ofof

oh

y of K

de R

form

validity of fundamental laws of physical chemistry andchemical thermodynamics, was an additional factor stim-

taken by renowned industrial laboratories, are ratherscarce and in some instances even condential.

The increasing practical interest in carbohydrate solu-tions caused a great demand for predictive methods ofdetermining physical properties of both aqueous and

.

* Corresponding author. Fax: +27 031 260 1118.E-mail address: starzak@ukzn.ac.za (M. Starzak).

Food Chemistry 96 (200

Food0308-8146/$ - see front matter 2005 Elsevier Ltd. All rights reserved1. Introduction

Aqueous solutions of carbohydrates have been studiedfor many years due to their both scientic and practicalimportance (molecular biology and biochemistry, foodchemistry and technology, sugar industry, etc.). Due tothe common availability of sugar and the ease of its puri-cation, the system sucrosewater was the subject ofcountless physicochemical studies, especially in the rsthalf of the 20th century. The fact that dilute sucrose solu-tions could be used as a model system to demonstrate the

ulating the popularity of these studies. However, the sys-tem later descended into obscurity being consideredthermodynamically almost ideal, and hence no longerchallenging from a scientic point of view. New researchin the area of physical properties of sugar solutions hasbeen observed over the last two decades with the rapiddevelopment of food technology and introduction ofnon-conventional carbohydrates to industrial processes.Routinely, this research involves also the sucrosewatersystem, as it is often studied for comparative purposes.Studies on concentrated sucrose solutions, mainly under-Abstract

A comprehensive experimental data analysis was performed to evaluate the eect of temperature on the water activity coecientand selected excess thermodynamic functions for aqueous solutions of sucrose. A four-sux Margules equation with temperature-dependent parameters was used to t thermodynamic data such as the vapor pressure, boiling point, osmotic coecient, freezingpoint, sucrose solubility, heat of dilution and specic heat of solution. The proposed equation gives an adequate representationof the available literature data on sucrose solutions for temperatures from 15 to +150 C and sucrose concentrations up to98% wt. The isotherms of water activity coecient exhibit a characteristic minimum at about 96% wt. sucrose which is then followedby a dramatic increase to values well exceeding 1, as it was suggested before by some theoretical models [Starzak, M., & Mathlouthi,M. (2002). Water activity in concentrated sucrose solutions and its consequences for the availability of water in the lm of syrupsurrounding the sugar crystal. Zuckerindustrie, 127, 175185; Van Hook, A. (1987). The thermodynamic activity of concentratedsugar solutions. Zuckerindustrie, 112, 597600]. The eect of temperature on water activity, almost negligible for dilute solutions,was found signicant for very concentrated solutions (above 80% wt. sucrose). The new water activity equation should nd numer-ous applications in the food technology and sugar industry. 2005 Elsevier Ltd. All rights reserved.

Keywords: Sucrosewater system; Activity coecient; Margules equation; Temperature dependence; Data regressionTemperature dependencesolutions

Maciej Starzak a,*, Ma School of Chemical Engineering, Universit

b Laboratoire de Chimie Physique Industrielle, Universite

Received 15 September 2004; received in reviseddoi:10.1016/j.foodchem.2005.02.052water activity in aqueoussucrose

amed Mathlouthi b

waZulu-Natal, 4041 Durban, South Africa

eims Champagne-Ardenne, 51687 Reims Cedex, France

16 February 2005; accepted 16 February 2005

www.elsevier.com/locate/foodchem

6) 346370

Chemistry

ood CNomenclature

a function dening the assumed temperaturedependence

ai constant coecients in Eq. (4)Aij interaction constants in Eq. (41)bi expansion coecients in Margules equation,

Eq. (3)DB thermochemical parameters in Eqs. (10) and

(14)ci expansion coecients in sucrose activity

equationcp specic heatCp molar heat capacity

GE excess molar Gibbs energyhi0 coecients of McMillan-Mayer expansion,

Eq. (19)HE excess molar enthalpyI performance indexm molality of solutionM molecular massM0, M1 matrices dened by Eqs. (A5) and (A6),

respectively

M. Starzak, M. Mathlouthi / Fnon-aqueous solutions involving various sugars. The lasttwo decades have brought a number of studies in thisarea. The general trend in recent prediction methods isthe development of a comprehensive group contributionmodel capable of predicting solvent and solute activitycoecients in multicomponent systems containing vari-ous types of sugars (mono- and polysaccharides) and su-gar-related polyols. The UNIFAC group contributionmethod is currently the most popular technique used topredict thermodynamic activity for a broad range ofchemical compounds, including sugars. Group interac-tion parameters characterizing sugar molecules havebeen determined by regressing colligative propertiesdata, such as the boiling point elevation (BPE), vaporpressure lowering, freezing point depression (FPD), os-motic pressure, heat of dilution and solubility, for solu-tions of various carbohydrates in water and othersolvents (Abed, Gabas, Delia, & Bounahmidi, 1992;Achard, Gros, & Dussap, 1992; Catte, Dussap, & Gros,1995; Ferreira, Brignole, & Macedo, 2003; Peres & Mac-edo, 1997; Spiliotis & Tassios, 2000). More recently,some of the UNIFAC parameters have even been pre-dicted theoretically with methods of molecular mechan-ics (Jonsdottir & Rasmussen, 1999). Data regressionstudies involving sugars were also performed using other

n number of terms in Margules equationni number of molesP total pressureP 0i saturated vapor pressureR universal gas constantT absolute temperaturew weighting factorx mole fractiona expansion coecients in water activity equa-

tionb expansion coecients in sucrose activity

equationc activity coecienth dimensionless temperatureki coecients dened by Eq. (A3)sij Boltzmann factors in UNIQUAC modelUc apparent molar heat capacity of sucrose

Subscripts

asym asymmetric conventionA waterB sucrosed dilution propertyf at freezing pointm at melting pointp at constant pressure

hemistry 96 (2006) 346370 347thermodynamic models such as ASOG (Correa, Come-sana, & Sereno, 1994) and UNIQUAC (Catte, Dussap,Achard, & Gros, 1994; LeMaguer, 1992; Peres & Mac-edo, 1996). However, the selection of experimental datain most of these studies was typically limited to 35 liter-ature sources for each data category. Moreover, data forhighly concentrated solutions were generally not in-cluded while thermochemical data, if used at all, were re-stricted to room temperatures only. Consequently, theresulting activity equations are valid in a relatively nar-row range of conditions, mainly at ambient temperaturesand low concentrations. This applies to the sucrosewater system in particular. Despite a vast amount of dif-ferent experimental data generated for this system overthe past decades, there is still no reliable water activityequation covering a suciently large temperature-com-position domain. Interestingly, ICUMSA a worldwideorganization establishing analytical standards for the su-gar industry, which has ocially adopted equations for anumber of physical properties of pure sucrose solutions(density, viscosity, refractive index and solubility) givesno recommendation for the water activity coecient.The two principal sources of information used at presentin the sugar industry such as Sugar Technologists Manual(Bubnik, Kadlec, Urban, & Bruhns, 1995) and Sugar

Superscripts

E excess propertym molal property1 at innite dilution

popular amongst food technologists, have been derivedfrom freezing point data and therefore can be used forlow temperatures only. In fact, they do not account for

348 M. Starzak, M. Mathlouthi / Foodthe temperature whatsoever, assuming that for relativelydilute solutions the eect of temperature on the wateractivity coecient can practically be neglected. In turn,Starzak and Peacock (1997) used the Margules equationwith a simple built-in expression for the temperature ef-fect. The equation was developed using a massiveamount of literature data and robust statistical analysis.Unfortunately, the experimental database comprised va-porliquid equilibrium and osmotic coecient data only(the latter from studies on isopiestic solutions).

