Water sorption isotherms of foods and foodstus: BET or GAB parameters? E.O. Timmermann a , J. Chirife b, * , H.A. Iglesias b a Facultad de Ingenier ıa, 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 dierences between the values of the monolayer and the energy parameters obtained by the regression of water sorption data by foods and foodstus 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 foodstus (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 con- stant, 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; 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 (Timmer- mann, 1989). By postulating that the states of water molecules in the second and higher layers are the same as each other but dierent from that in the liquid state, the GAB model introduced a second well-dierentiated sorption stage for water molecules. This assumption introduces an additional degree of freedom (an addi- tional 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 de- noted by x mG . The other two GAB constants, denoted Journal of Food Engineering 48 (2001) 19–31 www.elsevier.com/locate/jfoodeng * Corresponding author. Fax: +54-1-7943344. E-mail addresses: [email protected](E.O. Timmermann), [email protected] (J. Chirife). 0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 1 3 9 - 4
Transcript
Water sorption isotherms of foods and foodstu�s: BET or GABparameters?
E.O. Timmermann a, J. Chirife b,*, H.A. Iglesias b
a Facultad de Ingenier�õa, Universidad de Buenos Aires, and PRograma de INvestigaciones en S �Olidos (PRINSO), CITEFA-CONICET,
Zufriategui 4380, 1603 Villa Martelli, Provincia de Buenos Aires, Argentinab 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 andTeller (BET) sorption isotherm was the model that hadthe greatest application to water sorption by foods andfoodstu�s (Labuza, 1968; Iglesias & Chirife, 1976a),although it was known to hold only for a limited rangeof water activity (aw), up to only 0.3±0.4. Two familiarconstants are obtained from the BET model, namely themonolayer moisture content, xmB, and the energy con-stant, cB. Despite the theoretical limitations of the BETadsorption analysis, the BET monolayer concept wasfound to be a reasonable guide with respect to variousaspect of interest in dried foods (Karel, 1973; Iglesias &Chirife, 1982).
In more recent years, the Guggenheim, Anderson andde Boer (GAB) isotherm equation has been widely usedto describe the sorption behavior of foods (Bizot, 1983;
Weisser, 1985; Maroulis, Tsami, Marinos-Kouris, &Saravacos, 1988; Iglesias & Chirife, 1995). Having areasonable small number of parameters (three), theGAB equation has been found to represent adequatelythe experimental data in the range of water activity ofmost practical interest in foods, i.e., 0.10±0.90. TheGAB equation has been recommended by the EuropeanProject Group COST 90 on Physical Properties ofFoods (Wolf, Spiess, & Jung, 1985) as the fundamentalequation for the characterisation of water sorption offood materials.
Both isotherms (BET and GAB) are closely related asthey follow from the same statistical model (Timmer-mann, 1989). By postulating that the states of watermolecules in the second and higher layers are the sameas each other but di�erent from that in the liquid state,the GAB model introduced a second well-di�erentiatedsorption stage for water molecules. This assumptionintroduces an additional degree of freedom (an addi-tional constant, k) by which the GAB model gains itsgreater versatility. One of the three GAB constants is, asin the BET equation, the monolayer capacity now de-noted by xmG. The other two GAB constants, denoted
0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 2 6 0 - 8 7 7 4 ( 0 0 ) 0 0 1 3 9 - 4
by cG and k, are energy constants as the BET constantcB, but with slighly di�erent physical meanings. TheBET constant cB is related logarithmically to the dif-ference between the chemical potential of the sorbatemolecules in the pure liquid state and in the ®rst sorp-tion layer. On the other hand, the GAB constant cG isrelated to the di�erence of this magnitude in the upperlayers and in the monolayer, while the constant k isrelated to this di�erence in the sorbate's pure liquid stateand in the upper layers, and the product of both(cGk� cB�G�) represents the equivalent to cB of BET. It isto be mentioned that the third GAB constant k is,practically without exception, near to but less than unity(Chirife, Timmermann, Iglesias, & Boquet, 1992), a factwhich constitutes a de®nitive characteristic of this iso-therm (Timmermann, 1989).
