Physical Chemistry Labratry I CHEM 445 Experimet Partial Mlar lume (Revised, 0/3/03) lume is, t a gd apprximati, a additive prperty. Certaily this apprximati is used i preparig slutis whse ccetratis are t eeded with a high degree f accuracy. Hwever, the stadard methd f preparig slutis f accurately kw ccetratis is t the additi f a weighed amut f slid t a kw vlume f slvet, but the additi f slvet, with mixig, t a weighed amut f slid util the fial vlume reaches the desired value:.000 g{x}/0.00 ml H O ad.000 g{x}/0.00 ml sluti are t idetical slutis. If e adds salt t a full glass f water, des the water flw ver the rim f the glass? Accrdig t Miller [], it was reprted i 770 that the vlume decreases whe salts are added t a fixed vlume f water. A early thery f electrlyte slutis (which may eve be held by peple tday) is that whe salts disslve i water, they fill the empty space i the liquid. Obviusly that thery is t crrect. The geeral reducti i vlume addig a salt t water is attributed t electrstricti, the ctracti i vlume f the plar slvet arud the is. If e takes 00.0 g C H 5 OH (= 6.7 ml at 0.0 C) ad 900.0 g H O (= 90.6 ml at 0.0 C) ad mixes these liquids, e btais a sluti with a desity f 0.980 g/ml (all desity values frm CRC Hadbk f Chemistry ad Physics) ad a actual vlume f 08.3 ml, cmpared with the additive vlume f 08.3 ml. Csequetly there is a decrease i vlume f ~ 0 ml r ~ % mixig these tw liquids. {See Ref. fr experimets shwig the decrease i vlume mixig fr sme aqueus systems.} The fractial decrease i vlume is a idicati f the differeces betwee itermlecular iteractis f the tw species. Partial mlar quatities are key elemets i the thermdyamics f slutis. The mst imprtat f these is the partial mlar free eergy r chemical ptetial, G µ j () j T, P, j Hwever, the partial mlar vlume is the easiest t measure, ad perhaps, t uderstad: j () j T,P, j The stadard treatmet f partial mlar quatities csiders the differetial chage i that prperty with the additi f mre (ml) f material. Hece, fr vlume, e has the fllwig equati, d = d + d (3) The partial mlar vlume f each cmpet, j, is likely t be a fucti f ccetrati, but it des t deped the ttal umber f mles. Therefre, e ca represet the ttal vlume f the sluti by the fllwig equati,
= + (4) I the subsequet discussis, the slvet (water, i these experimets) is cmpet ad the salt (sample) is cmpet. Oe ca determie the vlume f sluti fr a fixed amut f e cmpet, ml f slvet, ad variable amuts f sample (i these experimets fr 000.0 g H O ad mlality, m, ml f sample). A aqueus sluti f mlality, m, ctais m mles f salt per 000 g f water. Frm the desity f this sluti, d, i g/ml, e may calculate the vlume (i ml) f a sluti that ctais 000 g f H O (r = 000/8.05 = 55.508 ml f water) ad m (= ) ml f salt f mlecular weight MW i g/ml: 000 + m *( MW) = (5) d The data fr vlume as a fucti f mlality, {m}, ca be fitted t a equati, geerally a pwer series i m r m ½. The derivative f this fucti with respect t (r m ml f sample) is the partial mlar vlume f the sample as a fucti f mlality, m. Agai, m is the umber f mles f sample fr 000 g f water. m = = (6) T,P, Rearrage Eq. (4) t slve fr, the partial mlar vlume f the slvet. The replace by its equivalet frm Eq. (6). Because we are usig the mlality f aqueus slutis, = m ad = 000/8.05 = 55.508 ml f H O. Oe btais the fllwig equati fr the partial mlar vlume f the slvet, : T, P ( ) = m m = (7) Because maipulati f umerical data was t always as easy as it is w ad because f histrical develpmet, the partial mlar vlume f the sample,, is als btaied frm ather quatity, the apparet mlar vlume f the sample, ϕ. The apparet mlar vlume f the sample, ϕ, is calculated with the assumpti that the mlar vlume f the slvet remais cstat (is the same i sluti as it is i the pure slvet) ad ay chage i vlume is due ly t the sample. This assumpti is false, but it fits a simplified picture f slvet/sample iteractis i which ly the sample prperties chage. The defiiti f remais the same (Equati 4); therefre, = + mφ (8) I this equati = m, because we are csiderig m mles f sample i 000 g f water ( = 000/8.05 = 55.508 ml fr all slutis; hwever, ϕ, because. That is, the partial mlar vlume f the slvet i the sluti,, is t the same as the partial mlar vlume f the slvet i the pure slvet, water i pure water. Hwever, Lim(ϕ{m}) as m 0 = ϕ =., which is the vlume ccupied by e mle f