The above brief review shows clearly that at presentthere is no accurate and statistically reliable empiricalmodel of thermodynamic activity for the sucrosewatersystem which would cover the entire composition rangeas well as temperatures from the freezing point to theboiling point of highly concentrated solutions. The pur-pose of this study is, therefore, to develop such a corre-lation by critically evaluating and regressing varioustypes of experimental data available from the literature.

2. Experimental database

Since activity coecients have been introduced in ther-modynamics to account for non-ideal behavior of mix-tures, they can be determined by analysing variouspieces of experimental data, especially those inwhich ther-modynamic non-ideality is manifested most. Typically,these are phase equilibrium data as well as thermochemi-cal data on mixing. It was, therefore, decided that the fol-lowing categories of data would be used in this study:

vaporliquid equilibrium data (boiling point eleva-tion, equilibrium vapor pressure above a solution,equilibrium relative humidity, as well as the osmoticcoecient determined from isopiestic vapor pressureTechnology (Van derPoel, Schwieck, & Schwartz, 1998)recommend polynomial equations for selected colligativeproperties of sucrose solutions. These were obtained byregressing data from single experimental studies, appar-ently regarded as being more reliable than other datasources. Bubnik et al. (1995) give a table of water activitydata with a reference to the well-known BFMIRA Tech-nical Report by Norrish (1967) which, in turn, is basedon an earlier study of the same author (Norrish, 1966).However, a closer inspection of this study shows thatthe water activity coecient correlation proposed thererests on 25 experimental measurements taken by severalauthors at one temperature (25 C) for sucrose contentsnever exceeding 60% wt.! More recent correlations ofChen (1989) and Miyawaki et al. (1997), particularlymeasurements); solidliquid equilibrium data (freezing point depres-sion and sucrose solubility);

thermochemical data (heat of dilution and specicheat of solution).

Osmotic pressure data were not included, since rigor-ous analysis of these data would require considering theeect of pressure on the activity coecient, which wasbeyond the scope of this study.

The vaporliquid equilibrium (VLE) data used inthis study come mainly from the database created byStarzak and Peacock (1997), being the result of anextensive literature search (Chemical Abstracts 19021995/1996, volumes 2124; Sugar Industry Abstracts19491995, volumes 1157; and the landmark data col-lection by Timmermans (1960)). New additions to thisdatabase include a few older studies unavailable at thetime as well as experimental results published since1996. Initially, the updated VLE database contained1603 experimental points from 64 dierent literaturesources. However, after rejection of evident statisticaloutliers this number was reduced to 1507. The rejectioncriteria are discussed elsewhere (Starzak & Peacock,1997). The accepted VLE data consist of various typesof measurements such as the boiling temperature ofsolution at a known total pressure, saturated vaporpressure of solvent at a known temperature, concentra-tions of isopiestic solutions at a known temperature(leading to osmotic coecients), and equilibrium rela-tive humidity of air above a solution at a known tem-perature and total pressure. A brief description of theultimate VLE data collection is given in Table 1. Be-cause of the natural technological progress, dierentexperimental methods used by dierent authors, as wellas the varying nature and objectives of these studies(scientic/industrial), it is obvious that the data sourceslisted in Table 1 are characterized by very diverse de-grees of precision. In our opinion, however, when con-sidering the quality of data the time factor should notbe overestimated. For example, early measurements ofBerkeley, Hartley, and Burton (1919), Scatchard, Ha-mer, and Wood (1938) or Robinson, Smith, and Smith(1942) are far more accurate than some published evenquite recently. A system of weighting factors speciallydeveloped to account for the quality of individualVLE data sets was discussed in detail by Starzak andPeacock (1997).

The solidliquid equilibrium (SLE) data includepoints characterizing two types of phase equilibria pos-sible in the sucrosewater system: equilibrium betweenice and sucrose solution, known as the freezing pointcurve, and equilibrium between sucrose crystals andthe solution, known as the sucrose solubility curve.The freezing point data collection consisted of 213experimental points from 13 dierent literature sources

Chemistry 96 (2006) 346370(freezing point depression and the corresponding su-

Table 1Collection of experimental VLE data

Original authors Other sources of information Typeof dataa

Numberof points

Rangeb Accepted datac

% wt. Sucrose Temperature or pressure Number of points Weight wVLE,i

Wullner (1858) Timmermans (1960) bpe 66 3360 26761 mmHg 53 0.1531Gerlach (1863) Prinsen Geerligs (1909); Holven (1936) bpe 14 1091 760 mmHg 12 0.0885Flourens and Freutzel (1876) Wohryzek (1914); Spencer and Meade (1929) bpe 13 1090 760 mmHg 11 0.2037Dupont (1885) Spengler et al. (1938) bpe 6 8694 760 mmHg 6 0.1434Beckmann (1890a, 1890b) Timmermans (1960) bpe 5 518 760 mmHg 5 0.2090Baroni (1893a, 1893b) Timmermans (1960) bpe 5 517 760 mmHg 5 0.2909Claassen and Freutzel (1893) Wohryzek (1914) bpe 60 1092 760 mmHg 58 0.2462Flourens (1897) Spengler et al. (1938) bpe 9 6791 760 mmHg 9 0.2624Mikhailenko (1901a, 1901b, 1901c) Timmermans (1960) vpl 6 1152 50 C 6 0.2676Juttner (1901) vpl 17 1050 100 C 16 0.2403Kahlenberg (1901) bpe 28 1774 739 mmHg 28 0.2342Smits (1902) vpl 12 139 0 C 12 0.4661Claassen (1904) Wohryzek (1914); Hugot and Jenkins (1986) bpe 20 594 760 mmHg 17 0.1804Perman (1905) vpl 2 1227 99, 100 C 2 0.2991Vivien (1906) Timmermans (1960) bpe 13 4490 760 mmHg 10 0.1626Tower (1908) vpl 3 626 15 C 3 0.4080Krauskopf (1910) vpl 3 1738 40 C 1 0.1663Perman and Price (1912) bpe 10 1964 213521 mmHg 7 0.2343Wood (1915) Timmermans (1960) bpe 26 4869 113527 mmHg 23 0.1930Berkeley et al. (1919) vpl 14 2571 0, 30 C 14 0.9170Boswell and Cantelo (1922) Timmermans (1960) vpl 2 2534 23 C 2 0.2924Perman and Saunders (1923) vpl 45 1277 70, 90 C 39 0.2108Swietosawski (1925) vpl 5 933 100 C 5 0.3758Mondain-Monval (1925) vpl 5 6069 32, 47 C 5 0.2970Downes and Perman (1927) vpl 101 776 4080 C 91 0.1803Fricke and Havestadt (1927) vpl 14 2173 0, 10 C 13 0.4118Jordan (1930) Holven (1936) bpe 11 3090 760 mmHg 9 0.1202Whittier and Gould (1930) erh 1 64 25 C 0 0.0792Sinclair (1933) vpl 10 632 25 C 10 0.4363Bukharov (1934, 1938) bpe 56 1590 93760 mmHg 52 0.2882Robinson and Sinclair (1934) iso 31 634 25 C 31 0.5755Thieme (1934a, 1934b) erh 6 6993 30 C 6 0.2266Plake (1935) Bartels et al. (1960) bpe 7 04 760 mmHg 5 0.1514Holven (1936) bpe 51 4494 760 mmHg 48 0.2235Van Ruyven (1936) Bartels et al. (1960) bpe 6 218 760 mmHg 6 0.2459Scatchard et al. (1938) iso 23 668 25 C 23 0.6285Spengler et al. (1938) bpe 95 1591 1761988 mmHg 91 0.2680Pouncy and Summers (1939) erh 1 64 23 C 1 0.0980Tressler et al. (1941) bpe 21 572 760 mmHg 21 0.3170Robinson et al. (1942) iso 33 967 25 C 33 0.7103Grover (1947) erh 7 6073 20 C 7 0.2873Jackson (1950) Nicol (1968) bpe 32 6180 60, 90 C 32 1.0000Money and Born (1951) erh 9 4271 20 C 8 0.3079Dunning et al. (1951) vpl 109 4583 6096 109 0.4559