Now, if both isotherms (BET and GAB) are usedfor regression analysis of sorption data, two sets ofvalues of the monolayer capacity and of the energyconstant are obtained, which should be comparable.However, it has been observed by several authors (vanden Berg, 1981; Kim, Song, & Yam, 1991; Duras &Hiver, 1993; Lagoudaki, Demertzis, & Kontominas,1993) that
xmB�BET� < xmG�GAB�; cB�BET� > cB�GAB�: �1�
That is, the monolayer capacity by BET is always lessthan the GAB value, while the energy constant cB by BETis always larger than the GAB value. These inequalitiesset up the dilemma, about which values resemble a betterphysical reality, a dilemma not solved so far.
Following a general approach given elsewhere by oneof us (Timmermann, 2000), it will be shown here thatthere exits mathematical (and physical) reasons for theinequalities set by Eq. (1) and that the GAB values arethe values of better physical reality. For this purpose,several experimental data for water sorption in foodsand foodstu�s are analysed and in each case, the in-equalities (Eq. (1)) are qualitatively and quantitativelyexplained. Hydration of proteins, in terms of the at-tachment of one water molecule to each polar group ofthe side chains of the aminoacids (Pauling, 1945), is alsodiscussed in terms of the BET and GAB monolayervalues.
2. BET regression vs GAB regression
2.1. The BET isotherm
The classical BET equation, giving the amount ofwater x(aw) sorbed by a unitary amount of sorbant interms of the water activity aw, is the following:
It is well known that the two constants, the monolayervalue xmB and the energy constant cB, are obtained fromthe so-called BET plots (Iglesias & Chirife, 1976a). Insynthesis, in these plots, the linearised form, F(BET), ofthis isotherm
is drawn in terms of aw. This function should be linear ifthe BET assumptions apply, and within the linear range,using a linear least-square analysis, F(BET) is adjustedby a linear polynomial
P�BET�i � a0 � a1xi �4�by minimising the squares sum over the n experimentalpoints (index i)Xi�n
i�1
F�BET�i� ÿ a0� � a1xi�
�2 � minimum; �5�
where xi stands for aw at the point i. The coe�cients a0
and a1 are given by the solutions of the system of normalequations associated to the extremum condition (5).According to Eqs. (3) and (4), the least-squares esti-mates of a0 and a1, a0 and a1, are related to the BETconstants by
a0 � 1=cBxmB; a1 � �cB ÿ 1�=cBxmB �6�and herefrom
xmB � 1= a0
�� a1
�; cB � a0
�� a1
�=a0 �7�
relations by which the BET constants are calculated. Itshould be noted that the energy constant cB is in-versely proportional to the intercept a0 of the linearregression polynomial P(BET) of Eq. (3) and, there-fore, cB is very sensitive to the value of a0, which isusually very low.
Usually, and this was observed by many workers, theBET plots give only an apparent linear plot at lowwater activities (0.05 < aw < 0.3±0.4) and over this rangethe BET regression is performed (Iglesias & Chirife,1976a). For aw > 0.3±0.4, always deviation from linear-ity is observed with an upswing of F(BET) indicatingthat at higher water activities less water is sorbed thanthat predicted by the BET equation as shown in Fig. 1for the BET plots of water sorption for various foodmaterials.
20 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31
where
cB�G� � cGk: �9�To determine the three constants of the GAB equation,several methods can be employed. In the present con-text, a linearisation method of the GAB isotherm ana-logue to that of the BET model (Eq. (3)) is the mostadequate; the other methods will be examined latter. Tolinearise the GAB isotherm, the following functionF(GAB) applies:
(Anderson, 1946; Gascoyne & Pethig, 1977; Timmer-mann, 1989) ± should be linear in aw, 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 aw; a too high k-value determines an upward curva-
ture 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 linearregression of Eq. (10) in terms of variable k determinesthe best k-value. Moreover, from the two linear regres-sion coe�cients of F(GAB), the other constants ± xmG
and cG ± can be obtained.The corresponding representation of F(GAB) vs aw is
also given in Fig. 1 (corresponding values of k have beentaken from Table 1). The linearisation of experimentaldata through Eq. (10) is possible within the range0.05 < aw < 0.8, which represents a much broader appli-cability range of the GAB isotherm compared with theBET equation. Furthermore, at higher water activities,these GAB plots present a downward deviation due tothe appearance of the third sorption stage (Timmer-mann & Chirife, 1991), an e�ect, which determines theupper limit of application of the GAB isotherm.
