Finitely Concentrated Partial Molar Excess Properties Of .
Korean J. Chem. Eng., 20(4), 745-754 (2003)Finitely Concentrated Partial Molar Excess Properties of Solvent/Polymer[poly(4-methylstyrene) (PMS), poly(vinylbenzyl chloride) (PVBC)] SystemsSang Soon Park* and Joong So Choi†Department of Chemical Engineering, Kwangwoon University, 447-1, Walgyedong, Nowonku, Seoul 139-701, South Korea*Institute of Technology, Chementech, Inc., Kwangwoon University Business Incubating Center, 447-1,Wolgyedong, Nowonku, Seoul 139-701, South Korea(Received 1 November 2002 accepted 28 January 2003) The finitely concentrated activity coefficients and partial molar excess properties of solvents were measAbstract ured with inverse gas chromatography (IGC) in polymer solutions containing a poly(4-methylstyrene) (PMS) or apoly(vinylbenzyl chloride) (PVBC). The experimental temperature ranges were 373.15 K to 413.15 K for PMS and353.15 K to 393.15 K for PVBC. They were over melting point or glass transition temperature of each polymer. Tenkinds of solvents (Acetone, n-Heptane, Cyclohexane, Chloroform, Methylisobutylketone, Trichlorobenzene, Benzene, Toluene, Ethylbenzene, Chlorobenzene) that are important in the chemical engineering field were arbitrarilychosen for binary polymer solutions. The external degree of freedom of original UNIFAC-FV model was empiricallymodified to give flexibility to itself as a C1 A BT from the experimental data in finite concentration. The UNIFACFV model included a new external degree of freedom as a function of temperature. The parameters (A, B) wereestimated by correlating the activities of solvent with the modified model and extended to predict the partial molarexcess properties of solvents in the finite-concentrated polymer solutions. The predicted values were compared withthem by original UNIFAC-FV as well as the experimental data. The results obtained with the revised model using thenew parameter showed the higher quality than the results obtained by original model.Key words: Inverse Gas Chromatography, Activity Coefficient, Partial Molar Excess Properties, External Degree of Freedom,UNIFAC-FVINTRODUCTIONexcess properties in finite-concentrated polymer solution systemshas not yet been established. Many researchers have been measuring them mainly for polymer solution in infinite dilution by IGCmethod, but due to technical problems, they have not progressedfor the method to measure and predict them in finite concentrationby IGC method.In this work, the IGC method was used to measure the finitelyconcentrated activity coefficients of each solvent in polymer solutions containing PMS or PVBC in the temperature ranges over glasstransition temperature of each polymers. The partial molar excessproperties will be also evaluated from the activity coefficients. Moreover, original UNIFAC-FV representing the activities of solvent inpolymer solutions will be empirically modified from the experimental data for finite-concentrated polymer solutions, and extended topredict the partial molar excess properties, such as partial molar excess enthalpy, partial molar excess Gibbs energy and partial molarexcess entropy, of each solvent in polymer solutions. The new parameters will be introduced to measure the fixed external degreesof freedom in UNIFAC-FV model as the temperature-dependentmolecular external degrees of freedom. The estimated new parameters from the experimentally measured activity coefficients will beused to predict the partial molar excess properties by UNIFAC-FV.The activity coefficients of each solvent play an important rolein chemical technology, namely in qualitative and quantitative analysis of processing and application of polymer and are basically applied to evaluate other thermodynamic properties in the polymersolutions. The partial molar excess properties among those properties play an important role to analyze an energy flow in the polymer manufacture process and are essential for thermodynamic analysis of separation processes in chemical engineering. Many researchers have used Inverse gas chromatography (IGC) to measure suchproperties by [Patterson, 1962; Schuster et al., 1984; Kim et al., 1996,1998] because it has the merit of reaching the phase equilibria ofpolymer solutions within a short time. The IGC method that hasbeen used to measure the finitely concentrated thermodynamic properties was suggested by Conder and Purnell [1968a, b, 1969a, b]and continued to be used by Brockmeier et al. [1972], Choi et al.[1995] and Patterson et al. [1983]. The group contribution modelsto describe the phase behavior of polymer solutions were typicallythe UNIFAC-FV [Oishi and Prausnitz, 1978], the modified ASOG[Choi et al., 1995], and GC-Flory EOS [Holten-Anderson et al.,1987]. Recently, Kim et al. [1998] modified the UNIFAC-FV torepresent the partial molar heat of mixing at infinite dilution in solvent/polymer solutions. In spite of the importance of partial molarexcess properties, a systematic predictive method for partial molarEXPERIMENTAL1. MaterialsThe special grades of poly(4-methylstyrene) and poly(vinylbenzylchloride) were supplied from Aldrich Chemical. The average†To whom correspondence should be addressed.E-mail: jschoi@kw.ac.kr745
