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Food Properties Handbook

M. Shafiur Rahman

Data and Models of Water Activity. I: Solutions and Liquid Foods

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781420003093.ch3 Piotr P. Lewicki Published online on: 28 May 2009

How to cite :- Piotr P. Lewicki. 28 May 2009, Data and Models of Water Activity. I: Solutions and Liquid Foods from: Food Properties Handbook CRC Press Accessed on: 29 Sep 2021 https://www.routledgehandbooks.com/doi/10.1201/9781420003093.ch3

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 CONTENTS Lewicki P. Piotr ß 54 62 Equations...... References...... Empirical 37 ...... Foods ...... Liquid Equations 3.4 and Semiempirical Solutions of Activity 3.3 Water 33 3.2 Introduction...... 3.1 h rsneo lcrlts oi neatosas cu n nuesm tutrn fwater of structuring In some reduced. induce is molecule and structure Water occur clathrate-like formed. in also as is interactions well solute ionic structure as hydrophobic electrolytes, clathrate-like state with of A hydration interaction presence water. and an the structure solvent hand, both the other in of of the mobility formation structure the On to the water. lead solute hydration affects hydrophilic and with Interactions water bonds. structure hydrogen are 2004). strong changes (Lewicki, form those material to all the a ability of storage. in dynamics is during water the food of changes and state course that many the thermodynamic undergoes show that the structure which to examples prove related state, of years above-mentioned the equilibrium the changes over is the conducted All emulsions cause Research from of changes. far forces coalescence those system, and surface of in dynamic foams chains. result and strongly of polymer macroscopic Destabilization mass of polymers a concentration. diffusion boundaries, between spatial translational interphase and constituents, Interactions rotational and with on storage. depend materials during and In to material change constituents, ability the structure, all of of texture, its properties Redistribution relax hence reactions. material, and and the chemical of higher creep properties facilitate with rheological can affects food enable water, turn and the especially in which substrates within diffusion, domains between in result creates contact gradients Concentration Processing passed constituents. are of storage. concentration which during lower changes, physical material and chemical the many causes within equilibrium of lack This equilibrium. 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 h hroyai tt fwtri odarises food in water of state thermodynamic The thermodynamic a in not usually is that system multiphase and multicomponent a is Food aaadMdl fWtrActivity. Water of Models and Data . INTRODUCTION 3.1 :SltosadLqi Foods Liquid and Solutions I: fi sl rmuuulpoete fwtradits and water of properties unusual from rstly CHAPTER fl uence .48 3 Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 hr ocnrto ean osat h ata oa ib reenergy free Gibbs molar partial The constant. remains concentration where where o nyo h eprtr n rsue u loo h muto ahcmoetpeeti the in present component depend each will of energy amount the free on Gibbs also the but pressure, system, work Hence and multicomponent only system. temperature which open the in an on composition only In constant not place. of system takes homogeneous expansion a of to applies equation above The ersnstecag nGbsfe nryo h ytmcue yadto foeml of mole one of addition by caused system the of energy free Gibbs in change the represents xrse yEuto . scle hmclptnilo component of potential chemical called is 3.6 Equation by expressed which in ß physical or is chemical water of of feasibility state of equation: thermodynamic criterion the The by a expressed work. is quantitatively is which the energy the energy, do free affects Gibbs free to also The Gibbs molecules transformation. interstices, foods, its or water for drops by in of common capillaries, expressed in quite ability surface is solvent the the which of reduces Curvature material, water. the of state of thermodynamic porosity Finally, molecules. fGbsfe nryi ecie yteequation: the by described is energy free Gibbs of ib reeeg ndfeeta form: differential in energy free Gibbs where reenergy: free d Substituting where 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 S T H V P E steetoy(J entropy the is stetmeaue(K) temperature the is stevlm (m volume the is (Pa) pressure the (J) is energy internal the is nry( energy steetap (J) enthalpy the is n 1 , i n eoe htcmoetweecnetaincagsand changes concentration where component that denotes 2 ,..., TS H ¼ ) n d n E d stenme fmlso opnn ,2 , . . . 2, 1, component of moles of number the is = G þ 3 )adcnb nesoda oa nry( energy total as understood be can and K) ) ¼ P d @ @ V G P þ V T d , n P j n d and G d d P G ¼ d þ ¼ f G E ( @ G @ T ¼ d ¼ G T @ @ , H ¼ T P n G V i d , P H n d S , n T 1 P T j – , , P d n , P n S 2 TS d j , d S T ... V d T þ S il h ifrnilcag nteGibbs the in change differential the yield , d X n T n n n i j )( @ @ G n H iiihdb unavailable by diminished ) n j T ne hsstain change situation, this Under . , eoe l hs components those all denotes P , n j i n sdntdas denoted is and d n i m (3 (3 (3 (3 (3 3 i .It : : : : : : 4) 6) 5) 1) 2) 3) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where napr da gas ideal pure a In where aigtesbtnea nielgs ei bandceia oeta ntrso easily of terms in potential chemical obtained Lewis gas, ideal an as properties. measurable substance the Taking osat hnEuto . a ewitna follows: as written be can 3.3 Equation Then constant. component ß where hr h uesrpsrfrt ifrn hss iiigEuto . ytenme fmlsof moles of number the by 3.3 Equation Dividing phases. different to refer superscripts the where suf and component necessary of the potential system chemical any of for equality that shown has Gibbs component xsec fitroeua ocs(runt,16) h ai ewe uaiisi aldactivity, called is fugacities between ratio The 1969). a (Prausnitz, forces intermolecular of existence nara ytmpoete fgssdvaefo h da n n oacutfrtedvainLewis deviation the for account to and one ideal the fugacity, from called function deviate new gases a proposed of properties system real a In tcntn temperature constant At Denoting ec,Euto .4becomes 3.14 Equation Hence, . 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 p m s v i i i stemlretoyJ entropy molar the is stemlrvlm (m volume molar the is steiiilpesr ntesystem the in pressure initial the is steceia oeta fcomponent of potential chemical the is R stegscntn (J constant gas the is V = i, i d epn eprtr,ttlpesr,adnme fmlso l te components other all of moles of number and pressure, total temperature, keeping h olwn a ewritten: be can following the n i ¼ v f i i ¼ and p i h ata rsueo h a.Ten daiyo h a olw rmthe from follows gas the of ideality no The gas. the of pressure partial the , S = = d mlK) (mol 3 n = = i o ) nertn qain31 h olwn oml sobtained: is formula following the 3.12 Equation Integrating K). mol mol) ¼ s d v i qain39becomes 3.9 Equation , i G ¼ ¼ d d f RT m m n G m ec,Euto .3frara a ae h olwn form: following the takes gas real a for 3.13 Equation Hence, . d P V i i i m i m ¼ d i i I P d ¼ ¼ d m m m m i n d and V n i i i nalpae.Ta is That phases. all in i i v i m tsadr conditions standard at i ¼ S ¼ ¼ d ¼ d i II d P P v T RT RT ¼ RT i d m þ P m d i s ln ln ln S X i III n i ¼ d i p f T a f p d i i RT i T P m fi i d in odto o qiiru sthe is equilibrium for condition cient d n P i (3 (3 (3 (3 (3 (3 (3 (3 (3 : : : : : : 12) 15) 14) 13) 11) 10) : : : 8) 7) 9) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 For where Raoult follows and gas ideal an to analogously behaves component solution ideal An where ß it and 1 ambient to equal at be and to assumed 6%, be in by as can experienced saturation than written conditions at the be more vapor under can water food not Thus, activity, for 0.2%. by storage important than and gas less temperatures processing ideal is food of deviation the range the from the pressure, in and deviates temperature that vapor shows water 3.1 the Table processing, in presented data of Analysis fwtri oi rlqi odacrigt qain38 aigasse ossigo nideal an follows: of as consisting described system activity be a the can Taking as state 3.8. taken equilibrium Equation is the to 3.16 gas, according Equation ideal food equilibrium from an in liquid calculated and is or phase solution food solid gas that in of Assuming activity water known. is the of food food phase, in that gas activity over the water vapor with calculate water to of used pressure is partial 3.16 Equation when respectively. pressure, total and temperature 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 x k x steml rcino component of fraction mole the is constant the is i ¼ p w 1 and f i i na da ouini rprinlt h ocnrto fta opnn.Thus component. that of concentration the to proportional is solution ideal an in ¼ 180 160 140 120 100 80 60 40 20 10 0.01 with ( Equilibrium Temperature in Vapor Water of Activity and Fugacity 3.1 Table Source: p k w ¼ r h ao rsueo ae ntesse n fpr ae ttesame the at water pure of and system the in water of pressure vapor the are f i hence , dpe rmHs,J.L., Hass, from Adapted h iuda auainada rsue00 MPa 0.01 Pressure at and Saturation at Liquid the 8 )Pesr ka uaiy(P)Activity (kPa) Fugacity (kPa) Pressure C) f i solution i 1002.7 f 618.04 361.35 198.53 101.325 m i 47.362 19.920 i solution 7.376 2.337 1.227 0.611 f f eci.Csohm Acta Cosmochim. Geochim. f ¼ w w i ¼ ¼ x ¼ i kx p p ¼ m w w i a i vapor i solution 939.93 589.40 349.43 194.07 9860.9855 0.9912 0.9950 99.856 46.945 19.821 .5 0.9974 0.9988 0.9992 0.9995 7.357 2.334 1.226 0.611 4 2,1970. 