(continued on next page)

M.Starza

k,M.Mathlouthi/FoodChem

istry96(2006)346370

349

Table 1 (continued)

Original authors Other sources of information Typeof dataa

Numberof points

Rangeb Accepted datac

% wt. Sucrose Temperature or pressure Number of points Weight wVLE,i

Hawaiian Sugar Technologists (1955) Meade and Chen (1977) bpe 12 3092 760 mmHg 11 0.2151Heiss (1955) Norrish (1966) erh 7 4470 25 C 7 0.3182Jones (1959) Norrish (1967) bpe 20 6099 760 mmHg 20 0.2802Rother (1960) vpl 47 1672 050 C 40 0.2316Robinson and Stokes (1961) iso 28 1166 25 C 28 0.6267Pidoux (1962) bpe 1 86.5 760 mmHg 1 0.1734Ingleton (1965) bpe 9 4095 760 mmHg 9 0.1834Hoynak and Bollenback (1966) bpe 18 1099 760 mmHg 16 0.2016Norrish (1966) erh 5 5767 25 C 5 0.3460Tuzhilkin and Kaganov (1969a, 1969b) bpe 69 5595 149760 mmHg 67 0.4666Batterham and Norgate (1975) bpe 20 5176 54282 mmHg 20 0.3603Sarka et al. (1978) bpe 21 2181 760 mmHg 21 0.4058Ruegg and Blanc (1981) erh 26 2470 25 C 25 0.4108Modell and Reid (1983) bpe 40 7595 551760 mmHg 38 0.1639Crapiste and Lozano (1988) bpe 18 2373 55525 mmHg 18 0.3207Abdera and Bounahmidi (1994) bpe 11 587 760 mmHg 9 0.1127Lehnberger and Schliephake (1999) bpe 164 2780 375750 mmHg 156 0.3626Velezmoro et al. (2000) erh 38 245 2535 C 32 0.1163Comesana et al. (2001) erh 15 2667 35 C 15 0.5913Cooke et al. (2002) vpl 9 2872 25 C 9 0.3685a bpe boiling point data, vpl vapor pressure data, erh equilibrium relative humidity data, iso data from the isopiestic method.b Rounded-o values given.c Data nally accepted after the completion of the iterative elimination of outliers.

350M.Starza

k,M.Mathlouthi/FoodChem

istry96(2006)346370

crose concentration in solution), while the solubilitydata collection included 265 data points from 34 authors(sucrose concentration in saturated solution and the cor-responding temperature). Both the freezing point andsolubility data required additional enthalpy data on sol-vent and solute, respectively, which will be introducedlater. The SLE data used in this study were summarizedin Tables 2 and 3.

Thermochemical data provide more information onthe eect of temperature on activity coecients thanany other type of data. They typically include mea-surements of the heat of dilution and specic heatof solution. Although dilution of sucrose solutions isan exothermic process, it is associated with a relativelysmall heat eect. Two groups of experiments for thedetermination of the heat of dilution are in commonuse. In the rst, a comparatively large amount of su-crose solution is diluted with a small amount of water,

data selected for this study included 283 experimentalpoints taken from 10 dierent authors (Table 4). Acertain lack of consistency was observed acrossindividual data sets regarding the eect of temperature

Table 3Collection of experimental sucrose solubility data

Authors Numberof points

Temperaturerange (C)

Courtonne (1877) 2 12, 45Herzfeld (1892) 6 599Horsin-Deon (1902); after Vavrinecz (1961) 3 070Osaka (1903) 1 25Scott-Mace (1906);after Hruby and Kasjanov (1938)

10 10100

Lebede (1908) 3 3070Horiba (1917) 7 060Van der Linden (1919) 2 30, 50Kremann and Eitel (1923) 2 14, 16Jackson and Gillis-Silsbee (1924) 3 2350Mondain-Monval (1925) 4 130Grube and Nussbaum (1928) 9 0100

Vernon (1939) 14 1891Hruby and Kasjanov (1938/1939, 1940) 2 50, 80Verhaar (1940, 1941) 3 30Lyle (1941) 9 080Benrath (1942) 6 107144Taylor (1947, 1948) 11 6481Young and Jones (1949) 5 (14)6Dubourg et al. (1952) 1 87Wise and Nicholson (1955, 1956) 59 1890Kelly (1958) 7 0100Charles (1960) 37 186Wagnerowski et al. (1960) 5 4080Vavrinecz (1962) 14 (13)0Vasatko et al. (1966) 6 2070Smelik et al. (1968) 9 66126Abed et al. (1992) 1 70

Table 4Collection of experimental heat of dilution data

Authors Numberof points

Rangea

% wt.Sucrose

Temperature(C)

Porter (1917) 7 1558 20Pratt (1918) 64 2940 033Hunter (1926) 34 1456 1230Naude (1928) 4 315 18Vallender and Perman (1931)b 65 1969 2080Gucker et al. (1939) 62 26 20, 30Lange and Markgraf (1950) 9 06 18Savage and Wood (1976) 13 241 25Stroth and Schonert (1977) 9 02 25Wood and Hiltzik (1980) 16 635 25

a Rounded-o values given.

M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370 351whereas in the second, a small amount of solution isdiluted with a large amount of water. In both cases,the initial and nal concentrations of the solutionare recorded and the accompanying heat eect is mea-sured at a constant temperature. The data is then nor-mally reported either as the integral heat of dilution,being the amount of heat released during dilutionper mole of sucrose in solution or as the dierentialheat of dilution, being the amount of heat releasedper unit mass of water added. While the integral heatcan be directly obtained from the experiment, the dif-ferential heat is calculated by dividing a relativelysmall amount of measured heat by also a small (butnite, not innitesimal) amount of added water. Thismakes the resulting value of dierential heat of dilu-tion approximate. The experimentalist must, therefore,use an optimal amount of diluting water to compro-mise between the precision of the heat measurementand the accuracy of calculation. The heat of dilution

Table 2Collection of experimental freezing point depression data

Authors Numberof points

Rangea

% wt. Sucrose

Guthrie (1876) 11 551Pickering (1893a, 1893b) 75 064Abegg and Muller (1894) 4 1031Raoult (1897) 6 126Ewan (1899) 5 2342Jones and Getman (1904a, 1904b) 10 575Young and Sloan (1904) 12 224Morse et al. (1906) 10 326Kremann and Eitel (1923) 13 963Babinski (1924a, 1924b) 12 2162Young and Jones (1949) 29 575Neau et al. (1989) 13 963de Cindio et al. (1995) 13 1076a Rounded-o values given.Reinders and Klikenberg (1928) 3 2545Nishizawa and Hachihama (1929) 3 2575Ahrens (1930) 6 0100Grut (1936/1937, 1937/1938) 6 2080dOrazi (1938) 5 4090Scatchard et al. (1938) 1 25b Data at 0 C were rejected as unreliable.

3. Composition and temperature dependence of activity

coecients

The main objective of this study was to establishempirical equations describing the activity coecientsof water and sucrose, cA and cB, as functions of the su-crose mole fraction xB and the absolute temperature T.Amongst various possible mathematical forms used in

0 2 4 62

0

1

1 Galema et al. (1993)

22 Darros Barbosa et al. (2003)3 Di Paola & Belleau (1977)4 Banipal et al. (1997)5 Gucker & Ayres (1937)

Molality

Mol

ar e

xce

ss h

eat c

apac

ity a

t 25

C, J/

mo

Fig. 1. Molar excess heat capacity of sucrose solutions at 25 Caccording to dierent literature sources (calculated from smoothedheat capacity data).

