Short Communication Frequency Factor-Activation ... - Science Direct

Journal of Food Engineering 17 ( 1992) 143- 15 1

Short Communication Frequency Factor-Activation Energy Compensation Relations for Viscosity of the Fruit Juices Mustafa Food Engineering

Ozilgen

Department,

& Levent

Bayindirli

Middle East Technical University, 065 3 1 Ankara, Turkey

(Received 23 March 1991; revised version received 22 July 199 1; accepted 3 1 July 199 1)

ABSTRACT Activation energy-frequency factor compensation was studied for the viscosity of sour cherry, apple and grape juices. According to Eyrings theory, which relates liquid viscosity to a chemical reaction, liquid molecules are activated to escape from a cage formed by their nearest neighbours in order to slide over each other. Activation entropies of this process were positive. This result implied that the activated complex had higher vibrational and rotationalfreedom than the other liquid molecules and verified Eyring’s viscosity model for fruit juices. Sour cherry juice had an apparent isokinetic temperature (460 K) showing that activation occurred via formation of structurally similar complexes at all concentrations,

NOTATION fl

E AG;: AG* h AH” k

N* r

Distance travelled by jump (m) Activation energy (J/mol) Activation Gibbs free energy at rest (J/mol) Activation Gibbs free energy when flowing (J/mol) Planck constant (J/s) Activation enthalpy (J/mol) Reaction rate constant ( 1 /s) Avagadro constant ( 1/mol) Correlation coefficient 143

Journal of Food Engineering 0260~8774/92/$05.00 Publishers Ltd. England. Printed in Great Britain

- 0

1992

Elsevier

Science

144

R AS’ t T T, ux hl

M. &i&en,

L. Bayindirli

Gas constant (J/mol K) Activation entropy (J/K mol) Time (s) Temperature (K or “C) Isokinetic temperature (K) Velocity in x direction (m/s) Molecular volume (m’/mol) Constant in eqn ( 11) (mol s/m”) Constant in eqn ( 11) (Pas) Constantineqn(12)(1/K) Boltzman constant (J/K) Distance between the layers of molecules (m) Viscosity (Pas) Frequency factor (Pas) Shear stress (N/m’) Constant in eqn ( 12) (J/K mol)

INTRODUCTION According to Eyring’s theory, when no shear stress is applied, molecules of a pure liquid vibrate in a ‘cage’ formed by its nearest neighbours. This ‘cage’ represents an energy barrier of AG(T/N,. The liquids at rest continuously undergo re-arrangements in which the liquid molecule may jump from the ‘cage’ into an adjoining ‘hole’. The frequency of these activated jumps may be calculated by anology with the rate constants of first order reactions (Glasstone et al., 1941; Bird et al., 1960):

With shear effects, the energy barrier is distorted by the work done: -AG”=

-AG:

f

(2)

The frequency of forward jumps, i.e. moving in the direction of the applied shear stress (positive sign), and that of backward jumps (minus sign) may be calculated after substituting - A G ’ for - A Gc in eqn ( 1). The net velocity of the activated jumps is the difference between the

Frequency factor-activation energy compensation relations

145

forward and backward jump frequencies multiplied by the average jump distance. The velocity gradient is then calculated by dividing the net velocity by the thickness of the molecular layers:

(3) When ( atyx V,/( 21RT) is small, eqn (3) may be rearranged as (Glasstone et al., 1941; Bird et ai., 1960): (4) Equation (3) becomes the same as Newton’s law of viscosity when:

(5) The ratio I. /a is taken as unity in most applications Gibbs free energy of activation at rest is: AG;: =AH’ - TAS x

(Bird et al., 1960). (6)

After substituting eqn (6) into eqn (5) and taking A/a unity: p=Fexp($$)

exp( -$)

(7)

Temperature effects on viscosity are generally simulated with the Arrhenius expression (Saravacos, 1970; Rao et al., 1984; Constenla et al., 1989): E ~~

