Water activity in aqueous solutions of sucrose: an improved temperature dependence

  Maciej STARZAK School of Chemical Engineering University of Kw aZulu-Natal Durban, South Africa

  W ater activity in aqueous W ater activity in aqueous

solutions of sucrose: an improved

solutions of sucrose: an improved temperature dependence temperature dependence Mohamed MATHLOUTHI

  Laboratoire de Chimie Phy sique Industrielle Université de Reims, Champagne-Ardenne

  Outline Outline

  1. Introduction

  2. Previous studies

  3. Experimental database

  

4. Selection of the water activity model

  

5. Relationships between experimental

variables and activity coefficients

  6. Data regression

  7. Results

  8. Recommended equation for water activity coefficient

  Water activity formulae popular amongst food technologists

  2 ln x

  Norrish, 1966 γ = α

  A B

  1000 A

• M m

  γ = A

  Chen, 1989 n

  M m A Bm 1000 ( )

  A

  3 ln x x

  Miyawaki, 1997 γ = α β

• 2

  A B B

  

Models used to predict

water activity coefficient

n

  Redlich-Kister expansion - empirical

(includes Margules equation)

n

  UNIQUAC - phenomenological (two-fluid theory) n group contribution methods (UNIFAC, ASOG)

  Starzak & Peacock, 1997 Q

  2

  2 ln x 1 bx cx

  γ =

  A B B B ( )

  RT

Based on 1197 experimental points (56 data sets),

mainly VLE data (BPE, vapour pressure, ERH, isopiestic solutions)

1.4 Water activity coefficient

  0.5

  1.4 Water activity coefficient Starzak & Peacock, 1997 UNIQUAC Margules eq.

  1.3

  1.2

  1.1

  1

  0.9

  0.8

  0.7

  0.6

  0.4

  0.5

  Catté et al., 1994

  1.3

  1.2

  1.1

  1

  0.9

  0.8

  0.7

  0.6

  0.4

  Water activity coefficient (boiling curve)

  1.8

  1.6

  1193 points Starzak & Peacock, 1997

  1.4

  1.2

  1

  0.8

0.6 Water activity coefficient on the boiling curve

  

Theoretical models of activity

for highly concentrated sucrose solutions

Van Hook, 1987:

  §

• - sucrose hydration - sucrose association (clustering) Starzak & Mathlouthi, 2002:

  §

• - water association - sucrose hydration - sucrose association (clustering)

  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 equation

q wide range of temp. & concentrations

q large database

  

Thermodynamic data typically used

to determine the activity coefficient

  VLE data (BPE, VPL, ERH, q osmotic coeff. by isopiestic method) q SLE data (FPD, sucrose solubility)

q Termochemical data (heat of dilution,

excess heat capacity)

EXPERIMENTAL DATABASE

  Experimental points Literature sources Data type

  VLE 1507

  64 FPD

213

  13 Solubility 265

  34 Heat of dilution 283

  10 Heat capacity

  

70

  4

  Distribution of experimental data

  Distribution of experimental data

• -suffix Margules equation Advantages: linear with respect to its coefficients

  

2

ln n k

  A k B k x

  γ α =

  = ∑

  Selection of water activity model n

  

Drawbacks: lack of sound theoretical foundation

  2

  3

  1

  2

  3

  4

  5 ( )

ln

k k k k k k k

  α α θ α α θ α θ α θ α θ θ = + + + + +

  

General temperature dependence

Disadvantage: large number of parameters where T T

  

θ =

  

Simplified temperature dependence

Advantage: reduced number of parameters

  2

  2 ln ( ) n k

  A k B k a b x

  γ θ − =

  = ∑

  2

  3

  1

  2

  3

  4

  5 ( ) ln a a a a a a a

  θ θ θ θ θ θ

  = + + + + + Disadvantage: temperature effect pattern independent of composition

  

Sucrose activity coefficient

n k ln x

  γ = β B k A

  ∑ k

  2 =

  