2.3. The relation between F(BET) and F(GAB)
As already stated, the results of both regressions forthe same set of experimental data lead to the inequalitiesstated by Eq. (1). This dilemma may be tackled in thefollowing way. F(GAB) is related to F(BET) by
and by introducing here the expression (10) of F(GAB)and multiplying out the resulting expression, a secondrelationship for F(BET), named F�(BET), is obtained,now in terms of the three constants of the GAB iso-therm. It results
where Eq. (9) has been used to introduce cB�G�. Thissecond expression for F(BET) shows that, if k < 1,F(BET) will not be linear in aw, but will present an hy-perbolic behaviour
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.
E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 21
Tab
le1
BE
Ta
nd
GA
Bco
nst
an
tsfo
rw
ate
rso
rpti
on
info
od
sa
nd
foo
dst
u�
sa
BE
T(r
an
ge:
0:0
56
a w6
0:4
)G
AB
(ran
ge:
0:0
56
a w6
0:8
)
nB
xm
B(e
xp
)c B
(exp
)x
mB
(calc
)c B
(calc
)n
Gx
mG
c Gk
c Gk
To
ma
to(A
)4
14
.5�
1.1
61.8
�67.5
14.3
�0.5
68.8
�39.5
916.6
�0.6
31.4
�9.3
0.8
3�
0.0
626.1
�9.6
30°C
/NE
F5
.49
(0.1
13±0.4
32)b
2.4
82.3
8(0
.113±0.8
36)b
Co
rnb
ran
(A)
66
.12
�0
.36
9.5
�1.3
5.9
2�
0.1
410.3
�0.6
97.2
1�
0.7
59.8
�4.5
0.7
6�
0.2
07.5
�5.4
25°C
/NE
F3
.10
(0.0
63±0.3
79)b
1.3
63.3
8(0
.063±0.7
16)b
Fis
h¯
ou
r4
4.9
1�
0.6
55.0
0�
1.1
4.5
6�
0.1
95.8
�0.5
85.8
0�
1.2
35.1
�3.7
0.8
1�
0.3
34.1
�4.7
(25
°C)/
NE
F4
.17
(0.1
15±0.4
43)b
1.3
25.5
8(0
.115±0.8
48)b
Po
tato
sta
rch
(na
tiv
e)(A
)
20°C
/NE
F
10
8.0
7�
0.2
426.9
�4.8
8.2
2�
0.4
525.0
�7.5
16
9.7
9�
0.2
720.4
�3.4
0.7
5�
0.0
515.4
�3.6
3.8
4(0
.0399±0.4
027)b
2.5
23.1
1(0
.0399±0.8
887)b
Wh
eat
sta
rch
(na
tiv
e)(A
)
20°C
/NE
F
57
.93
�0
.17
32.8
�4.7
7.9
5�
0.5
438.0
�20.9
11
9.8
9�
0.2
126.7
�3.8
0.6
8�
0.0
418.1
�2.2
2.5
7(0
.0404±0.4
013)b
3.6
41.9
4(0
.0404±0.8
663)b
Wh
eat
(D)
57
.64
�0
.41
21.7
�5.7
7.7
3�
0.5
922.7
�8.9
910.2
4�
0.1
18.6
�1.0
0.6
2�
0.0
211.4
�0.9
25°C
/NE
F5
.50
3.0
45.4
0
Ch
ick
en(c
oo
ked
)(D
)
19
.5°C
/NE
F
56
.81
�0
.06
22.6
�1.1
7.0
�0.2
21.5
�3.5
97.7
5�
0.3
118.7
�0.8
0.8
6�
0.0
716.1
�1.9
0.8
81.3
11.7
6
22 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31
Tu
rkey
(co
ok
ed)
(A)
22°C
/NE
F
55
.36
�0
.31
7.7
�0.9
5.3
5�
0.2
17.8
�0.7
10
6.2
9�
0.2
67.4
1�
1.2
20.8
2�
0.0
76.1
�1.7
3.4
90.3
73.8
7(0
.05±0.9
)b
Tu
rkey
(co
ok
ed)
(D)
22°C
/NE
F
56
.24
�0
.44
7.0
�1.0
6.2
0�
0.3
07.2
�0.7
10
6.9
2�
1.5
97.5
0�
0.4
50.7
9�
0.1
05.9
�1.0
3.9
90.9
53.7
4(0
.05±0.9
)b
Co
rn¯
ou
r(d
eger
med
)5
7.9
3�
0.3
879
�61.4
7.9
1�
0.6
4138
�314
910.2
7�
0.1
642.4
�5.4
0.5
9�
0.0
325.1
�4.3
25°C
/NE
F7
.47
5.8
01.0
7
Ric
e(A
)4
7.7