929, 34, , ’ 0.9374 0.9537 0.9670 0.9775 a.Fgct fa of Fugacity law. s (3 (3 (3 (3 : : : : 19) 18) 17) 16) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 pl uc concentrate juice Apple juice Apple and ciiyo ae nslto a ecluae rmEuto .0btteatvt coef activity the but 3.20 Equation from calculated be can solution in water of Activity ewe emnn ioe,fre fatato,adrplinbtennnoa oeue and molecules nonpolar forces between induction repulsion forces, and cient electrostatic attraction, from of arise forces speci and strong dipoles, are permanent ideality between from deviations liquids, In ß Product Foods Liquid Some of Activity Water 3.2 Table oue,adtmeaue nms rs od,i scoet n t esrmn rsnssome presents measurement its and 1 to close is it foods, fresh most dif In temperature. and solutes, undetermined. usually is and composition and temperature of function a is this However, known. be hryjuice Cherry 60.7 juice currant Black concentrate juice Aronia concentrate juice Apple ofebeverage, Coffee concentrate freeze-dried 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 fi ae ciiyo ouin n iudfosdpnso ocnrto,ceia aueof nature chemical concentration, on depends foods liquid and solutions of activity Water ute.Wtratvt fsm ouin n iudfosi rsne nTbe . through 3.2 Tables in presented is foods liquid and solutions some of activity Water culties. fi g neatossc shdoe od.T con o hs neatos natvt coef activity an interactions, these for account To bonds. hydrogen as such interactions c a nrdcd Thus introduced. was . AE CIIYO OUIN N IUDFOODS LIQUID AND SOLUTIONS OF ACTIVITY WATER 3.2 Concentration 40 71.2 62.4 70.1 66.2 66.2 65.4 66.0 66.6 63.9 60.6 65.0 65.0 64.5 40 30 20 10 5 (%) – m 20.91 42 i solution a m i solution i 0.986 0.986 0.815 0.803 0.846 0.822 0.812 0.810 0.823 0.734 0.821 0.739 0.791 0.795 0.798 0.792 0.964 0.978 0.990 0.991 0.996 solution a – w ¼ 0.93 g ¼ i x RT i ln Temperature g 23.3 24.2 23.8 23.3 24.3 24.4 23.7 23.8 24.1 23.4 23.5 23.7 24.3 24.2 i x ( 8 i C) hrf n er Fontan Ferro and Chirife acy ta.(1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk (1987a) Chen hn(1987a) Chen (1987a) Chen (1987b) Chen (1987a) Chen (1987b) Chen Fontan Ferro and Chirife (1982) (1982) Reference fi cient ( continued g w (3 (3 must 3.6 : : 21) 20) fi - ) . Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 ik whole Milk, whole Milk, pasteurized Milk, fat 1.5% Milk, juice Maracuja lvrdmilks Flavored fat 40% Cream, syrup Corn rnejuice Orange juice Orange solution NaCl Molasses condensed Milk juice Lemon ß Product Foods Liquid Some of Activity Water (continued) 3.2 Table ri juices Fruit oe,rape Honey, rpfutjuice Grapefruit 20 concentrate juice Grape juice Grape juice Grape 66% Glucose, iepl juice Pineapple oe,bcwet17.6% buckwheat Honey, Honey concentrate concentrate sweetened concentrate concentrate concentrate 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 Concentration 54 7 ae 0.62 water 17% 50 40 15 0.799 0.877 0.927 salt 23.41% 0.977 salt 15.45% 0.76 salt 10.16% salt 2.18% water 26% 40 65 65.0 60 55 17.4% 59 30 10 61 72 68 7 ae 0.54 water 17% water water – 42 (%) – – 19.5% 20.9% a a 0.994 0.995 0.995 0.995 0.88 0.90 0.9899 0.979 0.892 0.908 0.982 0.988 0.833 0.9901 0.802 0.824 0.84 0.88 0.844 0.962 0.976 0.986 0.983 0.78 0.84 0.735 0.79 w w ¼ ¼ 0.060 0.203 – – – – – – – a 0.932 0.87 0.90 w 0.001 0.0017 0.995 0.905 0.835 0.860 0.0007 0.0017 þ þ 0.026 0.02 x x Temperature 32Jrzke l (1995) al. et Jarczyk 23.2 ( 8 C) hrf n er Fontan Ferro and Chirife (1989) al. et Esteban Fontan Ferro and Chirife (1987a) Chen (1987a) Chen hrf n er Fontan Ferro and Chirife hn(1987a) Chen (1987a) Chen Fontan Ferro and Chirife (1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and Chuang (1983) al. et Favetto Fontan Ferro and Chirife (2006) Bakier sea ta.(1990a) al. et Esteban hn(1987a) Chen (1987a) Chen (1987a) Chen (2006) Bakier hn(1987b) Chen (1987b) Chen (1987a) Chen Fontan Ferro and Chirife (1986) Richardson (1990a) al. et Esteban hn(1987a) Chen (1987a) Chen (1987a) Chen hn(1987b) Chen (1982) (1982) (1982) (1982) (1982) (1982) Reference Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 aper juice Raspberry o sauce Soy juice cherry Sour solution milk Skim juice currant Red juice Raspberry Product Foods Liquid Some of Activity Water (continued) 3.2 Table ß tabryjuice Strawberry Sucrose oaopse triple paste, Tomato ketchup Tomato juice Strawberry hycheese Whey purée Tomato outbeverages Yogurt uretboh(Difco) broth Nutrient microbiology food in used media Culture ri er nuinbroth infusion heart Brain Staphyloccocus atyatetat50% extract yeast Malt atyatetat70% extract yeast Malt NaCl) (25% broth Halophylic concentrate concentrate concentrate concentrated concentrate (Oxoid) o 1 (Difco) 110 no. lcs broth glucose lcs rcoebroth fructose glucose 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 medium Concentration 530.793 0.816 0.796 65.3 0.756 64.3 0.796 67.4 0.788 66.2 0.971 64.2 0.982 0.990 67.0 0.997 40 30 0.825 20 0.811 10 0.828 62.3 65.6 0.903 61.0 53.6 62 0.839 0.86 0.90 66.29 0.944 66 0.982 59 0.993 40.78 15.29 5.87 600.838 56.0 (%) 0.988 0.810 0.959 0.991 0.978 0.934 0.996 0.9876 0.997 0.995 0.936 0.876 0.740 0.816 a w 0.018 .1 0Jkbe (1983) Jakobsen 20 0.012 0.0013 .2 0Jkbe (1983) Jakobsen 20 0.029 0.000 0.0008 Temperature 48Jrzke l (1995) al. et Jarczyk 24.8 (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) 23.3 al. et Jarczyk 23.6 (1995) al. et Jarczyk 23.2 23.5 40Jrzke l (1995) al. et Jarczyk (1995) al. et Jarczyk (1995) al. et Jarczyk 24.0 (1995) al. et Jarczyk 24.3 (1995) al. et Jarczyk 23.4 24.1 24.0 24Jrzke l (1995) al. et Jarczyk 22.4 0Etbne l (1990b) al. et Esteban 20 0Etbne l (1990b) al. et Esteban 20 0Etbne l (1990b) al. et Esteban 20 0Etbne l (1990b) al. et Esteban 20 0Etbne l (1990b) al. et Esteban (1990b) al. et Esteban 20 20 ( 8 C) hn(1987b) Chen (1987b) Chen (1987b) Chen (1987b) Chen Fontan Ferro and Chirife hagadTld (1976) Toledo and Chuang (1986) Richardson (1983) al. et Favetto (1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and Chuang (1978) Stoloff hrf n er Fontan Ferro and Chirife aet ta.(1983) al. et Favetto sea ta.(1989) al. et Esteban sea ta.(1990a) al. et Esteban (1982) (1982) Reference Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 Xylose acid Tartaric ucncacid Succinic Sorbitol Ribose acid Oxalic Mannose Mannitol Maltose acid Malonic acid Lactic Glycine acid Glucuronic acid Gluconic Galactose Arabinose ß (1940) Nicol and Grover (1940) Nicol and Grover (1940) Nicol and Grover (1976) Toledo and 15 Chuang (1940) Nicol and Grover (1940) Nicol and Grover 0.275 (1976) Toledo and 0.446 Chuang 0.587 (1940) Nicol and (1987b) Grover Chen 0.737 (1940) Nicol and (1987b) Grover Chen 0.814 (1987b) 5 Chen (1976) Toledo and 0.839 Chuang (1940) Nicol and Grover 0.887 0.923 0.944 (1987b) (1987b) Chen Chen (1987b) 0.955 Chen (1976) Toledo and L Chuang (1940) Nicol and 0.93 Grover 0.971 0.935 Solute 0.983 (1987b) 0.961 Chen 0.982 (1987b) Chen (1987b) Chen 0.92 83.0 75.0 0.96 0.988 66.29 0.992 60.0 0.993 0.985 50.0 0.991 40.78 40 35.0 0.996 30 25.0 0.992 20 15.29 15 15 0.995 15 0.997 Reference 10 0.999 8 5.87 Glucose 5 Glycerol 5 5 Sucrose 3 Fructose 1 Lactose Compounds Weight Molecular Low of (%) Solutions Concentration of Activity Water 3.3 Table -Alanine 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 0.98 0.97 0.97 0.97 0.98 0.98 0.99 0.96 0.98 0.98 0.98 0.97 0.98 0.98 0.96 0.97 0.97 ocnrto (%) Concentration 0.95 0.96 0.95 0.96 0.96 0.96 0.96 0.96 0.95 0.94 0.94 0.96 0.96 0.96 0.96 .6Ce n ams(1980) Karmas and Chen 0.96 (1980) Karmas and Chen 0.98 hnadKra (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen hnadKra (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen (1980) Karmas and Chen Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 al 3.7 Table yai prah n qain ae nslto hois(eeoe l,2001). al., et (Sereno theories thermo- the solution by on derived based equations equations composition, and solution approach, ideal on dynamic based an equations of empirical includes those This from solution coef real activity a of the properties by coef of expressed deviation is the solution, function, excess the Hence, xesfntosaetemdnmcpoete fsltos hc r necs ftoeo nideal Thus an concentration. of and those of pressure, The excess temperature, in function. same are excess the which thermodynamic solutions, at the of solution by properties characterized thermodynamic be are functions can excess solution liquid the of Nonideality ß 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 fi in fcomponent of cient .7 .6 .7 .6 .7 .8 .5 0.979 0.984 0.258 0.988 0.217 0.990 0.980 0.176 0.995 0.985 0.134 0.998 0.270 0.989 0.091 0.228 0.991 0.046 0.965 0.186 0.995 Source: 0.971 0.142 0.998 0.274 0.978 0.096 0.229 0.985 0.049 0.962 0.186 0.989 0.970 0.142 0.995 0.272 0.977 0.096 0.229 0.983 0.049 0.186 0.990 0.142 0.995 0.096 0.049 Wa Solutions Sugar of Activity Water 3.5 Table oe W Note: eea prahshv enue ocluaeatvt coef activity calculate to used been have approaches Several . Source: 60.0 55.0 50.0 45.0 40.0 35.0 30.0 Glucose ocnrto (%) Concentration Solutions Glucose of Activity Water 3.4 Table Glucose dpe rmVlzoo . erle,AJ,adVtl,A,in A., Vitali, 7 and A.J., ICEF Meirelles, C., Valezmoro, from Adapted A148. egtfato o solute. for fraction weight , dpe rmVe u,A,Mn gyn . n ulr J., Muller, and H., Nguyen, 2003. Minh 243, A., 57, Bui, Viet from Adapted w oit .