Gibbs-Duhem equation

d d ln ln

  γ γ A B x x

  

=

A B d x d x

  A B

  

Expansion coefficients of sucrose activity

( 1) ,

  2 n k k l l k l A k k

  β   = −  

    ∑

  ³ ³

  1

  1 , 2,3,...,

  1 l l l n n l A l n l

• = − = − =

  A α α α

  where

  2

  3

  2

  2

  2

  2

  2

  8

  35

  5

  8

  3

  15

  5

  35

  9

  4

  2

  24

  

14

  5

  35

  6 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 = + + + + +

  = − − − − − = + + + = − − − = +

  7

  3

  2

  7

  3

  

4

  5

  6

  7

  3

  3

  4

  5

  6

  4

  7

  4

  5

  6

  7

  5

  5

  6

  7

  6

  6

  

Expansion coefficients of sucrose activity, n = 7

  

Expansion coefficients of sucrose activity

Matrix representation

=

  ß M a

  1

  1

  1

• =

  1 , 1, 2,...,

  1 n k k l l l

  M k n β α

  −

  = −

  ∑ =

  Processing experimental data

  VLE data ln

  

⇒ γ

A

  FPD data ln

⇒ γ

  A Solubility ln

  

⇒ γ

B E

  

Heat of dilution H RT

/

  ⇒ asym E

  Specific heat C / R

⇒ pasym ln

  VLE data A

  γ ⇒

Modified Raoult’s law

  (1 ) ( ) A B A P x P T

  γ =

−

ERH 100

  

1

A B x

  γ = −

Equilibrium relative humidity (ERH):

BPE, vapour pressure, osmotic coeff. :

  Assumed

  H T C T B T T RT R T C T T T B T B T x R T T

  C T C T B T T R R ∆ ∆ = + ∆ −

  ( ) ( ) ( )

pA pA m

A m

     ∆       ∆

     

  = − − + −   

    ∆

  γ   ∆ ∆

  A m A m B m m

  Freezing point data ln A

  A m f pA m f f

  2 fA m pA m A m m

  1 ln(1 )

  2 ( ) ln

  1

  ( ) ( ) ln

  γ ⇒ Solid-liquid equilibrium

• − ∆ + − − −            

  Sucrose solubility data ln ⇒ γ

  % B

  

Solid-liquid equilibrium (asym. convention)

∞

  C T ( )

  ∆ ( T )  H ( ) T B T  T T

    pB

  γ ∆ ∆

  1 γ = − −

• B m dB

  − B

  ∞   

  B m   ln

     RT R T T

  2 γ

  B m

       

  C ( ) T ∆   T B T  T 

   pB  ∆

  B T ln 1 ln( x ) − ∆ − − −

• B m

  B B

        R T T

  2 m m

       

C T C T

  ( ) ( ) Assumed ∆ ∆

pB pB

  B T T ( )

  = + ∆ − B for sucrose:

  

R R

  E H asym

  Heat of dilution data ⇒

  RT

Two-step procedure

q fitting each set of isothermal data to

  McMillan-Mayer expansion (evaluating coefficients of expansion) q data generated from the expansion used as input data for regression

  Types of experimental heats of dilution

I. Amount of heat liberated per one mole of sucrose in solution after addition of a known amount of pure water

  H k k

  ∆

  1

  1 I − −

  (J/mol solute) h m (f ) m (i)   = − k

  

∑

  n k

  2 B

=

  

Types of experimental heats of dilution

  II. amount of heat liberated per one mole of water in original solution after addition of a known amount of dilute sucrose solution H

  ∆

  II (J/mol solvent)

  = n (i)

  A

  n (i) n (a) m (f ) n (a) m (a) n (i) m (i) − −

• k k k

  A A A A [ ] h k

  ∑ n (i) k

  2 =

  A

  

Types of experimental heats of dilution

  III. amount of heat liberated per one gram of water added after addition of a small amount of pure water (differential heat of dilution) H

