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

Water activity formulae popular amongst food technologists

Norrish, 1966

ln

γ

_A

=

α

x

_B2

Chen, 1989

1000

1000

(

)

A

A _n

A

M m

M m A

Bm

γ

=

+

+

+

Models used to predict

water activity coefficient

n

Redlich-Kister expansion - empirical

(includes Margules equation)

n

UNIQUAC - phenomenological

(two-fluid theory)

(

)

2 2

ln

_A

Q

x

_B

1

bx

_B

cx

_B

RT

γ =

+

+

Starzak & Peacock, 1997

0 20 40 60 80 100 0.4

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

% mas. sucrose

Water activity coefficient

Catté et al., 1994

0 20 40 60 80 100

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

% mas. sucrose

Water activity coefficient

Starzak & Peacock, 1997

0 10 20 30 40 50 60 70 80 90 100 0.4

0.6 0.8 1 1.2 1.4 1.6 1.8

% mass sucrose

Water activity coefficient on the boiling curve

1193 points

Water activity coefficient (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

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

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)

4 70

Heat capacity

125 2338

Total

10 283

Heat of dilution

34 265

Solubility

13 213

FPD

64 1507

VLE

Literature sources Experimental points

Data type

2

ln

n

k

A k B

k

x

γ

α

=

=

∑

Selection of water activity model

n-suffix Margules equation

Advantages: linear with respect to its coefficients

2 3 0

1 2 3 4 5

( )

k

ln

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

0

Simplified temperature dependence

Advantage: reduced number of parameters

2 2

ln

( )

n

k

A k B

k

a

b

x

γ

θ

₋

=

=

∑

2 3 0

1 2 3 4 5

( )

a

ln

a

θ

a

a

θ

a

θ

a

θ

a

θ

θ

=

+ +

+

+

+

2

ln

n

k

B k A

k

x

γ

β

=

=

∑

Sucrose activity coefficient

ln

_A

ln

_B

A B

A B

d

d

x

x

d x

d x

γ

=

γ

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

+

+

=

−

=

−

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 7

2 2

2 2 2

8 35

5 8

3 3

15 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

β α

= + + + + +

= − − − − −

= + + +

= − − −

= +

= −

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

ln

ln

ln

/

/

A A B

E asym E

pasym

H

RT

C

R

γ

γ

γ

⇒

⇒

⇒

⇒

⇒

VLE data

FPD data

Solubility

Heat of dilution

ln

VLE data

⇒

γ

_A

Modified Raoult’s law

0

(1

)

( )

A

B A

P

x P T

γ =

−

ERH 100

1

A

B

x

γ =

−

ln

Freezing point data

⇒

γ

_A

Solid-liquid equilibrium

( ) ( )

ln 1

2

( )

ln 1 ln(1 )

2

fA m pA m _{A m m}

A

m f

pA m f _{A m} f

A m B

m m

H T C T _{B T T}

RT R T

C T T _{B T} T

B T x

R T T

γ = − _∆ − ∆ + ∆ _ _ − _    _{  } ∆ _{   }   ∆ + _ − ∆ _{  } + _ − −_ −       ( ) ( ) ( )

pA pA m

A m

C T C T

B T T

R R

∆ ∆

= + ∆ −

Assumed

ln

Sucrose solubility data

⇒

γ

%

_B

Solid-liquid equilibrium (asym. convention)

0 0 ( ) ( ) ( ) pB pB B

C T C T

B T T

R R ∆ ∆ = + ∆ − Assumed for sucrose: 0

0 0 0

0 0 0 ( ) ( ) ( ) ln 1 2 ( )

ln 1 ln( )

2

pB

B m dB B m

B

B m

pB _{B m}

B B

m m

C T

T H T B T T T

RT R T T

C T _{T B T T}

B T x

R T T

Heat of dilution data

E asym

H

RT

⇒

Two-step procedure

q fitting each set of isothermal data to

McMillan-Mayer expansion (evaluating coefficients of expansion)

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 1

I

0 2

(J/mol solute)

_k k

(f )

k

(i)

k B

H

h

m

m

n

− −

=

∆

=



−







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

[

]

II

0 2

(J/mol solvent)

(i)

(i)

(a)

(f )

(a)

(a)

(i)

(i)

(i)

A

k k k

A A A A

k

k _A

H

n

n

n

m

n

m

n

m

h

n

=

∆

=

+

−

−

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)

III

0 2

1

(J/g solvent added)

(i)

1000

(f )

(i)

k k

k A

H

h m

w

m

m

=

∆

= −

∑

Heat of dilution data

E asym

H

RT

⇒

Excess enthalpy of solution

from McMillan-Mayer expansion

0 2

(J/mol)

1000

k k

E _k

asym

A

h m

H

m

M

=

=

+

/

Specific heat data

⇒

C

_{p asym}E

R

Apparent molar heat capacity of sucrose:

1 ( )

1000 ( )

B

p pA

c

M

m c m c

m

m

 ₊  ₋

 

 

Φ =

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

[

]

1

2 1

2 1 2

(

)

ln

( )

(

)

( )

