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

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

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= +

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)

( )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)

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.

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

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)

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

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)

470Heat capacity

1252338Total

10283Heat of dilution

34265Solubility

13213FPD

641507VLE

Literature sourcesExperimental pointsData type

EXPERIMENTAL DATABASE

Distribution of experimental data

Distribution of experimental data

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

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 =

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

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

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

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

Expansion coefficients of sucrose activityMatrix representation

=ß Ma

1

1 11

, 1, 2,..., 1n

k kl ll

M k nβ α−

+ +=

= −∑ =

Processing experimental data

lnlnln

/

/

A

A

B

Easym

Epasym

H RT

C R

γγγ

⇒⇒⇒

⇒

⇒

VLE dataFPD dataSolubility

Heat of dilution

Specific heat

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. :

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:

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

∞

∞

∆ ∆ ∆ = − − + − ∆ ∆

+ − ∆ − − −

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

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− −

=

∆ = − ∑

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=

∆=

+ − −∑

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=

∆= − ∑

@

Heat of dilution dataEasymH

RT⇒

Excess enthalpy of solutionfrom McMillan-Mayer expansion

02(J/mol)1000

kk

E kasym

A

h mH

m M==

+

∑

/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

=+

×

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

∞

∞

−−

−= = =

=

+ −−∑∑ ∑

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 ∞∂

= +∂

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

′ = − + + + +

=

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′ ′′+ = + + +

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

= −

+ − + −

+ − + −

∑

∑ ∑

∑ ∑

% %

Results of data regression: correlation diagrams

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

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

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

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

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

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

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

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

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

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

AcknowledgementAcknowledgement

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