Fr aqueus slutis at 5.0 C,, is the vlume f 000 g f pure water: 00.96 ml = 000 g/(0.997044 g/ml). Csequetly, ϕ, i ml, is very easy t calculate. 00. 96 φ = (9) m Because is geerally apprximately 000 ml/ml, the apparet partial mlar vlume, ϕ, is determied as a small differece betwee tw large umbers. Csequetly, very accurate data fr (ad, therefre, f d) are required t btai precise values fr ϕ. Oe ca fit the data umerically t btai ϕ{m} (that is, ϕ as a fucti f m), geerally as a pwer series i m r m ½. If e differetiates {m} frm Equati (8) with respect t m, e btais a equati t calculate the partial mlar vlume f the sample frm ϕ{m}, φ φ m m = = + (0) m T, P T, P Oe ca calculate, the partial mlar vlume f the slvet, frm ϕ as a fucti f mlality, m, by cmbiig Equatis (4), (8), ad (0), Remember that ad m are the same, that = 000/8.05 = 55.508 ml, ad that = m ϕ m = 00.96/55.508=8.069 ml/ml at 5.0 C I the past, calculatis with large umbers f decimal places were difficult ad it was advatageus t simplify the arithmetic t reduce rudig errrs. Nw, calculatis with a large umber f decimal places are easily de with a calculatr, a spreadsheet, r ather mathematical prgram. It is easy t carry ut the idividual steps with mre decimal places tha are warrated ad t rud back at the ed t the accuracy warrated by the data. The partial mlar vlumes f the salts,, are fuctis f ccetratis f the slutis. Usig Debye-Huckel thery, e may extraplate these values t ifiite diluti, where iic iteractis are lger bserved ad the extraplated values fr are related t prperties f the slvated is ad their effects the structure f water. Ufrtuately, extraplatis t ifiite diluti require very precise determiatis f desities at lw ccetratis f salts because the desities are t very differet frm the desity f pure water ad the ucertaities i the measuremets fte give sigificat scatter t the data. Partial mlar vlumes ca be measured ly fr salts, t idividual is, but with mdels, e ca assig values t idividual is. Oe cveti t btai values fr idividual is is t assig + {H,aq} = 0 ml/ml. The values fr iic sizes are t the same as thse btaied frm crystallie radii because the is are slvated i sluti. The partial mlar vlumes at ifiite diluti, i fr the alkyl ammium is, C H + NH + 3 icrease with icreasig chai legth ad hece iic size, as e wuld expect. The partial mlar vlumes at () 3
ifiite diluti f the alkali metal is icrease with icreasig atmic umber as e wuld expect frm treds i the peridic table, but 0 + { Li } ad { N a + } are egative! The partial mlar vlumes at ifiite diluti f mst multiply charged catis are egative. The partial mlar vlumes f mst ais at ifiite diluti are psitive. [] Experimetal Prcedure: Determie the desities f slutis f differet ccetratis f NaCl. The desities f these salt slutis ad {m}icrease mtically with icreasig ccetrati, m r M. CuSO 4 r CuSO 4.5H O is a mre iterestig salt t use because {m} is t a mtically icreasig fucti f ccetrati, but mre, ad mre accurate, data are eeded t defie the {m} curve accurately fr cpper sulfate. Yu will measure the desities by tw methds. If the experimets are de prperly, the desities btaied by the tw methds will fit a sigle curve. A. Use a 00 ml Cassia vlumetric flask t determie the desities f the slutis. This type f vlumetric flask has graduatis the eck f the flask at 0.0 ml itervals, like a buret (ad csts ~ $90, s be careful!). Csequetly, yu may read vlumes f slutis ver a rage. The vlumetric flask has bee calibrated by the maufacturer at 0.0 C; hwever, yu shuld calibrate it at 5.0 C. Yu will determie the vlume f the flask frm the weight (ad, therefre, vlume) f distilled water ctaied. (If yu use mre tha e Cassia flask, keep the stppers separate. They will t have idetical weights.). The flask shuld have bee dried i the ve previus t yur use t esure that there is liquid iside. T get the weight f the dry flask that is represetative f these experimets, wet the utside f the flask with distilled water ad the dry the utside f the flask thrughly ad weigh. Weigh the dry flask (with