P = 41 exp

i

Comparing

1

eqns (7) and (8) suggests the relationships: E=AH++RT

and

N,h “=

2.72 V, exp

(1Oj

Generally linear relationships are observed between the activation energy-frequency factor, and the activation enthalpy-activation entropy of the family of related chemical reactions. A set of chemical reactions

146

M. &ilgen,

L. Bayindirli

occurring under slightly different experimental conditions, i.e. with media of different solids contents, may be regarded as a family of related reactions. The compensation relationships suggest that the activation energy and the frequency factor or the activation enthalpy and the activation entropy are not independent of one another and any change in the activation energy is compensated by changes in the frequency factor. In other words, any change in the activation entropy is compensated by changes in the activation enthalpy. There are numerous reports verifying these compensation relations in various fields of chemistry and biology (Barnes et al., 1969; Lumry & Rajender, 1970; Elizodo & Labuza, 1974; Uden & Vidal-Leiria, 1976; Labuza, 1980;; Aguerre et al., 1986; Rhim et al., 1990). Similar empirical relations may also be formulated for the viscosity of the fruit juices: lny,,=aE+/?

(11)

and AS” = 6AHX + I#

(12)

The relationships between E and AH’, and p(, and AS’ were given in eqns (9) and ( 10). Therefore verifying eqn (11) will imply that eqn ( 12) is also true. Viscosity is one of the major physical properties of fruit juices. It determines the agitation and pumping power requirements of processes as well as product characteristics. The compensation relations may help in process design.

MATERIALS

AND METHODS

Clear natural sour cherry juice, a commercially-available beverage, was purchased in the local supermarket. The total solids content of the juice was adjusted by vacuum evaporation at 40°C to 13.8, 19.2, 22.8, 26.1 and 29.9”Brix. The viscosities were measured at least twice with an Oswald-Fenske viscometer in the range of 20°C to 80°C. Almost the same values were obtained in the replicate experiments. Experimental data of Constenla et al. (1989) (clarified apple juice in the range 20-80°C and 12-685”Brix) and Rao et al. (1984) (apple juice in the range 41-O-73.5”Brix and grape juice in the range of 41-68*3”Brix, in the temperature range - 15-40°C) were also used in the analysis. These fruit juices were Newtonian fluids (Rao et al., 1984; Constenla et al., 1989).

Frequency factor-activation

energy compensation relations

147

Equation (8) was linearized to give: (13) Values of parameters In ,M()and E were evaluated from eqn (13) (Fig. 1). AS’ was calculated from eqn (lo), using previously determined values of ,M,,.Values of p(,, E and AS’ are given in Table 1.

RESULTS

AND DISCUSSION

It can be seen from Table 1 that the activation energies generally tend to increase with concentration (“Brix) of the fruit juices. Dissolved solids may influence viscosity by bond formation between the dissolved solid and liquid molecules. Activation energies from the data of Rao et al. ( 1984) were greater than the activation energies calculated with the other data. However, they obtained some of their data at very low temperatures and reported that when fruit juices were close to the frozen state there were larger ‘holes’ or more space and hence more activation energy would be required for flow. The compensation relationship between the parameters p(, and E is given in eqn ( 11). Values of parameters a and B were determined from Fig. 2. This figure clearly shows that the frequency factor-activation energy compensation exists for the viscosity of the fruit juices. Compensation relations may be used for obtaining information about the mechanism of the reaction described by Eyring’s theory. In a chemical reaction reactants first make an activated complex which is converted into the products. A related family of activated jump reactions occurring in the fruit juices with different “Brix may have the same rate at a certain temperature. This is referred to as the isokinetic temperature. While studying the kinetics of virus inactivation Barnes et al. (1969) concluded that having an isokinetic temperature indicates that the related family of reactions occurs through the same mechanism. The term ‘same mechanism’ may imply that structurally similar activated complexes are involved in the reactions. Equation (13) may be stated at the isokinetic temperature as: lnpu,,= --

E RT,

+lnpc

(14)

After comparing eqns ( 11) and ( 14), the coefficients of eqn ( 11) may be expressed as a = - l/RTc and /3=ln ,uc. From the parameter a, the

148

M. c)zilgen, L. Bayindirli TABLE 1

Variation of the Numerical Values of the Kinetic Parameters with the Experimental Conditions

