¼ k1(T)(c1 c0) (5:62)
Again, if we know how temperature T changes with time t, as well as the dependence of k1on T, the concentration change can be calculated for any temperature profile using numerical integration.
So, in conclusion, variable temperature kinetics can be done for every model, provided we know the time–temperature profile and the temperature dependence of the parameters, either via Arrhenius or Eyring relations, or via empirical relations such as Equation 5.31. This is especially very useful for accelerated shelf life testing, and to study the effect of varying temperature in a food chain during distribution and storage. The same approach can be applied to variable temperature kinetics for microbial kinetics; in such cases the Arrhenius=Eyring equation does not make sense because the processes studied in microbial growth and death are not single reaction events like a chemical reaction is. This is further discussed in Chapters 12 and 13.
5.10 Effect of Pressure
High-pressure treatment of foods is an emerging technology. It implies that foods are brought under high pressure (up to 1000 MPa), and as a result bacteria can be killed and some enzymes inactivated. Because this can be done at relatively low temperature, there is little or no heat damage and the preserved foods retain their freshness. This research area is strongly in development, and here we state only some basic principles.
From a thermodynamic point of view, the effect of pressure can be dealt with as follows. First, we start with considering a reaction in the gaseous phase, and we state the general reaction as in Equation 3.1
nAAþ nBB !nPPþ nQQ
The equilibrium constant KPexpressed via partial pressures is (cf. Equation 3.102)
KP¼
Pois the standard state pressure of 1 bar. As a reminder, based on Equation 3.113 the following relation holds
DrG¼ RT ln KP (5:64)
This equation shows that KPrefers to standard states at Po¼ 1 bar, so it is by definition independent of pressure. This does not mean, however, that pressure has no effect on equilibria. This should become clear from the following reasoning. We can express the equilibrium constant also in terms of mole fractions, KX
KX¼(XP)nP(XQ)nQ
(XA)nA(XB)nB (5:65)
For a perfect gas the following relation holds between partial pressure Piand total pressure P:
Pi Po¼ Xi
P
Po (5:66)
Combining Equation 5.65 with Equation 5.66 gives
Combining Equation 5.63 with Equation 5.67 results in (see also Equation 3.104)
KX ¼ KP
By increasing the pressure, KXmust change by a factor (P=Po)Dnin order to keep KPconstant. So, the result of this exercise is to show that the mole fractions of the components in the equilibrium mixture do depend on the total pressure P, despite the fact that KPdoes not. Just another way of expressing this is by differentiating Equation 5.68 with respect to pressure
@ ln KX
Using the perfect gas law PDV ¼ DnRT, and using the fact that KPis independent of pressure so that the first term on the right-hand side of Equation 5.67 equals zero, Equation 5.70 can also be written as
@ ln KX
This equation is not only valid for equilibria in the gas phase but also for equilibria in solution.DV is the volume change accompanying 1 mol of reaction with all substances in the standard state. This is actually a statement of Le Châtelier’s principle: equilibrium phenomena that are subject to a decrease in volume upon reaction will move from left to right upon an increase in pressure and vice versa. IfDV can be considered pressure independent, it can be estimated from the dependence of KXon pressure P according to Equation 5.71, analogous to the van’t Hoff equation for temperature effects (Equation 5.2).
High-pressure treatment thus favors reactions that result in a volume decrease. It affects mainly noncovalent bonds, which implies that low-molecular weight compounds are not really sensitive to pressure but high-molecular weight components (biopolymers) are because they are stabilized by noncovalent bonds. Also dissociation of weak acids in water is enhanced by pressure. Consequently, pressurization of acetate, citrate, and phosphate buffers is accompanied by a large negative volume change upon dissociation and this leads to a significant acidification of such buffer solutions. Further-more, crystallization phenomena are also affected by pressure. Especially hydrophobic bonds and ionic bonds are very sensitive to pressure, hydrogen bonds less so. This means that the quaternary and tertiary structure of proteins will be disrupted but not so much the secondary structure when subject to high pressure. In other words, proteins (and hence enzymes) will denature, which could also be the basis for microbial inactivation as a result of high-pressure treatment. Also, aggregation and gelation of proteins may occur as a result of pressure-induced denaturation.
When applying pressure, its transmission is uniform and virtually instantaneous, independent of vessel size and geometry (the isostatic principle), unless gas is present because then the gas can be compressed.
Furthermore, an increase in pressure (at constant T) results in an increase in the degree of ordering of molecules (the microscopic ordering principle).
For this book, we are mainly interested in describing the effect of high pressure on the resulting kinetics.
By analogy with the effect of temperature, the activation volumeDV# (the difference in molar volume of the activated complex and the reactant) and the preexponential factor APare introduced
k¼ APexp DV#P RT
!
(5:72)
Analogous to the Arrhenius equation, a reference rate constant at a reference pressure can be chosen to eliminate the preexponential factor
k¼ Krefexp DV#
R (P Pref)
!
(5:73)
The termsDV#, DH#, DS#, DG# are related
DV#
@T
!
P
¼ @DS#
@P
!
@DH#
T ¼ DV# T @DV#
@T
!
P
(5:74)
Currently, most studies are done on inactivation of microorganisms via high pressure. We will come back to this in Chapter 13. Furthermore, the effect of high pressure on proteins and enzymes has been studied extensively. As an example, Figure 5.25 shows pressure-induced inactivation of an enzyme,
0 0.2 0.4 0.6 0.8 1
0 50 100 150
Time (min)
[A]/[A]0
FIGURE 5.25 Inactivation of polygalacturonase in tomato juice at 208C at 400 MPa (^) and 450 MPa (D) according to afirst-order model (solid lines). Dataset in Appendix 5.1, Table A.5.9.
polygalacturonase (PG) in tomato juice. Figure 5.26 shows how Equation 5.73 can be used to estimate the activation volumeDV# for the same example.
Since both pressure and temperature have an effect on protein conformation, it has also been studied how the two conditions combine. A mathematical model that takes this into account is the following, based on thermodynamic and kinetic considerations:
ln k¼ ln krefDV#0
RT (P Pref)þDS#0
RT (T Tref)Da#
RT (P Pref)(T Tref)
DB#
2RT(P Pref)2þDCp
RT T ln T Tref 1
þ Tref
(5:75)
In this equationDa# is the thermal expansivity factor (cm3mol1K1),DB# the compressibility factor (cm6J1mol1),DV#0 (cm3mol1), andDS#0 (J mol1K1) the volume and entropy change between native and denatured states at Tref and Pref, respectively, DCp the heat capacity at constant pressure (J mol1K1). These parameters need to be estimated from experimental data. A possible problem is the large number of parameters to be estimated and therefore a simplified version of Equation 5.75 has been proposed (by simply omittingDB# and DCp)
ln k¼ ln krefDV#0
RT (P Pref)þDS#0
RT (T Tref)Da#
RT (P Pref)(T Tref) (5:76) With this model it can be predicted what the combined effects of temperature and pressure are.
Figure 5.27 gives a so-called isorate contour plot, again for the example of inactivation of polygalactur-onase. It shows the temperature–pressure combinations that lead to the same extent of inactivation of the enzyme.
−9
−8
−7
−6
−5
300 400 500 600
P (MPa)
ln k
FIGURE 5.26 Example of the use of (the logarithm of) Equation 5.73 applied to inactivation of polygalacturonase in tomato juice at 158C and Pref¼ 525 MPa. The activation volume is estimated at 43.5 cm3mol1, and krefat 0.002 s1. Dataset in Appendix 5.1, Table A.5.10.