Colligative Properties

Transcription

1 Chapter 5 Collgatve Propertes 5.1 Introducton Propertes of solutons that depend on the number of molecules present and not on the knd of molecules are called collgatve propertes. These propertes nclude bolng pont elevaton, freezng pont depresson, and osmotc pressure. Hstorcally, collgatve propertes have been one means for determnng the molecular weght of unknown compounds. In ths chapter we dscuss usng collgatve propertes to measure the molecular weght of polymers. Because collgatve propertes depend on the number of molecules, we expect, and wll show, that collgatve property experments gve a number average molecular weght. 5.2 Bolng Pont Elevaton Fgure 5.1 shows the vapor pressure of a lqud for pure lqud and for a soluton wth that lqud as the solvent. In an deal soluton, the vapor pressure of the solvent, P A, s reduced from the vapor pressure of a pure lqud, PA, to X APA where X A s the mole fracton of lqud A. Ths reducton s reflected n a shft to the rght of the vapor-pressure curves n Fg By defnton, bolng pont s the temperature at whch the vapor pressure of the lqud reaches 1 atm. Thus, the rght-shft caused by the dssoluton of component B n solvent A causes the bolng pont to ncrease. Ths ncrease, T b, s the bolng pont elevaton effect. A well known result from ntroductory chemstry s that the bolng pont elevaton s proportonal to the molar concentraton of solute partcles T b = K b m (5.1) where m s the molalty of solute molecules and K b s the bolng pont elevaton coeffcent that s a functon of only the solvent. Molalty s the number of moles of component B per 1000 grams of solvent. If we prepare a soluton of an unknown compound of molecular weght B at a concentraton 65

2 66 CHAPTER 5. COLLIGATIVE PROPERTIES Vapor Pressure Vapor Pressure Pure Solvent Soluton T 1 atm Temperature T b Fgure 5.1: Bolng pont elevaton effect s a consequence of the effect of solute molecules on the vapor pressure of the solvent. c n g/cm 3, then m = 1000c M B ρ where ρ s the densty of the solvent (n g/cm 3 ). Substtutng nto the expresson for T b gves or (5.2) M B = 1000K bc ρ T b (5.3) T b c = 1000K b ρm B (5.4) For a gven solvent (e.g., water where K b = 0.52 and ρ = 1.00) and concentraton (c), all terms n Eq. (5.4) are known except for M B. Thus, measurng T b can be used to determne the molecular weght M B. We can also express bolng pont elevaton n terms of mole fracton. Mole fracton s X B = cv M B ρv M A + cv M B cm A ρm B (5.5) where V s total volume and M A s molecular weght of the solvent. The bolng pont elevaton becomes T b = 1000K b M A X B (5.6) To apply bolng pont elevaton to polymers, we begn by usng soluton thermodynamcs to derve an expresson for T b. At equlbrum, the chemcal potental of the vapor s equal to the chemcal potental of the lqud µ vap A = µlq A = µ A + RT ln X A or µ vap A µ A RT = ln X A (5.7)

