Water sorption isotherms of foods and foodstu s: BET or GAB parameters?

Transcription

1 Journal of Food Engineering 48 (2001) 19±31 Water sorption isotherms of foods and foodstu s: BET or GAB parameters? E.O. Timmermann a, J. Chirife b, *, H.A. Iglesias b a Facultad de Ingenierõa, Universidad de Buenos Aires, and PRograma de INvestigaciones en S Olidos (PRINSO), CITEFA-CONICET, Zufriategui 4380, 1603 Villa Martelli, Provincia de Buenos Aires, Argentina b Departamento de Industrias, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina Received 3 March 2000; accepted 8 August 2000 Abstract The aim of the present work is to solve the dilemma about the di erences between the values of the monolayer and the energy parameters obtained by the regression of water sorption data by foods and foodstu s using the Brunauer, Emmett and Teller (BET) two-parameter isotherm or the Guggenheim, Anderson and de Boer (GAB) three-parameter isotherm. It is shown that the GAB values are more general and have more physical meaning, and that the two BET parameters can be calculated in terms of the three GAB-parameters. Furthermore, the marked dependency of the BET constants on the regression range as well as the typical upswing at higher water activities observed in the so-called BET plots are explained. It is also shown that the rough agreement early reported by L. Pauling, J. Am. Chem. Soc. 67 (1945) 555±557 between monolayer values and number of polar groups in the aminoacid side chain in several proteins is enhanced if the former are evaluated by means of the GAB sorption equation. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Sorption isotherms; Monolayer values; Energy constants; BET equation; GAB equation; Proteins 1. Introduction In the past, the well-known Brunauer, Emmett and Teller (BET) sorption isotherm was the model that had the greatest application to water sorption by foods and foodstu s (Labuza, 1968; Iglesias & Chirife, 1976a), although it was known to hold only for a limited range of water activity (a w ), up to only 0.3±0.4. Two familiar constants are obtained from the BET model, namely the monolayer moisture content, x mb, and the energy constant, c B. Despite the theoretical limitations of the BET adsorption analysis, the BET monolayer concept was found to be a reasonable guide with respect to various aspect of interest in dried foods (Karel, 1973; Iglesias & Chirife, 1982). In more recent years, the Guggenheim, Anderson and de Boer (GAB) isotherm equation has been widely used to describe the sorption behavior of foods (Bizot, 1983; * Corresponding author. Fax: addresses: etimmer@citefa.gov.ar (E.O. Timmermann), jchirife@satlink.com (J. Chirife). Weisser, 1985; Maroulis, Tsami, Marinos-Kouris, & Saravacos, 1988; Iglesias & Chirife, 1995). Having a reasonable small number of parameters (three), the GAB equation has been found to represent adequately the experimental data in the range of water activity of most practical interest in foods, i.e., 0.10±0.90. The GAB equation has been recommended by the European Project Group COST 90 on Physical Properties of Foods (Wolf, Spiess, & Jung, 1985) as the fundamental equation for the characterisation of water sorption of food materials. Both isotherms (BET and GAB) are closely related as they follow from the same statistical model (Timmermann, 1989). By postulating that the states of water molecules in the second and higher layers are the same as each other but di erent from that in the liquid state, the GAB model introduced a second well-di erentiated sorption stage for water molecules. This assumption introduces an additional degree of freedom (an additional constant, k) by which the GAB model gains its greater versatility. One of the three GAB constants is, as in the BET equation, the monolayer capacity now denoted by x mg. The other two GAB constants, denoted /01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S ( 0 0 )

