Water activity in aqueous solutions of sucrose: an...
Transcript of Water activity in aqueous solutions of sucrose: an...
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Maciej STARZAKSchool of Chemical EngineeringUniversity of KwaZulu-NatalDurban, South Africa
Water activity in aqueous Water activity in aqueous solutions of sucrose: an improved solutions of sucrose: an improved
temperature dependencetemperature dependence
Mohamed MATHLOUTHILaboratoire de Chimie Physique IndustrielleUniversité de Reims, Champagne-ArdenneReims, France
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OutlineOutline
1. Introduction2. Previous studies3. Experimental database4. Selection of the water activity model5. Relationships between experimental
variables and activity coefficients6. Data regression7. Results8. Recommended equation for water
activity coefficient
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Water activity formulae popular amongst food technologists
Norrish, 1966 2ln A Bxγ α=
Chen, 19891000
1000 ( )A
A nA
M mM m A Bm
γ+
=+ +
Miyawaki, 1997 2 3ln A B Bx xγ α β= +
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Models used to predictwater activity coefficient
n Redlich-Kister expansion - empirical (includes Margules equation)
n UNIQUAC - phenomenological(two-fluid theory)
n group contribution methods(UNIFAC, ASOG)
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( )2 2ln 1A B B BQ
x bx cxRT
γ = + +
Starzak & Peacock, 1997
Based on 1197 experimental points (56 data sets),mainly VLE data (BPE, vapour pressure, ERH,isopiestic solutions)
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0 20 40 60 80 1000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
% mas. sucrose
Wat
er a
ctiv
ity c
oeff
icie
ntCatté et al., 1994
0 20 40 60 80 1000.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
% mas. sucrose
Wat
er a
ctiv
ity c
oef
ficie
nt
Starzak & Peacock, 1997
UNIQUAC Margules eq.
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0 10 20 30 40 50 60 70 80 90 1000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
% mass sucrose
Wat
er a
ctiv
ity c
oef
ficie
nt o
n th
e b
oili
ng
cu
rve
1193 points
Water activity coefficient (boiling curve)
Starzak & Peacock, 1997
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Theoretical models of activityfor highly concentrated sucrose solutions
§ Van Hook, 1987:- sucrose hydration - sucrose association (clustering)
§ Starzak & Mathlouthi, 2002:- water association- sucrose hydration - sucrose association (clustering)
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Previous studies (small exptl. databases)
Ø Le Maguer, 1992 (UNIQUAC)Ø Caté et al., 1994 (UNIQUAC)Ø Peres & Macedo, 1996 (UNIQUAC)Ø Peres & Macedo, 1997 (UNIFAC)Ø Spiliotis & Tassios, 2000 (UNIFAC)
Objective of this study:
q an empirical activity equationq wide range of temp. & concentrationsq large database
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Thermodynamic data typically used to determine the activity coefficient
q VLE data (BPE, VPL, ERH, osmotic coeff. by isopiestic method)
q SLE data (FPD, sucrose solubility)
q Termochemical data (heat of dilution,excess heat capacity)
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470Heat capacity
1252338Total
10283Heat of dilution
34265Solubility
13213FPD
641507VLE
Literature sourcesExperimental pointsData type