3 5.2. BOILING POINT ELEVATION 67 where we have assumed an deal soluton. Dfferentatng both sdes gves ( ( ) µ A ) GA T T = = 1 ( ) GA G A T T T T P T 2 = S A T H A T 2 + S A T whch s used to get Hvap A H A RT 2 = H A T 2 (5.8) = d dt ln X A (5.9) where H vap A H A s the heat of vaporzaton of the solvent or H vap. Now consder the process of formng a soluton. As the polymer s added, the mole fracton of A wll go from 1 at the start to X A whch s the mole fracton of the fnal soluton. Durng the process, the bolng pont wll go from T b to T where T b s the bolng pont of the pure lqud and T s the bolng pont of the soluton. Integratng over ths process gves T T b H XA vap RT 2 dt = d ln X A (5.10) 1 The ntegrals are easly evaluated f we assume that H vap s ndependent of temperature over the small temperature range from T b to T. The result s ( H vap 1 R T 1 ) = ln X A (5.11) T b We can smplfy ths result usng T b = T T b, T T b T 2 b, and ln X A = ln(1 X B ) X B. These smplfcatons apply when X B s small (whch occurs when the soluton s dlute) and when T b s small. In general, T b wll be small when the soluton s dlute. The prevous equaton smplfes to or H vap T b RT 2 b T b = = X B (5.12) RT 2 b H vap X B (5.13) Comparson of ths result to Eq. (5.6) gves a theoretcal expresson of K b : K b = The result s often derved n physcal chemstry books. M ART 2 b 1000 H vap (5.14) In applyng bolng pont elevaton to polymer solutons, we should realze that polymer solutons are really solutons of many components. The varous components are the polymer speces of dfferent molecular weghts. Because bolng pont elevaton s a collgatve property, we can wrte the bolng pont elevaton of a polymer soluton as a sum over the mole fractons of each molecular weght component: H vap T b RT 2 b = X (5.15)

4 68 CHAPTER 5. COLLIGATIVE PROPERTIES where X s the mole fracton of polymer wth molecular weght M. We more convenently rewrte X n terms of concentraton: X = c V M ρv M A + c V M c M A ρm (5.16) where c s the concentraton n weght/unt volume (e.g., g/cm 3 ) of polymer wth molecular weght. The approxmaton n ths expresson s vald for dlute solutons n whch the number of moles of solvent s much greater than the total number of moles of polymer. Summng the mole fractons, X, results n where X = cm A ρ c M c c = = cm A ρ The fnal expresson for the bolng pont elevaton becomes w M = cm A ρm N (5.17) c (5.18) T b c = M ART 2 b ρ H vap M N (5.19) It s common to express the bolng pont elevaton n terms of the latent heat of vaporzaton, l vap, defned as energy or vaporzaton per unt weght or l vap = H vap M A = J/mole = heat of vaporzaton n J/g (5.20) g/mole The bolng pont elevaton becomes T b c = RT 2 b ρl vap M N (5.21) Except for ncorporaton of polydspersty, there s nothng new about the bolng pont elevaton expresson for polymer solutons vs. the comparable expresson for small molecule solutons. In polymers, however, the soluton s more lkely to be non-deal. For ths equaton to apply we wll probably need to use very low concentratons or technques to extrapolate to very low concentratons. For an example, let s consder a soluton of polystyrene n benzene. For benzene ρ = g/cm 3, T b = 55 C, and l vap = 104 cal/g. We assume a relatvely concentrated soluton of c = 1 g/cm 3 of a polymer wth molecular weght M N = 20, 000. The change n the bolng pont elevaton for ths soluton s T b = C. Ths bolng pont elevaton s very small. It s probably beyond the accuracy of most temperature measurng equpment. The small change arses despte relatvely deal condtons of a farly concentrated soluton and a low molecular weght polymer. More dlute solutons or hgher molecular weght polymers would gve an even smaller T b. The problem wth polymer solutons s that for a gven weght of materal, the polymer soluton wll