4�
0.2
332.9
�7.4
7.9
4�
0.6
031.1
�16.6
811.0
�0.5
319.2
�5.3
0.5
8�
0.0
811.2
�4.6
25°C
/NE
F2
.38
(0.1
±0.4
)b2.8
32.0
4(0
.1±0.8
)b
Co
rn(D
)5
7.3
9�
0.2
622.6
�4.0
7.4
9�
0.5
523.3
�8.9
99.7
8�
0.2
118.8
�2.3
0.6
33
�0.0
37
11.9
�2.1
30°C
/NE
F3
.67
2.9
61.2
77
Wh
eat
glu
ten
(A)
45
.33
�0
.68
21.9
�14.8
5.3
6�
0.2
421.7
�4.9
86.3
8�
0.5
416.4
�7.6
0.7
8�
0.1
512.9
�8.5
3°C
/NE
F1
.02
(0.1
±0.4
)b1.5
24.4
0(0
.1±0.8
)b
Sk
imm
ilk
(A)
43
.92
�0
.20
52.6
�31.7
3.8
7�
0.1
167.3
�28.9
94.2
7�
0.1
138.0
�10.2
0.8
76
�0.0
533.0
�13.1
34°C
/NE
F4
.68
(0.1
±0.4
)b1.3
22.9
8(0
.1±0.9
)b
aA
:a
dso
rpti
on
,D
:d
eso
rpti
on
;x
m:
ing
H2O
/10
0g
dry
matt
er;
nB,
nG
:n
um
ber
of
exp
erim
enta
lp
oin
ts;
(exp
):o
bta
ined
usi
ng
the
dir
ect
regre
ssio
nb
yF
(BE
T);
(calc
):ca
lcu
late
din
term
so
fth
eG
AB
con
sta
nts
usi
ng
F� (
BE
T);
NE
F:
no
rmali
zed
erro
rfu
nct
ion
(Eq
.(2
1))
.S
ourc
eof
sorp
tion
data
:T
om
ato
:K
iran
ou
dis
,M
aro
uli
s,T
sam
i,an
dM
ari
no
s-K
ou
ris
(1993);
Co
rnb
ran
:D
ura
san
dH
iver
(19
93);
Fis
h¯
ou
r:L
ab
uza
,K
aan
an
e,a
nd
Ch
en(1
98
5);
Po
tato
starc
h(n
ati
ve)
:van
den
Ber
g(1
981);
Wh
eat
starc
h(n
ati
ve)
:id
em;
Wh
eat:
Hu
bb
ard
,E
arl
e,an
dS
enti
(1957);
Ch
ick
en(c
oo
ked
):T
aylo
r
(19
61);
Tu
rkey
(co
ok
ed):
Kin
g,
Lam
,a
nd
Sa
nd
all
(19
68);
Co
rn¯
ou
r(d
eger
med
):K
um
ar
(1974);
Ric
e:N
emit
z(1
963);
Co
rn:
Hu
bb
ard
etal.
(1957);
Wh
eat
glu
ten
:B
ush
uk
an
dW
ink
ler
(1957);
Sk
imm
ilk
:B
erli
n,
An
der
son
,an
dP
all
an
ach
(1970).
bO
ther
evalu
ati
on
ran
ge
tha
nth
at
sta
ted
inth
eh
ead
ings.
E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 23
Conversely, if k� 1, Eqs. (3), (13) and (14) becomeidentical as C (k� 1)� 0.
Eq. (13) readily explains qualitatively and quantita-tively (Eq. (14)) the usually observed upswing in theBET plots at aw > 0.4±0.5, if k < 1. In the graphs ofFig. 1, F�(BET) has been represented in terms of thecorresponding GAB constants taken from Table 1. It isshown that this function reproduces quite well the (non-linear) experimental F(BET) within the whole GABapplicability range 0.05 < aw <0.8.
Furthermore, it is evident that, if F(BET) responds toEq. (13) but analysed using Eq. (3), the so-obtained (andassociated to the BET isotherm) values of xmB and cB
will certainly be functions of the three GAB constantsxmG, cG and k through Eqs. (12)±(14) and of the aw in-terval over which the regression is performed. And thisfunctional dependence determines the di�erence betweenthe BET and the GAB sets for the monolayer capacity(xm) and the principal energy constant (cB) and, there-fore, the inequalities stated by Eq. (1).