(d) Shef (Ed.), R. Jowitt, , i nteslto.Atvt coef Activity solution. the in m i solution Wa Fructose .3 .3 .3 0.841 0.870 0.895 0.914 0.839 0.930 0.869 0.945 0.893 0.957 0.913 0.837 0.930 0.867 0.944 0.891 0.956 0.912 0.835 0.929 0.865 0.943 0.891 0.955 0.910 0.927 0.942 0.954 20 m i ideal fi l cdmcPes Shef Press, Academic eld w fi in.Tepolmrsle to resolves problem The cient. ae ciiya eprtr ( Temperature at Activity Water ¼ RT .4 0.948 0.958 0.445 0.967 0.399 0.974 0.349 0.292 ln 25 Wa a x Sucrose i i ¼ fi RT insfrsm uasaegvnin given are sugars some for cients ln w fi g l,U,19,p.A145 pp. 1997, UK, eld, 30 i niern odat Food & Engineering fi inso iudsolutions. liquid of cients .Fo Eng Food J. 8 Wa C) Maltose fi dn h activity the nding 35 ., w – (3 : 22) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 . 0.8493 7.2271 6.0778 6.0 . 0.8634 0.8776 0.8917 5.5 0.9057 5.0339 5.0 0.9193 4.5 4.0 0.9328 3.9199 0.9477 3.5 3.1491 0.9442 0.9457 3.0 2.7466 2.563 2.5 2.3865 .0 .610.8758 0.8631 5.551 5.508 . 0.9581 0.9617 2.379 0.9599 2.0 0.9719 1.9976 1.926 0.9806 1.5212 1.4 1.0727 1.0 0.7513 where . 0.9907 0.9982 0.617 0.5 0.1 ß Molality Solutions Sugars of Activity Water 3.6 Table obnn h oa xesGbseeg ihteGibbs the with energy Gibbs excess total the combining 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 usrpswadsdnt ae n oue respectively solute, and water denote s and w subscripts A q z o ipetocmoetmxue h oleutosfrbt opnnsaeotie by obtained are components both for equations Wohl the mixture, two-component simple a For , steefcievlm rcino opnn nsolution in component of fraction volume effective the is steefciemlrvlm fcmoeti solution in component of volume molar effective the is B r constants are 55 550 25 50 25 urs ( Sucrose 8 )Guoe( Glucose C) ln ln .720.8878 0.8722 .400.9517 0.9400 .750.9851 0.9735 g g w s ¼ ¼ a w z z w 2 s 2 A B 8 þ þ )Futs ( Fructose C) 2 2 z z w s A B q q q q 0.841 0.870 0.897 0.924 0.938 0.947 0.954 0.963 0.971 0.979 0.985 – w w s s ue equation. Duhem 25 B A .0 orae l (1994b) al. et Correa (1994b) al. et Correa 0.005 0.005 .0 orae l (1994b) al. et Correa 0.005 (1994b) al. et Correa (1994b) al. 0.005 et Correa (1994b) al. 0.005 et Correa 0.005 (1994b) al. et Correa 0.005 .0 orae l (1994b) al. et Correa (1994b) al. 0.005 et Correa (1994b) al. 0.005 et Correa (1994b) al. 0.005 et Correa 0.005 8 C) oisnadSoe (1965) Stokes and Robinson (1975) Ross os(1975) Ross (1965) Stokes and Robinson (1965) Stokes and Robinson (1965) Stokes and Robinson (1965) Stokes and Robinson (1965) Stokes and Robinson (1965) Stokes and Robinson (1975) Ross (1965) Stokes and Robinson (1975) Ross os(1975) Ross (1965) Stokes and Robinson (1965) Stokes and Robinson (1975) Ross oisnadSoe (1965) Stokes and Robinson oisnadSoe (1965) Stokes and Robinson (1965) Stokes and Robinson Reference (3 : 23) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 . .0 1.189 1.104 1.169 1.083 1.150 1.061 1.130 1.306 1.109 1.329 1.088 1.298 1.068 1.039 1.266 1.048 1.233 1.032 1.200 1.166 1.025 1.131 1.031 1.018 1.025 1.013 1.097 1.019 1.009 1.084 1.014 1.007 1.006 1.288 coef Activity 1.071 1.010 1.005 1.265 6.0 1.004 1.242 5.5 1.059 1.006 1.003 1.220 5.0 1.005 1.002 1.199 1.176 4.5 1.047 1.003 1.002 1.179 1.163 4.0 1.002 1.001 1.158 1.150 3.5 1.036 1.001 1.001 1.138 1.138 3.0 1.031 1.001 1.120 1.126 2.5 1.026 1.000 1.103 1.114 2.0 1.021 1.000 1.088 1.103 1.9 1.017 1.000 1.074 1.091 1.8 1.013 1.000 1.061 1.081 1.7 1.009 1.049 1.070 1.6 1.006 1.040 1.061 1.5 1.004 1.030 1.052 1.4 1.002 1.023 1.044 1.3 1.016 1.036 1.2 1.010 1.027 1.1 1.003 1.020 1.0 1.014 0.9 1.010 0.8 1.007 0.7 1.003 0.6 0.5 0.4 0.3 0.2 0.1 ciiycoef Activity ciiycoef Activity suigta oeue r iia nsz,saeadceia aue( nature chemical and shape size, in similar are molecules that Assuming Molality Coef Activity 3.7 Table ß coef activity for equation Wohl The r qa,tetwo-suf the equal, are ciiycoef Activity nwtrsolution water in nwtrsolution water in nmloesolution maltose in nxls solution xylose in 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 fi fi fi fi inso xylose of cients inso maltose of cients inso water of cients inso water of cients ta. 1983a) al., et (Miyajima Maltose fi fi aglseutosaeotie Pasize l,1999) al., et (Prausnitz obtained are equations Margules x insfrSchrdsa 25 at Saccharides for cients ln ln ln ln ta. 1983a) al., et Maltotriose where where where where (Miyajima g g g g w s w s ¼ ¼ fi ¼ ¼ in fwtri simpli is water of cient 0.02814 0.007270 x x m m ln 1.282 2.518 steml rcino h solute the of fraction mole the is solute the of fraction mole the is stemolality the is stemolality the is g w x x m ¼ 2 2 m þ x ta. 1983b) al., et þ 9.093 60.71 s 2 0.003432 D (Miyajima ( 8 0.02727 -Glucose A C þ x x 3 3 kx þ þ m m w 302.5 65.61 )( 2 2 fi 0.003207 0.0006427 dt h olwn form: following the to ed x x 4 4 ta. 1983b) al., et , , D (Miyajima -Mannose m m 3 , 3 q , 1 ¼ q 2 n that and ) ear n Uedaira and Uedaira Uedaira and Uedaira ear n Uedaira and Uedaira ear n Uedaira and Uedaira (1969) (1969) (1969) (1969) ta. 1983b) al., et D -Galactose (Miyajima A and 3 : 24) B Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 h oa volume molar The can as which mixture equation, binary following a the 3.8. for developed Table written and in be models collected local-composition are applied solutions (1964) sugar Wilson for equations above the for parameters selected The equations: Laar van 2001). al., et (Poling Three-suf follows as are derivations these of expansions Further xs o isle opnns ec h ai ftermlrvlmsi sdadteeuto is equation the and used is volumes molar their of ratio the hence simpli components, dissolved for exist ß 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 ouintere eeue odvlpmlrecs ib nryadatvt coef activity and energy Gibbs excess molar develop to used were theories Solution fi dt h olwn form: following the to ed fi aglsequations: Margules x ol(qain32)2.585 Source: 3.24) (Equation Wohl 3.27) (Equation Laar van agls(qain32)2.1819 3.26) (Equation Margules 3.25) (Equation Margules Sucrose 3.24) (Equation Wohl 3.27) (Equation Laar van 3.26) (Equation Margules 3.25) (Equation Margules Glucose Sugar Equation for 3.27 through 3.24 Equations of Parameters 3.8 Table v ln w L g and 12 w ln ¼ ¼ v dpe rmd ido . orr,S,adHf,V., Hoff, and S., Correra, Eng B., Cindio, de from Adapted g s v v w ee opr iud iha ciiyo .Apr iudpaede not does phase liquid pure A 1. of activity an with liquids pure to refer w ouin d idoe l,1995) al., et Cindio (de Solutions s n( ln ,2,45 1995. 405, 24, ., ¼ e ( ln x E g w ws ln w = þ RT ln ! ¼ ln L ) X g k ; g (2 ¼ ws n w s 1 ln B x ln ¼ ¼ L x s g ) k 21 g L B A þ w s A w ¼ ¼ ¼ x ) k 2.064 5.998 4.240 8.494 4.240 2.217 1 1 s x v ABk v RT RT þ þ s 2 w A A s þ x þ e w B B A A ( x x 1 þ 2( w 2 s 2 x x L x x E w w sw s s A ws L = X k ws RT ¼ n 2 2 1 x ) B 3.948 2.064 3.866 2.977 s P ) x x i n s 3 ¼ x k s L 1 þ x w L i k L L sw ik sw .Food J. 9.916 2.526 x w fi cients. (3 (3 (3 (3 (3 : : : : : 27) 26) 25) 29) 28) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 14)uigaltieter.Tedvlpdeuto ecie reeeg hnedrn mixing during change energy polymer. free amorphous a and describes equation solvent developed a The of theory. lattice a was using mixing (1942) of entropy the in model reduction Recently, segment-based hard-sphere-chain The perturbed 1982). (1994). the systems. al. al., and polymer et (1993) Song et for Chen by state (Chen by NRTL proposed of developed The was solutions equations was 3.9. model develop electrolyte Table NRTL to in to polymer made volatiles extended been food have some was for attempts model presented are composition equation local NRTL the of Parameters where ty ctt 024. 1. .84LeadKm(1995) Kim and Lee 0.2834 699.9 618.1 1443.6 2241.3 25 71 1-Propanol 40 50 Methanol Methanol 100 acetate Ethyl acetate Ethyl Ethanol acid Acetic eo n Prausnitz and Renon ß ( Temperature Water Mixtures in Binary for Equation NRTL Substance the of Parameters 3.9 Table h bv qaini ai o uhplmrcnetain twihplmrcan nelc in interlace chains polymer simpli which suf 3.31 at is Equation concentrations solution other, polymer the such When for solution. valid the is equation above The r lobsdo h ocp flclcmoiin h RLeuto o iaymxueis mixture binary a for equation NRTL The composition. local of concept as the expressed on based also are 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 n x n iigo ihmlclrwih oye ihsletcue h nrp hne This change. entropy the causes solvent with polymer weight molecular high of Mixing 2 1 steFoyitrcinprmtrdntn nemlclrinteractions intermolecular denoting parameter interaction Flory the is stemlrvlm fpolymer of volume molar solvent the of is volume molar the is 88 65 ’ 16)eutoskona h o-admtolqi qain (NRTL) equations liquid two non-random the as known equations (1968) s – – – – 618.5770 .90 ane n adraa(2004) Valderrama and Faúndez 0.39304 1094.4 573.90 727.01 1881.15 95 100 76 1 183.82 119 ln m G b g ws fi ¼ ¼ w 8 st h form: the to es C) m ¼ ¼ m n n 1 2 0 ¼ x ( exp x þ s 2 "# m RT t 0 A sw fi þ ws fi a inl iue n h hisaentitrcigec with each interacting not are chains the and diluted ciently n(1 ln s omltdidpnetyb lr 14)adHuggins and (1942) Flory by independently formulated rst RT 12 x w t ws þ n(1 ln G ) x sw t n 0.20246FúdzadVlerm (2004) Valderrama (1995) and Kim Faúndez and Lee 0.29426 0.2442 (1995) Kim 208.12 and Lee 280.6 0.1803 391.8 s ws G 2150322FúdzadVlerm (2004) Valderrama and Faúndez 0.30282 12.175 (2004) Valderrama and Faúndez 0.39688 47.032 2 A ) sw sw þ ¼ n 2 A RT 2 ) ws 1 þ þ n ( 2 x b 1 s a þ þ 12 t n ws xn 2 x G þ w 2 2 G ws xn ws 2 2 ) 2 Reference (3 (3 (3 : : : 30) 32) 31) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 bandbteneprmna n rdce ausof values predicted to 25 (1994a) and at al. et urea experimental Correa and by between sugars used of obtained was solutions method of ASOG method activities The this water forces. to calculate intermolecular According in (1979). and shape Tochigi and and Wilson size Kojima by and developed coef (1969), was activity Deal (ASOG) the and Groups Derr of (1962), Solution Deal Analytical and The groups. functional standard 3.10. Table in presented are calculation Flory The where ß follows: as is solution polymer the in activity water for rewritten 3.32 Equation ciiycoef activity respectively. 