2 20 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 by c G and k, are energy constants as the BET constant c B, but with slighly di erent physical meanings. The BET constant c B is related logarithmically to the difference between the chemical potential of the sorbate molecules in the pure liquid state and in the rst sorption layer. On the other hand, the GAB constant c G is related to the di erence of this magnitude in the upper layers and in the monolayer, while the constant k is related to this di erence in the sorbate's pure liquid state and in the upper layers, and the product of both (c G k ˆ c B G ) represents the equivalent to c B of BET. It is to be mentioned that the third GAB constant k is, practically without exception, near to but less than unity (Chirife, Timmermann, Iglesias, & Boquet, 1992), a fact which constitutes a de nitive characteristic of this isotherm (Timmermann, 1989). Now, if both isotherms (BET and GAB) are used for regression analysis of sorption data, two sets of values of the monolayer capacity and of the energy constant are obtained, which should be comparable. However, it has been observed by several authors (van den Berg, 1981; Kim, Song, & Yam, 1991; Duras & Hiver, 1993; Lagoudaki, Demertzis, & Kontominas, 1993) that x mb BET < x mg GAB ; c B BET > c B GAB : 1 That is, the monolayer capacity by BET is always less than the GAB value, while the energy constant c B by BET is always larger than the GAB value. These inequalities set up the dilemma, about which values resemble a better physical reality, a dilemma not solved so far. Following a general approach given elsewhere by one of us (Timmermann, 2000), it will be shown here that there exits mathematical (and physical) reasons for the inequalities set by Eq. (1) and that the GAB values are the values of better physical reality. For this purpose, several experimental data for water sorption in foods and foodstu s are analysed and in each case, the inequalities (Eq. (1)) are qualitatively and quantitatively explained. Hydration of proteins, in terms of the attachment of one water molecule to each polar group of the side chains of the aminoacids (Pauling, 1945), is also discussed in terms of the BET and GAB monolayer values. 2. BET regression vs GAB regression 2.1. The BET isotherm The classical BET equation, giving the amount of water x(a w ) sorbed by a unitary amount of sorbant in terms of the water activity a w, is the following: BET : x a w ˆx mb c B a w = 1 a w 1 c B 1 a w Š: 2 It is well known that the two constants, the monolayer value x mb and the energy constant c B, are obtained from the so-called BET plots (Iglesias & Chirife, 1976a). In synthesis, in these plots, the linearised form, F(BET), of this isotherm F BET a w = 1 a w xa w Š ˆ 1=c B x mb c B 1 =c B x mb Ša w 3 is drawn in terms of a w. This function should be linear if the BET assumptions apply, and within the linear range, using a linear least-square analysis, F(BET) is adjusted by a linear polynomial P BET i ˆ a 0 a 1 x i 4 by minimising the squares sum over the n experimental points (index i) X iˆn iˆ1 F BET i a 0 a 1 x i 2 ˆ minimum; 5 where x i stands for a w at the point i. The coe cients a 0 and a 1 are given by the solutions of the system of normal equations associated to the extremum condition (5). According to Eqs. (3) and (4), the least-squares estimates of a 0 and a 1, ^a 0 and ^a 1, are related to the BET constants by ^a 0 1=c B x mb ; ^a 1 c B 1 =c B x mb 6 and herefrom x mb ˆ 1= ^a 0 ^a 1 ; c B ˆ ^a 0 ^a 1 =^a 0 7 relations by which the BET constants are calculated. It should be noted that the energy constant c B is inversely proportional to the intercept ^a 0 of the linear regression polynomial P(BET) of Eq. (3) and, therefore, c B is very sensitive to the value of ^a 0, which is usually very low. Usually, and this was observed by many workers, the BET plots give only an apparent linear plot at low water activities (0.05 < a w < 0.3±0.4) and over this range the BET regression is performed (Iglesias & Chirife, 1976a). For a w > 0.3±0.4, always deviation from linearity is observed with an upswing of F(BET) indicating that at higher water activities less water is sorbed than that predicted by the BET equation as shown in Fig. 1 for the BET plots of water sorption for various food materials The GAB isotherm On the other hand, the GAB equation is GAB : x a w ˆx mg c G ka w = 1 ka w 1 c G 1 ka w Š; 8