5 5.3. FREEZING POINT DEPRESSION 69 have many less molecules than the comparable small molecule soluton. When there are a small number of molecules, the change n bolng pont (a collgatve property) s small. The problem wth the bolng pont elevaton method appled to polymer solutons s that t s not senstve enough. It has found some use wth polymers but t s lmted to polymers wth relatvely low molecular weghts. (e.g., M N less than 20,000 g/mol). 5.3 Freezng Pont Depresson A smlar analyss (but wth sgn changes) can be appled to the freezng pont depresson of a polymer soluton. The fnal result s T f c = RT 2 f ρl f M N (5.22) where T f s the freezng pont of the solvent and l f s the latent heat of fuson. We consder the same example of polystyrene n benzene wth T f = 5.5 C, l f = cal/g for the freezng pont of benzene. For a c = 1 g/cm 3 soluton of polystyrene wth molecular weght M N = 20, 000, the change n the freezng pont of the soluton s T f = C. Lke the bolng pont elevaton effect, the freezng pont depresson effect s too small. The technque s nsenstve and only useful for low molecular weght polymer (e.g., M N less than 20,000 g/mol). 5.4 Osmotc Pressure Another collgatve property s osmotc pressure. Fgure 5.2 llustrates the osmotc pressure effect. Imagne a pure solvent and a soluton separated by a sempermeable membrane. An deal sempermeable membrane wll allow the solvent molecules to pass but prevent the solute molecules (polymer molecules) from passng. The dfferent concentratons on the two sdes of the membrane wll cause an ntal dfference n chemcal potental. At equlbrum, ths dfference n potental wll be counteracted by an effectve pressure across the membrane. As shown n Fg. 5.2, t can be magned that solvent molecules pass from the pure solvent sde to the soluton sde. The excess heght n the column of lqud above the soluton sde s related to the osmotc pressure by = ρgh. Here s the osmotc pressure, ρ s the densty of the soluton, g s the acceleraton of gravty (9.81 m/sec 2 ) and h s the heght of the column of lqud. We begn wth a thermodynamc analyss of osmotc pressure. potental n the soluton wll be equal to the chemcal potental n the pure solvent: µ solvent A At equlbrum the chemcal = µ soluton A = µ A + RT ln a A (5.23) where µ solvent A s the chemcal potental of the pure lqud or µ solvent A = µ A (5.24)

6 70 CHAPTER 5. COLLIGATIVE PROPERTIES = ρgh Membrane h Solvent Soluton Fgure 5.2: A schematc vew of osmotc pressure across a sempermeable membrane. The only way the chemcal potentals wll be equal wll be f the actvty of component A n the soluton s equal to 1. The actvty can be rased to 1 by applyng pressure. That s, actvty s a pressure dependent quantty. By applyng the correct pressure, the osmotc pressure, the actvty n the soluton can be changed to 1. The osmotc pressure that gets appled occurs naturally by the tendency to approach equlbrum. To get the pressure dependence of actvty, we consder the pressure dependence of the chemcal potental dµ A dp = d dg = d dg dp dn A dn A dp = dv = V A = RT d dn A dp ln a A (5.25) Rearrangng and ntegratng the left hand sde from the orgnal actvty a A to the fnal actvty 1 and the rght hand sde from the ntal pressure 0 to the fnal pressure the osmotc pressure results n: 1 whch ntegrates to Now, n an deal soluton a A d ln a A = 0 ln a A = V A RT V A dp (5.26) RT (5.27) ln a A = ln X A X B. (5.28) Ths last approxmaton follows because ln(1 x) x for small x. Fnally, as n the analyss of bolng pont elevaton, we replace X B by X whch was derved n Eq. (5.17) to be gvng X = cm A ρm N (5.29) cm A = V A ρm N RT (5.30)