stpper) twice because this is a critical umber that is used i all f yur calculatis, Weight. Use the aalytical balace fr these measuremets. There may be differeces i the last decimal place (± 0. mg r s), but yur duplicate weights fr the flask shuld t differ by mre tha ± mg.. Fill the flask t the 00 ml mark (r slightly higher) with distilled water that is at a temperature slightly belw 5.0 C. Immerse the flask (with stpper) i the bath fr ~ 5 miutes. Recrd the vlume after the sluti has equilibrated at 5.0 C (which must be 00 ml) t ~ ± 0.0 ml (by iterplati betwee the 0.0 ml graduati marks as yu did whe readig burets fr titratis i QUANT). Dry the utside f the flask ad weigh the aalytical balace, Weight. Weigh twice: the tw values shuld agree withi a few mg, althugh perhaps t t ± mg. The differece betwee Weight ad Weight is the weight f water ctaied by the vlumetric flask. Use the desity f water f 0.997044 g/ml at 5.0 C t calculate the vlume f the flask. This value shuld be very clse t 00.00 ml (r whatever vlume yu read frm the graduatis the eck f the flask), but it is ulikely t be exactly that value. Frm the weight 4
f the water, ~ 00.xxx ± 0.00y g, the vlume f the vlumetric flask is kw t a accuracy f better tha /0,000. 3. Fill the vlumetric flask t ~ 05 ml, immerse i the 5.0 C bath fr ~ 5 miutes, dry, ad weigh (twice). Read the vlume after equilibrati t ~ ± 0.0 ml. Calculate the vlume frm the weight f water ad its desity as i. Make a tw-pit calibrati curve r equati: lume{calculated frm desity} vs. lume{read frm flask}. 4. Yu are t measure the desities f five slutis f NaCl f differet ccetratis frm ~ 0. M t ~ 3.0 M. The desities f each sluti shuld be reprted t a accuracy f ~ /0,000, that is.00xx g/ml. The mlarities f the slutis shuld be reprted t fur decimal places (the accuracy f the aalytical balace):.xxxx ml/kg Prepare the slutis by addig NaCl t the vlumetric flask ad weighig (aalytical balace, ± 0. mg) the addig ~ 00 g H O (abve the 00 ml mark) ad weighig agai (aalytical balace, ± 0. mg). Because the vlumetric flask will be wet iside, yu must btai the weight f the wet flask befre addig the salt. Distilled water is t a ctamiat here. Mix well util the salt disslves ad equilibrate at 5.0 C fr ~ 5 mi. Read the vlume f sluti t ± 0.0 ml ad calculate the crrect vlume frm the tw-pit calibrati curve btaied abve. Yu ca calculate bth the mlarity ad mlality directly frm these measuremets. Begi with the mst ccetrated sluti ad fiish with the least ccetrated sluti. The vlumetric flask must be thrughly rised with distilled water after each sluti is prepared ad its desity measured. The flask cat be dried the iside. Distilled water is t a ctamiat i preparig these slutis, but each precedig sluti is a ctamiat. D t rise with acete ad attempt t dry i the ve. If yu use mre tha e Cassia vlumetric flask, yu must calibrate each e. Be careful t remember which flask ad which stpper yu use fr each desity determiati. The flasks ad stppers are t idetical. Whe yu have cmpleted the experimets, rise the Cassia vlumetric flask(s) several times with distilled water. That is, fill partially ad shake. Put it (them) it the ve t dry s that they will be clea ad dry fr use i subsequet labs! D t rise with acete. If yu d t rise the Cassia flasks thrughly, a depsit f salt will appear the iside f the flask ad greatly ay thse wh d the experimet after yu. B. Use a 5 ml pycmeter t measure the desities f slutis. The pycmeter has a small capillary with a grud glass jit that fits it a small vial. Use a small amut f stpcck grease the tapered jit f the capillary.. Determie the vlume f the pycmeter as yu determied the vlume f the Cassia vlumetric flask. Weigh the dry pycmeter ad capillary the aalytical balace twice ad use the average value. The weigh (twice) the pycmeter ad capillary filled with water 5