3 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 21 where c B G c G k: 9 To determine the three constants of the GAB equation, several methods can be employed. In the present context, a linearisation method of the GAB isotherm analogue to that of the BET model (Eq. (3)) is the most adequate; the other methods will be examined latter. To linearise the GAB isotherm, the following function F(GAB) applies: F GAB a w = 1 ka w xa w Š ˆ 1= c G kx mg Š c G 1 =c G x mg Ša w : 10 Thus the so-called GAB plots ± i.e., F(GAB) vs a w (Anderson, 1946; Gascoyne & Pethig, 1977; Timmermann, 1989) ± should be linear in a w, if the correct k- value is used for the experimental F(GAB). In practice, one looks for the k-value which best linearises F(GAB) vs a w ; a too high k-value determines an upward curvature in these plots as in the BET plots and a too low k- value determines a downward curvature. Analytically, the minimum of the sum of the least squares of the linear regression of Eq. (10) in terms of variable k determines the best k-value. Moreover, from the two linear regression coe cients of F(GAB), the other constants ± x mg and c G ± can be obtained. The corresponding representation of F(GAB) vs a w is also given in Fig. 1 (corresponding values of k have been taken from Table 1). The linearisation of experimental data through Eq. (10) is possible within the range 0.05 < a w < 0.8, which represents a much broader applicability range of the GAB isotherm compared with the BET equation. Furthermore, at higher water activities, these GAB plots present a downward deviation due to the appearance of the third sorption stage (Timmermann & Chirife, 1991), an e ect, which determines the upper limit of application of the GAB isotherm The relation between F(BET) and F(GAB) As already stated, the results of both regressions for the same set of experimental data lead to the inequalities stated by Eq. (1). This dilemma may be tackled in the following way. F(GAB) is related to F(BET) by F BET ˆa w = 1 a w xa w Š ˆ 1 ka w = 1 a w ŠF GAB 11 and by introducing here the expression (10) of F(GAB) and multiplying out the resulting expression, a second relationship for F(BET), named F (BET), is obtained, now in terms of the three constants of the GAB isotherm. It results F BET 1 ka w = 1 a w Š 1= kc G x mg c G ˆ 1=c B G x mg =c B G x mg aw 1 1 =c G x mg a w Š c B G k c B G 1 k 1 21 k =c B G x mg aw = 1 a w Š; 12 where Eq. (9) has been used to introduce c B G. This second expression for F(BET) shows that, if k <1, F(BET) will not be linear in a w, but will present an hyperbolic behaviour F BET ˆA Ba w Ca w = 1 a w Š ˆ A C B C a w C= 1 a w ; 13 where Fig. 1. (a) Fish our; (b) corn barn; (c) wheat starch BET and GAB plots for food materials. Note that GAB plots are displaced 0.02 units downwards to avoid overlapping. Values of the k-constant for sh our, corn bran and wheat starch are 0.81, 0.76 and 0.68, respectively. A 1=c B G x mg ; B c B G k =c B G x mg ; C 1 k c B G 1 k =c B G x mg : 14

4 22 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 Table 1 BET and GAB constants for water sorption in foods and foodstu s a BET (range: 0:05 6 aw 6 0:4) GAB (range: 0:05 6 aw 6 0:8) nb xmb (exp) cb (exp) xmb (calc) cb (calc) ng xmg cg k cgk Tomato (A) C/NEF 5.49 (0.113±0.432) b (0.113±0.836) b Corn bran (A) C/NEF 3.10 (0.063±0.379) b (0.063±0.716) b Fish our (25 C)/NEF 4.17 (0.115±0.443) b (0.115±0.848) b Potato starch (native) (A) 20 C/NEF (0.0399±0.4027) b (0.0399±0.8887) b Wheat starch (native) (A) 20 C/NEF (0.0404±0.4013) b (0.0404±0.8663) b Wheat (D) C/NEF Chicken (cooked) (D) 19.5 C/NEF

5 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 23 Turkey (cooked) (A) 22 C/NEF (0.05±0.9) b Turkey (cooked) (D) 22 C/NEF (0.05±0.9) b Corn our (degermed) C/NEF Rice (A) C/NEF 2.38 (0.1±0.4) b (0.1±0.8) b Corn (D) C/NEF Wheat gluten (A) C/NEF 1.02 (0.1±0.4) b (0.1±0.8) b Skimmilk (A) C/NEF 4.68 (0.1±0.4) b (0.1±0.9) b a A: adsorption, D: desorption; x m:ingh2o/100 g dry matter; nb, ng: number of experimental points; (exp): obtained using the direct regression by F(BET); (calc): calculated in terms of the GAB constants using F (BET); NEF: normalized error function (Eq. (21)). Source of sorption data: Tomato: Kiranoudis, Maroulis, Tsami, and Marinos-Kouris (1993); Corn bran: Duras and Hiver (1993); Fish our: Labuza, Kaanane, and Chen (1985); Potato starch (native): van den Berg (1981); Wheat starch (native): idem; Wheat: Hubbard, Earle, and Senti (1957); Chicken (cooked): Taylor (1961); Turkey (cooked): King, Lam, and Sandall (1968); Corn our (degermed): Kumar (1974); Rice: Nemitz (1963); Corn: Hubbard et al. (1957); Wheat gluten: Bushuk and Winkler (1957); Skimmilk: Berlin, Anderson, and Pallanach (1970). b Other evaluation range than that stated in the headings.