Table 5Collection of experimental specic heat data

Authors Numberof points

Rangea

% wt.Sucrose

Temperature(C)

Vivien (1906) 19 867 15Gucker and Ayres (1937) 25 067 20, 25DiPaola and Belleau (1977) 13 331 25Banipal et al. (1997) 13 331 25

a Rounded-o values given.

oodon the heat of dilution. This might result from thefact that a considerable portion of data was availablein the form of less accurate dierential heats of dilu-tion. Analysis of the frequently cited study by Val-lender and Perman (1931), involving raremeasurements of dierential heats of dilution in abroad range of temperatures (080 C), shows thatthe error made by these authors when calculatingmean dierential heats of dilution from the measured(integral) amounts of heat was at least 1015%. Con-sequently, whenever sucient information was avail-able to reproduce the actual conditions of theexperiment, the reported dierential heats of dilutionwere converted back into the corresponding integralvalues.

Data on the specic heat of solution were used todetermine the excess heat capacity CEp , the propertywhich is directly linked to the temperature dependenceof chemical potential, and hence to the temperaturedependence of activity coecients. In general, reliabledeterminations of CEp can be obtained only fromhigh precision specic heat data. Unfortunately, manyolder measurements, especially those popular amongstsugar technologists, are of an industrial nature andnot accurate enough to serve this purpose. They typi-cally suggest a linear dependence of specic heat onmass concentration which implies zero excess heatcapacity. Moreover, even studies performed with a sci-entic rigour are often contradictory. For example,the specic heat data reported by Galema, Engberts,Hiland, and Frland (1993) as well as more recentdata of Darros-Barbosa, Balaban, and Teixeira(2003) produce negative CEp values at 25 C (Fig. 1),contrary to results obtained by Gucker and Ayres(1937), DiPaola and Belleau (1977) and Banipalet al. (1997) as well as those of Vivien (1906) at 15C. A closer inspection of the problem shows thateven small inaccuracies in the experimental gradientof the specic heat vs. concentration curve producelarge errors in CEp , to the extent that even the signof the excess function is falsied (as was the case inthe rst two studies mentioned above). It was decided,therefore, to reject at least those data that were evi-dently inconsistent with the specic heat of pure waterwhen extrapolated to innite sucrose dilution. As a re-sult, 70 experimental points from four dierent litera-ture sources were only accepted for the nal dataregression (Table 5).

Eventually, the entire database contained 2338experimental points fairly evenly distributed on theconcentration-temperature plane, with VLE data beingthe major contributor (Fig. 2). This allowed, in ouropinion, for a statistically signicant analysis of thecomposition and temperature dependence of waterand sucrose activity coecients in a wide range of

352 M. Starzak, M. Mathlouthi / Fconditions.0

2

4

34

5

l . K

Chemistry 96 (2006) 346370the literature to express the composition dependence in

5 wt. s

oints

M. Starzak, M. Mathlouthi / Food Cbinary systems, the conventional Margules equation wasfound to be particularly suitable for the data regressionapproach used in this study. The single-constant Mar-gules equation (also known as the two-sux Margulesequation) provides a satisfactory representation for the

0 10 20 30 40

0

50

100

150

200

%

Tem

pera

ture

, C

VLE FPD Solubility Heat of dilutionSpecific heat

Fig. 2. Distribution of all experimental pactivity coecient behavior only for binary liquid mix-tures containing constituents of similar size, shape, andchemical nature.However, formixtures of dissimilarmol-ecules, such as the sucrosewater system, a higher-orderexpansion (an n-sux Margules equation) is required.In particular, the water activity coecient cA is given by

ln cA Xnk2

akxkB; 1

where the expansion coecients ak are, in general, tem-perature dependent. The following mathematical formof the temperature dependence, commonly applied inthermochemistry, was assumed:

akh ak0h ak1 ak2 ln h ak3h ak4h2 ak5h3;

with h T=T 0; 2

where T0 = 298.15 K is a reference temperature chosenmerely for numerical convenience. Since the adoptionof Eq. (2) would result in an excessive number of param-eters to be estimated, for the sake of simplicity, it wasassumed that all expansion coecients ak(h) depend ontemperature in the same way, i.e., ak(h) = a(h)bk2. Con-sequently, the proposed Margules equation could bewritten as:ln cA ahXnk2

bk2xkB; 3

where

a0 2 3

0 60 70 80 90 100ucrose

on the concentration-temperature plane.

hemistry 96 (2006) 346370 353ah h a1 a2 ln h a3h a4h a5h 4

and the resulting new expansion coecients bk are tem-perature independent (with b0 = 1 by denition). Obvi-ously, by producing an identical temperature trendirrespective of composition, Eq. (3) oers much less ex-ibility in terms of data tting than the more general Eq.(2). There is no convincing experimental evidence, how-ever, indicating a dramatic change of the temperaturetrend with the increasing sucrose concentration. Theonly exception is a portion of the heat of dilution databy Vallender and Perman (1931) in the region of concen-trated solutions and elevated temperatures, where someheat of dilution isotherms (for 40, 60 and 80 C) inter-sect at about 6263% wt. sucrose, as shown in Fig. 3,and then exhibit a reverse temperature behavior (i.e.,for a given composition heat of dilution decreases withincreasing temperature). However, the validity of theseresults is disputable, as the authors of this study re-ported serious problems with strict temperature controlduring experimentation due to high viscosity of theinvestigated media. Since the other two isotherms (0and 20 C) did not follow the same pattern, we foundVallenders results inconclusive in this regard.

The sucrose activity coecient cB can also be ex-pressed in a polynomial form:

012 60C80Cat

er a

Mo

rom V

354 M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370ln cB Xnk2

bkxkA. 5

0 1 2 3

Fig. 3. Heat of dilution data f2

4

6

8

10

Hea

t of d

ilutio

n, k

J/kg

w14

16

18

0C20C40C

ddedThe expansion coecients bk are not independent, how-ever (see Appendix). For example, the following set ofrelations between bk and ak holds for the seven-suxMargules equation (n = 7):

b2 a2 3

2a3 2a4 5

2a5 2a6 7

2a7; 6a

b3 a3 8

3a4 5a5 8a6 35

3a7; 6b

b4 a4 15

4a5 9a6 35

2a7; 6c

b5 a5 24

5a6 14a7; 6d

b6 a6 35

6a7; 6e

b7 a7. 6f

4. Relations between activity coecients and measured

data

The entire study is based on the assumption that inneutral sucrose solutions (pH 7), the eect of sucroseinversion on the measured variables is negligible (Pea-cock & Starzak, 1995) and, consequently, the solutioncan be treated in modeling as a binary system involv-ing sucrose and water only. In order to avoid heavynon-linear and multi-parameter regression calcula-tions, the original data were transformed to new

4 5 6 7lality

allender and Perman (1931).experimental variables that could theoretically be ex-pressed as linear functions of the expansion coe-cients ak (or bk).

4.1. Vaporliquid equilibrium

All kinds of available VLE data were converted tocorresponding water activity coecients by using thegeneralized Raoults law:

cA P

1 xBP 0AT ; 7

where P is the measured pressure of water vapor abovethe solution and P 0AT is the saturated vapor pressure ofpure water at temperature T, calculated from the Wag-ner equation as described by Starzak and Peacock(1997). The pressure P is identical with the total pres-sure, except the measurements of equilibrium relativehumidity (% ERH) when it represents the partial pres-sure of water vapor in air, in which case Eq. (7) simpli-es to:

cA ERH=100

1 xB . 8

Eq. (7) is valid under the assumption that the vaporphase is thermodynamically ideal, the pressure P is mod-

crose) is non-volatile. All these assumptions were met in

point of sucrose.

one can easily arrive at the following formula linking

lnc xA DH fATm Tm T f 1

pA

30 to 0 C:

B

M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370 355DCpAT R

DCpATmR

DBAT Tm 10

with DCpA(Tm)/R = 4.555 and DBA = 0.045 K1. Then,A RTm T f RT f

Z TmT f

DCpA dT 1RZ TmT f

DCpAT

dT ; 9

where DHfA(Tm) is the enthalpy of fusion of solid sol-vent at its normal melting point temperature (accord-ing to Dickinson & Osborne (1915), for iceTm = 273.16 K and DHfA(Tm) = 6008 J/mol), andDCpA is the dierence between molar heat capacitiesof solvent in the liquid and solid state. For small tem-perature changes, DCpA can be assumed constant(temperature independent). However, in the case ofconcentrated sucrose solutions, reported DT valuesexceed 20 C and some correction is needed. Fromheat capacity data for supercooled water and ice(Dickinson & Osborne, 1915; Osborne, Stimson,& Ginnings, 1939), one can obtain a linearrepresentation of DC (T) valid approximately fromthe freezing point depression, DT = Tm Tf, to theactivity coecient of solvent, cA (symmetricconvention):4.2. Solidliquid equilibrium freezing point depression

Experimental values of the water activity coecientwere also determined from the freezing point data. Itis well known that a small amount of solute added toa liquid solvent aects the temperature Tf at which thepure solvent starts to separate out as a solid duringthe freezing process. This temperature is lower thanTm, the freezing (or melting) point of the pure solvent.The freezing point depression phenomenon can be de-scribed mathematically using a rigorous thermodynamicapproach. From the condition of thermodynamic equi-librium between the crystal of pure solvent and the solu-tion, after computing the Gibbs free energy of fusion,the investigated range of xPT conditions. The esti-mated saturated vapor pressure of sucrose was foundnegligible even at temperatures approaching the meltingerate (the Poynting factor equal to 1), and the solute (su-Eqs. (9) and (10) yield:ln cA DH fATm

RTm DCpATm

R DBATm

2

TmT f

1

DCpATmR

DBATm

lnT fTm

DBATm2

T fTm

1

ln1 xB

11Eq. (11) was used to determine experimental values ofthe water activity coecient cA from the freezing pointtemperature at dierent sucrose concentrations.

4.3. Solidliquid equilibrium sucrose solubility

The sucrose solubility data allow for the calculationof experimental values of the sucrose activity coe-cient cB. Thermodynamics of solubility is usually de-scribed by applying the same theoretical concepts asthose used for the freezing point depression. Unfortu-nately, because of the thermal decomposition of su-crose above 185 C, experimental values of itsmelting point temperature and, more importantly, fu-sion enthalpy are uncertain. As a result, the conditionof solidliquid equilibrium written in a form analo-gous to Eq. (9) becomes impractical in this case. Amore eective solution to this problem can be ob-tained by adopting the asymmetric convention foractivity coecients, i.e., by choosing the state of in-nite dilution for the solute and pure liquid for the sol-vent, at the equilibrium T and P, as a new referencesystem (Catte et al., 1994). Then, the solubility curveis dened by

lncBxB DHdBTm

RT1 T

Tm

1RT

Z TmT

DCpBdT 1RZ TmT

DCpBT

dT lnc1B Tm;12

where cB cB=c1B , and DHdB H1B H 0SB is the enthal-py of sucrose dissolution, i.e., the dierence betweenthe enthalpy of sucrose at innite dilution in waterand the enthalpy in its pure solid state. Similarly, forthe heat capacity we have DCpB C1pB C0SpB. Sinceexperimental values of H1B and C

1pB are usually avail-

able at ambient temperatures (typically at T0 = 298.15K) rather than at the melting point of sucrose, it ismore convenient to write Eq. (12) in the followingequivalent form:

lncBxBc1B

DHdBT 0

RT1 T

Tm

1RT

Z TT 0

DCpB dT 1RTm

Z TmT 0

DCpB dT

1Z Tm DCpB

dT ln c1Tm. 13R T T

1

peratures from the sucrose solubility data. The

B

i.e., c1T and c1T . By nding the limit x ! 1, one

Xn a Xn b

BXn Xn

356 M. Starzak, M. Mathlouthi / Food aT k2

M1M0bk1xkA aTm aT k2

bk2k 1.Hence, the proposed activity model produces the follow-ing expression for the left-hand side of Eq. (15):

ln~cB lnc1B Tmc1

cB

ln c1B T k2

k

k 1 aT k2

k2k 1 . 17B B m A

can easily show thatcorresponding theoretical values of ~cB were calculatedfrom Eqs. (5) and (6). The same equations were usedto evaluate the required values of c at innite dilution,~cB cB Tmc1B

cB. 16

The unknown constant DBB was optimized togetherwith the other parameters of the activity model. Eq.(15) was used eventually to determine experimental val-ues of the sucrose activity coecient ~cB at dierent tem-where DBBTm2

1 TTm

lnxB; 15DCpBT R

DCpBT 0R

DBBT T 0; 14

where DBB is a constant. Under this assumption, Eq.(13) leads to

ln~cB DHdBT 0

RT 0 DCpBT 0

R DBBT 0

2

TmT

1

T 0Tm

DCpBT 0R

DBBT 0

lnTTm

The heat capacity dierence DCpB was assumed to be alinear function of temperature:The required thermochemical data at T0 = 298.15 Kwere taken from Jasra and Ahluwalia (1984):

DHdBT 0 5760 J=mol;C1pBT 0 647 J=mol K;C0SpBT 0 427.8 J=mol K.184.4. Thermochemical data enthalpy of dilution

The enthalpy of dilution data selected for this studyrepresent various types of heat eect measurements.They have originally been reported as:

(i) the amount of heat, (DHI), liberated perone mole of sucrose in solution after the addi-tion of a known amount of pure water to thesystem,

(ii) the amount of heat, (DHII), liberated per onemole of water in the original solution after theaddition of a known amount of a dilute sucrosesolution to the system,

(iii) the amount of heat, (DHIII), liberated per onegram of water added after the addition of a smallknown amount of pure water to the system.

Although it was theoretically possible to expresseach of these observables in terms of the coecientsof the proposed activity model and t the data accord-ingly, a more uniform two-step data handling proce-dure was implemented instead. First, the McMillanMayer expansion a formula widely used in solutionchemistry to relate the excess enthalpy to the molalityof solution (Barone, Cacace, Castronuovo, & Elia,1981) was tted by multi-linear regression for eachset of isothermal heat of dilution data, in order to ob-tain the corresponding coecients of expansion. In thenext step, an equivalent number of smoothed molal ex-cess enthalpy values was generated from the resultingMcMillanMayer expansion, converted to molar val-ues, and then used as experimental input data in fur-ther analysis. According to the McMillanMayertheory, in the asymmetric convention, the molal excessenthalpy is given by

HEmasym J=kg solvent Xk2

hk0mk; 19

where m is the molality of solution and hk0 are temper-ature dependent expansion coecients. Then, the molarexcess enthalpy HEasym is

HEasymJ=mol HEmasym

m 1000=MA . 20

The following working formulae associate the threetypes of measured heats of dilution with the McMil-lanMayer expansion coecients hk0:

i DH InB

J=mol solute

Xk2

hk0mk1f mk1i; 21a

where m(f) and m(i) are the molality of the nal and ini-

Chemistry 96 (2006) 346370tial solution, respectively;

limit:

M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370 357ii DH IInAi J=mol solvent

Xk2

hk0nAi nAamkf nAamka nAimki

nAi ;

21bwhere m(f), m(i), and m(a) are the molality of the nal,initial and added solution, respectively, n(i) and n(a)are the number of moles of solvent in the initial andadded solution, respectively;

iii DH IIIwA

J=g solvent added

11000

Xk2

hk0mki; mf mi 21c

where wA is the mass of water added to dilute thesolution.