after it has equilibrated i the cstat temperature bath fr ~ 5 miutes (aalytical balace). Calculate the vlume f the pycmeter frm the weight f water ad the kw desity f water. The vlume will be clse t, but t exactly, 5.0000 ml. Each pycmeter has a umber. Each capillary has a umber. If yu use tw pycmeters, yu must calibrate each e. Remember t keep the capillary ad vial tgether. The weights ad vlumes f the pycmeters are t idetical. Reprt the vlume as 5.xxx ± 0.00x ml r 5.xxxx ± 0.000x ml, depedig the precisi f yur measuremets. Use this value i yur calculatis.. Prepare five slutis f NaCl i water f ccetratis similar t, but t the same as, the slutis i A. Prepare these by weight i a sap tp jar usig a aalytical balace, ~ 60 g f water (t ± 0. mg) ad a apprpriate weight f salt. The mlality is determied frm the weights f water ad salt. This amut f material will give eugh sluti t rise the pycmeter ad d tw determiatis f desity, if ecessary. Begi with the mst ccetrated sluti ad fiish with the least ccetrated sluti. Rise the pycmeter with small amuts f each ew sluti. Bth the precedig sluti ad distilled water are ctamiats i this experimet. Fill the pycmeter ad equilibrate at 5.0 C with slutis that are iitially at a temperature belw 5.0 C. Dry the utside f the pycmeter ad weigh (aalytical balace t ± 0. mg). Calculate the desity f each sluti. Whe yu have cmpleted the experimet, rise the pycmeter several times with distilled water t remve ay remaiig salt sluti ad place it the ve t dry. Calculatis ad Discussi: Preset yur primary data fr experimets frm the Cassia flask i a table that ctais the weight f salt, the weight f water, the vlume f the sluti, the mlarity, mlality (fur decimal places), ad desity fr each sluti (fur decimal places). Fr the pycmeter experimets, preset yur primary data i a table as weight f salt, weight f sluti, weight f water, vlume f pycmeter (t 3-4 decimals), mlality (fur decimals), ad desity (fur decimals). Plt desity vs. mlality usig a spreadsheet r ther data aalysis prgram. Fr the NaCl/H O system the desity is a mtically icreasig fucti f mlality, but t a liear fucti. Bth sets f data shuld fit t a sigle, smth curve. If yur data d t give a smth curve, check the calculatis. Make this plt befre the secd week fr this experimet. If yur data are bad r icsistet, repeat sme desity measuremets. Preset yur results i ather table that ctais mlality, m; = vlume f sluti ctaiig 000 g f water frm Equati (5); ad ϕ = apparet mlar vlume f the salt frm Equati (9) fr each sluti. Idetify the data frm the Cassia flask ad frm the pycmeter. Plt {m} ad ϕ{m} vs. m ad vs. m ½ ad fit the data t a pwer series i m ad i m ½ with a spreadsheet r ther prgram. {m} ad ϕ{m} are mtically icreasig, but t ecessarily liear fuctis f m r m ½. If {m} is t a mtically icreasig fucti f m, smethig is seriusly wrg with yur experimetal data. Irregularities i a plt f φ{m} vs. 6
m may ccur at the lwest ccetratis because very accurate desity measuremets are eeded t give reliable values fr φ. (See Eq. (9).) Whe yu fit the data t a equati, check the deviatis at each pit t see if there is systematic variati that suggests ather term i the pwer series. Oe uses the miimum umber f terms i a pwer series that will fit the data. The empirical equatis shuld be clearly idetified i the text. Give ucertaities i the cefficiets whe pssible. I ather table, preset yur results fr each sluti as mlality ad partial mlar vlume f salt ad f water calculated by the differet techiques give abve. That is, e may calculate bth ad, frm {m} ad frm ϕ{m}. The values fr each partial mlar vlume may be differet depedig the methd f aalyzig the data. Plt vs m ad m ½ ad extraplate t zer (plt ad data aalysis) t btai ifiite diluti. Cmpare yur results with literature values., the partial mlar vlume f NaCl at 7
The fllwig data are take frm a ld CRC Hadbk f Chemistry ad Physics fr the desity f aqueus NaCl slutis as a fucti f ccetrati. These desities were btaied at 0.0 C ad are t idetical t yur data. Calculate, tabulate, ad plt these data: d{m}, {m}, ϕ{m}, ad {m}. There is a relatively simple relati betwee M ad m fr ay sluti. I the fllwig equati, d = desity i g/ml (= kg/l) ad MW is the mlecular weight (relative mlar mass) i g/ml: M m = M MW d 000 Desity f Aqueus NaCl 0.0 C CRC Hadbk f Chemistry ad Physics, 47 th Ed, 966-67 Mlarity, ml/l Desity, g/ml 0 0.998.086.007.7.0053.346.05.53.095.703.067.885.0340.069.04.445.0559.734.0670.9.0857.637.008 3.056.6. F. J. Miller, Chem. Rev. 7, 47 (97). J. Olmsted, III, J. Chem. Ed. Refereces 8