6 24 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 Conversely, if k ˆ 1, Eqs. (3), (13) and (14) become identical as C (k ˆ 1) ˆ 0. Eq. (13) readily explains qualitatively and quantitatively (Eq. (14)) the usually observed upswing in the BET plots at a w > 0.4±0.5, if k < 1. In the graphs of Fig. 1, F (BET) has been represented in terms of the corresponding GAB constants taken from Table 1. It is shown that this function reproduces quite well the (nonlinear) experimental F(BET) within the whole GAB applicability range 0.05 < a w <0.8. Furthermore, it is evident that, if F(BET) responds to Eq. (13) but analysed using Eq. (3), the so-obtained (and associated to the BET isotherm) values of x mb and c B will certainly be functions of the three GAB constants x mg, c G and k through Eqs. (12)±(14) and of the a w interval over which the regression is performed. And this functional dependence determines the di erence between the BET and the GAB sets for the monolayer capacity (x m ) and the principal energy constant (c B ) and, therefore, the inequalities stated by Eq. (1) BET regression by F (BET) The second expression (13) of F(BET) may also be adjusted by the same linear polynomial (4), but now using an analytical formulation as F (BET) is known as a function of a w and not by a set of numerical data. The calculation implies the adjustment of a function of a known functional dependence of a higher degree than one to a straight line. This regression of F (BET) can be made either in a discrete form or in a continuous form, as it is shown elsewhere (Timmermann, 2000). In the discrete procedure, F (BET) given by Eq. (13) is explicited into condition (5), which becomes X iˆn iˆ1 f A C B C x i C= 1 x i Š a 0 a 1 x i g 2 ˆ minimum 15 and this expression can now be solved analytically for a 0 and a 1 in the usual way of least squares. As it is to be expected, it results (Timmermann, 2000) that a 0 and a 1 become functions of the constants A, B and C of Eq. (13) on one side, and of regression sums over the values of the independent variable x i on the other. The nal expressions are the following: a 0 ˆ A C C d 0=d ; a 1 ˆ B C C d 1=d ; 16a 16b where a 0 and a 1 are the minimum squares estimates in terms of F (BET). The functions d, d 0 and d 1 contain only the regression sums of a w over the employed regression interval with the following signs: d 0 /d <0, d 1 / d > 0 and d 0 /d + d 1 /d > 0 for a w < 1 (Timmermann, 2000). The corresponding relations are, where Gaussian brackets have been used, d 0 1= 1 a w Š a 2 w aw = 1 a w Š a w Š; 17a d 1 a w = 1 a w Šn 1= 1 a w Š a w Š; 17b d na 2 w aw Š 2 : 17c n is the number of data included in the regression. As the Eqs. (6) and (7) remain valid, the BET constants are now given by x mb ˆ 1= a 0 a 1 ˆ 1= f A B C d 0 d 1 =d 2Šg 18a and c B ˆ a 0 a 1 =a 0 ˆ f A B C d 0 d 1 =d 2Šg= f A C d 0 =d 1Šg: 18b Finally, by Eq. (14) A, B, C ˆ f(x mg, c G, k) and after some algebra, explicit expressions for x mb and c B in terms of the three GAB constants are obtained (Timmermann, 2000): x mb ˆ x mg= k =c B G R m 19a and c B ˆ c B G k =c B G R c ; 19b where the functions R m and R c are given by R m 1 1 k c B G 1 k = c B G 2 1 k d 0 d 1 =d 2Š 20a and R c R m = 1 1 k cb G 1 k d 0 =d 1Š : 20b These functions are always greater than unity (Timmermann, 2000). In consequence, Eqs. (19a) and (19b) reproduce the inequalities (1) if k <1. Eqs. (19a) and (19b) explicit and quantify the di erences between the BET set (x mb ) c B ) and the GAB set (x mg ) c B G ) of constants. These di erences