0.96%, water rpriso ua qeu ouin.Peitdwtratvte fbnr ytm water systems binary of thermodynamic activities predict water to Predicted (1996) solutions. Macedo aqueous and sugar and Peres by of for 0.28%, used properties activity was 0.21%, method water Chemical by of Quasi Universal values values and calculated measured that measured from (1997) the deviated al. et between systems Correa respectively. (Kawaguchi quaternary deviation 0.20%, by solutions and shown the electrolytes was to ternary, with it applied binary, method was and or method 1981) ASOG glycerol, The al., sugars, 1%. the et containing exceeding with not solutions values ternary predicted predicted and be binary can of activity urea water that showed (1994b) spcue sa grgt ffntoa rus n t hsclpoete r h u of sum the are properties physical its and groups, functional of molecule aggregate a method, an this In observed. as been has pictured method this is of development continued a and attention 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 usrpswadpaefrwtradplmr respectively polymer, and water for are p and w subscripts f h ehd ae ntegopcnrbto,vesmlclsa aeo eti ubrof number certain a of made as molecules views contribution, group the on based method, The nte prahbsdo h ru otiuincnetkona nvra functional universal as known concept contribution group the on based approach Another h ocp fgopcnrbto eeoe yArm n runt 17)konas known (1975) Prausnitz and Abrams by developed contribution group of concept The stevlm fraction volume the is = D futs,adwater and -fructose, – ugn oe a ple osltoso ipemlcls n h eut of results the and molecules, simple of solutions to applied was model Huggins fi in UIA)wspooe yFeesude l 17) hshsrcie more received has This (1975). al. et Fredenslund by proposed was (UNIFAC) cient fi in fcomponent of cient .60.041 0.021 0.011 0.83 0.56 0.28 0.14 al .0Prmtr fteFlory the of Parameters 3.10 Table 1.11 Molality 1.39 2.22 2.56 3.33 Source: = urs ifrdfo h esrdvle y02% .4,and 0.44%, 0.28%, by values measured the from differed sucrose ln giutrlUiest rs,Wra,1998, 7 Warsaw, pp. Press, University Agricultural M., in Foods G., in Popenda, Plenzler, Water and D., S., Napierala, Surma, from Adapted a w qainfrGuoeSolution Glucose for Equation ¼ i – nalqi itr sarsl fdfeecsbt nmolecular in both differences of result a is mixture liquid a in 13. ln f w þ eik,PP E.,Warsaw (Ed.), P.P. Lewicki, , 0.060 0.079 0.096 0.146 0.176 0.204 f 1 s n n 2 1 a f – w p Huggins naohrpbiain orae al. et Correa publication, another In . 8 þ .A vrg eito f04 was 0.4% of deviation average An C. rprisof Properties xf p 2 þ 27.6 77.0 x 7.1 4.3 2.6 1.55 0.40 0.07 0.11 = D -glucose, (3 : 33) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where h obntra ati eie rmtepr opnn rprissc sgopvlm and volume group as such properties component pure the from derived is part combinatorial The h eiulpr fteatvt safnto fgopae rcin n hi neatosi pure in interactions their and fractions area by group given of is function It a mixtures. is in activity and the components of part residual The equations: by described is part This constants. area where fgroup of ru eae nadfeetwyfo h loo Hgop ec,tetocci structures cyclic two the hence, The OH-group; molecule. alcohol sugar the a from in group way new different a a introduced in satisfactory who allowed behaves (1997) model group modi Macedo The Further and solutions. solutions. Peres sugar sugar by in activity nonaqueous water and of of modi aqueous prediction solutions (2000) in anomers aqueous Tassios and equilibriums model isomers and thus phase to Spiliotis account, equations into distinguished. taken UNIFAC be was the equilibrium can conformational used model, (1995) this In al. sugars. et Catté solutions. glucose water. is w ‘‘ R h opnn rafraction, area component the group ieadwt neatoso h tutrlgop Wlo n el 92.TeUIA model Hence UNIFAC contributions. The residual 1962). and Deal, combinatorial and the (Wilson from results groups activity structural water the molecular that of in assumes differences interactions with with associated interactions. and are molecular contributions size The as additively. well and as Contributions independently shape energy additive. and and size independent molecular are in contributions differences to The due groups. are the by made contributions ß oueo group of volume ouincnann nymlclso ae (w): water of molecules only containing solution PYR k 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 ¼ h NFCmdlwsue yConr n eMge 18)t rdc ae ciiisof activities water predict to (1986) Maguer Le and Choundry by used was model UNIFAC The ouini rae samxueo rus hc otiuet h ata oa xesfree excess molal partial the to contribute which groups, of mixture a as treated is solution A ( V ’’ m a G k mn = k , and 15.17); k X sagoprsda ciiyand activity residual group a is ; stegopitrcinconstant; interaction group the is m Q ‘‘ steml rcino group of fraction mole the is k FUR ln stegopae constant, area group the is a ln v k Q w k V ’’ ; G ) w stenme fgop ftype of groups of number the is c m x a etetda igegop.I ytm,D-glucose systems, In groups. single as treated be can k i ¼ ¼ ¼ ¼ r oefato fcomponent of fraction mole are P ln Q q P F w j n k Q "# x n q w m w 1 j Q x F X þ j n m i ; X 2 z ln ln stecmoetvlm fraction, volume component the is n q ; F ! a w ln w w X ln ) X m r a ¼ m F Q w ¼ G V w w ¼ P ¼ k w m X r Q m þ l w usrpsdenote Subscripts . k stegoprsda ciiyo group of activity residual group the is i P j n C ln k ¼ x r ¼ l w mk j w P v j a x z k w x w ( j = j A j ; ) 2( P ln c v k F = m j x þ r r G w m 2.5 i x w i X j k ln v ¼ ! m k m j fi i q ; a X nmolecule in daUIA oe nodrt predict to order in model UNIFAC a ed i ; X j ) w ¼ ln n z 10 k fi ) i P r scodnto number, coordination is aino NFCmdlwsdone was model UNIFAC a of cation G C x 9 ( V v j k w ); n r l k i mn i j m R V i R C k component n ¼ k ; 1); C km stegopvlm constant, volume group the is exp nm q A i i V ; k ¼ m V stevndrWasarea Waals der van the is k X steae rcinof fraction area the is a k – stevndrWaals der van the is T mn sucrose i ; v k k i , Q m k k , – nareference a in n z ae,water water, ¼ r groups, are ‘‘ OH-ring 10. (3 (3 (3 (3 Q : : : : 37) 35) 34) 36) i is ’’ Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where ß form and carbons coef asymmetric D Activity several ions. have with activity biochemicals for complexes the because electrolyte describe 0.37% needed to was and used groups juice was of groups apple assignment of for assignment 0.34% new a of the with 60% coef deviation model and average 10% UNIFAC between The an concentration juice. with sugar assuming grape total predicted 1999) a Macedo, were For and water. activities and (Peres water sugars honey of synthetic mixtures and are they concentrates that juice fruit of activities model water UNIFAC the of 3.11. results Table Some 0.6%. in than presented lower are deviation application relative with predicted was activity 0.3648 0.8286 1.7391 1.2166 2.3504 2.8039 Sucrose System the for Activities Water 3.11 Table eito RS)lwrta % For 1%. than lower (RMSD) deviation agrinine ecie ytefloigequation: following the by described application practical For dominant mixture. some ternary are or developed. there binary are multicomponent unless as equations are treated suitable semiempirical be and very foods can empirical not liquid solution are However, the models and solutions. contribution components, ternary group they cases the although and some good are mixtures in modeling models molecular and contribution group and binary on chemistry based consider quantum predictions using The 2002) development. under 1999, are Sandler, and 1997). 