746S. S. Park and J. S. ChoiTable 1. Average molecular weight and glass transition temperature of each polymerPolymersTg (K)MwPoly(4-methylstyrene)Poly(vinylbenzyl chloride)45.5 107.2 104366295molecular weight (Mw) and the glass transition temperatures (Tg) ofeach polymer were analyzed with GPC (Shimadzu, R4A) and DSC(TA Instruments, DSC 2010), respectively. The temperature repeatability of DSC in this work was 0.1 oC. The data are presented inTable 1. Solvents (acetone, chloroform, n-heptane, cyclohexane,trichloroethylene, benzene, ethylbenzene, chlorobenzene, methylisobutylketone, toluene) were used without further purification asspecial grades, also supplied from Aldrich Chemical.2. Measurement of Partial Molar Excess PropertiesThe preparation of a column is an important factor to accuratelymeasure the partial molar excess properties in polymer solutions.The method was described in detail in the work of Kim et al. [1998]where they carried out the experiment at infinite dilution. Therefore, it will be described briefly in this paper.The solid support was a Fluoropak 80 (40/60 mesh) and the coating ratios of packing materials were 8.11% and 7.90% for PMS andPVBC, respectively. The apparatus to measure the vapor-liquid equilibria (VLE) in this work and in the work of Kim et al. [1998] wasthe same except that a solvent supply flask was attached to the apparatus. The solvent was supplied through the gas diffuser attachedon the rounded flask as the method fully described in the work ofChoi et al. [1995]. Thereafter, the control of solvent concentrationswas carried out by ascending or descending temperature of the solvent supply flask. The retention times of air and solvent peak wereobtained from the recorder of VLE apparatus of polymer solutions.The procedures for evaluating the weight fractions were omitted inthis work because they were minutely described in the work of Conder and Purnell [1968a, b, 1969a, b] and Brockmeier et al. [1972].They showed that the distribution isotherm in column was givenby the following equation.j C V s VadCSq( P) ------ 0 ---------------m2 1 Ψ1was extended to 0.01-0.09.The retention volume (Vs Va) in Eq. (1) was typically determinedby substituting the retention time (ts ta) and the flow rate of carriergas (QHe), that are experimentally determined, into the followingequation.QHeT- (t t ) ---V s V a ----------1 Ψ s a Tf(3)Therefore q(P) is extended into the following equation to determinethe weight fraction of solvent (w1) in polymer solution.q( P)M1w 1 -----------------------1 q (P )M1(4)The weight fractions were used to determine the finitely concentrated activity coefficients of various solvents. Considering gasphase nonideality, we used the following equation of Chang andBonner [1975], to determine the activity coefficients.4 B 11( Ps1 P 1)A Po Ψ J 3-exp -----------------------------Ω1 ------1 ------------sRTw 1 w1 P 1(5)4In Eq. (5), J3, known as the James-Martin factor, refers to thepressure correction factor for the pressure between inlet and outletof the column and was represented by Eq. (6):nm ( Pi Po ) 1Jmn ---- -------------------------mn ( Pi Po ) 1(6)The activities evaluated by Eq. (1)-(5) were compared with thoseof Choi et al. [1995] for typical benzene(1)/polystyrene(2) systems(1)In this work, we used the elution on a plateau method which keptconstant concentration in the carrier gas to evaluate q(P). The criterion for constant concentration as given by Conder and Purnell[1969a] isy0(Pi P0)/P0 0.01(2)where 0.01 is the experimental uncertainty in Vs. If reliable data areto be obtained at high solvent mole fraction, the column pressuredrop must be held to a very low value. But, this restriction on pressure drop can be somewhat relaxed for the case of a nearly straightdistribution isotherm. With a straight isotherm, the contribution toretention time at the inlet of the IGC column (high C) is exactlyoffset by the loss at the outlet (low C) of the column. The pressuregradient must be nearly linear. (Pi/P0 1.7) All of q(P) values obtained from the change of Cs were almost linear in those solvent/polymer systems of this work. Experimental error can be generallyaccepted up to about 5%, in this work, so the limit set by Eq. (2)July, 2003Fig. 1. Comparison of the activities with the cited data for benzene(1)/polystyrene(2) system at 393.15 K.