Eqs. (21ac) were used to determine coecients hk0from heat of dilution data. Then, Eqs. (19) and (20) wereapplied to generate the corresponding HEasym values. Thelatter were nally used to estimate parameters of theproposed activity model, as demonstrated below. Bycombining the well-known expression for the molar ex-cess Gibbs energy,

GE RT xA ln cA xB ln cB 22with Eqs. (3), (5) and (6), one can arrive at:

GE

RT

Xnk2

akk 1

!xB

Xnk2

akk 1 x

kB. 23

In the symmetric convention, the molar excess enthalpy,HE, can be obtained as follows:

HE

RT T oG

E=RT oT

TXnk2

a0kT k 1

!xB

Xnk2

a0kT k 1 x

kB

" #; 24

where a0kT is the rst derivative of ak(T) with respect totemperature. In turn, the molar excess enthalpy in theasymmetric convention is (Catte et al., 1994):

HEasym HE RT 2xBo ln c1BoT

25

or after applying Eqs. (17) and (24):

HEasym RT 2Xnk2

a0kT k 1 x

kB. 26

For the temperature dependence given by Eq. (4), thisyields nally

HEasym Ta0T Xn bk2 xkB; 27RTk2 k 1Uc0 limm!0

Ucm MB1000

cpA dcpdm

m0

. 34

Clearly, even small errors in experimental values ofcp(m), inevitable in practice, and subsequent large inac-curacies in the slope determination have a tremendousimpact on CEP asym values. Even with the aid of reneddata smoothing techniques, the estimates of Uc(0) arewhere

Ta0T a0h a2 a3h 2a4h2 3a5h3; with

h T=T 0. 28Eqs. (27) and (28) were used to predict molar excessenthalpies at dierent temperatures and sucrose concen-trations from the activity model.

4.5. Thermochemical data excess heat capacity

The excess molar heat capacity CEp can be determinedfrom experimental specic heat data. The heat capacityof 1 kg of water and m moles of sucrose dissolved in it(note that m is also the molality of solution) is

Cp mUc cpA; 29where Uc is the apparent molar heat capacity of sucrosein solution (J/mol sucrose K), and cpA is the specic heatof pure water (J/kg water K). On the other hand, Cp canalso be calculated from the specic heat of solution, cp(J/kg solution K), as follows:

Cp 1 MB1000

m

cpm. 30

Combining these two equations results in the apparentmolar heat capacity of sucrose, Uc(m), in terms of theexperimental specic heat of solution, cp(m):

Ucm 1 MB

1000m

cpm cpAm

31

In turn, Uc(m) can be used to calculate the molal excessheat capacity, since in the asymmetric convention, thelatter is dened as:

CEmp asymJ=kg water K Ucm Uc0m. 32Similarly to the molar excess enthalpy, the molar excessheat capacity is then:

CEp asymJ=mol K CEmp asym

m 1000=MA . 33

The major problem with the accurate determination ofCEP asym from experimental data is the evaluation ofUc(0), the apparent molar heat capacity of sucrose atinnite dilution. Since Eq. (31) becomes indeterminateat m = 0, Uc(0) can be evaluated only as a mathematicalstill subject to signicant errors. Fig. 4 illustrates this

4 pure water (IAPWS, 1997)

c, the

358 M. Starzak, M. Mathlouthi / Foodproblem on a series of literature data which showrelatively small dierences in terms of observed cp val-ues, but surprisingly large discrepancies in terms of the

0 0.1 0.2 0.3 0.4

4.02

4.04

4.06

4.08

4.1

4.12

4.14

4.16

Molality

Spec

ific h

eat,

c p

, kJ

/kg.

K

25Ca

Fig. 4. Errors in specic heat measurements (a) and their eect on Umentioned references for further information.4.18

4.2

Gucker & Ayres (1937) Darros Barbosa et al. (2003)corresponding values of Uc(m), and Uc(0) in particular.Theoretical values of the excess heat capacity CEP asym

can be obtained by direct dierentiation of the excess en-thalpy (Eq. (27)) with respect to temperature:

CEP asymR

oHEasym=RoT

TXnk2

2a0k T a00kk 1 x

kB; 35

where a00kT is the second derivative of ak(T). For the as-sumed temperature dependence of activity coecientsthis results in

CEP asymR

T 2a0T Ta00T Xnk2

bk2k 1 x

kB; 36

where the temperature dependent term is

T 2a0T Ta00T a2 2a3h 6a4h2 12a5h3;with h T=T 0. 37

Eqs. (36) and (37) were used to calculate from the activ-ity model the molar excess heat capacity at requiredtemperatures and sucrose concentrations.

5. Data regression technique

The proposed model of thermodynamic activity inthe sucrosewater system was identied using the exper-imental data discussed in detail in previous sections. Thefollowing least-squares performance index, composed ofve dierent types of deviations, was used in data

0 0.5 1 1.5 2560

580

600

620

640

660

680

700

720

Galema et al. (1993)

Di Paola & Belleau (1977)

Banipal et al. (1997)

Gucker & Ayres (1937)

Molality

Appa

rent

mol

ar h

eat c

apac

ity o

f suc

rose

,

c , J/

mol

.

K

25C

b

apparent molar heat capacity of sucrose in solution (b). See above-

Chemistry 96 (2006) 346370regression:

I w2VLEXi

w2VLE;iln cA;i ln cexpA;i 2VLE w2FPD

Xi

ln cA;i ln cexpA;i 2FPD w2SOLXi

ln~cB;i

ln~cexpB;i 2SOL w2HEXi

HEasym;iRT i

HE;expasym;i

RT expi

" #2

w2CEXi

CEP asym;iR

CE;expP asym;i

R

" #2. 38

The weighting factors wVLE, wFPD, wSOL, wHE and wCEwere taken as inverted mean values of the correspondingexperimental variables and only wVLE,i were specic tothe given set of VLE data. The latter were estimatedbeforehand by considering the VLE data only andapplying the technique of iteratively reweighted leastsquares (IRLS) with the elimination of outliers and rig-orous statistical analysis. They were intended to reectthe performance of individual data sets as comparedto the performance of the entire VLE database. Moredetails on this technique can be found in Starzak andPeacock (1997). Values of the normalized weighting fac-tors wVLE,i are given in the last column of Table 1.

The expansion coecients of the water activity coef-cient equation, representing both the temperature and

parameters describing the temperature dependence

data were expressed in terms of the corresponding val-ues of boiling point elevation (BPE) and presented asa correlation graph in Fig. 6. It is clear from thisgraph that the new activity equation represents a sta-tistical compromise for scarce and rather inconsistentmeasurements from the region of high sucrose concen-trations. A much better data t was obtained for thewater activity coecient along the curve representingSLE between ice and sucrose solution (Fig. 7(a))which also resulted in an excellent match of the freez-ing point depression (FPD) data (Fig. 7(b)). For thesucrose activity coecient at the conditions of SLEbetween sucrose crystals and the solution, the datat is excellent only up to about 7075% wt. sucroseand deteriorates slightly at higher concentrations(Fig. 8(a)). A similar pattern exhibits the solubility

M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370 359and those associated with the concentration depen-dence. Consequently, in general, the problem of dataregression is highly non-linear and numerically com-plex. The type of temperature dependence assumed inthis study allowed for the total separation of thesetwo groups of parameters, so the resulting regressionproblem was at least partly linear. For example, forxed values of the composition dependence parametersbk, the water activity coecient can be considered alinear function of the temperature dependence parame-ters ai (see Eq. (3)). The same conclusion can be drawnafter inspecting other relationships derived in previoussections which express experimental variables in termsof activity coecients, such as those given by Eq. (5)with (6), Eq. (27) with (28), and nally Eq. (36) with(37). All these regression equations are of the followinggeneral form:

Y X5i0

fiX; bai; 39

where Y is the selected experimental output variable andfi are known functions of the vector of experimental in-put variables X and the vector of non-linear parametersb. This type of structure allows for a parameter optimi-zation technique in which the full data regression prob-lem is decomposed into two problems of a smaller size.The computational algorithm consists then of an innerand outer regression loop. The outer loop optimizesthe performance index for the group of non-linearparameters (bk), using the simplex method of optimiza-tion, while the inner loop performs multi-linear regres-sion on the group of linear parameters (ai). Thesolubility parameter DBB was optimized in the third out-ermost loop using a one-dimensional direct searchtechnique.