2001). proteins (Lin al., al., models as et solvation et such Curtis molecules (Kuramochi 1995; complex 5% al., of et than solution (Chen larger made of was been properties have RMSD thermodynamic the predict compounds, to Attempts studied other For 2.69%. -mannose, 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 usrpswadsdnt ae n oi,respectively solid, and water denote s and w subscripts n h modi The h eimiia qain r ae nRaoult on based are equations semiempirical The contribution segment and group and 2002) Klamt, and (Eckert cosmo-Rs as such methods New fi stenme fmoles of number the is insi iaysseso mn cd n etdsi ae n te iceias h new The biochemicals. other and water in peptides and acids amino of systems binary in cients HCl, Molality D glcoe ats,scoe raf sucrose, maltose, -galactose, fi dUIA oe rpsdb ee n aeo(97 a sdt rdc the predict to used was (1997) Macedo and Peres by proposed model UNIFAC ed L -histidine D-Glucose 3.4947 2.9503 1.8737 2.4489 1.0996 0.5542 C,NC,adsdu lcrnt h MDwsbten1 and 1% between was RMSD the glucuronate sodium and NaCl, HCl, . EIMIIA EQUATIONS SEMIEMPIRICAL 3.3 rdce rmUIA Model UNIFAC from Predicted PrsadMcd,1997) Macedo, and (Peres D L -Glucose a hdoyrln,sdu n oasu glutamate, potassium and sodium -hydroxyproline, w fi fi ¼ oe n C eepeitdwt otma square mean root with predicted were KCl and nose, insfrsltosof solutions for cients 0.92406 0.92350 0.92207 0.92385 0.92189 0.92098 n w – n þ Sucrose w ’ n a o nielslto.Wtratvt is activity Water solution. ideal an for law s s – ae t25 at Water a w DL 8 poie xylose, -proline, C Lle n utn 1991) Sutton, and (Lilley Measured 0.9251 0.9251 0.9251 0.9260 0.9260 0.9260 D -glucose, (3 : 38) L - Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where where where follows: as is equation glucose, The solutions. and sugar noticed. of Sucrose was activity values. 1.5% water exceeding on calculated not effect and deviation a glycerol sorbitol measured and and sucrose sucrose for For and B. obtained arabinose, and and was A sucrose solutes agreement between interaction good no is very there that assumption the with neatwt ae oeue.Framxueo w oue h qainis equation the solutes two of mixture a For molecules. water with interact n solutions, sucrose For on ovn e ouemlcl.Frra ouin h yrto ubri ie ythe by given is number hydration the of solutions molecules real of number For average molecule. the solute equation: following denotes per number solvent hydration bound average equilibrium, solvation fslto fsvrlsltsbsdo nw aafrsnl oue a developed was solutes single activity for predicting data equation known and on (1966) based Robinson solutes and several Stokes of by solution up of taken was approach above The ß 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 ¼ a m s At moieties. hydrated of formation to leads molecule solute with molecules water of Interaction c c A h D oizoadcwres(01 eeoe neuto conigfrhdainwtradits and water hydration for accounting equation an developed (2001) coworkers and Poliszko and 3 s w m steconcentration the is steaeaehdaindegree hydration average the is sthe is ¼ ¼ stemolality the is stestrto concentration saturation the is stewtractivity water the is steaeaeecs hmclptnilo yrto water hydration of potential chemical excess average the is Ka 1 þ K w fl Ka ¼ þþ cuto amplitude uctuation .2.The 0.720. w þ ( Ka n ( Ka n w ) ¼ 2 w n þþ 1and 11 ) n au a sue ob qa ooye ie namlcl beto able molecule a in sites oxygen to equal be to assumed was value and m A 55 D RT K þ m a : K ( 51 w Ka h h steslaineulbimconstant equilibrium solvation the is m ¼ ¼ ¼ ¼ B 55 h w .9.Frglucose, For 0.994. ) ¼ exp 1 D m RT ¼ n : 51 m c 1 55 s h c ¼ a m : s D RT w 51 a 1 1 m w þ a (1 þ w A a 1 hc m w cos a A þ c w m h ) a A A S s w 2 þ þ D p n c ¼ m m B B and 6 h B K ¼ .8 n o glycerol for and 0.786 (3 (3 (3 (3 : : : : 40) 39) 42) 41) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 Source: Xilitol Maltose Glucose suigta oeaon fwtri on ihteslt,Shatbr 17)pooe to proposed (1976) Schwartzberg solute, the with bound is water Raoult of modify amount some that Assuming where h ouewsintroduced was solute the Raoult where Fructose ß 3.12. Table in presented are sugars some for 3.42 Equation of Parameters 3.42 Sugar Equation of Parameters 3.12 Table where xrse stedfeec ewe h auscluae o nielslto n h mutof amount the and solution solutes ideal and an solvent for between calculated interactions values from the arising between overestimation difference the was as It on expressed 1980). depending 950.1, Karmas, and and 821.2 between (Chen was proteins Chicago) Soya, composition. soy (Central its isolated Promine for and EMW salts, that found inorganic acids, coef activity organic calculate acids, to used was equation above The 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 x M x M steefciemlclrwih fslt (kg solute of weight molecular effective the is EMW E Raoult are(93 eie neuto ae nRaoult on based equation an derived (1983) Caurie w steslt otn nteslto (kg solution the in content solute the is stertoo h oeua egto ae omlclrwih ftesolute the of weight molecular to water of weight molecular the of ratio the is stemlclrwih fwtr(kg water of weight molecular the is and ’ b a a rsne yPlikradHlmn(90 nwihefciemlclrwih of weight molecular effective which in (1970) Heldman and Palnitkar by presented was law s ..(d) aswArclua nvriyPes asw 01 p 12 pp. 2001, Warsaw, Press, University Agricultural in Warsaw S., Poliszko, (Ed.), and P.P. D., Klimek-Poliszko, H.M., Baranowska, From steaon fwtrbudb ntwih fsld(kg solid of weight unit by bound water of amount the is ’ x a rte nms rcin ilstefloigequation: following the yields fractions mass in written law s s r h aso ae n oue epciey(kg respectively solute, and water of mass the are ’ a n aeteexpression: the gave and law s c s 0.600 0.380 0.470 0.824 (g = g) a w ¼ a m w s a ¼ 55 þ w a ¼ : ( 55 w 5 = x w mol) 1 = ¼ : = 5 E gsolution) kg 0.1680 0.2520 0.1755 0.0979 M 1 1 ¼ ) x A ( x m 1 þ M M w x s x = w ( s þ bx þ x M m fi x s ’ s = 55 a nwihwtratvt faslto is solution a of activity water which in law s amino and amines polyols, sugars, for cients Ex ) bx þ EMW) = mol) : Ex 5 (1 = gsolution) kg a rpriso ae nFoods in Water of Properties w )( 0.670 1.632 1.128 0.214 h = – gsld) Modi solids). kg 20. fi D ainof cation m Lewicki, , (J 1226 (3 (3 (3 (3 406 144 285 3 = mol) : : : : : 47) 44) 46) 45) 43) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 ihteRs 17)euto a sdt aclt ae ciiyi enr n quaternary and ternary in activity NaCl water as such calculate solutions to electrolyte In used 1990). NaCl (Chen, was solutions equation nonelectrolyte and (1975) in electrolyte given Ross are the solutes some with for constants of Values where ihteeprmna aawti .1ui of unit 0.01 within data experimental the with where qaini sfollows: as is equation where ß ae nteeprmna ato iero erlna eainhpbtentemllte fmost of molalities the between relationship linear near or linear of fact experimental dif the be on would based measurement equilibrium since interest, in presented are mixtures ciiis aigEutos34 n .5tefloigi obtained: is following the 3.45 and 3.44 solution. Equations a Taking in activities. water of fraction unbound and available, free, of qainpeit ae ciiyo ierneo oue iha cuayof accuracy an with solutes of range wide a of activity water predicts equation 3.13. Table ifrdfo h rdce ausb oe0.002 some by values predicted the from differed cuayo tleast at of accuracy 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 0.01 b m acltdwtratvte rmteaoeeuto pt ua ocnrto f4%agreed 40% of concentration sugar to up equation above the from activities water Calculated rdcino ae ciiyi ie ouino neatn opnnsi fparticular of is components interacting of solution mixed a in activity water of Prediction hn(97)pooe odrv ciiycoef activity derive to proposed (1987a) Chen hn(99 eeoe ipeeuto opeitwtratvt fsnl ouin.The solutions. single of activity water predict to equation simple a developed (1989) Chen and stemolality the is þ Lactose D D 3.48 Sugar Equation of Parameters 3.13 Table Maltose Sucrose reedidsi ik(10% milk skim Freeze-dried reedidcfe eeae(5% beverage coffee Freeze-dried ocnrtdoag juice orange Concentrated Source: (average) juice Fruit m y -Glucose -Fructose concentration) concentration) a KNO ¼ w s n o sucrose For . stemllcnetaino h oue twscnlddta ae ciiyi measure a is activity water that concluded was It solute. the of concentration molal the is x r constants are = (1 3 NaCl , dpe rmCe,C.S., Chen, from Adapted x ). þ 0.005 þ KCl lcs n sucrose and glucose al 3.15 Table þ a w il ae ciiywspeitdwt nacrc ihrthan higher accuracy an with predicted was activity water LiCl, – h itr fscoeadNC ile ae ciiis which activities, water yielded NaCl and sucrose of mixture The . 40% a – w . 40% LWT g ¼ ¼ 1 0 4 1987a. 64, 20, , þ 1 0 þ : a 018( 1 w lcrl ae ciiycnb rdce ihan with predicted be can activity water glycerol, þ qainprmtr o uasaecletdin collected are sugars for parameters Equation . fi al 3.14 Table Eby 1 y in ycmaio fieladra solution real and ideal of comparison by cient b a ( w E þ fi ae ciiiso oemulticomponent some of activities Water . 2 uti o mosbe h eeomn is development The impossible. not if cult Bm b ) n 0.053 0.100 0.100 0.053 0.053 0.054 0.052 0.075 0.089 ) h bv qaini combination in equation above The . m Eb 0.21 0.15 0.10 0.21 0.26 0.11 0.00 0.15 0.20 0.001 – – – – – – – – 0.25 0.18 0.46 0.26 0.30 0.13 0.02 0.20 a þ w (3 (3 The . KCl, : : 49) 48) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where urs:alwtr2:06 .4 hagadTld (1976) Toledo and Chuang 0.744 20:20:60 Sucrose:NaCl:water ß (1) Sucrose (1) Sucrose (1) Sucrose 25 at Systems Multicomponent System of Activity Water 3.15 Table the relation (1966) 1973): Robinson al., and et Stokes (Chen the proposed and was rule equation Zdanorskii the following on Based solutions. isopiestic rdce ae ciiisfr5 etdmxue eitdfo h esrdvle yls hn0.5%. than less by values measured the from deviated mixtures tested 51 for activities water Predicted Mixture 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 m h m h 2 1 o2 o1 steaeaehdainnme fslt tagvnvleof value given a at 2 solute of number 1 hydration solute average of the number is hydration average the is stemllt fbnr ouino oue2 solute of solution 1 binary solute of of molality solution the binary is of molality the is Source: NaCl Sucrose lcs .440.926 0.0424 1 Glycerol Glucose 3.49 Solution Equation of Parameters 3.14 Table þ þ þ lcrl()200 .65090Ce (1990) Chen (1990) Chen 0.930 0.926 1.3615 0.5542 2.0405 2.8039 (2) glycerol (2) glucose (2) NaCl rmCe,C.S., Chen, From 1.868 1 1 b .28343 .3 hn(1990) Chen (1990) Chen 0.930 (1990) Chen 0.925 3.4537 0.857 3.4947 (1990) 0.3278 Chen (1990) 0.2424 Chen (1990) 0.3648 Chen 0.939 0.950 5.5290 0.986 4.467 4.357 1.4285 4.32 1.2240 3.419 0.2943 0.6830 0.5082 0.2209 m .Fo Sci Food J. ocnrto (%) Concentration 1 m m Molalities 2 1 ¼ (( (( 0.0582 0.1136 0.0250 ,5,11,1989. 