7 5.5. PRACTICAL ASPECTS OF OSMOTIC PRESSURE 71 But, M A /ρ s the grams per mole of solvent dvded by the grams per cm 3 of solvent. The grams cancel and we have cm 3 per mole of the solvent or the partal molar volume of component A V A. Substtutng nto the osmotc pressure equaton thus gves: c = RT (5.31) M N Rewrtng the osmotc pressure equaton gves a result that s smlar to the deal gas law crt = N M N or N M RT = N M V N (5.32) whch smplfes to V = N RT (.e. P V = nrt ) (5.33) For an example, let s consder the soluton of polystyrene n benzene that was used for examples of bolng pont elevaton and freezng pont depresson;.e., a soluton of polystyrene n benzene wth M N = 20, 000 and a concentraton of c = 1 g/cm 3. For the correct unts we use R = ergs/k/mol and calculate = dynes/cm 2. Ths pressure wll be measured by a dfference n heghts of lquds n columns. The heght dfference comes from = ρgh or h = dynes/cm g/cm 3 2 = 14.3 cm (5.34) 981 cm/sec Ths heght dfference s large and s an easly measurable quantty. In fact we expect to be able to measure dstances at least 100 tmes smaller than ths result. Thus osmotc pressure can, n prncple, be used to determne molecular weghts n polymers wth M N up to 2,000,000 g/mol. 5.5 Practcal Aspects of Osmotc Pressure Osmotc pressure measurements appear to be a sutable method for measurng number average molecular weghts n polymers. It s therefore worthwhle consderng practcal aspects of polymer characterzaton by osmotc pressure. The frst practcal consderaton s that we expect polymer solutons to devate from deal behavor and thus the osmotc pressure expresson wll need to be corrected. In the lmt of zero concentraton, the soluton wll eventually become deal. We can therefore take a seres of measurements and extrapolate back to zero concentraton to get the deal result. In other words lm c 0 c = RT (5.35) M N The queston whch remans s how do we extrapolate? A common approach n thermodynamcs s to use a vral expanson. We thus wrte c as a sum of many terms: c = RT M N + RT A 2 c + RT A 3 c 2 + (5.36)

8 72 CHAPTER 5. COLLIGATIVE PROPERTIES or c = RT ( 1 + Γ2 c + Γ 3 c 2 + ) M N (5.37) Here A 2, A 3,... and the related Γ 2, Γ 3,... are called the vral coeffcents. If we nclude enough vral coeffcents we wll always be able ft expermental data. But, how many of these terms do we need? Furthermore, how do we analyze expermental data when vral expanson terms are requred? We consder two approaches to ths problem. In the frst approach, we assume that only the second vral coeffcent A 2 or Γ 2 wll be needed. Then /c s predcted to be lnear n concentraton: c = RT + RT A 2 c (5.38) M N A set of data for /c vs. c can be plotted. If the results are lnear, the assumpton n the frst approach s vald. When the data s lnear, the ntercept of the data at zero concentraton wll be RT/M N and thus can be used to determne M N. Besdes an ntercept, we can measure the slope whch s equal to RT A 2. In other words the slope of the /c vs. c plot s proportonal to the second vral coeffcent A 2. We can make use of the Flory-Huggns theory to get a physcal nterpretaton of the second vral coeffcent. The Flory-Huggns theory ncludes non-deal nteractons through the Flory nteracton parameter, χ. Let s use the Flory-Huggns theory to develop an osmotc pressure theory for nondeal solutons. We begn wth an early osmotc pressure formula: = RT ln a A V A = µ A µ A V A (5.39) The term µ A µ A s found by dfferentatng the free energy of mxng d G mx dn A = µ A µ A (5.40) To use the Flory-Huggns theory we dfferentate the G mx from that theory. In performng the ntegraton we must realze that v A and v B also depend on n A. The work s left as an exercse to the reader. The result s µ A µ A = RT [ ( ln v A ) ] v B + χvb 2 x (5.41) Substtutng nto the osmotc pressure formula and at the same tme usng the approxmaton ln v A = ln(1 v B ) v B + v 2 B /2 (Note that n ths approxmaton to ln(1 v B) we keep one more term than we have used n the past. The reason for the extra term s that the µ A µ A expresson already ncludes terms wth vb 2 ), the osmotc pressure becomes: = RT V A [ ( ) ] vb 1 x + 2 χ vb 2 + (5.42)