Fruit juice

AS%

PO

(J/m01 K)

(Ws) 13*8”B, sour cherry 19*2”B, sour cherry 22W’B, sour cherry 261”B, sour cherry 29*9”B, sour cherry 12”B, apple” 2O”B, apple” 3O”B, apple” 40”B, apple” 50”B, apple” 60”B, apple” 6%5”B, apple” 41W’B, apple* 73WB, apple” 695”B, apple* 73.5“B, apple’ 45.1”B, apple’ 50_4“B, apple’ 55*2”B, appleb 60.1”B, apple* 64*9”B, appleh 68.3”B, apple” 4 1“B, grape* 74*5”B, grape’ 73.5”B, grape’ 43.1”B, grape” 49+2”B, grapeb 54.0”B, grape’ 59.2”B, grape’ 64*5”B, grape’ 68*3”B, grape”

3.31 x 10-h 4.42 x 1.73 x 1.50 x 1.50 x 1.32 x 1.56 x 1.18 x 7.60x 5.92 x 4.22 x 6.21 x 8.09 x 1.61 x 7.95 x 6.55 x 3.38 x 1.18 x 2.70 x 3.91 x 7.93 x 1.15 x 3.94 x 1.52 x 9.57 x 8.17 x 1.07 x 9.16 x 1.25 x 1.34 x 6.06 x

lo-” 10-h 10-h lo-” 10-h lo-” lo-h 1O-7 1O-7 lo-” lo- “’ lo-’ lo-l4 lo- I3 lo- I5 1O-7 10-7 1O-y lo- 1” lo-‘? lo- ‘? 10-x lo- IJ lo- I5 10-x 10-x lo- I” lo- “’ 10-l” lo-‘?

1.48 1.46 1.75 1.82 1.85 1.71 1.73 1.91 2.17 2.42 3.38 4.81 2.80 7.73 6.60 7.98 2.51 2.88 3.93 4.46 5.68 6.40 3.05 7.94 7.90 2.93 3.55 4.30 4.93 5.14 6.06

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

lo4 lo4 10“ 10“ lo4 104 104 10“ 10” 10J lo4 104 lo4 104 104 104 10” 10” 104 lo4 lo4 10” 1OJ 10” IOJ lo4 104 1OJ 1OJ 104 104

7.5 5.1 12.9 14.1 14.1 15.2 13.7 16.1 19.7 21.8 43% 78.8 38.3 166.5 134.2 174.1 26.5 35.2 66.6 82.7 115.1 131.1 44,3 167.1 171.0 38.3 55.1 75.6 92.1 91.6 117.3

“Source: Constenla et al. ( 1989). bSource: Rao et al. (1984).

numerical value of T, was calculated as 460 K for the sour cherry juice and 479 K for apple juice from the data of Constenla et al. (1989). The isokinetic temperature was calculated as 377 K and 387 K for apple and grape juice, respectively, from the data of Rao et al. (1984). If the other

Frequency factor-activation

-5

1

-62

;; g

-6.6

energy compensation relations

149

6

I c

-70

-7

-70

4

2.6

3.0

32

(I/T)

34

IO3 (K-‘I

Fig. 1. Variation of the sour cherry viscosity with temperature and soluble solids content: (0) 29.9”Brix, (+) 26.1”Brix, (0) 22.8”Brix, ( X ) 19.2”Brix, (A ) 13.8”Brix.

-14 -12.3

.

-12.5

-18

.

-12.7

-22

-12.9

I ;;

-26 cl

-13.3 -13.1 -13.5

Ed I.4

-3c

. .

15

I6

1.7

1.8

1.9

-34

a”

E

1o-4 ( Joules

/ mole ) -

Fig. 2. Variation of parameter In p. with activation energy: (a) sour cherry juice: equation of the line: In p’. = - 8.64-261 x IO-‘E (r= - 0.98). (b) Apple juice: original data were taken from Constenla et al. ( 1989); equation of the line: In p,, = - 8.80-2.5 1 X 10 -‘E (r= 099). (c) Apple juice: original data were taken from Rao et al. (1984); equation of the line: In p,, = - 7.09-3.19 x 10m4E (r= 0.96). (d) Grape juice: original data were taken fromRaoetaI.(1984);equationoftheline:In~,,= -7.31-3.10x10-‘E(r=0.99).