are directly related to k < 1 through the factor (1 ) k) present in these equations and hence, explain the inequalities set in Eq. (1). They become more important with decreasing values of k as well as with an increase of the regression interval. On the other hand, if k ˆ 1, all these expressions coincide with the classical results shown before. Eqs. (16a), (16b), (19a) and (19b) have been used to calculate the linear correlation of F(BET) GAB in terms of the corresponding GAB constants and for the same water activity interval used for the empirical BET equation (numerical values are stated in Table 1); the results are shown in Fig. 1(a±c). The reproduction of the empirical BET constants and of the inequalities stated in

7 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 25 Eq. (1) are quite acceptable. For the case of sh our (Fig. 1(a)), a slight di erence between the linear F(BET) exp and the linear F(BET) GAB is observed, a difference which is determined by the empirical dispersion of the experimental sorption data. In the case of corn bran (Fig. 1(b)), this dispersion is less and the coincidence of F(BET) exp and F(BET) GAB is better. Finally, in the case of wheat starch (Fig. 1(c)), this dispersion is minimal and F(BET) exp and F(BET) GAB become practically undistinguishable. Thus, the approach given here solves the dilemma of the inequalities stated by Eq. (1). It also explains the restricted range of linearity of the BET regression, the dependence of the values of the BET constants on the activity range used for the regression, and the upswing of the BET plots at higher water activities. 3. Analysis of water sorption data for various foods and food materials Two groups of sorption systems were analysed, (1) various foods and foodstu s, and (2) proteins. The second group corresponds exclusively to the comprehensive data set of water sorption by proteins due to Bull (1944), where this author tested the applicability of the BET isotherm to these sorption systems. The results are presented in Tables 1 and 2, and in Fig. 2. Table 2 Monolayer moisture contents for various proteins a;b Protein BET (range: 0:05 6 a w 6 0:3) GAB (range: 0:05 6 a w 6 0:8) N B x mb (exp) c B (exp) x mb (calc) c B (calc) n G x mg c G k c G k Collagen NEF ˆ 0.94 NEF ˆ 0.92 NEF ˆ 3.01 Gelatin NEF ˆ 3.22 NEF ˆ 1.01 NEF ˆ 4.31 Seroalb NEF ˆ 1.75 NEF ˆ 0.75 NEF ˆ 2.57 Elastin c NEF ˆ 11.1 NEF ˆ 2.2 NEF ˆ 9.2 Wool NEF ˆ 1.87 NEF ˆ 0.99 NEF ˆ 2.42 a=b-pseudo Globulin NEF ˆ 1.90 NEF ˆ 0.75 NEF ˆ 3.28 c-pseudo Globulin NEF ˆ 2.32 NEF ˆ 0.76 NEF ˆ 2.84 Lactoglob Crist. NEF ˆ 1.90 NEF ˆ 0.59 NEF ˆ 1.93 Lactoglob f.dried NEF ˆ 0.49 NEF ˆ 0.60 NEF ˆ 3.39 Eggalbum Coag. d NEF ˆ 0.99 NEF ˆ 0.76 NEF ˆ 2.70 Egg album f.dried NEF ˆ 1.99 NEF ˆ 0.72 NEF ˆ 3.06 Egg album Not f.dried NEF ˆ 1.05 NEF ˆ 0.76 NEF ˆ 1.68 c-zein NEF ˆ 0.83 NEF ˆ 0.64 NEF ˆ 4.42 b-zein NEF ˆ 1.43 NEF ˆ 0.64 NEF ˆ 1.88 Silk NEF ˆ 2.40 NEF ˆ 0.86 NEF ˆ 2.62 Salmin e NEF ˆ 3.27 NEF ˆ 0.10 NEF ˆ 1.54 a n B, n G, number of experimental points; x m :ingh 2 O/100 g dry matter; (exp): obtained using the direct regression by F(BET); (calc): calculated in terms of the GAB constants using F (BET); NEF: normalized error function (Eq. (21)). b Experimental data: Bull (1944). c BET range: 0.05±0.4. d GAB range: 0.05±0.9. e The isotherm presents two branches which are incompatible with each other; at a w ˆ 0:05±0:3 BET applies, but at a w ˆ 0:3±0:8, the application of GAB is quite questionable (see error gures) and the BET and GAB monolayer and energy values cannot be related.