EXPERIMENTAL DATABASE
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Distribution of experimental data
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Distribution of experimental data
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2
lnn
kA k B
k
xγ α=
= ∑
Selection of water activity model
n-suffix Margules equation
Advantages: linear with respect to its coefficients
Drawbacks: lack of sound theoretical foundation
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2 301 2 3 4 5( ) lnk
k k k k k kα
α θ α α θ α θ α θ α θθ
= + + + + +
General temperature dependence
Disadvantage: large number of parameters
0where T Tθ =
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Simplified temperature dependence
Advantage: reduced number of parameters
22
ln ( )n
kA k B
k
a b xγ θ −=
= ∑2 30
1 2 3 4 5( ) lna
a a a a a aθ θ θ θ θθ
= + + + + +
Disadvantage: temperature effect patternindependent of composition
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2
lnn
kB k A
k
xγ β=
= ∑
Sucrose activity coefficient
ln lnA BA B
A B
d dx x
d x d xγ γ
=
Gibbs-Duhem equation
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Expansion coefficients of sucrose activity
( 1) , 2n
kk l
l k
lA k
kβ
= −
∑³
³
1
1, 2,3,..., 1l l l
n n
lA l n
lA
α α
α
++
= − = −
=
where
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2 2 3 4 5 6 7
3 3 4 5 6 7
4 4 5 6 7
5 5 6 7
6 6 7
7 7
3 5 72 2
2 2 28 35
5 83 3
15 359
4 224
145
356
β α α α α α α
β α α α α α
β α α α α
β α α α
β α α
β α
= + + + + +
= − − − − −
= + + +
= − − −
= +
= −
Expansion coefficients of sucrose activity, n = 7
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Expansion coefficients of sucrose activityMatrix representation
=ß Ma
1
1 11
, 1, 2,..., 1n
k kl ll
M k nβ α−
+ +=
= −∑ =
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Processing experimental data
lnlnln
/
/
A
A
B
Easym
Epasym
H RT
C R
γγγ
⇒⇒⇒
⇒
⇒
VLE dataFPD dataSolubility
Heat of dilution
Specific heat
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lnVLE data Aγ⇒
Modified Raoult’s law
0(1 ) ( )AB A
Px P T
γ =−
ERH 1001A
Bxγ =
−
Equilibrium relative humidity (ERH):
BPE, vapour pressure, osmotic coeff. :
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lnFreezing point data Aγ⇒
Solid-liquid equilibrium
( ) ( )ln 1
2
( )ln 1 ln(1 )
2
fA m pA m A m mA
m f
pA m f fA mA m B
m m
H T C T B T TRT R T
C T T TB TB T x
R T T
γ ∆ ∆ ∆
= − − + −
∆ ∆+ − ∆ + − − −
( ) ( )( )pA pA m
A m
C T C TB T T
R R
∆ ∆= + ∆ −Assumed
for pure water:
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lnSucrose solubility data Bγ⇒ %
Solid-liquid equilibrium (asym. convention)
00
( ) ( )( )pB pB
B
C T C TB T T
R R
∆ ∆= + ∆ −
Assumedfor sucrose:
00 0 0
0
00
( )( ) ( )ln 1
2
( )ln 1 ln( )
2
pBB m dB B mB
B m
pB B mB B
m m
C TT H T B T T TRT R T T
C T B TT TB T x
R T T
γγ
γ
∞
∞
∆ ∆ ∆ = − − + − ∆ ∆
+ − ∆ − − −
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Heat of dilution dataEasymH
RT⇒
Two-step procedure
q fitting each set of isothermal data toMcMillan-Mayer expansion (evaluatingcoefficients of expansion)
q data generated from the expansion usedas input data for regression
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I. Amount of heat liberated per one mole of sucrose in solution after addition of a known amount of pure water
Types of experimental heats of dilution
1 1I0
2
(J/mol solute) (f ) (i)k kk
