Temperature and Pressure Effects
5.4 Arrhenius ’ Law
Arrhenius’ law was derived empirically, but it has proven to be very worthwhile in chemical kinetics. Arrhenius’ law states that
−37.5
−37
−36.5
−36
−35.5
−35
−34.5
−34
2.35 2.4 2.45 2.5 2.55 2.6 2.65
103/T (K–1) ln (khp/kBT)
FIGURE 5.5 Eyring plot depicting the temperature dependence of deamidation reactions of caseinate solutions;
results are shown for three different initial concentrations. Dataset is given in Appendix 5.1, Table A.5.3.
k¼ A exp Ea
RT
(5:12)
where A is a so-called preexponential factor (sometimes called the frequency factor), and Ea is the activation energy. The dimension of A should be the same as that of the rate constant k; it therefore does have units of frequency only in the case of a first-order reaction. It is very instructive to compare Arrhenius’ law, Equation 5.12, with the expression derived from transition state theory Equation 5.11.
Obviously, Earelates to the activation enthalpyDHzand the exact relationship is found as follows. From Equation 5.8 it follows that
ln k¼ ln kB
hP
þ ln T þ ln Kz (c)1m
(5:13)
hence
d ln k
d(1=T)¼ T þd ln Kz=(c)1m
d(1=T) (5:14)
and combining the temperature effect on Kzas displayed in Equation 5.2 gives d ln k
d(1=T)¼ T DHz
R (5:15)
From the Arrhenius Equation 5.12 it follows that d ln k d(1=T)¼ Ea
R (5:16)
−20
−15
−10
−5 0 5 10 15 20
0 50 100
T⬘(⬚C) Energy (kJ mol−1)
T∆S
∆Η
∆G
FIGURE 5.6 Highly schematic drawing of the change in free energy (DG), the change in enthalpy (DH), and the change in entropy (TDS) with temperature for hydrophobic bond formation.
and consequently combining Equation 5.15 with Equation 5.16 results in
DHz¼ Ea RT (5:17)
The observant reader may find this result unexpected because of the definition of enthalpy given in Equation 3.17
H¼ E þ PV (5:18)
From the general gas law it follows that
PDV ¼ RTDn (5:19)
withDn the change in number of moles, so that it would follow that
DHz¼ Eaþ DnRT (5:20)
However, Dn should be seen here as the change in number of moles in going from reactants to the activated state. Hence, for bimolecular reactionsDn ¼ 1, so that Equation 5.17 is indeed seen to be correct. For monomolecular reactions, Dn ¼ 0, so that in that case the activation energy equals the activation enthalpy.
As mentioned, the activation energy can be seen as the energy barrier that molecules need to overcome in order to be able to react. As shown in Figure 5.4, the proportion of molecules able to do that increases with temperature, which qualitatively explains the effect of temperature on rates. The Eyring and Arrhenius equation give a quantitative account. The preexponential factor A is seen to be related to the activation entropyDSz A¼kBT
R exp DSz R
and this comparison makes the preexponential factor A much more comprehensible. The physical meaning of A as such seems to be experienced as somewhat vague, which probably accounts for the fact that the factor A very often is not reported as a result in food science literature. It gives, however, as much useful information as Eadoes. An interpretation of A is that it represents the rate constant at which all molecules have sufficient energy to react (i.e., Ea¼ 0). In principle one could even envisage a negative activation energy, namely if molecules attract each other.
This could be the case for positively and negatively charged reactants.
Another difference between Arrhenius’ and Eyring’s expressions is that the temperature T appears in the preexponential factor in Eyring’s equation (Equation 5.11). This has a consequence in the way results are presented and analyzed. Very often, Arrhenius law is presented as a plot of ln k versus 1=T, which should result in a straight line (if the relationship holds). With Eyring’s relationship, ln(k=T) versus 1=T should be plotted (as is done in Figure 5.5). We would like to remark here that it is not a good idea to derive the activation energy parameters from linear regression of ln k or ln(k=T) versus 1=T because of the weighting of data points through logarithmic transformation; rather, nonlinear regression should be used, as is discussed in Chapter 7 on kinetics and statistics. Another remark in this respect is that the two-step procedure of first deriving rate constants and then regressing them versus temperature results usually in very wide confidence intervals if only three to four temperatures have been studied, as is frequently the case. A better approach is to substitute the rate constant in the appropriate rate equations using Equation 5.11 or Equation 5.12 and perform a nonlinear regression. In this way, all data are used to estimate the activation parameters at once and an estimate of these parameters of much higher precision is obtained. This is called globalfitting and will also be demonstrated in Chapter 7. It probably remains a good idea to present Arrhenius’ or Eyring’s expression in the form of a plot of ln k or ln(k=T) versus 1=T because any deviation of the data from these expressions becomes immediately apparent. In doing so,
however, the values of the parameters estimated by nonlinear regression should be used to construct the plot. Discrepancies between experimental data and Arrhenius’ and Eyring’s relationship are indeed possible and it is the responsibility of the researcher to check this. In the case that they are not applicable (for instance, because an undetected change in mechanism occurs at the higher temperatures), the resulting parameter estimates are worthless. So, thefirst step should always be to check the validity of the laws of Arrhenius=Eyring, and only if they appear to be correct the next step would be the estimation of the activation parameters. Obvious as this may seem, this rule is not always obeyed.