9 5.5. PRACTICAL ASPECTS OF OSMOTIC PRESSURE 73 The volume fracton of polymer, v B, s equal to the concentraton of B n g/cm 3 dvded by the densty of polymer ρ B (v B = c/ρ B ). The osmotc pressure s then c = RT + RT ( ) 1 V A xρ B V A ρ 2 B 2 χ c + (5.43) The frst term s the deal soluton result (whch can be deduced by notng that M N = V B ρ B and V B = xv A n the lattce soluton model).. The second term, whch s proportonal to concentraton, gves the second vral coeffcent We return now to the slope of the lnear ft of /c vs. c whch gves the second vral coeffcent. From the Flory-Huggns analyss, the second vral coeffcent s: A 2 = 1 ( ) 1 V A ρ 2 B 2 χ or A 2 = M 0 ρ B ( ) 1 2 χ (5.44) where M 0 = M N /x s the monomer molecular weght. When χ s large and negatve, the second vral coeffcent wll be large and postve and the slope of /c vs. c wll be large and postve. A negatve nteracton parameter also mples a favorable nteracton ( G mx more negatve) and therefore a good solvent wll gve a large postve slope. In fact the slope of the osmotc pressure data can be thought of as a drect measure of the solvent qualty the hgher the slope the better the solvent. The second vral coeffcent from the Flory-Huggns result s also proportonal to 1/ρ B or rather s proportonal to the specfc volume of polymer. Ths result suggests an excluded volume effect. When the excluded volume effect s absent, the soluton wll act as f the specfc volume of the polymer s zero (1/ρ B = 0) and the second vral coeffcent wll therefore be zero. The excluded volume effect dsappears n a theta solvent and as a result the osmotc pressure data slope wll be zero n a theta solvent. In other words a theta solvent acts as an deal soluton to farly hgh concentratons. The fact that a zero slope s a low slope llustrates a result from earler n the course theta solvents n general are not very good solvents. Although workng n theta solvents would smplfy data analyss (.e., gve results that obey deal soluton laws) the fact that theta solvents are poor solvents makes workng wth them dffcult. It s usually more convenent to work n good solvents and make use of extrapolaton technques. The observaton of zero slope n osmotc pressure data, however, s a useful method for determnng theta solvent condtons. Unfortunately, plots of /c vs. c are often not lnear. We thus need a second approach to analyss of data from nondeal solutons. The obvous approach s to nclude both the second and the thrd vral coeffcents. In other words we assume that devatons from lnearty are caused by the thrd vral coeffcent no longer beng nsgnfcant. Let s take g as the rato of the thrd vral coeffcent to the second vral coeffcent squared (Γ 3 = gγ 2 2 ) and let s gnore terms beyond the thrd vral coeffcent. Then /c becomes c = RT M N ( 1 + Γ2 c + gγ 2 2c 2) (5.45)

10 74 CHAPTER 5. COLLIGATIVE PROPERTIES We now have two parameters Γ 2 and g. Dervng two parameters for osmotc pressure data wll be more complcated than dervng the slope and ntercept of smple lnear fts. It requres more advanced curve-fttng technques. We can smplfy the process by ntroducng some theoretcal calculatons about g. For hard spheres, g can be calculated to be g = 5/8. For polymer molecules, g has been estmated to be g = 0.25 to The actual value of g depends on varous propertes such as the expanson coeffcent α, the characterstc rato, etc.. Fortunately, however, g s restrcted to a relatvely narrow range for most polymers. Because g must be postve and a polymer cannot be more mpenetrable than hard spheres, g must be between 0 and 5/8. If we pck a value for g than we are left wth only one parameter (Γ 2 ) and we calculate M N and Γ 2 by smpler curve fttng analyses. Fortunately t has been found that the results are not very senstve to the exact value of g. Because polymers have g s calculated to be near 0.25, we wll assume g = The choce of g = 0.25 s desrable because t completes the square and the data analyss can agan be done by lnear fts (Scentsts, especally scentsts that worked before computers, lke lnear theores): c = RT (1 + Γ 2 c + 14 ) M Γ22c 2 N or ( c = RT 1 + Γ ) 2c M N 2 (5.46) When g can be assumed to be 0.25, a plot of /c vs. c should be lnear. The slope wll gve Γ 2 and the ntercept wll gve RT/M N. The advantage of settng g = 0.25 s that the data can be analyzed wth a smple lnear ft. Ths advantage was mportant before computers were readly avalable. Now we can easly treat g as a second parameter and do a two parameter ft to the data. You wll try ths type of analyss n one of the class labs and be able to dscuss whether the added complexty mproves or weakens the nterpretaton of the results. 5.6 Expermental Aspects of Membrane Osmometry A smple type of osmometer s llustrated n Fg The soluton s placed n a cell wth membranes on ether sde (one or two membranes, but two gves more area and faster equlbraton). The entre assembly s then mmersed n pure solvent. The heghts of the lquds n the capllares are read and the heght dfference gves the osmotc pressure. Ths apparatus s called a block type osmometer. It s the type of osmometer used to get the data that wll be gven to you n a lab. Ths osmometer uses a small cell and a large membrane. The membrane s supported by stanless steel plates wth holes. By supportng the membrane, the membrane can be made larger; wth larger membrane area equlbrum wll be reached sooner. Block osmometers are called statc osmometers because they wat for the natural development of equlbrum. The problem wth statc osmometers s that t can take hours (12-24 hrs) to reach