kB
Hh m m
n− −
=
∆ = − ∑
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Types of experimental heats of dilution
II. amount of heat liberated per one moleof water in original solution afteraddition of a known amount of dilutesucrose solution
[ ]
II
02
(J/mol solvent)(i)
(i) (a) (f ) (a) (a) (i) (i)(i)
A
k k kA A A A
kk A
Hn
n n m n m n mh
n=
∆=
+ − −∑
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Types of experimental heats of dilution
III. amount of heat liberated per onegram of water added after additionof a small amount of pure water(differential heat of dilution)
III0
2
1(J/g solvent added) (i)
1000
(f ) (i)
kk
kA
Hh m
w
m m=
∆= − ∑
@
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Heat of dilution dataEasymH
RT⇒
Excess enthalpy of solutionfrom McMillan-Mayer expansion
02(J/mol)1000
kk
E kasym
A
h mH
m M==
+
∑
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/Specific heat data Epasym
C R⇒
Apparent molar heat capacity of sucrose:
1 ( )1000( )
Bp pA
c
M m c m cm
m
+ − Φ =
Excess heat capacity of solution
[ ]( ) (0)(J/mol K)
1000E c c
pasymA
m mC
m MΦ − Φ
=+
×
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Experimental variables& Margules expansion coefficients
SLE (asym. convention) - solubility
[ ]1
21
2 1 2
( )ln
( ) ( ) ( )1
B mB
B
n n nk k
k l l A mk l k
T
ba T M b x a T a T
k
γγ
γ
∞
∞
−−
−= = =
=
+ −−∑∑ ∑
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Experimental variables& Margules expansion coefficients
Excess enthalpy of solution
Derived from free Gibbs energy:
( )ln lnEA A B BG RT x xγ γ= +
( / )E EH G RTT
RT T∂
= −∂
2 lnE E Basym BH H RT x
Tγ ∞∂
= +∂
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Experimental variables& Margules expansion coefficients
Excess enthalpy of solution
2
2
( )1
E nasym kk
Bk
H bT a T x
RT k−
=
′=−∑
2 302 3 4 5
0
( ) 2 3
where
aT a T a a a a
T T
θ θ θθ
θ
′ = − + + + +
=
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Experimental variables& Margules expansion coefficients
Excess heat capacity of solution
By definition: ( )E Ep asymasym
C H R
R T
∂=
∂Hence:
[ ] 2
2
2 ( ) ( )1
En
pasym kkB
k
C bT a T Ta T x
R k−
=
′ ′′= +−∑
[ ] 2 32 3 4 52 ( ) ( ) 2 6 12T a T Ta T a a a aθ θ θ′ ′′+ = + + +
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DATA REGRESSIONPerformance index
2 2 2VLE VLE, , , VLE
2 2 2 2FPD , , FPD SOL , , SOL
22 ,,, , , ,2 2
HE CE
( , ) (ln ln )
(ln ln ) (ln ln )
a b expi A i A i
i
exp expA i A i B i B i
i i
E E expE E expp pasym i asym i asym i asym i
expi ii i
I w w
w w
C CH Hw w
RT RT R R
γ γ
γ γ γ γ
= −
+ − + −
+ − + −
∑
∑ ∑
∑ ∑
% %
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Results of data regression: correlation diagrams
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0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
BPE experimental, ºC
BP
E p
red
icte
d, º
C
Boiling point elevation – predicted vs experimental
1507 points
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0 20 40 60 800.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
% wt. sucrose
Wat
er a
ctiv
ity c
oeff
icie
nt,
γ A
0 20 40 60 800
5
10
15
20
25
30
35
40
% wt. sucrose
Fre
ezin
g p
oin
t dep
ress
ion
[°C
]
Results of FPD data regression
213 points
213 points
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0 50 100 150-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Temperature [°C]
ln [