  1 ∆ k

  III (J/g solvent added) h m (i)

  = − k

  ∑ w

  1000 k

  2 A = m (f ) @ m (i)

  E H asym

  Heat of dilution data ⇒

  RT Excess enthalpy of solution from McMillan-Mayer expansion k h m k

  ∑ E k

  2 =

  H (J/mol) =

  1000 m M

• asym

  A

  / Specific heat data E p asym

  C R ⇒

  Apparent molar heat capacity of sucrose: 1 ( ) 1000

  M m c m c m m

   

• + −

      Φ =

  ( ) B p pA c

  Excess heat capacity of solution [ ]

  ( ) (0) (J/mol K) 1000

  E c c p asym

• ×

  A m m C m M

  Φ − Φ =

  Experimental variables & Margules expansion coefficients SLE (asym. convention) - solubility ∞

  ( T )   γ

  B m ln

  γ = B

  ∞   γ

  B   n n n

  1 − b

  k

  2 − a T ( ) M b x a T ( ) a T ( )

• k

  − k l l A m

  1 − k

  [ ] ∑∑

  ∑

  1 k l k

  2

  1 2 −

  = = =

  

Experimental variables

& Margules expansion coefficients

Excess enthalpy of solution

  Derived from free Gibbs energy: ( ) ln ln

  E A A B B G RT x x

  γ γ = + ( / )

  E E H G RT T RT T ∂ = −

  ∂

  2 ln

  E E B asym B

  H H RT x γ

  ∞ ∂ = +

  Experimental variables & Margules expansion coefficients Excess enthalpy of solution E n

  H asym b k k

  2 −

  ( ) T a T ′ x

  = B

  ∑

  1 RT k k

  

2 −

= a

  2

  3 T a T ( ) a a 2 a 3 a ′ = − + + θ θ θ

  2

  3

  4

  5 θ where T T

  θ =

  12 T a T Ta T a a a a θ θ θ ′ ′′

  − ∑

  6

  2

  5 2 ( ) ( )

  4

  3

  

2

  3

  2

  [ ]

  − = ′ ′′ = +

  Experimental variables & Margules expansion coefficients Excess heat capacity of solution By definition: ( )

  R k

  B k C b T a T Ta T x

  1 E n p asym k k

  2 2 ( ) ( )

  2

  [ ]

  ∂ Hence:

  C H R R T ∂ =

  E E p asym asym

• = + + +

  DATA REGRESSION Performance index exp

  2

  2

  2 I w w ( , ) a b (ln ln )

  = γ − γ

i A i A i

  VLE

  

VLE, , ,

  VLE ∑ i exp exp

  2

  2

  2

  2 w (ln ln ) w (ln ln )

  γ − γ γ % − γ % + + FPD A i , A i , FPD SOL B i , B i , SOL

  ∑ ∑ i i

  2

  2 E E exp , E E exp

  ,  

  C C H H p p   asym i , asym i , asym i , asym i ,

  2

  2 w w  

  − + + −

  HE CE

   exp  ∑

  ∑ RT RT R R i i i i

         

  Results of data regression: correlation diagrams

  

Boiling point elevation – predicted vs experimental

  70 1507 points

  60

  50

  40

30 BPE predicted, ºC

  20

  10

  Results of FPD data regression

  1.02

  40

  1

  213 points

  35

  0.98 A

  30

  γ

  0.96

  25

  0.94

  0.92

  20

  0.9

  15

  0.88

  10 Water activity coefficient,

  Freezing point depression [°C]

  0.86

  5

  213 points

  0.84

  Results of solubility data regression

• 0.5

  100

  265 points

• 1

  95 Solubility curve

  90

• 1.5

   ] ∞ B

  85

  γ

• 2

  ) / m

  80

  ( T ∞ B

  γ

• 2.5

  B γ

  % wt. sucrose

  75

  ln [

• 3

  70

• 3.5

  65

  265 points

  0.1

20 E

  2

  2

  1.5

  1

  20 30°C

  10

  2

  1.5

  1

  20 25°C

  10

  1.5

  20 33°C

  1

  20 20°C

  10

  2

  1.5

  1

  20 15°C

  10

  