11 5.6. EXPERIMENTAL ASPECTS OF MEMBRANE OSMOMETRY 75 Capllares Outer flange Membrance Inner flange Soluton cell Outer flange Exploded Vew SIDE VIEW FRONT VIEW Fgure 5.3: A block type, statc osmometer. an accurate equlbrum. The tme depends on many factors such as the membrane area and the speed of transport through the membrane. To qucken osmotc pressure experments, dynamc osmometers are sometmes used. Recall that osmotc pressure develops for the purpose of rasng the actvty of the solvent n the soluton to 1. By applyng a pressure t s possble to do the same thng. You wll know when you have appled the correct pressure by montorng flow across the membrane. When you apply enough pressure to stop the flow you have artfcally reached equlbrum. The pressure requred can be used to get the equlbrum osmotc pressure. Ths quck method, unfortunately, s less accurate. Fnally, we make a few comments about what makes a good sempermeable membrane. The membrane must be permeable to solvent and mpermeable to polymer. Ths requrement lmts the low-end applcablty of osmometry to M N of 20,000 g/mol or more. Note that we really requre all polymers to be above 20,000 g/mol otherwse the low molecular weght tal wll pass through membrane and the measured M N wll be too hgh (do you see why t would be too hgh?). Therefore, polydsperse polymers probably requre M N greater than about 50,000 g/mol; for monodsperse polymers t mght be possble to go down to 20,000 g/mol. There are also some materal concerns for the membrane. An obvous concern s that the membrane not be soluble n the solvent. Perhaps the most common membrane materal s gel cellulose.

12 76 CHAPTER 5. COLLIGATIVE PROPERTIES Other membranes nclude cellulose hydrate, cellulose acetate, cellulose ntrate, polyurethanes, and poly(chlorotrfluoroethylene). Problems 5 1. In analyzng osmotc pressure data why s a plot of (/c) versus c sometmes used rather than a plot of (/c) versus c? 5 2. The followng are data from osmotc pressure measurements on a soluton of polyester n chloroform at 20 C. The results are n terms of centmeters of solvent. The densty of HCCl 3 s 1.48 g/cm 3. Plot /c versus c and fnd M N under the assumpton that you can neglect terms beyond the second vral coeffcent. c (g/dl) h (cm of HCCl 3 ) a. Suppose that n a dfferent unverse that the bolng pont elevaton was gven nstead by H vap T b R T 2 b = w BM B = 1 w M (5.47) M 0 M 0 where w s the weght fracton of polymer wth molecular weght M and M 0 s the monomer molecular weght. By ths law, what molecular weght average could by found from bolng pont elevaton measurements and gve a formula for calculatng that molecular weght average. b. The measurement of bolng pont elevaton s not very useful for fndng the molecular weght of hgh molecular weght polymers. If bolng pont elevaton was gven nstead by the formula n part a, would t be a more or less useful approach to fndng the molecular weght of hgh molecular weght materals?