Temperature dependence of complex reactions. It is essential to realize that the concept of transition state theory or Arrhenius law is strictly speaking only valid for elementary reactions. In the case of a complex reaction in which an observed rate constant is actually composed of several elementary ones the meaning of an activation energy determined from such rate constants becomes a bit blurred. Let us consider, for instance, the reaction
Aþ B ! k1
k2
C Cþ D !k3 P
(5:21)
If this is the correct mechanism, the rate of formation of product P, d[P]=dt is d[P]
dt ¼ k3[C] [D] (5:22)
However, if we can determine only component P, we are not in the position to estimate rate constant k3
because then we would need to determine compounds C and D as well (if we can do that we can apply multiresponse modeling to be discussed in Chapter 8). Let us suppose for the moment that we can only observe the formation of component P, and we could try to model it with a pseudo-first-order reaction, for instance. The observed rate constant will then be a composite of the three elementary rate constants k1, k2, and k3. By postulating that the intermediate compound C is following a steady state after the initial start up of the reaction, we can investigate how the observed rate constant is related to the elementary ones. The steady state implies that d[C]=dt 0
d[C]
dt ¼ k1[A] [B] k2[C] k3[C] [D]¼ 0 (5:23) Hence
[C]¼ k1[A] [B]
k2þ k3[D] (5:24)
Substituting Equation 5.24 in Equation 5.22 yields d[P]
dt ¼k1k3[A] [B] [D]
k2þ k3[D] (5:25)
Now we can investigate some possibilities to see how that works out for the rate equation. First, let us assume that k2 k3[D]. It then follows that
d[P]
dt ¼ k1[A] [B] (5:26)
We then have a ‘‘normal’’ second-order rate equation, and the rate of formation of P is completely determined by the rate of disappearance of components A and B. The observed rate constant will correspond to the elementary rate constant k1 and the temperature dependence should be Arrhenius-like. Another possibility could be that k2>> k3[D]. It then follows that
d[P]
and we end up with a third-order rate equation where the rate constant k0is clearly a composite one. The question is now how temperature affects this composite rate constant. If the three elementary rate constants k1, k2, k3each obey Arrhenius law, we can apply Equation 5.16 tofind
d ln k0
In words, the activation energy determined from the observed rate constant is composed of the activation energies for the underlying elementary reactions. This can lead to unexpected results such as that a reaction decreases in rate with increasing temperature. A negative activation energy is not well conceiv-able for an elementary reaction (unless it concerns two oppositely charged reactants), but for a composite reaction this can happen if Ea2> Ea1þ Ea3according to Equation 5.28. The question is, of course, when it will happen that a situation occurs as in the above example that k2>> k3[D]. In order to investigate this, we performed some simulations based on the mechanism depicted in Equation 5.21. The requirement Ea2> Ea1þ Ea3implies that the preexponential factor must be very high to compensate for the higher activation energy (otherwise k2will be of no significance). Table 5.1 shows a possible condition where this would happen.
The reaction was simulated via numerical integration of the differential equations representing the rate equations, using the values shown in Table 5.1 to calculate the rate constants via Arrhenius’ equation.
The initial concentrations of A and B were arbitrarily set at 100 units and that of component D at 150 units. It was indeed found that the intermediate C showed steady-state behavior and with the numerical values chosen it was also true that k2>> k3 [D]. It was then assumed that only compound P was experimentally accessible, so that the simulated experiment consisted of concentration–time profiles for various temperatures; Figure 5.7 shows some results.
These profiles were fitted to a first-order rate equation; a reasonable fit was obtained (but not perfect because the actual mechanism is more complex than afirst-order reaction as shown in Equation 5.21).