Finitely Concentrated Partial Molar Excess Properties of Solvent/Polymer (PMS, PVBC) Systemsto verify the accuracy of experimental data. The results were plotted in Fig. 1. As shown in Fig. 1, solvent activities in a variety ofconcentrated polymer solutions can often be estimated with an uncertainty of no more than 5% in comparison with typical experimental results. Therefore, other experimental data were assumed tobe correct within those experimental error ranges. The activity coefficients were extended to evaluate the partial molar excess propEEEerties (H1 , G1 , S1 ) of each solvent by the fundamental thermodynamic relations applied at constant pressure and constant weightfraction.E2 lnΩ2 ln ΩH1 RT --------------1 RT ---------------1 T p, w T p, w(7)G1 RTln Ω1(8)H 1 G1ES1 ----------------T(9)EEEThe second term of Eq. (7) was determined from the third onewhich represents the slope of a plot between activity coefficient andtemperature, because there is linear relationship between ln Ω1 andT. The example of plots was representatively shown in Fig. 2 forthe benzene(1)/PVBC(2) systems at constant pressure and weightfractions.sThe saturated vapor pressures (P1) of pure solvents were estimated by Wagner equation and Antoine equation [Reid et al., 1987]and the second virial coefficient (B11) was calculated from the equation of Tsonopoulos [1974]. Furthermore, J, P1 and Ψ were calculated by the method of Conder and Purnell [1969a].Fig. 2. Temperature dependence on the activity coefficients basedon the weight fraction for benzene(1)/poly(vinylbenzyl chloride)(2) system.747RESULTS AND DISCUSSION1. Partial Molar Excess PropertiesPartial molar excess properties were defined as the differencebetween the actual property of a component in real solution andthe value that it would have in an ideal solution at the same temperature, pressure, and composition. The partial molar excess properties obtained from Eqs. (7) to (9) are shown in Table 2. In general, partial molar excess Gibbs energy is an elementary and essential property in fluid phase equilibria. As shown in Table 2, thepartial molar excess Gibbs energy decreased with temperature andweight fraction of solvents in all systems. But partial molar excessenthalpy and entropy for PMS increased with temperature and weightfraction of all solvents except for cyclohexane. Cyclohexane, whichis a ring compound, for PMS showed that it absorbed heat becausethe partial molar excess enthalpy decreased with weight fraction ofit when it was compared with other solvents. In addition, partialmolar excess enthalpy and entropy for PVBC were the same trendas that of PMS except for benzene. Benzene, which was aromaticcompound, for PVBC showed the same trend as cyclohexane forPMS.On the other hand, the experimental partial excess properties ofeach component in polymer solutions are important factors to determine the total excess properties of polymer solutions. The totalexcess properties are usually used to describe the behavior of polymer solutions. Most interesting properties among those excess proEEEperties are H , G , and S . They are evaluated from partial molarexcess properties of each component in polymer solutions. To conEEsider the mutual relation between those experimental data, H1 , G1 ,ES1 as a function of the composition changes were representativelyplotted in Figs. 3 to 6 for eight polymer solution systems at 373.15 K.EFrom Figs. 3 to 6, G1 values of PMS solutions were larger thanEEthose of PVBC solutions. H1 and S1 of PVBC solutions were larEger than those of PMS solutions. The tendency of G1 was also sameEEas the former but H1 and S1 