2.4. BET regression by F�(BET)
The second expression (13) of F(BET) may also beadjusted by the same linear polynomial (4), but nowusing an analytical formulation as F�(BET) is known asa function of aw and not by a set of numerical data. Thecalculation implies the adjustment of a function of aknown functional dependence of a higher degree thanone to a straight line. This regression of F�(BET) can bemade 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
Xi�n
i�1
A� ÿ C � �Bÿ C�xi � C=�1ÿ xi�� ÿ a0 � a1xi� �f g2
� minimum �15�
and this expression can now be solved analytically for a0
and a1 in the usual way of least squares. As it is to beexpected, it results (Timmermann, 2000) that a0 and a1
become functions of the constants A, B and C of Eq.(13) on one side, and of regression sums over the valuesof the independent variable xi on the other.
The ®nal expressions are the following:
a�0 � �Aÿ C� � C�d0=d�; �16a�
a�1 � �Bÿ C� � C�d1=d�; �16b�
where a�0 and a�1 are the minimum squares estimates interms of F�(BET). The functions d, d0 and d1 containonly the regression sums of aw over the employed re-gression interval with the following signs: d0/d < 0, d1/d > 0 and d0/d + d1/d > 0 for aw < 1 (Timmermann, 2000).
The corresponding relations are, where Gaussianbrackets have been used,
� �ÿ �aw�2: �17c�n is the number of data included in the regression.
As the Eqs. (6) and (7) remain valid, the BET con-stants are now given by
x�mB � 1=�a�0 � a�1�� 1= Af � B� C �d0� � d1�=dÿ 2�g �18a�
and
c�B � a�0ÿ � a�1
�=a�0
� Af � B� C �d0� � d1�=dÿ 2�g= Af � C d0=d� ÿ 1�g:�18b�
Finally, by Eq. (14) A, B, C� f(xmG, cG, k) and aftersome algebra, explicit expressions for x�mB and c�B interms of the three GAB constants are obtained (Tim-mermann, 2000):
x�mB � xmG= 1�� � 2�1ÿ k�=cB�G�
�R�m �19a�
and
c�B � cB�G� 1� � 2�1ÿ k�=cB�G�
�R�c ; �19b�
where the functions R�m and R�c are given by
R�m � 1� �1ÿ k� cB�G�ÿ� � 1ÿ k
�= cB�G�ÿ
� 2�1ÿ k�� d0�� � d1�=dÿ 2� �20a�and
R�c � R�m= 1� � �1ÿ k� cB�G�
ÿ � �1ÿ k� d0=d� ÿ 1��:�20b�
These functions are always greater than unity (Tim-mermann, 2000). In consequence, Eqs. (19a) and (19b)reproduce the inequalities (1) if k < 1.
Eqs. (19a) and (19b) explicit and quantify the di�er-ences between the BET set (xmB ) cB) and the GAB set(xmG ) cB�G�) of constants. These di�erences are directlyrelated to k < 1 through the factor (1 ) k) present inthese equations and hence, explain the inequalities set inEq. (1). They become more important with decreasingvalues of k as well as with an increase of the regressioninterval. On the other hand, if k� 1, all these expres-sions coincide with the classical results shown before.
Eqs. (16a), (16b), (19a) and (19b) have been used tocalculate the linear correlation of F(BET)GAB in terms ofthe corresponding GAB constants and for the samewater activity interval used for the empirical BETequation (numerical values are stated in Table 1); theresults are shown in Fig. 1(a±c). The reproduction of theempirical BET constants and of the inequalities stated in
24 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31
Eq. (1) are quite acceptable. For the case of ®sh ¯our(Fig. 1(a)), a slight di�erence between the linearF(BET)exp and the linear F(BET)GAB is observed, a dif-ference which is determined by the empirical dispersionof the experimental sorption data. In the case of cornbran (Fig. 1(b)), this dispersion is less and the coinci-dence of F(BET)exp and F(BET)GAB is better. Finally, inthe case of wheat starch (Fig. 1(c)), this dispersion isminimal and F(BET)exp and F(BET)GAB become prac-tically undistinguishable.
Thus, the approach given here solves the dilemma ofthe inequalities stated by Eq. (1). It also explains therestricted range of linearity of the BET regression, thedependence of the values of the BET constants on
the activity range used for the regression, and the up-swing 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. Thesecond group corresponds exclusively to the compre-hensive data set of water sorption by proteins due toBull (1944), where this author tested the applicabilityof the BET isotherm to these sorption systems. Theresults are presented in Tables 1 and 2, and in Fig. 2.