8 26 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 Fig. 2. Experimental and calculated water sorption isotherms for (a) foods/foodstu s and (b) proteins (Bull). Note that the isotherms have been displaced upwards a certain amount of units for a better view. The left-side sections of Tables 1 and 2 present the sets of BET constants obtained by (a) the direct regression by F(BET) (Eqs. (6) and (7)) and (b) in terms of the GAB constants using F (BET) [Eqs. (14), (16a)±(18b)]; and the right-side section contains the set of GAB constants obtained using a parabolic regression (Eq. 22) (see also Fig. 3). The general regression ranges are given in the headings, with exceptions indicated in each case. The error of each numerical value has been calculated using the regression covariance of each parameter. Finally, the normalised error function (NEF), de ned as "!, # 1=2, X iˆn 2 NEF ˆ 100 x exp x calc n x i m 21 iˆ1 is also given. This function is related to, but simpler than the relative percentage root, mean square value often used in the literature. It can be seen (see also the top graphs of Fig. 4) that GAB monolayer values are about 10±40% higher than the BET value, while the energy constant c B G is much lower (35±50% and even more) than the BET value. In the same way, the errors of the energy constant values are much stronger (15±25% up to 60±70% and more) than that of the monolayer (4±8%). It is also to be noted that the error of the third GAB constant, k, is in the order 10±15% and therefore, the value of this constant should be given with only two signi cant gures. For the constant k, the values already stated elsewhere (Chirife et al., 1992) for proteins and protein foods (k 0.8, range 0.78±0.85) and for starchy foods (k 0.7, range 0.65±0.75) are con rmed. A lower value of k indicates a much less structured state of the sorbate in the layer following the monolayer, the so-called GAB layers, as in the sorbate's pure liquid state (Timmermann, 2000). In food science studies, preferred or almost exclusive attention is paid to the monolayer value (Karel, 1973; Iglesias & Chirife, 1976b). However, the values of the energy constants should not be overlooked nor ignored because they are simultaneous outputs of the regression processes and they in uence the sigmoidal shape of the isotherms, i.e., the form of the normalised Ôx/x m vs a w Õ plot, since c B and c G determine the more or less pronounced form of the `knee' at the lower water activity range. On the other hand, the third GAB constant k determines the pro le of the isotherm at the higher water activity range, regulating the upswing after the ÔplateauÕ at medium water activity range. Higher values of k determine a more pronounced upswing. This can be readily observed in Fig. 2; proteins and protein-foods (k 0.8) present a much more noticeable upswing than starchy foods (k 0.7). Finally, the function NEF is a measure of the experimental dispersion of the sorption data; good (mean) values of NEF are in the order 2±5%. If this dispersion is homogeneous over the whole GAB range, NEF has coincident values for the BET as well as for the GAB regressions indicating the much better ability of the GAB equation to represent the data as it embraces a much broader range of water activity. But if this dispersion is heterogeous (in the BET region di erent than in the GAB region), then NEF oscillates about the same values being in some cases the BET values lower than the GAB values and in others the opposite is observed, but always within the range 2±5%. A case markedly beyond this range is the protein elastin, the sorption data of which present NEF values of 9% (GAB) to 11% (BET). Morover, the NEF values of the calculated BET parameters by Eqs. (19a) and (19b) using the GAB constants are due to the intrinsic hyperbolic curvature of F (BET) (Eqs. (12) and (13)) over the BET range of activities. The experimental NEF values of the BET regression include this e ect and therefore, by the

9 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 27 Fig. 3. (a) Inverse and (b) parabolic plots for various foods, foodstu s and proteins. Symbols: experimental data. The arrows in the graphs indicate the upper limit of each equation. experimental dispersion of the data, the NEF values uctuate (upwards or downwards) about the `theoretical' values indicated by the former. Fig. 2 shows calculated food isotherms obtained by the BET and GAB regressions as well as by the BET constants calculated in terms of the GAB parameters, using the constants given in Tables 1 and 2. The limited range of applicability of the BET equation and the ability of the GAB equation to represent the experimental data up to a w 0.85 is observed in all cases; as well as the good agreement between calculated BET isotherm by the GAB constants with the BET curve obtained directly by the regression of the experimental data. To determine the GAB constants, a simple method was used which is straightforward; it uses the so-called transformed form of the GAB equation (Schaer & Ruegg, 1988), i.e., the following parabolic expression, which is easily derived from Eq. (8): a w =x ˆ a ba w ca 2 w ; 22 where a 1=x mg c G k; b c G 2 =x mg c G ; 23a 23b c c G 1 k=x mg c G : 23c The three constants a, b and c are readily determined by a least-square regression of this second degree polynomial and from these, the three GAB constants are calculated by k ˆ f 1=2 b =2a; 24a x mg ˆ f 1=2 ˆ 1= b 2ka ˆk= k 2 a c ; 24b c G ˆ 1 c=k 2 a ˆ 2 b=ka ˆ f 1=2 =ka; 24c where f b 2 4ac: 24d The constants stated in Tables 1 and 2 were obtained by this method. The upper limit of the regression or