were reversely shown in the remaining twelve polymer solution systems. And, solubility parameters[Brandrup and Immergut, 1989] of four solvents (acetone, n-heptane, chloroform, chlorobenzene) used in Figs. 3 to 6 exhibited atendency to above 19.0 (MPa)1/2 but those of remaining six solvents were shown as below 19.0 (cal/cm3)1/2. And, solubility parameters of PMS or PVBC were known as above 19.0 (MPa)1/2.In general, the greater the difference in solubility parameters between two liquids, the larger the heat of mixing becomes, and thusthe two liquids become less miscible. For this reason the best solvent for a substance is the solute itself.From the viewpoint of solubility parameters, polymer (PMS orPVBC) solutions of four solvents become more miscible than thoseof six solvents because the difference in solubility parameters issmall. From the viewpoint of partial excess enthalpy, each heat ofmixing of polymer (PMS or PVBC) solutions consisting of foursolvents becomes larger than that of heat of mixing of polymer (PMSor PVBC) solutions consisting of six solvents. Therefore, we couldobtain the result that polymer solutions consisting of four solventsbecome more miscible than polymer solutions consisting of remaining six solvents from the solubility parameters and heat of mixingdata. And exact dissolution between the polymer solutions cannotbe described by solubility parameters but miscibilities of polymerKorean J. Chem. Eng.(Vol. 20, No. 4)
748July, 2003Table 2. Experimental thermodynamic properties of each solvent in solvent(1)/polymer(2) systems373.15 KSolventsw1Ω1E1H383.15 KE1GSE1Ω1E1H393.15 KE1GE1SΩ1E1H403.15 KE1GSE1413.15 S. S. Park and J. S. 480.21080.11060.18490.25010.19410.25050.3019
Table 2. Continued373.15 KSolventsw1Ω1E1H383.15 KE1GSE1Ω1E1H393.15 KE1GE1SΩ1E1H403.15 KE1GSE1413.15 340.97859.57274.16509.85967.60635.40100.4 nzyl 034.3749Korean J. Chem. Eng.(Vol. 20, No. 0210.28060.34580.40980.27310.34590.4166Finitely Concentrated Partial Molar Excess Properties of Solvent/Polymer (PMS, PVBC) SystemsΩ1E1
750S. S. Park and J. S. ChoiFig. 3. Partial molar excess properties as a function of weight fraction of solvent for acetone/polymer systems at 373.15 K.Fig. 5. Partial molar excess properties as a function of weight fraction of solvent for chloroform/polymer systems at 373.15 K.Fig. 4. Partial molar excess properties as a function of weight fraction of solvent for n-heptane/polymer systems at 373.15 K.Fig. 6. Partial molar excess properties as a function of weight fraction of solvent for chlorobenzene/polymer systems at 373.15K.solutions could be confirmed by them. It showed that four solventshad better dissolution feasibility than those of remaining six solvents for PMS or PVBC.July, 20032. Representation of ActivitiesThe UNIFAC-FV model of Oishi and Prausnitz [1978] has been
Finitely Concentrated Partial Molar Excess Properties of Solvent/Polymer (PMS, PVBC) SystemsFig. 7. Temperature dependence on the external degree of freedomfor benzene(1)/poly(vinylbenzyl chloride)(2) system.typically used to predict the activities of solvent in polymer solutions. Its representation of activity based on weight fraction wasexpressed as Eqs. (10) to (18) as follows:lnA1 lnA C1 lnAR1 lnAFV1(10)θ' zθ'zlnAC1 ln φ '1 φ 2' ---M1q'1ln-----1 --- M1q'1 1 -----1 φ 1' 2φ 1' 2(11)r'iw iq'i wi-, θ i' --------------φ i' ------------- r'jwj q'j wj(12)11()()r'i ------ νki Rk , q'i ------ νki Q kMi kMi k(13)lnAR1 ν (k1) [ln Γ k lnΓ(k1) ](14) Θ m' Ψ km ln Γk Mk Q'k 1 ln Θ 'm Ψ mk --------------------mmΘ 'n Ψnm n(15)Q'm Wma mnQ- , Ψ mn exp -----, Q'k ------kΘ 'm ------------------TMk Q'nWn(16)jjknable