Table 2
Monolayer moisture contents for various proteinsa ;b
Protein BET (range: 0:056 aw ÿ 6 0:3) GAB (range: 0:056 aw6 0:8)
NB xmB (exp) cB (exp) xmB (calc) cB (calc) nG xmG cG k cGk
a nB, nG, number of experimental points; xm: in g H2O/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 aw � 0:05±0:3 BET applies, but at aw � 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.
E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 25
The left-side sections of Tables 1 and 2 present the setsof BET constants obtained by (a) the direct regressionby F(BET) (Eqs. (6) and (7)) and (b) in terms of theGAB constants using F�(BET) [Eqs. (14), (16a)±(18b)];and the right-side section contains the set of GABconstants obtained using a parabolic regression (Eq.22) (see also Fig. 3). The general regression ranges aregiven in the headings, with exceptions indicated ineach case. The error of each numerical value has beencalculated using the regression covariance of eachparameter. Finally, the normalised error function(NEF), de®ned as
NEF � 100Xi�n
i�1
x exp
ÿ "ÿ xcalc
�2
i
!,n
#1=2,xm �21�
is also given. This function is related to, but simpler thanthe relative percentage root, mean square value oftenused in the literature.
It can be seen (see also the top graphs of Fig. 4) thatGAB monolayer values are about 10±40% higher thanthe BET value, while the energy constant cB�G� is muchlower (35±50% and even more) than the BET value. Inthe same way, the errors of the energy constant valuesare much stronger (15±25% up to 60±70% and more)than that of the monolayer (4±8%). It is also to be notedthat the error of the third GAB constant, k, is in theorder 10±15% and therefore, the value of this constantshould be given with only two signi®cant ®gures. Forthe 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 in-dicates a much less structured state of the sorbate in thelayer following the monolayer, the so-called GAB lay-ers, as in the sorbate's pure liquid state (Timmermann,2000).
In food science studies, preferred or almost exclusiveattention is paid to the monolayer value (Karel, 1973;Iglesias & Chirife, 1976b). However, the values of theenergy constants should not be overlooked nor ignoredbecause they are simultaneous outputs of the regressionprocesses and they in¯uence the sigmoidal shape of theisotherms, i.e., the form of the normalised Ôx/xm vs awÕplot, since cB and cG determine the more or less pro-nounced form of the `knee' at the lower water activityrange. On the other hand, the third GAB constant kdetermines the pro®le of the isotherm at the higherwater activity range, regulating the upswing after theÔplateauÕ at medium water activity range. Higher valuesof k determine a more pronounced upswing. This can bereadily observed in Fig. 2; proteins and protein-foods(k� 0.8) present a much more noticeable upswing thanstarchy foods (k� 0.7).
Finally, the function NEF is a measure of the ex-perimental dispersion of the sorption data; good (mean)values of NEF are in the order 2±5%. If this dispersion ishomogeneous over the whole GAB range, NEF hascoincident values for the BET as well as for the GABregressions indicating the much better ability of theGAB equation to represent the data as it embraces amuch broader range of water activity. But if this dis-persion is heterogeous (in the BET region di�erent thanin the GAB region), then NEF oscillates about the samevalues being in some cases the BET values lower thanthe GAB values and in others the opposite is observed,but always within the range 2±5%. A case markedlybeyond this range is the protein elastin, the sorptiondata of which present NEF values of 9% (GAB) to 11%(BET). Morover, the NEF values of the calculated BETparameters by Eqs. (19a) and (19b) using the GABconstants are due to the intrinsic hyperbolic curvature ofF�(BET) (Eqs. (12) and (13)) over the BET range ofactivities. The experimental NEF values of the BETregression include this e�ect and therefore, by the
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.
26 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31
experimental dispersion of the data, the NEF values¯uctuate (upwards or downwards) about the `theoreti-cal' values indicated by the former.
Fig. 2 shows calculated food isotherms obtained bythe BET and GAB regressions as well as by the BETconstants calculated in terms of the GAB parameters,using the constants given in Tables 1 and 2. The limitedrange of applicability of the BET equation and theability of the GAB equation to represent the experi-mental data up to aw� 0.85 is observed in all cases; aswell as the good agreement between calculated BETisotherm by the GAB constants with the BET curveobtained directly by the regression of the experimentaldata.