150

M. bilgen, L. Bayindirli

physical properties of the fruit juices had not changed, at the isokinetic temperature all the related set of fruit juices would have the same viscosity. Unfortunately, fruit juices boil before reaching the isokinetic temperature. The isokinetic temperatures were substituted in eqn (13) with appropriate values of parameters p,, and E, then numerical values of parameter ,u, were calculated. These numerical values were compared with the ones obtained from the relation /!I= In ,+ The agreement was better than 13% with sour cherry juice. Differences of up to 35% and 42% were calculated for the apple juice data of Constenla et al. (1989) and Rao et al. ( 1984), respectively, while a 76% difference was observed with the grape juice data of Rao et al. ( 1984). Small differences (such as 13%) may be attributed to the experimental errors associated with the measurements and show that the activation process involved in sour cherry juice viscosity may be actually occurring through the same mechanism. The larger deviations observed with the apple and grape juice may not be considered merely as experimental errors and imply that different activation mechanisms are involved in apple or grape juice viscosity. The data of Constenla et al. ( 1989) and Rao et al. ( 1984) were obtained over a much larger concentration range than those of the sour cherry juice experiments. The major cause of having structurally different activated complexes might be the very large concentration range of these experiments. Positive values of AS ‘, as given in Table 1, indicate that the activated complex has higher vibrational and rotational freedom than the other liquid molecules. Positive values of AS’ confirm that Eyring’s theory is valid for the viscosity of fruit juices.

ACKNOWLEDGEMENTS This study was supported by the applied research fund No. 89-03-14-o 1 of the Middle East Technical University. Mr Giiven Kaqaman helped to obtain the experimental data.

REFERENCES Aguerre, R. J., Suarez, C. & Viollaz, P. E. (1986). Enthalpy entropy compensation in sorption phenomena: Application to the prediction of the effect of temperature on food isotherms. J. Food Sci., 5 1, 1547-9. Barnes, R., Vogel, H. & Gordon, I. (1969). Temperature of compensation: Significance for virus inactivation. Proc. Nat. Acad. Sci. (Wash.), 62,263-70.

Frequency factor-activation energy compensation relations

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Bird, R. B., Stewart, W. E. & Lightfoot, E. N. ( 1960). Transport Phenomena. John Wiley and Sons, New York, pp. 26-8. Constenla, D. T., Lozano, J. E. & Crap&e, G. H. (1989). Thermophysical properties of clarified apple juice as a function of concentration and temperature. J. Food Sci., 54,663-g. Elizodo, H. & Labuza, T. P. (1974). Death kinetics of yeast in spray drying. Biotechnology and Bioengineering, 16, 1245-9. Glasstone, S., Laidler, K. J. & Eyring, H. ( 1941). Theory of Rate Processes, McGraw-Hill, New York, pp. 477-5 16. Labuza, T. P. (1980). Enthalpy/entropy compensation in food reactions. Food Technol., 34,67-77.

Lumry, R. & Rajender, S. (1970). Enthalpy-entropy compensation phenomena in water solutions of proteins and small molecules: A ubiquitous property of water. Biopolymers, 9, 1125-6. Rao, M. A., Cooley, H. J. & Vitali, A. A. (1984). Flow properties of concentrated juices at low temperatures. Food Technol., 38 (3), 113- 19. Rhim, J. W., Jones, V. A. & Swartzel, K. R. (1990). Kinetic compensation effect in heat denaturation of whey protein. J. Food Sci., 55,589-592. Saravacos, G. D. (1970). Effect of temperature on viscosity of fruit juice and purees. J. FoodSci., 35,122-5. Uden, N. V. & Vidal-Leiria, M. M. (1976). Thermodynamic compensation in microbial thermal death studies with yeasts. Archives of Microbiology, 108, 293-8.