1318, 54, ., a a 01:0080Cun n oeo(1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and 0.890 Chuang (1976) Toledo and 0.879 Chuang 0.827 10:10:80 0.846 20:10:70 0.794 15:15:70 30:10:60 25:15:60 Bn w w .69084Ce (1990) Chen (1990) Chen (1990) Chen (1990) 0.874 Chen 0.890 0.890 0.914 0.6689 0.2287 0.2666 0.2510 558 .6 hagadTld (1976) Toledo and Chuang 0.963 15:5:80 = = ::0096Cun n oeo(1976) Toledo and Chuang (1976) Toledo and Chuang (1976) Toledo and Chuang 0.976 0.973 0.968 8:2:90 7:3:90 6:4:90 m 1 1 2 a a w w ) ) 8 þ þ C h h 1.618 0.955 0.855 1 2 ) ) a a w w a w natraymixture ternary a in oaiyRange Molality < < < < 14 Reference 6 6 7.5 (3 : 50) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 follows: where qain35,Rs 17)dvlpda qaindsrbn ae ciiyo itr fdifferent of mixture a of activity water describing equation solutes an developed (1975) Ross 3.51, Equation Gibbs the integrating By where where ncnlso,i a esae httedvlpetmd yCui a omn de many so had predicting Caurie by in made solution. (1988). useless development Chirife complex was the and a that Kitic equation by stated of the criticized be activity were may equation water it Ross conclusion, the overestimates to In equation (1985) Caurie Ross by the made Corrections that shows correction The where hswsciiie yCui 18)adteRs qainwscretd o three-component a For corrected. was above equation the Ross is of the equation derivation and corrected In (1985) the solute products. Caurie mixture and interactions food by small criticized the for relatively was usual that are This ranges assumed 3.52 concentration Equation was by the it calculated in equation activities 1% water exceed in not errors do the that shown was It ß xrse iia opinion. similar a expressed h Gibbs the 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 usrpsw ,2 , . . . 2, 1, w, subscripts a n y a n m usrps1 ,3 , . . . 3, 2, 1, subscripts m ae ciiyo togeetoye a ecluae rmteomtccoef osmotic the from calculated be can electrolytes strong of activity Water hnei ciiyo ouindet hne niscmoiini ioosfr sgvnby given is form rigorous a in composition its in changes to due solution of activity in Change i w i steml rcino component activity a the of is fraction mole the is stenme fcomponents of number the is stenme fin nowihec ouedissociates solute each which into ions of number the is stemllcnetaino component a of concentration molal the is stemolality the is stewtractivity water the is – ue equation: Duhem a w ¼ a n w1 w – n a dln n ue qainfroeslt n usiuigi noitga omof form integral into it substituting and solute one for equation Duhem w2 eoesltsa h aecnetaina ntecmlxsolution complex the in as concentration same the at solutes denote eoewtr opnn ,cmoet2 component , . . . 2, component 1, component water, denote a a w3 w þ n a a 1 n w dln w ( m ¼ a ¼ f 1 w m x ( exp a ¼ a 1 fsml n utcmoetmxue.Ce (1990) Chen mixtures. multicomponent and simple of 2 w1 þ þ a (55 n w2 55 m 2 f dln 1 : a : 5) m 1ln 51 w3 y 0 i 3 2 – m , : a þ ouecne nteaeaei h mixture. the in average the on cancel solute 018 ... i w2 a m , w , m ... 2 a m i w y 3 n , i ) )( n þ n dln ( n þ a n (55 1) ¼ m : 5) 0(3 1 m 3 2 m 3 fi in de cient fi inisthat ciencies n fi e as ned (3 (3 (3 3 : : : : : 51) 52) 54) 55) 53) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where where fteslto otiscytliesgrte h urs ocnrto nteslto auae with saturated solution the in concentration sucrose 20 the at then sucrose sugar crystalline contains solution the If where miia qaino h form: the of equation empirical ß coef osmotic The ouin safnto ftmeaueadfudta h ifrnebtenpeitdadmeasured and 0.002 salt predicted of saturated between order difference several the the of that of activities found Using was water and solutions. values temperature predicted those of (1986) of function al. activity a et as water solutions Kitic predict to approach, used presented successfully above be the can 3.56 and 3.55 Equations ial,teeuvln urs ocnrto scluae rmteequation: not the from was in calculated listed solution solutions are is sugar factors concentration sugar to conversion sucrose added Equivalent equivalent investigated substances mixture. the the nonsugar of Finally, of soluble pressure pressure that vapor shown relative vapor was lower It relative temperature. the on dependent that showed Experiments (g 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 = I f a b I f c B A z y ae) eaiepesr speitdb h bv qainwt nacrc of accuracy an with equation above the by predicted is pressure Relative water). g rvr(97 sdata aafrrltv ao rsue fsgrsltost eeo an develop to solutions sugar of pressures vapor relative for data actual used (1947) Grover etn 0dfeeteetoye sauossltos emru ta.(99 hwdthat showed (1979) al. et Benmergui solutions, aqueous as electrolytes different 30 Testing M stecneso factor conversion the is f MX steincstrength ionic the is ¼ ¼ stecnetain(g concentration the is ¼ stecntn qa o12fralsolutions all for 1.2 to equal constant the is stecntn qa o2fralsolutions all for 2 to equal constant the is and steDebye the is E 1 y s ¼ = M ¼ A 2 z b f þ S s X j MX m p þ 8 y r epciecagsi lcrncunits electronic in charges respective are a ecluae rmteequation: the from calculated be can C i X I ffiffi z 1.3 þ = i 2 1 fi and f b i in sgvnb izr(93 as (1973) Pitzer by given is cient þ – þ MX I üklcoef Hückel b 0.8 y p 1 exp M I ffiffi ¼j g j and ÀÁ and = z water) g s M a y ¼ z ,i s, X p X . MIIA EQUATIONS EMPIRICAL 3.4 fi 1 a j r h ubro n ions X and M of number the are I ffiffi a and , in o h soi ucineult .9 t25 at 0.392 to equal function osmotic the for cient : w w 994 nyi n ae h ifrnews0.009. was difference the case, one in Only . ¼ f þ 1 g : r h ocnrtoso urs,ivr ua,adglucose and sugar, invert sucrose, of concentrations the are 04 m 0 : E 339( 2 s y 0 ¼ M y : b g 10 y X MX X þ E , i s ) B cf b þ þ MX I MX 0 0 : and , þ : 0045 038( m 2 E 2 g C s 2 p þ MX ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i y ) constants 2 M y y X ) 3 C MX 8 C al 3.16 Table (3 (3 (3 (3 0.5%. : : : : 56) 59) 57) 58) . Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where where hroyai rud.Teeuto sa follows: as is equation. equation above The the grounds. of thermodynamic precision the on information no however, is, There ß 85% to 45% from solutions sucrose 60 over of pressure temperatures vapor the water between calculate concentration to equation empirical An o lcrlts h eito rmlnaiyi bevd n tcnb lmntdb nrdcinof introduction by eliminated be can it and observed, is intercept. linearity an from deviation the electrolytes, For 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 m m T k ors 16)drvda qaint rdc ae ciiyo oeetoyesltoson solutions nonelectrolyte of activity water predict to equation an derived (1966) Norrish w s steeprclconstant empirical the is stetmeaue(K) temperature the is steml rcino sucrose of fraction mole the is and eai,csi . rvr(1947) (1947) Grover Grover (1947) Grover 1.3 1.3 1.0 casein Gelatin, sugar Invert lactose Sucrose, ur 2 sy rti ih11 salts) 1.1% with protein (soya 620 Supro syrup Corn Glycine Hexaglycerol Sorbitol Fat chloride Sodium salts Glycerol their and acid, citric acid, Tartaric etc. pectin, Gums, Starch Confectioners al .6Cneso atr o h rvrEuto 3.58 Equation Grover the Substance for Factors Conversion 3.16 Table m s r h oefatoso ae n oue respectively solute, and water of fractions mole the are ’ lcs solids glucose log a w ¼ 0 : 4343 m a log w s þ ¼ m a w w 8 m Cand95 0 w ¼ : 4721 x ( exp km s 2 km þ þ 8 1.56 0 9.07 9.0 2.49 4.0 2.5 0.8 0.8 0.8 1.11 0.91 3.60 1.70 a eie yDnige l (1951) al. et Dunning by derived was C 713 s 2 b f T )( m s 2 1 : 32 m ushe l (1987) al. et Munsch (1947) Grover (1987) al. et Munsch (1947) Grover (1987) al. et Munsch (1947) Grover (1947) Grover (1947) Grover (1947) Grover (1947) Grover ushe l (1987) al. et Munsch (1987) al. et Munsch (1987) al. et Munsch (1987) al. et Munsch s 3 Reference (3 (3 3 : : : 61) 60) 62) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where where nte a oueteNriheuto,we oeua egto h ouei o nw,was known, not is equation: solute following the the of deriving weight (1997) molecular Perera when and equation, Rahman Norrish by the proposed use to way Another ß (0.9% Glycerol Glycerol Glycerol Glucose Glucose Galactose Fructose Fructose Dextrose DE) (42 syrup Corn acid Citric glycol 1,3-Butylene a b Solute Coef 3.17 Table modi was 3.61 Equation known, not is solute the of weight molecular when case the In .7oesol ecrflbcuesm uhr s aua oaihsadohr s decimal use others and logarithms natural Table use from authors values of Using some values 3.17. because the Table careful Then in logarithms. collected be are should 3.63 one and 3.17 3.61, 