2

  1.5

  10

  1

  20 10°C

  5

  25°C ____ ____ experimental predicted

  10

  2

  1

  10 25°C

  5

  2

  1

  50 80°C

  10

  50 100 60°C

  2

  10

  5

  50 100 40°C

  10

  5

  50 100 20°C

  

10

  5

  20 25°C

  10

  1

  10

  0.2

  20

  4

  2

  40 22°C

  20

  4

  2

  50 18°C

  4

  2

  40 16°C

  4

  2

  2

  40 12°C

  20

  

4

  2

  0.2 30°C

  0.1

  0.2

  0.1

  0.2 20°C

  0.1

  50 100 26°C

  4

  2

  1.5

  1.5

  1

  /RT [-] 5°C

  10

  2

  1.5

  1

  20 0°C

  10

  2

  1

  50 30°C

  50 20°C

  4

  2

  2 18°C

  1

  1

  0.5

  0.2 18°C

  0.1

  

0.2

  0.1

20 Molality

  Prediction of enthalpy of dilution 140

  100

  6 mol/kg (67%)

  90

  40°C

  120

  20°C

  80 /RT [-] /RT [-]

  100 E E

  5 mol/kg (63%) 10°C

  70

  60°C 5°C

  60

  80

  0°C

  50

  4 mol/kg (58%)

  60

  40

  30

  3 mol/kg (51%)

  40

  80°C

  20 1000 x Enthalpy of dilution, H

  1000 x Enthalpy of dilution, H

  2 mol/kg (41%)

  20

  10

  1 mol/kg (26%)

  Regression of 5 heat capacity data sets

  Gucker & Ayres, 1937 Gucker & Ayres, 1937 Banipal et al., 1997

  0.4

  0.3

  0.04 T = 20°C T = 25°C

  T = 25°C

  0.25

  0.3

  0.03

  0.2

  0.2

  0.15

  0.02

  0.1

  /R [-]

  0.1

  0.01 E

  0.05

  2

  4

  6

  2

  4

  6

  0.5

  1

  1.5 DiPaola & Belleau, 1977 Vivien, 1906

  0.05

  0.4 T = 25°C

  T = 15°C

  0.04

  0.3

  0.03

  0.2

0.02 Excess heat capacity, C

  0.1

  0.01

  0.5

  1 0°C

  5°C 10°C

  20°C 40°C 60°C 80°C

  Excess heat capacity, C E /R [-]

  0.2

  0.4

  0.6 1 mol/kg 2 mol/kg 3 mol/kg 4 mol/kg 5 mol/kg 6 mol/kg

  

Excess heat capacity, C

E

  

/R [-]

Prediction of excess heat capacity of solution

• 1
• 0.5
• 1
• 0.8
• 0.6
• 0.4
• 0.2

  Prediction of water activity coefficient

  1.6 ISOTHERMS BETWEEN 0°C and 150°C A

  γ

  1.4

  1.2

  1

  0.8 Water activity coefficient,

  0.6

  Prediction of water activity coefficient

1 A

  0.9 γ

  0.8 80°C 100°C

  0.7 150°C

  60°C

0°C 20°C 40°C

Water activity coefficient,

  0.6

  0.5

  Final water activity coefficient equation

  γ θ θ θ θ θ θ

  1 = -1.9831 b

  4 = -277.88 b

  3 = 1399.9 a

  2 = -1102.3 a

  1 = -861.42 a

  = -268.59 a

  = + + = + + + + a

  A B B B a b x b x x a a a a a a

  2

  4 ln ( ) (1 ) ( ) ln

  3

  2

  1

  2

  2

  1

  2

  2 = 0.92730

  Acknowledgement Acknowledgement Association Andrew van Hook for the Advancement of the Knowledge on Sugar, Reims, France