To determine the GAB constants, a simple methodwas used which is straightforward; it uses the so-calledtransformed form of the GAB equation (Schaer & Ru-egg, 1988), i.e., the following parabolic expression,which is easily derived from Eq. (8):
aw=x � a� baw � ca2w; �22�
where
a � 1=xmGcGk; �23a�b � �cG ÿ 2�=xmGcG; �23b�
c � ÿ�cG ÿ 1�k=xmGcG: �23c�The three constants a, b and c are readily determined bya least-square regression of this second degree polyno-mial and from these, the three GAB constants are cal-culated by
k � �f 1=2 ÿ b�=2a; �24a�
xmG � f ÿ1=2 � 1=�b� 2ka� � k=�k2aÿ c�; �24b�
cG � 1ÿ c=k2a � 2� b=ka � f ÿ1=2=ka; �24c�where
f � b2 ÿ 4ac: �24d�The constants stated in Tables 1 and 2 were obtainedby this method. The upper limit of the regression or
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.
E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 27
applicability range of the GAB isotherm (the lower limitis, as in the BET case, aw� 0.05) is determined with theso-called inverse plot (Timmermann, 1989). At high aw,for strongly sorbing substances (cG� 1), both isothermsbecome very simple for the inverse of x(aw):
BET : 1=x � �1=xmB��1ÿ aw�; �25a�
GAB : 1=x � �1=xmG��1ÿ kaw�: �25b�These relations indicate that 1/x is linear at highenough aw for both isotherms and that the limits for1/x� 0 (x!1) are at the points (aw� 1, 1/x� 0; BET)
and (aw� 1/k (>1), 1/x� 0; GAB), respectively. Thus, ifthe linear part at higher aw of the inverse plot 1/x vs aw
do not extrapole to aw� 1 for 1/x� 0 (BET condition),it is readily concluded that k < 1 (see Eq. (25b)) andthat the GAB equation applies. The extrapolation to1/x� 0 gives 1/k directly as the intercept with theaw-axis (Timmermann, 1989). Hence, these plotsreadily illustrate which isotherm applies. Furthermore,if after the linear part the graph becomes curveddownwards (usually at aw� 0.85±0.9), then this is adirect evidence of the presence of the third sorptionstage (Timmermann, 1989; Timmermann & Chirife,
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.
28 E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31
1991), these points should not be used for the GABregression. These procedures (inverse and parabolicplots) are shown in Fig. 3 for various food materials.The calculated BET and GAB isotherms are alsodrawn and the upper limits of both equations areshown by arrows.
An alternative method to obtain the GAB constantsconsists in a non-linear least-squares regression of theGAB equation. It has been claimed (Schaerr & Ruegg,1988) that this method and that of the parabolic trans-form give di�erent results, but it can be shown (Tim-mermann et al., 1991) that if the points of the thirdsorption stage are not included, both regressions giveidentical results. Accordingly, it has already been statedthat, if points belonging to the third sorption stage areincluded in the GAB regression, NEF increases verysharply this being another criterion to ®x the upper limitof the GAB regression.
The results contained in Tables 1 and 2 are illustratedin Fig. 4; the BET monolayer value (xmB) and energyconstant (cB) are plotted against the GAB monolayervalue (xmG) and the value by GAB (cB�G� � cGk), re-spectively (Figs. 4(a)). These plots illustrate the in-equalities stated by Eq. (1); i.e., that the BET evaluationalways underestimates the monolayer, while it overesti-mates the energy constant. Figs. 4(b) shows the plot ofthe two BET constants against the respective valuescalculated in terms of the GAB constants. Within thecorresponding error intervals, the `experimental' BETvalues are well reproduced by the calculated ones, andthe scatter is much lower for the monolayer capacitythan for the energy constant.
It is therefore straightforward to conclude that theGAB constants are to be taken as the representativeparameters of the multilayer sorption. A much moreprecise description of multilayer sorption of water byfood materials can be achieved if the analysis is madewith 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 thewater sorption by proteins, Pauling (1945) published anow classical paper about the hydration of proteins. Headvanced that the water sorption monolayer of proteinscan be thought in terms of the attachment of one watermolecule to each polar group of the side chains of theaminocids in the protein. In his analysis, Pauling utilisedBET monolayer values reported by Bull (1944). Theagreement of these BET monolayer values with thenumber of polar groups of the proteins was roughlysatisfactory in as much both values were of the sameorder of magnitude. However, it is worth noticing that
the monolayer values were in most cases lower than thenumber of polar groups.