6. Results of data regression

Expansions of dierent lengths have been tested forboth the temperature and composition dependence tond the most suitable mathematical representation ofthe activity model. Finally, in order to avoid over-parametrization of the model, it was decided to dropthe cubic term in the temperature dependence (bycomposition dependence, a = [a0, a1, . . . , a5]T and

b = [b0, b1, . . . , bn 2]T, respectively, were the estimated

parameters of the model. In addition, the parameterDBB of the solubility model was also optimized. As apolynomial, the Margules equation is linear with re-spect to its expansion coecients. However, for tem-perature dependent coecients, the equation becomesparametrically non-linear because of the crossover ofassuming a5 = 0 in Eq. (4)) and consider only threenon-zero terms (n = 4) in the composition dependentpart of Eq. (3). This resulted nally in a four-suxMargules equation with temperature dependentparameters. The optimized parameters of the modelminimizing the weighted sum of all experimental devi-ations are given in Table 6. The optimal value of thesolubility parameter DBB was 0.698 K

1.Contributions from dierent data categories to the

global performance index were shown in Fig. 5 in theform of correlation diagrams. The largest discrepanciesbetween predicted and measured values occur forhighly concentrated sucrose solutions, regardless ofthe type of data considered. This was expected, asthe experimental results in this region are not highlyreliable (mainly due to strict temperature control prob-lems when working with very viscous media) and showsometimes serious inconsistencies when comparedamongst dierent literature sources. Also in this region,the system exhibits particularly strong non-ideal behav-ior that cannot be fully described using a relativelysimple non-mechanistic model of thermodynamicactivity.

For the purpose of illustration, the original VLE

Table 6Estimated parameters of the proposed activity coecient model

Estimated parameters(water activity coecient)

Parameterscalculated as:c =M1M0b (sucroseactivity coecient)

a0 268.59 c0 0.12001a1 861.42 c1 0.48972a2 1102.3 c2 0.92730a3 1399.9a4 277.88a5 (assumed) 0b0 (by denition) 1b1 1.9831b2 0.92730curve (Fig. 8(b)). The results of data regression for

D dat

nts

ood Chemistry 96 (2006) 3463700.5

VLE data: ln(A)

1507 points 0

FP

213 poi

360 M. Starzak, M. Mathlouthi / Fthe heat of dilution can be considered satisfactory aswell, except in the case of the highest studied temper-ature (80 C) at which the model underpredicts the en-thalpy. Fig. 9 presents these calculations for selecteddata sets at dierent temperatures. Figs. 10(a) and

5 0 0.5

5

0

0.15 0.1

0.15

0.1

0.05

10 6 10 4 10 2

10 6

10 4

10 2

Heat of dilution data: HasymE /RT

Dilute solutions

0 0.0

0.05

0.1

experi

pred

icted

Heat of dilution

283 points

Fig. 5. Correlation diagrams for dieren

0 10 20 30 40 50 60 700

10

20

30

40

50

60

70

Fig. 6. Correlation of VLE data presented in terms of BPE (boilingpoint elevation).(b) show predicted values of the dimensionless excessenthalpy plotted against sucrose concentrationand temperature, respectively. Interestingly, for each

0.05 0

a: ln(A)

4 3.5 3 2.5 2 1.54

3.5

3

2.5

2

1.5Solubility data: ln(~B)

265 points

05 0.1mental

data: HasymE /RT

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

Heat capacity data: Cp,asymE /R

70 points

t categories of experimental data.solution composition the excess enthalpy shows amaximum with respect to temperature. In fact,because of the simplied temperature/concentrationrelationship assumed in this activity model, all pre-dicted maxima occur at the same temperature ofabout 35 C (Fig. 10(b)). The t for heat capacitydata is generally acceptable, although for dilutesolutions (below 1 mol/kg water), when the excessheat capacities are relatively small, the predicted val-ues are well below some of the observed data (Fig.11). This was mainly due to the evident discrepancybetween the heat capacity data published by Guckerand Ayres (1937) and those reported more recently(Banipal, Banipal, Ahluwalia, & Lark, 1997; DiPaola& Belleau, 1977). Following the temperature trendalready observed for excess enthalpy, the excess heatcapacity assumes positive values at temperaturesbelow 40 C and negative values above this tempera-ture (Fig. 12). A zero excess heat capacity at 40 C,regardless of the sucrose concentration, is again aresult of the specic mathematical form of the activitymodel assumed. Due to the lack of direct experimentaldata, the results of heat capacity calculations for non-ambient temperatures are rather uncertain. Theyshould be seen as predictions deduced indirectly fromthe temperature eect on heat of dilution rather thanderived directly from the specic heat data.

0 20 40 60 800.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

% wt. sucrose

Wa

ter a

ctiv

ity c

oeffi

cient

a

0 20 40 60 800

5

10

15

20

25

30

35

40

% wt. sucrose

Fre

ezi

ng p

oint

dep

ress

ion,

C

b

Fig. 7. SLE between ice and aqueous solution of sucrose predicted (solid line) and experimental data (circles): (a) water activity coecient;(b) freezing point depression.

0 50 100 150

5

5

5

5

Temperature, C

ln

~ B

a

0 50 100 15060

65

70

75

80

85

90

95

100

Temperature, C

% w

t. su

cros

e

b

Fig. 8. SLE between sucrose crystals and aqueous solution of sucrose predicted (solid line) and experimental data (circles): (a) logarithm of thesucrose activity coecient ~cB; (b) sucrose solubility.

M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370 361

Pra

362 M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 346370Pratt, 1918The water activity coecient cA predicted by theproposed activity model exhibits a characteristic mini-mum with regard to the sucrose content in the region

0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

Dim

ensi

onle

ss m

olar

exc

ess

enth

alpy

, HE /

RT

Hunter, 1926

30C

1 1.5 2

0.004

0.006

0.008

0.0100C

1

0.004

0.006

0.008

0.01010C

0 20

0.02

0.04

0.06

0.08

0.1

Mo

Vallender &

40C

Fig. 9. Predicted (solid lines) and experimental (broken lines) dim

0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0C

5C

20C

40C

60C

80C

Molality

Dim

ensi

onel

ss m

olar

exc

ess

enth

alpy

, H

E /

RT

a

Fig. 10. Model predictions of dimensionless molar excess entt, 19180.08

Vallender & Perman, 1931of very high concentrations (Fig. 13(a)). According tothe model, the minimum occurs at about 96% wt. su-crose and its location does not depend on tempera-

1.5 2 0 2 4 60

0.02

0.04

0.0620C

4 6lality

Perman, 1931

0 2 4 6 80

0.02

0.04

0.06

0.08Vallender & Perman, 1931

60C

ensionless molar excess enthalpy at selected temperatures.

0 20 40 60 800

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 mol/kg

2 mol/kg

3 mol/kg

4 mol/kg

5 mol/kg

6 mol/kg

Temperature, C

Dim

ensi

onle

ss m

olar

exc

ess

enth

alpy

, H

E /

RT

b

thalpy as a function of: (a) molality; (b) temperature.

ker &

M. Starzak, M. Mathlouthi / Food C0.4Guc

0.4Vivien, 1906ture. The latter is, however, the result of a specicmathematical form of the temperature dependenceadopted in this study for the activity coecient. Con-

0 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Mo

20C

0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35M

olar

exc

ess

heat

cap

acity

, C p

E /R

15C

Fig. 11. Predicted (solid lines) and experimental (circles) dimensi

0 2 4 6 8

0

0.5

1

0C

10C

20C

40C

60C

80C

Molality

Mol

ar e

xcess

hea

t cap

acity

, C p

E /R

Fig. 12. Dimensionless molar excess heat capacity as a function ofmolality and temperature model predictions.7. Comparison with the UNIQUAC and UNIFAC models

Since in a number of previous works on activitycoecients in the sucrosewater system the UNI-QUAC model was used to correlate experimental datasequently, the water activity coecient evaluated forthe conditions of isobaric VLE (when the boilingpoint temperature of solution varies together withthe composition) also show a minimum at the samelocation (Fig. 13(b)).