γ B γ
∞ B( T
m) /
γ∞ B
]
0 50 100 15060
65
70
75
80
85
90
95
100
Temperature [°C]
% w
t. s
ucr
ose
Solubility curve
Results of solubility data regression
265 points
265 points
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0 0.1 0.20
0.1
0.2
20°C
0 0.1 0.20
0.1
0.2
30°C
0 2 40
20
40
12°C
0 2 40
20
40
16°C
0 2 40
50
18°C
0 2 40
20
40
22°C
0 2 40
50
10026°C
0 2 40
5030°C
0 0.1 0.20
0.1
0.218°C
0 0.5 10
1
218°C
0 2 40
5020°C
1 1.5 20
10
200°C
1 1.5 20
10
20
100
0 x
Hea
t of d
ilutio
n,
HE/R
T [
-]
5°C
1 1.5 20
10
2010°C
1 1.5 20
10
2015°C
1 1.5 20
10
2020°C
1 1.5 20
10
2025°C
1 1.5 20
10
2030°C
1 1.5 20
10
20
33°C
0 1 20
10
20
25°C
0 5 100
50
100
20°C
0 5 100
50
100
40°C
0 5 100
50
100
60°C
0 5 100
50
80°C
0 1 20
5
1025°C
0 1 20
10
20
Molality
25°C
Regression of 26 heat of dilution data sets
________
experimental
predicted
![Page 41: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/41.jpg)
0 2 4 6 80
20
40
60
80
100
120
140
0°C
5°C
10°C
20°C
40°C
60°C
80°C
Molality, mol/kg
1000
x E
nth
alp
y o
f dilu
tio
n, H
E /RT
[-]
0 20 40 60 800
10
20
30
40
50
60
70
80
90
100
1 mol/kg (26%)
2 mol/kg (41%)
3 mol/kg (51%)
4 mol/kg (58%)
5 mol/kg (63%)
6 mol/kg (67%)
Temperature, °C
1000
x E
nth
alp
y o
f dilu
tio
n, H
E /RT
[-]
Prediction of enthalpy of dilution
![Page 42: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/42.jpg)
0 2 4 60
0.1
0.2
0.3
0.4
Exc
ess
heat
cap
acity
, C
E/R
[-]
T = 20°C
0 2 4 60
0.05
0.1
0.15
0.2
0.25
0.3
T = 25°C
0 0.5 1 1.50
0.01
0.02
0.03
0.04
T = 25°C
0 0.5 1 1.50
0.01
0.02
0.03
0.04
0.05
Molality, mol/kg
T = 25°C
0 2 4 60
0.1
0.2
0.3
0.4
T = 15°C
Gucker & Ayres, 1937 Gucker & Ayres, 1937 Banipal et al., 1997
DiPaola & Belleau, 1977 Vivien, 1906
Regression of 5 heat capacity data sets
![Page 43: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/43.jpg)
0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
0°C 5°C
10°C
20°C
40°C
60°C
80°C
Molality, mol/kg
Exc
ess
hea
t cap
acit
y, C
E/R
[-]
0 20 40 60 80-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1 mol/kg
2 mol/kg
3 mol/kg
4 mol/kg
5 mol/kg
6 mol/kg
Temperature, °C
Exc
ess
hea
t cap
acit
y, C
E /R [
-]
Prediction of excess heat capacity of solution
![Page 44: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/44.jpg)
70 75 80 85 90 95 100
0.6
0.8
1
1.2
1.4
1.6
% wt. sucrose
Wat
er a
ctiv
ity
coef
fici
ent,
γ A
ISOTHERMS BETWEEN 0°C and 150°C
Prediction of water activity coefficient
![Page 45: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/45.jpg)
85 90 95 1000.5
0.6
0.7
0.8
0.9
1
% wt. sucrose
Wat
er a
ctiv
ity
coef
fici
ent,
γ A
150°C
0°C 20°C 40°C60°C
80°C100°C
Prediction of water activity coefficient
![Page 46: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/46.jpg)
Final water activity coefficient equation
2 21 2
201 2 3 4
ln ( ) (1 )
( ) ln
A B B Ba b x b x xa
a a a a a
γ θ
θ θ θ θθ
= + +
= + + + +
a0 = -268.59a1 = -861.42a2 = -1102.3a3 = 1399.9a4 = -277.88
b1 = -1.9831b2 = 0.92730
![Page 47: Water activity in aqueous solutions of sucrose: an …eurofoodwater.eu/pdf/2004/Session1/download.php?file=...Maciej STARZAK School of Chemical Engineering University of KwaZulu-Natal](https://reader033.fdocuments.net/reader033/viewer/2022041507/5e25d6511151ad429b4461a2/html5/thumbnails/47.jpg)
AcknowledgementAcknowledgement
Association Andrew van Hook forthe Advancement of the Knowledge on Sugar, Reims, France