for all polymer solutions. First, we predicted the finitely concentrated activity coefficients according to a recommendation (C1 1.1) of Oishi and Prausnitz by using Eqs. (10) to (18) but could notobtain satisfactory results because C1 was set equal to 1.1 in the model as suggested by earlier results [Beret et al., 1975] in spite of thepresence of different values of C1 for other solvents containing largemolecules. In dependence on the external degree of freedom, Highand Danner [1990] also have shown that the C1 parameter relatingthe number of external degrees of freedom resulting from the rotation and vibration of the molecules was found to be a linear function of temperature. Chen et al. [1990] also have suggested a newcorrelation for the C1 parameter and introduced the simple and lineartemperature dependency of the C1 parameter of the Holten-Andersen et al. [1987] model. Therefore, the preceding results offer a greatdeal of important information about determination of the externaldegree of freedom from the experimental data in our work. To givethe flexibility to the original UNIFAC-FV model, the external degrees of freedom were determined from the experimental data. Thatis to say, we plotted representatively the external degree of freedoms according to a variation of temperature on Fig. 3 for benzene(1)/PVBC(2) system. As shown in Fig. 3, the relation betweenthe external degree of freedom and the temperature showed a linear form at each constant concentration. From those results, the external degree of freedom could be obtained as following form.C1 A BT(19)The parameters (A, B) of Eq. (19) were estimated by correlatingthe finitely concentrated activities of each solvent in the polymersolutions with Eq. (10) and extended to predict the partial molar excess properties. The mathematical algorithm to estimate the parameters was the Marquardts method and the estimated results shown onTable 3. In the computational procedures, the molar group volume(Rk), the group area (Qk), and UNIFAC group interaction parameter (amn) were cited from the results of Gmehling et al. [1982]. Theliquid molar volumes (νi) were obtained from Rackett equation fromthe work of Reid et al. [1987] for each solvent and also done fromthe method of Elbro et al. [1991] for the PMS and the PVBC.3. Prediction of Partial Molar Excess PropertiesThe models containing two parameters (A, B) to describe thetemperature-dependent molecular external degree of freedom parameter (C1) to predict the partial molar excess properties at constant pressure and weight fraction could be evaluated by substituting Eq. (10) to Eqs. (7) to (9). Therefore, the model that could preEdict the partial molar excess enthalpy (H1 ) was expressed by substituting Eq. (10) for Eq. (7) as following. ln ΩR lnΩFVE1 H 1 RT 2 --------------1 --------------- T P, w T P, w(20)Eν̃ 1ν̃1 1 1 ---- 1 1 ------lnAFV1 3C1 ln --------------- C 1 ν̃ν̃ 1ν̃11 3 M(17)ν1ν 1 w1 ν 2 w 2-, ν̃ ---------------------------------------------ν̃1 -------------------15.17br'1 M 15.17b( r'1w 1 r'2 w2 )(18)1 311 3M751The UNIFAC-FV model showed good results for the averageerrors between the experimental and the predicted values in the workof Oishi and Prausnitz [1978] but had a limitation to be not avail-EThe partial molar excess Gibbs energy (G1 ) and entropy (S1 )could be also expressed by substituting Eq. (10) to Eq. (8) and (9),respectively. The estimated parameters (A, B) were extended to preEEEdict the partial molar excess properties (H1 , G1 , S1 ) of each solvent in polymer solutions containing PMS or PVBC. Each rangeof deviation between the values evaluated by experiment and values predicted by original UNIFAC-FV, and of the former and predicted values of the partial molar excess properties by the correlation in this work are listed on Table 4. As shown in Table 4, thereKorean J. Chem. Eng.(Vol. 20, No. 4)