3.60, Equations in Parameters hagadTld 17)t h following: the to (1976) Toledo and Chuang -Amino- -Alanine 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 E X n s M s w w ¼ s stegaso ouei 0 solution g 100 in solute of grams the is stemlso ae n10go solution of g 100 in water of moles the is stemlclrwih fsolute of weight molecular the is and M n btrcacid -butyric w = X M s – r h asso ae n oue respectively solute, and water of masses the are s 56.3%) stertoo oeua egto ae omlclrwih fsolute of weight molecular to water of weight molecular of ratio the is fi insfrteNrihEquation Norrish the for cients a w ¼ k X ifrb 2.302585. by differ w X þ w log log EX n a n a s w w w w "# qain3.61 Equation 2.52 0.505 1.16 0.38 2.92 2.25 2.24 2.82 0.70 0.70 2.31 6.17 0.20 2.59 exp ¼ ¼ k Ks k "# k .7Ciiee l (1980) al. et Chirife 0.37 .4Ciiee l (1980) al. et Chirife 0.14 s 2 M s þ s 1 s b 2 X þ w b X þ w EX s 2 hagadTld (1976) Toledo and Chuang (1984) Labuza (1966) Norrish (1982) al. et Chirife (1984) Labuza (1990) al. et Leiras (1982) al. et Chirife (1966) Norrish (1966) Norrish (1966) Norrish (1984) Labuza (1966) Norrish Reference fi (3 (3 dby ed : : 64) 63) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 Product al(0.5% NaCl Solute Xylose acid Tartaric (3.3% Sucrose Sucrose Sucrose Sorbitol Sorbitol glycol Propylene L L NaCl NaCl Mannitol Maltose acid Malic Lysine Lactulose Lactose acid Lactic Glycine Glycine Solute Coef (continued) 3.17 Table ß tyeegyo 0.113 glycol Ethylene ocluaewtratvt fteslto ae ntefezn on ersin hn(1987) Chen depression. point freezing the on based in solution collected the are of they activity and water (1987), al. calculate to et Munsch by derived rpln lcl0.113 glycol Propylene Glycerol -Proline -Ornithine 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 miia oyoileutosfrpeitn ae ciiyo oefo uetnswere humectants food some of activity water predicting for equations polynomial Empirical – 25.9%) – 68.9%) 0.113 0.111 0.113 0.113 0.111 0.113 0.111 a w Range – – – – – – – – – .2 50.290 0.290 25 15 0.927 0.954 .0 50.195 0.195 0.195 35 0.290 25 15 0.908 35 0.927 0.954 0.908 .0 50.237 0.237 0.237 35 25 15 0.908 0.927 0.954 fi insfrteNrihEquation Norrish the for cients Temperature 7.578 kB ( 8 C) þ þ qain3.64 Equation qain3.62 Equation qain3.61 Equation 17.48 10.20 1.54 4.68 2.735 6.47 2.60 1.65 0.85 0.20 7.60 0.91 4.54 1.82 9.3 8.00 1.59 2.02 0.87 3.9 6.4 k 0.3 . hrf ta.(1980) al. et Chirife (1980) al. et Chirife 0.1 0.4 EK 0.33 0.0045 þ þ þ .22002Rha n eea(1997) Perera and Rahman (1997) Perera and Rahman 0.012 (1997) Perera and Rahman 0.009 0.4242 (1997) Perera and Rahman 0.012 0.6534 (1997) Perera and Rahman 0.006 0.6549 (1997) Perera and Rahman 0.006 0.0016 0.011 0.2642 0.3472 .12007Rha n eea(1997) Perera and Rahman (1997) Perera and Rahman 0.027 (1997) Perera and Rahman 0.014 0.5152 0.014 0.0639 0.0531 al 3.18 Table S Reference RSE hagadTld (1976) Toledo and Chuang aua(1984) Labuza (1984) Labuza (1976) Toledo and Chuang (1984) Labuza (1966) Norrish (1984) Labuza (1966) Norrish (1966) Norrish aua(1984) Labuza (1966) Norrish (1984) Labuza (1984) Labuza (1984) Labuza (1980) al. et Chirife (1984) Labuza (1984) Labuza (1984) Labuza (1980) al. et Chirife (1984) Labuza oeatmt made attempts Some . Reference Reference Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 twssonta h ifrnebtenmaue n acltdwtratvte i o exceed not did activities water calculated and measured between difference the 0.01 that shown was It where ifrne ewe esrdadcluae ae ciiiswr esta 0.01 than less were activities water calculated and measured between differences where ntemperature in ß Raoult on based equation solids: following dissolved the derived ytm n obndi ihteRbno n tkseuto o h eainhpbetween relationship the for equation equation: Stokes following the and deriving Robinson thus activity, the water with and point it freezing combined and systems olwn eainhp r used: are relationships coef following osmotic calculate to 18)pooe h olwn qaint aclt ae ciiyo ouinkoigisfreezing its knowing solution of Chirife activity and water Ferro-Fontan calculate measurement. depression: to temperature point equation solution following accurate the the the of proposed agitation for (1981) continuous essential and was measured properly cooling be during should depression point freezing that h qainpoie ipeadacrt a ocluaewtratvt faslto ihnthe within solution a temperature. of point activity water freezing 0 calculate at to from measured way range accurate that and is temperature simple equation a above provides equation the The from calculated activity Water ciiywsls hn0.01 than less was activity 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 C D eiie l 18)ue h Clapeyron the used (1983) al. et Lerici ae ciiyo h ouini eae oisomtcpesr.Hne oeatmt eemade were attempts some Hence, pressure. osmotic its to related is solution the of activity Water t ¼ a stefezn on ersin(K) depression point freezing the is w. T 5 stefezn eprtr K.Maueet oewt lcrchgoee hwdthat showed hygrometer electric with done Measurements (K). temperature freezing the is 10 5 K 0.2 uetn Equation Sucrose Humectant oe x Note: Source: NaCl Glycerol Sorbitol syrup Corn Humectants Food for Equations Regression 3.18 Table 8 2 euti ae ciiyvariation activity water in result C o ice for fi 8 a in rmomtcpesr n hnt siaewtratvt.The activity. water estimate to then and pressure osmotic from cient Cto sgsolute g is ln ln w dpe rmMnc,MH,Crir . n Chiasson, and A., Cormier, S., M.H., Munsch, from Adapted – h cuayo reigpitmaueeti motn n errors and important is measurement point freezing of accuracy The . ae system water a a aes.Ws.uTechnol. Wiss.-u Labensm. w t24 at w ¼ ¼ 40 a 8 9 27 C = w : 8 water. g 6934 .Tedfeec ewe esrdadcluae water calculated and measured between difference The C. : ¼ 622 0.1 1 8 þ C a a a a a – 0 w w w w w 10 luiseuto o solid for equation Clausius 528 : ¼ ¼ ¼ ¼ ¼ 0097 0.997 0.9862 1.0010 1.0092 1.0001 3 : 373(1 D 1 t D ’ a n fetv oeua egtof weight molecular effective and law s þ t 0 1,1987. 319, 20, , þ 0.520408 4 = 0.17637 0.12325 0.080314 0.077625 : T C 0.022 761 ) D t 2 4 : x x x a 10 7 ln 579 x x w þ þ þ þ . 0.400727 0.001109 0.00538 0.002582 0.002187 6 D t T 2 – x ao n solid and vapor 2 x x 2 2 x x 2 2 a w twsstressed was It . – (3 (3 (3 liquid : : : 67) 65) 66) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 2 where . ,7 7 9 5 2 724 933 724 1,020 758 1,013 790 1,034 776 1,820 1,372 0.4 0.3 0.2 0.1 2 1.5 1 6 6 4 oedt o urs ouin r rsne nTbe3.19. Table in presented are solutions sucrose for data Some 0.370 0.282 0.192 0.098 Solutes Some of Solutions (mol Water Concentration of Pressure Osmotic 3.19 Table ß Molality 0.825 0.757 0.685 0.610 0.533 0.453 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 10 V p usrpssadwdnt oueadwtr respectively water, and solute denote w and s subscripts m 10 10 10 10 10 w 10 steomtcpesr (Pa) pressure osmotic the is smlrfraction molar is 4 sprilmllvlm fwtr(m water of volume molal partial is 3 3 4 4 4 3 alGuoeFuts urs ats Lactose Maltose Sucrose Fructose Glucose NaCl 1 1 9 0 5 455 214 455 214 503 248 496 253 517 259 917 462 = L) soi rsuea 25 at Pressure Osmotic soi rsuea 20 at Pressure Osmotic 3 ln = mol) a f w ¼ ¼ fn p RT 2700 2400 2120 1840 1560 1290 1030 m m 771 513 262 8 m w n w (kPa) C s m V w s 8 (kPa) C a -Casein . asuaand Matsuura and Matsuura 6.0 and Matsuura 4.0 and Matsuura 2.9 and Matsuura 2.2 and Matsuura 1.6 and Matsuura 1.0 0.5 asuaand Matsuura and Matsuura and Matsuura and Matsuura orrjn(1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan or (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore (1962) Moore Reference ( continued (3 : 68) ) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 . 2361,6 asuaand Matsuura and Matsuura and Matsuura and Matsuura and Matsuura and Matsuura and Matsuura 12,866 9,128 and Matsuura 5,695 and Matsuura 5,061 38,335 and Matsuura 4,447 29,875 6.0 and 22,326 Matsuura 5.0 3,840 and 15,651 Matsuura 4.0 3,241 9,784 3,587 and Matsuura 3.0 2,668 8,701 3,101 2.0 2,379 7,646 2,564 1.8 2,103 6,612 2,310 1.6 5,612 2,067 1.4 1,827 4,640 1.2 1,551 4,158 1,824 1.0 1,282 3,682 1,611 1,744 0.9 3,213 1,307 1,517 0.8 2,744 1,293 0.7 2,282 0.6 0.5 ß Molality Solutes Some of Solutions Water of Pressure Osmotic (continued) 3.19 Table where and investigated also were substances weight in molecular their high experiments, some In molecules. small rs-ikdadfrsaglntok soi rsueo nielslto sdsrbdb van by described is solution ideal an is polymer of the pressure state, Osmotic second network. the In gel diluted. a is forms solution Ross The and other. use cross-linked each to interpenetrate or not 1987) do Chen, polymer 1976; (Schwartzberg, water bond for account to (1980), in this Hoff 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 R m T h bv-icse oest rdc ae ciiyo ouin r eae ora ouin of solutions real to related are solutions of activity water predict to models above-discussed The oyesi h ouincneiti w tts The states. two in exist can solution the in Polymers ’ stetmeaue(K) temperature the is stegscntn (J constant gas the is a,wihsae that: states which law, s stemllt ftesolution the of molality the is fl fl ec a ocluaeeuvln oeua egta rpsdb hnadKarman and Chen by proposed as weight molecular equivalent calculate to was uence ec nwtratvt fteslto a oie.Hwvr h nywyt con for account to way only the However, noticed. was solution the of activity water on uence alGuoeFuts urs ats Lactose Maltose Sucrose Fructose Glucose NaCl = mlK)) (mol soi rsuea 25 at Pressure Osmotic p ¼ mRT 8 (kPa) C fi s tt ssc httecan fthe of chains the that such is state rst a -Casein asuaand Matsuura and Matsuura orrjn(1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan orrjn(1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan (1986) Sourirajan ’ equation. s (3 : 69) ’ t Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 where irh ta.