1

B m B B

n n n

k _k

k l l A m

k l k

T

b

a T

M b x

a T

a T

Experimental variables

& Margules expansion coefficients

Excess enthalpy of solution

Derived from free Gibbs energy:

(

ln ln

)

E

A A B B

G = RT x γ + x γ

( / )

E E

H G RT

T

RT T

∂ = −

∂

2 ln

E E _B

asym B

H H RT x

T

γ ∞

∂

= +

Experimental variables

& Margules expansion coefficients

Excess enthalpy of solution

2 2

( )

1

E _n

asym _k k

B k

H

b

T a T

x

RT

k

− =

′

=

−

∑

2 3

0

2 3 4 5

0

( )

2

3

where

a

T a T

a

a

a

a

T T

θ

θ

θ

θ

θ

′

= −

+ +

+

+

Experimental variables

& Margules expansion coefficients

Excess heat capacity of solution

By definition:

(

)

E _E

p_asym asym

C H R

R T

∂ =

∂

Hence:

[

]

2

2

2 ( ) ( )

1 E

n

p asym _k k

B k

C _b

T a T Ta T x

R k

− =

′ ′′

= +

−

∑

[

]

2 3

2 3 4 5

2 ( ) ( ) 2 6 12

DATA REGRESSION

Performance index

2 2 2

VLE VLE, , , VLE

2 2 2 2

FPD , , FPD SOL , , SOL

2 2 , , , , , , 2 2 HE CE

( , ) (ln ln )

(ln ln ) (ln ln )

a b _{i A i} exp_{A i}

i

exp exp

A i A i B i B i

i i

E E exp

E E exp

p p

asym i asym i asym i asym i

exp

i i i i

I w w

w w

C C

H H

w w

RT RT R R

0 10 20 30 40 50 60 70 0

10 20 30 40 50 60 70

BPE experimental, ºC

BPE predicted, ºC

Boiling point elevation – predicted vs experimental

0 20 40 60 80 0.82

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02

% wt. sucrose

Water activity coefficient,

γ A

0 20 40 60 80

0 5 10 15 20 25 30 35 40

% wt. sucrose

Freezing point depression [°C]

Results of FPD data regression

213 points

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 50 100 150

60 65 70 75 80 85 90 95 100

Temperature [°C]

% wt. sucrose

Solubility curve

Results of solubility data regression

265 points

0 0.1 0.2 0

0.1 0.2

20°C

0 0.1 0.2 0

0.1 0.2

30°C

0 2 4

0 20 40

12°C

0 2 4

0 20 40

16°C

0 2 4

0 50

18°C

0 2 4

0 20 40

22°C

0 2 4

0 50 100

26°C

0 2 4

0 50

30°C

0 0.1 0.2 0

0.1 0.2

18°C

0 0.5 1 0

1 2

18°C

0 2 4

0 50

20°C

1 1.5 2 0

10 20

0°C

1 1.5 2 0

10 20

1000 x Heat of dilution, H

E /RT [-]

5°C

1 1.5 2 0

10 20

10°C

1 1.5 2 0

10 20

15°C

1 1.5 2 0

10 20

20°C

1 1.5 2 0

10 20

25°C

1 1.5 2 0

10 20

30°C

1 1.5 2 0

10 20

33°C

0 1 2

0 10 20

25°C

0 5 10 0

50 100

20°C

0 5 10

0 50 100

40°C

0 5 10 0

50 100

60°C

0 5 10

0 50

80°C

0 1 2

0 5 10

25°C

0 1 2

0 10 20

Molality

25°C

Regression of 26 heat of dilution data sets

____

0 2 4 6 8 0 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 Enthalpy of dilution, H

E /RT [-]

0 20 40 60 80

0 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 Enthalpy of dilution, H

E /RT [-]

0 2 4 6 0 0.1 0.2 0.3 0.4

Excess heat capacity, C

E /R [-]

T = 20°C

0 2 4 6

0 0.05 0.1 0.15 0.2 0.25 0.3

T = 25°C

0 0.5 1 1.5

0 0.01 0.02 0.03 0.04

T = 25°C

0 0.5 1 1.5

0 0.01 0.02 0.03 0.04 0.05

Molality, mol/kg

T = 25°C

0 2 4 6

0 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

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

Excess heat capacity, 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

Excess heat capacity, C

E /R [-]

70 75 80 85 90 95 100 0.6

0.8 1 1.2 1.4 1.6

% wt. sucrose

Water activity coefficient,

γ A

ISOTHERMS BETWEEN 0°C and 150°C

85 90 95 100 0.5

0.6 0.7 0.8 0.9 1

% wt. sucrose

Water activity coefficient,

γ A

150°C

0°C 20°C 40°C 60°C

80°C

100°C

Final water activity coefficient equation

2 2

1 2

2 0

1 2 3 4

ln

( ) (1

)

( )

ln

A

a

b x

B

b x

B

x

B

a

a

a

a

a

a

γ

θ

θ

θ

θ

θ

θ

=

+

+

=

+ +

+

+

a

0 = -268.59

a

1 = -861.42

a

2 = -1102.3

a

3 = 1399.9

a

4 = -277.88

b

1 = -1.9831

b

Acknowledgement

Acknowledgement

Association Andrew van Hook for