752S. S. Park and J. S. ChoiTable 3. Parameters estimated by modified UNIFAC-FV model for solvent(1)/polymer(2) HeptaneCyclohexaneChloroformMethylisobutyl ABpoly(vinylbenzyl chloride)** 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.01 0.01 0.02 0.01 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.03 0.04 0.06 1.60 1.89 2.34 0.04 0.04 .93124.650145.530193.890264.96096.35120.100158.350 0.07 0.08 0.09 0.06 0.07 0.07 0.15 0.16 0.18 0.07 0.08 0.10 0.21 0.32 0.52 0.11 0.14 0.19 0.06 0.06 0.06 0.19 0.24 0.32 0.35 0.46 0.63 0.24 0.30 0.40*poly(4-methylstyrene) experiment temperature (K) is 373.15-413.15.**poly(vinylbenzyl chloride) experiment temperature (K) is 353.15-393.15.Ewas quite satisfactory improvement for predictive value of H1 inEEall most systems and satisfactory improvement for G1 and S1 compared with original UNIFAC-FV. The excellent improvement forEH1 was primarily due to prediction of the partial molar excess enthalpy through the estimation of parameters (A, B) by the linearleast-squares analysis instead of fixing external degree of freedomE(C1) of original UNIFAC-FV as 1.1. The improvement for G1 andEEEES1 was the same as the case of H1 except that G1 and S1 were subEjective functions of H1 and activity coefficient in all systems. Furthermore, it was guessed that other factor of errors was the derivativeof the free volume term of the UNIFAC-FV with temperature in theassumption that the liquid molar volumes of the UNIFAC-FV wereconstant in spite of t
predict the partial molar excess properties, such as partial molar ex-cess enthalpy, partial molar excess Gibbs energy and partial molar excess entropy, of each solvent in polymer solutions. The new pa-rameters will be introduced to measure the fixed external degrees o
The partial and excess partial molar volumes for each component at infinite dilution have been appraised and reported. Keywords: Speed of sound, viscosity, density, excess parameters, polynomial equation, a reagent partial and excess parti
An A-algebra Bis finitely presented if it is the quotient of a polynomial ring A„X1;:::;Xn"by a finitely generated ideal. A ring homomorphism A!Bis finite, and Bis a finite3 A-algebra, if Bis finitely generated as an A-module. If A!Band B!Care finite ring homomorphisms, then so also is their composite A!C. Let kbe a field and Aa k .
Partial molar property is defined by . Similarly, partial molar excess property is defined by From Euler’s theorem Similarly, the excess property is given by . Without the correction for the enthalpy of mixing,
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Molar Volume General Definition of Molar Volume In chemistry, the molar volume of a substance is the volume of one mole of that substance. It can be computed as the substance's molecular (or atomic) weight divided by its density. The SI unit for volume is the cubic meter, m3. Thus, the SI unit of molar
Partial Molar Enthalpy or Partial Molar Heat Content In order to derive the expression for partial molar enthalpy, consider a system that comprises of n types of constituents with n. 1, n. 2, n. 3, n. 4 moles.
partial molar enthalpy 11 22" ,, ji ii i Tpn i U UUnUnUnU n iii nU partial molar energy _ Let’s compare µ of a pure ideal gas to µ in a mixture of ideal gases. Chemical potential in a pure (1-component) ideal gas F
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