(97 hwdta hre n nhre esas olwapwrlwbhvo and behavior law power a follow also gels uncharged substitute and to charged that proposed showed (1997) al. et Mizrahi n ml oeua egtcnttet Tshe l,19) ae ciiyo uhaglcnbe can gel a such of activity Water 1999). contain- al., equation: hydrogels et following on the (Tesch extended by the was constituents and approach expressed weight network scaling pressure molecular The the exponent. larger small scaling the ing cross-linking, the of of degree value the the greater higher the that shown also was It where where arx h bevdomtcpesr ssalrta hto oye ouin ec (Horkay hence the solution, in polymer a embedded of liquid that on than smaller pressure 1982) is mechanical Zrinyi, pressure and exerts osmotic observed chains The polymer matrix. cross-linked by formed aigtesaigapoc sabsswtratvt fadltducagdplmr h ouinis solution the polymer, uncharged diluted a of 1997) activity al., water et basis a (Mizrahi as approach scaling the Taking o elslto h cln prahwsue codn owihomtcpesr funcharged of equation: pressure following osmotic the which by to according expressed used is was solution approach scaling polymer the solution real a For ß odtos h tt fwtraie rmtefloigphenomena: following the from arises water of is state point the and syneresis solution. conditions, changes solute composition the gel the below the of point gel that that practically the Above solutes is of pressure. by activity osmotic determined activity water gel is water original activity the pressures, water by the network determined pressure comparison high in network At solutes low of and concentration. concentration of polymers high that of relatively to with concentration compared systems pressure to in osmotic Hence, high matrix. relatively polymeric exert the solutes weight molecular low that showed 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 c A . . . n oefospsesntv ln tutr n el r nati ultro.Udrthese Under turgor. full in intact are cells and structure plant native possess foods Some stecnetaino oye (g polymer of concentration the is stesaigexponent scaling the is steconstant the is ugrpressure matrix Turgor insoluble with water of Interactions solutes to due pressure Osmotic V p w os s smlrvlm fwtr o eta solutions dextran For water. of volume molar is somtcpesr xre ysalmlcls h nlsso h bv equation above the of analysis The molecules. small by exerted pressure osmotic is p os p net ¼ ln Bc a w n ln = ¼ noEuto .2 ec,anweuto sobtained is equation new a Hence, 3.72. Equation into cm a w RT V ln ln 3 w ) ¼ p a a w w ÀÁ RT V ¼ p w ¼ ¼ os Ac ( p AV BV þ n os RT RT w w p c c os s n n p n net a on ob ..I e,anetwork a gel, a In 2.1. be to found was )( p net (3 (3 (3 (3 3 : : : : : 74) 72) 71) 73) 70) Downloaded By: 10.3.98.104 At: 10:12 29 Sep 2021; For: 9781420003093, chapter3, 10.1201/9781420003093.ch3 h soi n arxptnil r nerltdaduulytesmo ohptnil measured potentials potential. both water of measured sum the the to usually equal and not interrelated is are separately potentials matrix and osmotic The hn ..Wtratvt-ocnrto oesfrsltoso uas at n acids. and salts sugars, of solutions for models activity-concentration Water C.S. Chen, eedo h ufc ftetsu arx h feto yrpii esi yols ntewater the on cytoplasm in gels forces hydrophilic these of All effect signi The hydration. matrix. be and tissue can adsorption, the equilibrium capillarity, of surface of the forces on in depend originates potential matrix The hn ..Peitn ae ciiyi ouin fmxdsolutes. mixed of solutions in activity water Predicting C.S. Chen, systems. food of depression point freezing and activity water coef between Relationship activity C.S. Chen, and activity water of Calculation C.S. equation. Chen, Ross corrected A M. Caurie, qain37 eue to reduces 3.75 Equation twsi h ag f0.72 of range the in was it Go hn ..AsgetbsdlclcmoiinmdlfrteGbseeg fplmrsolutions. polymer of energy Gibbs the for model composition local segment-based A C.C. Chen, sugars. Raoult of note. solutions research aqueous A for M. model Caurie, UNIFAC chemical physical A J.-B. Gros, and G.-G., Dussap, M., Catté, ß the for expression new A mixtures: liquid of thermodynamics Statistical J.M. Prausnitz, and D.S. Abrams, 1967) Rose, and (Wilson Hence tissue. plant intact in term water the potential, of chemical to analogy the On fsoaea 2 at storage of air .Caatrsiso ae tt nsm hsntpso oe on nPoland. in found honey of types chosen some in state water of Characteristics S. Bakier, aaosa .. lmkPlsk,D,adPlsk,S feto yrto ae ntepoete of properties the on water hydration of Effect S. Poliszko, and D., Klimek-Poliszko, H.M., Baranowska, emru,EA,FroFna,C,adCiie .Tepeito fwtratvt nauossltosin solutions aqueous in activity water of prediction The J. Chirife, and C., Fontan, Ferro E.A., Benmergui, 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 ł ci(94 esrdwtrptnilfrtreapevreisgoni oadadsoe that showed and Poland in grown varieties apple three for potential water measured (1994) acki ae potential Water Science Science eesitlWisncatudTechnologie und Weissenschaft Lebensmittel Science Equilibria 1990. 515, Equilibria Phase Fluid Technology Food of Journal xesGbsfe nryo atyo opeeymsil systems. miscible completely or partly of Journal energy Engineering free Gibbs excess () 1 7(1), acaiesltos in solutions, saccharide oncinwt nemdaemitr od.I. foods. moisture intermediate with connection rs,Wra 01 p 12 pp. 2001, Warsaw Press, – 2 433 52, , 4 1318 54, , 8 648 48, , ,2006. 9, 8 3 301 83, , C. ¼ – – – 1987b. 435, 4,1983. 649, soi potential osmotic 31 1989. 1321, – 1,1993. 312, 1 116 21, , fi at hntetsu setrl ild h rsuepotential pressure the killed, entirely is tissue the When cant. – 0,1 105, , rpriso ae nFoods in Water of Properties .2Mai coe n nrae o1.05 to increased and October in MPa 0.92 – ’ 4 625 14, , a,wtratvt n osueaalblt nsolutions. in availability moisture and activity water law, s 20. – – 2,1975. 128, 5 1995. 25, ora fFo Science Food of Journal p – þ 3,1979. 637, REFERENCES þ C arxpotential matrix C t ¼ ¼ 0 64 20, , ‘‘ ¼ p ae potential water p þ a RT V w þ t w eik,PP d,Wra giutrlUniversity Agricultural Warsaw Ed., P.P. 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Tassios, and N. Spiliotis, ee,AM n aeo ..Peito ftemdnmcpoete sn modi a using properties thermodynamic of Prediction E.A. Macedo, and A.M. and Peres, Correlation solutions: modi aqueous A in E.A. Macedo, sugars and of A.M. Peres, properties Thermodynamic E.A. coef Macedo, activity and solutions, A.M. the glucose Peres, for in activity equation water An of R.S. study The Norrish, G. Plenzler, and S., Surma, M., Popenda, D., Napierala, ß tks ..adRbno,RA neatosi qeu oeetoyesltos .Sltsletequilibria. Solut-solvent I. solutions. nonelectrolyte aqueous in Interactions R.A. Robinson, and R.H. Stokes, eo,H n runt,JM oa opstosi hroyai xesfntosfrlqi mixtures. liquid for functions excess thermodynamic in compositions Local J.M. Prausnitz, and H. Renon, tlf,L airto fwtratvt esrn ntuet n eie:Claoaiestudy. Collaborative devices: and instruments measuring activity water of Calibration L. Stoloff, ihrsn .EHo ofcinr odproducts. food confectionary of ERH sorption T. water Richardson, the predict to models Norrish and GAB the of Evaluation C.O. Perera, and S.M. Rahman, E.G. Azevedo, and R.N., Lichtenthaler, J.M., Prausnitz, equations. general and basis Theoretical I. electrolytes. of Thermodynamics K.S. Pitzer, ec,R,Rmn . ayhnk,I,Chn . n irh,S ae opiniohr fsolution of isotherm sorption Water S. Mizrahi, and Y., Cohen, I., Ladyzhinski, O., Ramon, R., Tesch, oisn ..adSoe,R.H. Stokes, and R.A. Robinson, J.M. Prausnitz, O and J.M., Prausnitz, B.E., Poling, itBi . ihNue,H,adMle,J rdcino ae ciiyo lcs n acu chloride calcium and glucose of activity water of Prediction J. Muller, and H., Nguyen, Minh A., Bui, Viet coef Activity H. Uedaira, and H. Uedaira, os ..Etmto fwtratvt nitreit osuefoods. moisture intermediate in activity water of Estimation K.D. Ross, isn ..adDa,CH ciiycoef Activity C.H. Deal, and G.M. Wilson, food Vapour containing G.M. Wilson, solutions of activity water of Prediction A. Vitali, and A.J., Meirelles, C., Valezmoro, eeo .. uigr .. oeaa .. n ora .Peito fwtratvt fosmotic of activity water of Prediction A. Correa, and J.F., Song, Comesaña, M.D., Hubinger, A.M., food. Sereno, of thawing and freezing the for capacities heat Effective H.G. Schwartzberg, isn .adRs,CW h opnnso efwtrptnil .Omtcadmti potentials. matric and Osmotic I. potential. water leaf of components The C.W. Rose, and W. Wilson, 08b alr&FacsGop LLC. Group, Francis & Taylor by 2008 ua solution. sugar fl plcto osgridsra systems. industrial sugar to application sugars. containing solutions non-aqueous and aqueous modi a using prediction syrups. confectionery 7 pp. 1998, in h ora fPyia Chemistry Physical of Journal The mrcnIsiueo hmclEgneigJournal Engineering Chemical of Institute American Shef soito fOf of Association stem nfos in foods, in isotherms 1999. NJ, Cliffs, Englewood Prentice-Hall, 1969. 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