10 28 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 Fig. 4. Comparison of (a) experimental BET and GAB constants and (b) experimental and calculated BET constants: (from data shown in Tables 1 and 2). Error bars are indicated in each case. applicability range of the GAB isotherm (the lower limit is, as in the BET case, a w ˆ 0.05) is determined with the so-called inverse plot (Timmermann, 1989). At high a w, for strongly sorbing substances (c G 1), both isotherms become very simple for the inverse of x(a w ): BET : 1=x ˆ 1=x mb 1 a w ; 25a GAB : 1=x ˆ 1=x mg 1 ka w : 25b These relations indicate that 1/x is linear at high enough a w for both isotherms and that the limits for 1/x ˆ 0(x!1) are at the points (a w ˆ 1, 1/x ˆ 0; BET) and (a w ˆ 1/k (>1), 1/x ˆ 0; GAB), respectively. Thus, if the linear part at higher a w of the inverse plot 1/x vs a w do not extrapole to a w ˆ 1 for 1/x ˆ 0 (BET condition), it is readily concluded that k < 1 (see Eq. (25b)) and that the GAB equation applies. The extrapolation to 1/x ˆ 0 gives 1/k directly as the intercept with the a w -axis (Timmermann, 1989). Hence, these plots readily illustrate which isotherm applies. Furthermore, if after the linear part the graph becomes curved downwards (usually at a w 0.85±0.9), then this is a direct evidence of the presence of the third sorption stage (Timmermann, 1989; Timmermann & Chirife,

11 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19± ), these points should not be used for the GAB regression. These procedures (inverse and parabolic plots) are shown in Fig. 3 for various food materials. The calculated BET and GAB isotherms are also drawn and the upper limits of both equations are shown by arrows. An alternative method to obtain the GAB constants consists in a non-linear least-squares regression of the GAB equation. It has been claimed (Schaerr & Ruegg, 1988) that this method and that of the parabolic transform give di erent results, but it can be shown (Timmermann et al., 1991) that if the points of the third sorption stage are not included, both regressions give identical results. Accordingly, it has already been stated that, if points belonging to the third sorption stage are included in the GAB regression, NEF increases very sharply this being another criterion to x the upper limit of the GAB regression. The results contained in Tables 1 and 2 are illustrated in Fig. 4; the BET monolayer value (x mb ) and energy constant (c B ) are plotted against the GAB monolayer value (x mg ) and the value by GAB (c B G ˆ c G k), respectively (Figs. 4(a)). These plots illustrate the inequalities stated by Eq. (1); i.e., that the BET evaluation always underestimates the monolayer, while it overestimates the energy constant. Figs. 4(b) shows the plot of the two BET constants against the respective values calculated in terms of the GAB constants. Within the corresponding error intervals, the `experimental' BET values are well reproduced by the calculated ones, and the scatter is much lower for the monolayer capacity than for the energy constant. It is therefore straightforward to conclude that the GAB constants are to be taken as the representative parameters of the multilayer sorption. A much more precise description of multilayer sorption of water by food materials can be achieved if the analysis is made with a set of experimental data, which span over the `quasi'-complete water activity range. 4. The stoichiometry of water sorption by proteins: Paulings (1945) hypothesis In 1945, short after Bull's (1944) paper about the water sorption by proteins, Pauling (1945) published a now classical paper about the hydration of proteins. He advanced that the water sorption monolayer of proteins can be thought in terms of the attachment of one water molecule to each polar group of the side chains of the aminocids in the protein. In his analysis, Pauling utilised BET monolayer values reported by Bull (1944). The agreement of these BET monolayer values with the number of polar groups of the proteins was roughly satisfactory in as much both values were of the same order of magnitude. However, it is worth noticing that the monolayer values were in most cases lower than the number of polar groups. In view of the results stated in the previous sections, it is straightforward to compare Pauling's data with the monolayer capacity obtained here by the GAB evaluation, as this equation is directly related and is an improvement of the original BET formulation. Pauling (1945) expressed his numerical data in terms of the number of polar groups