4 6lality

Ayres, 1937

0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

25C

Gucker & Ayres, 1937

onless molar excess heat capacity at selected temperatures.

hemistry 96 (2006) 346370 363(Catte et al., 1994; LeMaguer, 1992; Peres & Macedo,1996), it is worth comparing this model with the activ-ity model developed in our study. Equations describ-ing the molar excess Gibbs energy as well as otherthermodynamic functions according to the modiedUNIQUAC model (Larsen, Rasmussen, & Fredensl-und, 1987) are not presented here as they can be allfound elsewhere (Catte et al., 1994). The temperaturedependence of this model is represented by the Boltz-mann factors:

sij expAijT ; i; j 1; 2; i 6 j 40

where the interaction parameters Aij(T) have the mathe-matical form similar to Eq. (4):

AijT Aij;0T Aij;1 Aij;2 ln T Aij;3T . 41The required Van der Waals molecular volume and sur-face area parameters for sucrose and water were takenfrom Catte et al. (1994), while the constants Aij,k (k =1, 2, 3) were estimated by regressing the experimental

00.75

0.8

0.85

0.9

0.95

1

ctiv

ity c

oeffi

cient

a

ucros

364 M. Starzak, M. Mathlouthi / Food Chemistry 96 (2006) 34637070 80 90 100.55

0.6

0.65

0.7 0C20C

40C

60C

80C

% wt. sucrose

Wa

ter a

Fig. 13. Model predictions of water activity coecient as a function of s760 mmHg.data used in the main part of this study. The Van derWaals parameters as well as optimized values of Aij,kare listed in Table 7.

Calculations show that the UNIQUAC model givesan adequate description of the sucrosewater systemonly for concentrations not exceeding 90% wt. su-crose. In some instances, such as the solubility curveand heat of dilution, it even results in a slightly betterdata t than the Margules model, which might implythat the temperature dependence built-in to the UNI-QUAC model is more versatile than the restrictiveexpression employed in this study. On the other hand,the UNIQUAC model predicts a monotonic decline ofthe water activity coecient cA with increasing sucrose

Table 7Parameters of the UNIQUAC model

Van der Waalsmolecular parameters(Catte et al., 1994)

Estimated interaction constants

rA 0.9200 A12,0 227.30qA 1.4000 A12,1 8.3359rB 14.550 A12,2 1.2520qB 14.310 A12,3 5.0513 104

A21,0 224.72A21,1 12.554A21,2 1.8257A21,3 4.4100 1040 20 40 60 80 1000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

% wt. sucrose

Wa

ter a

ctiv

ity c

oeffi

cient

at i

soba

ric V

LE

b

e concentration and temperature: (a) selected isotherms; (b) for VLE atconcentration for all investigated temperatures. A sim-ilar monotonic behavior was also reported in the pre-vious studies involving this model. In fact, theinability of the UNIQUAC model to reproduce astrongly pronounced minimum of cA(xB) in the regionof highly concentrated solutions has been demon-strated numerically in one of our earlier studies (Star-zak & Peacock, 1998).

In a few earlier studies (Starzak & Mathlouthi,2002; Starzak, Peacock, & Mathlouthi, 2000; VanHook, 1987), it was demonstrated that the intriguingminimum of the water activity coecient could bepredicted theoretically from relatively simple chemicalmodels of thermodynamic activity by consideringcompetitive watersucrose (sucrose hydration) and su-crose-sucrose (sucrose clustering) interactions in thesolution. Therefore, for comparative purposes, we alsoinvestigated the behavior of a more recent A-UNI-FAC model developed by Ferreira et al. (2003) foraqueous solutions of sugars. This model is particularlyinteresting since it attempts to incorporate the eect ofhydrogen-bonded molecular association caused by thepresence of numerous hydroxyl groups in the system.The results of simulation showed, however, no mini-mum of the water activity coecient in the concentra-tion region of interest (9098% wt. sucrose).Interestingly, a very sharp minimum was foundnumerically at a non-realistic sucrose content

occurrence of a minimum of the water activity coe-cient at about 96% wt. sucrose and also to convincingly

Knowledge on Sugar, Reims (France), is greatlyacknowledged.

to one another through the GibbsDuhem equationwhich represents a dierential relation between the

ood Chemistry 96 (2006) 346370 365demonstrate that the UNIQUAC/UNIFAC models areinadequate. Nevertheless, these data do show a statisti-cally signicant trend which indicates the existence ofsuch an intriguing feature. Moreover, some mechanisticmodels of water activity in the sucrosewater system can(>99.5% wt.). This shows that the A-UNIFAC modelhas the theoretical capacity to reproduce the minimumprovided its parameters are determined using a largernumber of experimental points from the region of veryhigh sucrose concentrations.

8. Conclusions

The empirical four-sux Margules equation with anextended temperature dependence developed in thisstudy gives a satisfactory representation of all experi-mental measurements currently available for this sys-tem including VLE, SLE, as well as thermochemicaldata. The new equation predicts the existence of asharp minimum of the water activity coecient occur-ring at about 96% wt. sucrose, conrming some trendsobserved earlier in the relatively scarce experimentaldata for highly concentrated solutions. This predictionis also in line with theoretical results derived from theanalysis of specic hydrogen-bonded interactionsinvolving molecules of sucrose and water in solution.The proposed model is based on the simplifyingassumption that the temperature dependence of theactivity coecient can be separated from its composi-tion dependence. This was done in order to reducethe number of estimated parameters of the model aswell as to simplify the data regression procedure. Con-sequently, some model predictions, especially thoseregarding the temperature eect on the excess enthalpyand excess heat capacity, are perhaps less reliable thanothers. However, since some discrepancies are also ob-served amongst various pieces of the original thermo-chemical data, adopting a more elaborate activityequation (by releasing the restrictive assumption men-tioned above and hence increasing the number of esti-mated parameters) was not considered an option forfurther model renement.

Well-established methods of predicting activity coe-cients in liquid phase such as UNIQUAC, UNIFAC orASOG can produce models which are exible enough todescribe thermodynamic properties of the sucrosewatersystem for dilute and relatively high concentrated solu-tions (up to 90% wt. sucrose). At extreme sucrose con-centrations, predictions of these models and theproposed empirical correlation are substantially dier-ent. The experimental data presently available in this re-gion are insucient to prove beyond any doubt the

M. Starzak, M. Mathlouthi / Fexplain this hypothetical phenomenon. We agree, how-activity coecients of all the components in the solu-tion. In binary systems, this leads to unique relations be-tween the expansion coecients bk and ak. For thisspecial case, the GibbsDuhem equation may be writtenas:

xAd ln cAdxA

xB d ln cBdxB

. A1On substituting Eqs. (1) and (5) to Eq. (A1), one can ar-rive at:

bk 1kXnlPk

l

k

kl; k P 2; A2

where

kl al l 1l al1; l 2; 3; . . . ; n 1;kn an. A3In the more transparent matrix form:Appendix

The partial molar excess Gibbs energies (chemicalpotentials) of the components in a mixture are relatedever, that the issue can be resolved only by direct exper-imentation which is unfortunately not easy bearing inmind extreme conditions such as temperatures close tothe melting point of sucrose, high viscosity causing poormixing and heat transfer, the risk of sucrose crystalliza-tion, etc.

The empirical activity equation developed in thisstudy represents the most reasonable compromise be-tween the model complexity and the quality of theexperimental data available at present for the sucrosewater system. It should nd applications in variousareas of the sugar and food industries whenever theknowledge of water as well as sucrose activities in aque-ous sugar solutions is required in a wide range of oper-ating conditions. The proposed equation can also beused as a predictive tool for high temperatures and largesucrose concentrations not yet explored experimentally,these being of particular interest to the sugar confection-ery technologist.

Acknowledgment

Partial nancial support of this study from the Asso-ciation Andrew van Hook for the Advancement of theb M1M0a ahM1M0b A4

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