The so-derived rate constants were then plotted according to the Arrhenius equation (see Figure 5.8).
It is indeed found that the rate constants decrease with increasing temperature at the higher temper-atures. Admittedly, this may be a rather extreme example because of the requirement that k2>> k3[D].
The purpose is, however, to show that the activation energy determined from observed reaction rate constants may not reflect a single energy barrier. Thus, one should be aware of this when determining activation parameters from observed rate constants that could be composed of several elementary rate constants, certainly in foods where it is difficult to study elementary reactions.
TABLE 5.1 Values Used in the Arrhenius Equation for Simulation of the Complex Reaction in Equation 5.21
Reaction Step 1 Reaction Step 2 Reaction Step 3
A 13 1010dm3mol1s1 13 1026s1 13 1010dm3mol1s1
Ea 80 kJ mol1 150 kJ mol1 60 kJ mol1
Reparameterization. It is possible to reparameterize the Arrhenius or Eyring equation; it is actually desired from a statistical point of view as will be discussed in Chapter 7. A very simple reparameterization is to introduce a reference temperature Tref. The basis for this arises from the application of Equation 5.12 at two temperatures T1 and T2, assuming that the preexponential factor and Ea do not depend on temperature
Time (arbitrary units)
[P] (arbitrary units)
0 10 20 30 40 50 60 70 80 90
0 5 10 15 20 25
T = 353 K T = 413 K
T = 333 K
T = 313 K
T = 293 K
FIGURE 5.7 Simulated concentration profiles for compound P in the reaction mechanism displayed in Equation 5.21 using the numerical values shown in Table 5.1.
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−3
−2
−1 0
0.002 0.0025 0.003 0.0035 0.004 1/T (K−1)
ln k
FIGURE 5.8 Arrhenius plot of the first-order rate constants found from the simulated concentration profiles shown in Figure 5.7 using the numerical values displayed in Table 5.1.
k1¼ A exp Ea
If one arbitrarily chooses a reference temperature, say T2¼ Tref, one can combine these two equations:
k
The actual result of this is that the preexponential factor is replaced by a rate constant at some reference temperature. The reference temperature should preferably be chosen in the middle of the studied temperature regime.
Magnitudes of rate constants. It is perhaps instructive to return briefly to the actual values that rate constants can take. Table 5.2 shows orders of magnitude for rate constants, depending on conditions. The very large effect of activation energy on the rate of a reaction is apparent. In fact, without activation barriers, reactions would be so fast that foods would spoil immediately, and there would be no such thing as food technology, or life for that matter. It is just another way of saying how important kinetics is for processes that are studied by life sciences, including of course food science.
Generally, the temperature effects as laid down in the Arrhenius and Eyring relationships seem tofit quite well to chemical reactions in foods. As mentioned above, it should be realized, however, that observed rate constants are more often than not reflecting more than one elementary reaction, and one has to be careful with interpretation. Some typical values for activation enthalpies=energies and entropies for possible reactions in foods are given in Table 5.3.
Several physical reactions are less temperature-dependent and often diffusion controlled; diffusion-controlled reactions are discussed in Chapter 4. Transition state theory does not apply actually to physical reactions (such as coalescence, aggregation) because there are no molecular rearrangements. However, physical phenomena do usually have an energy barrier (due to, for instance, electrostatic repulsion), which provide stability to colloidal systems. Hence the concept of a kind of activation energy does apply but not with a temperature dependence as occurs for chemical reactions. The effect of temperature will be for the larger part on the rate of encounters. Sometimes, activation energies are reported for physical phenomena such as the temperature dependence of diffusion or viscosity. This seems to be impossible,
TABLE 5.2 Orders of Magnitude for Rate Constants of Bimolecular Reactions in Aqueous Solutions at 258C
Conditions
Order of Magnitude of Rate Constant (dm3mol1s1) No diffusion limit and no barriera 1014
Diffusion limit, no activation energy 1010 No diffusion limit
Activation energy 25 kJ mol1 1010
Activation energy 50 kJ mol1 105
Activation energy 100 kJ mol1 104
a This is in fact the value of the preexponential factor in the Arrhenius equation, corresponding to the hypothetical situation that T! 1.
since there is nothing to activate and there is no reaction. As discussed earlier, the point is that the temperature dependence of, for instance, diffusion apparently obeys Arrhenius’ law in several systems, but the parameter that comes out of it does not have the physical meaning of an activation energy.