or moles of water per 10 5 gof protein. We retain here these units, the monolayer values given as grams of water per 100 g of proteins in Table 2 are to be multiplied simply by 55.5 ( ˆ 1000/18) mol H 2 O/g. Table 3 shows the results of this new analysis of Pauling's hypothesis. The BET monolayer values stated in Table 2 are slightly di erent from those reported by Bull (1944) in his original paper, data which are given within parenthesis in the same table. This is likely due to some di erences in the water activity interval used for the BET regression (interval which is not stated exactly in Bull's paper) and to the fact that Bull's values are the mean between the values at 25 C and at 40 C. As in Pauling's (1945) original paper, the second value of the second column (Table 3) corresponds to the value obtained by taking also into account the aminoacids proline and hydroxiproline. Data within [ ] are the number of polar groups reported by other authors found in a rapid and not-exhaustive search of the literature, as a more profound revision of Pauling's values in terms of modern literature of protein aminoacid composition, is beyond the scope of this paper. The inspection of Table 3 shows that the rough agreement noted by Pauling is certainly improved when the GAB monolayer values are used for comparison. This conclusion becomes even more evident when the BET and GAB monolayers are graphically plotted against the number of polar groups (Fig. 5). The agreement is quite better for collagen, gelatin, serumalbumin, lactoglobulin, c- and b-zein. For silk, the BET value seems to be better and, on the other side, for salmin, the GAB value correlates surprisingly well, although its isotherm presents two branches (see Fig. 2 and note 4 of Table 2), a fact observed already by Pauling (1945) and by Bull (1944) themselves. Furthermore, in the case of egg albumin and wool, the agreement is improved when the number of polar groups reported in more modern publications are considered instead. In the case of collagen and gelatin, Pauling (1945) noted that the BET value failed to reach the value of the number of polar groups including the proline and hydroxiproline, and advanced some possible explanation for this discrepancy. However, when the GAB monolayer value is used, a close agreement is observed. Casein, a protein not considered by Pauling in his paper and not stated in Table 2, has been included in the present analysis. The sorption data due to Schaerr and Ruegg (1988) were evaluated elsewhere (Timmermann

12 30 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 Table 3 Comparison between number of polar groups and BET or GAB monolayer values a Protein Number of polar groups (mol/10 5 g) Monolayer capacity (mol/10 5 g) Pauling b BET GAB k (GAB) Collagen 328± Gelatin Idem Seroalbumin Wool 303, 341, 420 c Lactoglob.crist. 472, Idem, f.dried. Idem Eggalbum, coag 277, 313, 380 d Idem, f.dried Idem Idem, not f.dried Idem c-zein 305, b-zein Idem Silk 219± Salmin 611± Casein 416 e, 456, 521 f 306 g 343 g 0.89 a In Fig. 5, the underlined values of the polar group number are represented in the abscissa axis. b Reported by Pauling (1945). c Value reported by Windle (1956). d Value reported by Fogiel and Heller (1966). e Value reported by Ruegg and Hani (1975). f Values reported by McLaren and Rowen (1951). g Calculated by Timmermann et al. (1991). et al., 1991) using the BET and GAB equations and the corresponding monolayer capacities are stated directly in Table 3. It is observed that casein also ts well into the picture given by Fig. 5. Finally, it is interesting to consider also the case of another biopolymer, namely starch. The BET and GAB monolayer values for potato and wheat starch (Table 1) were 0.45 and 0.55 mol H 2 O/100 g, respectively. Since the polar group number (one water molecule per anhydroglucose monomer) is 0.62, as reported by McLaren and Rowen (1951), it follows that again the GAB monolayer correlates much better than the BET value. Thus, it is to be concluded that the present ndings that GAB parameters are more representative than the corresponding BETs ones, obtains additional support when Pauling's hypothesis of initial hydration of proteins, is considered. Acknowledgements The nancial support from University of Buenos Aires and CONICET (Argentina) are greatfully acknowledged. References Fig. 5. Comparison of BET and GAB monolayer values with number of polar groups in various proteins. Sil: silk; zei: b- and c-zein; egg: egg albumin; cas: casein; woo: wool; ser: serum albumin; lac: lactoalbumin; col: collagen; gel: gelatin; sal: salmin. The underlined values of the number of polar groups are used (see Table 3); in the cases of silk and lactoglobulin, both values are represented. Anderson, R. B. (1946). Modi cations of the Brunauer, Emmett and Teller equation. Journal of the American Chemical Society, 68, 686± 691. Berlin, E., Anderson, B. A., & Pallanach, M. J. (1970). E ect of temperature on water vapour sorption by dried milk powders. Journal of Dairy Science, 53, 146±149.

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