.7 0.935.636 0.70 43.94 3.6 0.553 0.74.845.060 0.5 0.96.759 0.80 48.4 9.89 0.46 0.85.079.09 0.6 0.887.04 0.90 53.45 5.38 0.6 0.903.6.003 0.085 0.875.768 0.93 55.4 3.83 0.6 0.93.30.00 0.060 0.874.707 0.96 56.87.3 0.094 0.96.344.00 0.035 0.884.957 0.98 58.0.3 0.048 0.980.376.000 0.07 0.87 4.560 59. 0 0.000.000.000 0.000
hem 453/544 Fall 003 /05/03.0 Fg. 6- Fg. 6- Actvty a Actvty oecet γ 0.8 0.6 0.4 a a 0. 0.0 4 γ 3 γ 0.0 0. 0.4 0.6 0.8.0 Mole Fracto Ethaol Gbbs Ecess G e /R 0.3 0. 0. 0.0 0.0 0. 0.4 0.6 0.8.0 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 0.-0.8 0.974.664.7 0.05-0.96.03.780.73 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 #.
hem 453/544 Fall 003 /05/03 Actvty oecet γ Actvty oecet γ 4 3 5 Fg. 6-3 Redlch-Kster Fts va Laar Fts 4 3 0.0 0. 0.4 0.6 0.8.0 Mole Fracto Ethaol Gbbs Ecess G e /R Gbbs Ecess G e /R 0. 0. 0.0 0.3 0. 0. Fg. 6-4 0.3 RK Fts v Laar Fts 0.0 0.0 0. 0.4 0.6 0.8.0 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. 6-4. he best t parameters were A.8 B 0.73. 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 } 4 3 4 A comparso o the observed ad calculated actvty coecets s provded Fg. 6-3. 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