In view of the results stated in the previous sections, it isstraightforward to compare Pauling's data with themonolayer capacity obtained here by the GAB evalua-tion, as this equation is directly related and is an im-provement of the original BET formulation. Pauling(1945) expressed his numerical data in terms of thenumber of polar groups or moles of water per 105 g ofprotein. We retain here these units, the monolayer valuesgiven as grams of water per 100 g of proteins in Table 2 areto be multiplied simply by 55.5 (� 1000/18) mol H2O/g.
Table 3 shows the results of this new analysis ofPauling's hypothesis. The BET monolayer values statedin Table 2 are slightly di�erent from those reported byBull (1944) in his original paper, data which are givenwithin parenthesis in the same table. This is likely due tosome di�erences in the water activity interval used forthe BET regression (interval which is not stated exactlyin Bull's paper) and to the fact that Bull's values are themean between the values at 25°C and at 40°C. As inPauling's (1945) original paper, the second value of thesecond column (Table 3) corresponds to the value ob-tained by taking also into account the aminoacids pro-line and hydroxiproline. Data within [ ] are the numberof polar groups reported by other authors found in arapid and not-exhaustive search of the literature, as amore profound revision of Pauling's values in terms ofmodern literature of protein aminoacid composition, isbeyond the scope of this paper.
The inspection of Table 3 shows that the roughagreement noted by Pauling is certainly improved whenthe GAB monolayer values are used for comparison.This conclusion becomes even more evident when theBET and GAB monolayers are graphically plottedagainst the number of polar groups (Fig. 5). Theagreement is quite better for collagen, gelatin, serumal-bumin, lactoglobulin, c- and b-zein. For silk, the BETvalue seems to be better and, on the other side, forsalmin, the GAB value correlates surprisingly well, al-though its isotherm presents two branches (see Fig. 2and note 4 of Table 2), a fact observed already byPauling (1945) and by Bull (1944) themselves. Further-more, in the case of egg albumin and wool, the agree-ment is improved when the number of polar groupsreported in more modern publications are consideredinstead. In the case of collagen and gelatin, Pauling(1945) noted that the BET value failed to reach the valueof the number of polar groups including the proline andhydroxiproline, and advanced some possible explana-tion for this discrepancy. However, when the GABmonolayer value is used, a close agreement is observed.Casein, a protein not considered by Pauling in his paperand not stated in Table 2, has been included in thepresent analysis. The sorption data due to Schaerr andRuegg (1988) were evaluated elsewhere (Timmermann
E.O. Timmermann et al. / Journal of Food Engineering 48 (2001) 19±31 29
et al., 1991) using the BET and GAB equations and thecorresponding monolayer capacities are stated directlyin Table 3. It is observed that casein also ®ts well intothe picture given by Fig. 5.
Finally, it is interesting to consider also the case ofanother biopolymer, namely starch. The BET and GABmonolayer values for potato and wheat starch (Table 1)were 0.45 and 0.55 mol H2O/100 g, respectively. Sincethe polar group number (one water molecule per anhy-droglucose monomer) is 0.62, as reported by McLarenand Rowen (1951), it follows that again the GABmonolayer correlates much better than the BET value.
Thus, it is to be concluded that the present ®ndingsthat GAB parameters are more representative than thecorresponding BETs ones, obtains additional supportwhen Pauling's hypothesis of initial hydration of pro-teins, is considered.
Acknowledgements
The ®nancial support from University of BuenosAires and CONICET (Argentina) are greatfully ac-knowledged.
References
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.
Table 3
Comparison between number of polar groups and BET or GAB monolayer valuesa
Protein Number of polar groups
(mol/105g)
Monolayer capacity
(mol/105g)
Paulingb BET GAB k (GAB)
Collagen 328±609 535 638 0.80
Gelatin Idem 470 572 0.78
Seroalbumin 424 362 419 0.81
Wool 303, 341, 420c 373 470 0.71
Lactoglob.crist. 472, 508 367 428 0.81
Idem, f.dried. Idem 328 408 0.80
Eggalbum, coag 277, 313, 380d 288 350 0.78
Idem, f.dried Idem 313 382 0.80
Idem, not f.dried Idem 348 413 0.78
c-zein 305, 390 212 242 0.83
b-zein Idem 227 263 0.77
Silk 219±228 230 278 0.77
Salmin 611±707 330 755 0.89
Casein 416e, 456, 521f 306g 343g 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).
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
