Effect of Temperature on Reaction Rate

the following stoichiometry: C₆H₁₂O₆ + 6 O₂ → 6 CO₂ + 6 H₂O. But this does mean that six oxygen molecules simultaneously collide with one glu-cose molecule, form an active intermediate, then break down to form carbon dioxide and water. The reaction is not sixth order with respect to oxygen.

Similarly a reaction with forward and backward processes occurring simultaneously might give an unexpected order (Fig. 3.4). A real effort to develop a kinetic test for a chemical mechanism is an involved process requiring many series of measurements at different concentrations of each of the reagents and catalyst (see several of the books in the bibliography for more details). For most physical changes though the kinetics are used in a more descriptive manner and only very rarely to understand the mechanism.

3.5 Effect of Temperature on Reaction Rate

The rate of a reaction may change with time as the concentration of reagents change, but the rate constant will remain constant as long as the reac-tion mechanism and the condireac-tions (e.g., temper-ature and pH) do not alter. The effect of tempera-ture on the rate constant is often modeled using the empirical Arrhenius equation:

(3.3)

where k is the rate at absolute temperature T, E_a is the activation energy of the reaction R is the gas constant (= 8.314 J·K⁻¹ mol⁻¹) and k₀ the fre-quency factor. By measuring rate as a function of temperature, the constants in Eq. 3.3 can be experimentally determined and used to calculate rates at other temperatures. Some of the strengths and weaknesses of this approach are illustrated in the following examples.

Example: Effect of Temperature on Color Loss Kinetics in Peas

Ryan-Stoneham and Tong (2000) repeated their measurements of chlorophyll deg-radation in peas over a series of different

0exp E^a

k k

RT

 

= − 

3.5 Effect of Temperature on Reaction Rate

k

k

Fig. 3.4 A possible mechanism for the collapse or forma-tion of a foam. In the forward reacforma-tion, two bubbles merge to form a larger one and in the reverse reaction, one large bubble is broken into two small ones

temperatures. The rate of loss increased with temperature but the general shape of the concentration–time curves was similar, suggesting the same first-order mechanism was responsible in each case and a the rate constant was calculated at each tempera-ture. Equation 3.3 was used in its linearized form ( ln k = ln k₀ −E_a/R · 1/T) to calculate the activation energy of the reaction from a plot of the logarithm of the measured rate constants against reciprocal absolute tem-perature (Fig. 3.5; note in the figure, the x-axis is reciprocal temperature so the hot-test sample, 100 °C is to the left). The slope of the line is −E_a/R and can be used to calcu-late the activation energy (= 68.1 kJ mol^{− 1}) and the intercept is ln k₀, the frequency fac-tor (= 20.1, k₀ = 5.4 × 10⁸ min⁻¹).

Knowing E_a and k₀, it is possible to calculate the rate constant at any temperature. This is particu-larly important in shelf-life testing of products.

Often the desired shelf-life may be several years and it is not practical to test formulations over that period before they are brought to market. An alternative is to measure the kinetics of decay at several higher temperatures and use an Arrhenius approach to calculate the rate constant at lower temperatures. So using the data in Fig. 3.5, we could estimate the rate constant at room tem-perature (300 K) as 0.0007 min⁻¹ and use this in the first-order rate equation from Table 3.1 to

estimate the time for, for example 50 % of the chlorophyll to be lost at room temperature (16 h).

This type of approach is fraught with difficul-ties associated with extrapolation. First, there is considerable error associated with extrapolating a best-fit line over a wide temperature range. A small uncertainty in the values of slope and in-tercept for the line will lead to larger errors in the extrapolated ln( k). Second, the reactions im-portant at one temperature may not be relevant at another. As a simple example, fresh-cut fruit will brown by an enzymatic mechanism and baked apple will also go brown via the Maillard reac-tion. Trying to use measurements of browning ki-netics at one temperature to predict the browning rate at another, radically different temperature would be fruitless, as the mechanism responsible has changed. One indication of a changing mech-anism is a change of slope in the Arrhenius plot as illustrated in the following example.

Example: Kinetics of Milk Protein Denatur-ation

α-lactalbumin is an important protein in the whey fraction of milk. It can be dena-tured (i.e., unfolded from its physiological structure—see Chap. 7 for more details) during thermal processing. Anema and McKenna (1996) used gel chromatog-raphy to separate the proteins in heated reconstituted whole milk and measured the concentration of each by staining the gel with a protein-sensitive dye. The rate of α-lactalbumin loss due to denaturation was shown to follow first-order kinetics and the effect of temperature on the rate constant was plotted Fig. 3.6) as an Arrhe-nius relationship but there was no simple linear relation between ln( k) and 1/T. The authors interpreted their data to suggest that there are two mechanisms important for protein denaturation. Both mechanisms have the same measured consequences (i.e., loss of native protein) but the mecha-nism important at high temperatures has a lower activation energy (slope of the best-fit line) than the mechanism important at



Fig. 3.5 Arrhenius plot showing the effect of tem-perature on the rate of chlorophyll degradation in peas. The line shown is a best fit to the experimen-tal ( open) points and is extrapolated to estimate the rate constant at room temperature ( filled point).

(Data from Ryan-Stoneham and Tong (2001))

47

low temperatures. If this type of data were to be used in an accelerated test, then the measurements of high temperature rates constants would overestimate of the rate constant at lower temperatures if the low temperature line were simply extrapolated.

The Arrhenius approach is empirical, but it bears a close relation to our understanding of equilibrium energy distributions described by the Boltzmann distribution (Eq. 1.4). Returning to our thermody-namic basis for kinetics (Fig. 3.1), we could argue that the rate of the forward reagents → products reaction is proportional to the fraction of the re-agent molecules with energies greater than ∆E^‡f. In effect, the Arrhenius relationship is telling us that this fraction, and thus the rate of the reac-tion, increases with temperature according to the Boltzmann distribution. The frequency factor is the rate at infinite temperature where all of the molecules have sufficient energy to clear the bar-rier and the rate is limited only by the rate at which they encounter one another. A more theoretically based approach to the problem was developed by Eyring and others who argued that there was a transition state between reagents and products that exists very briefly: A few molecular oscillations, before breaking down to product or returning to the reagents. The measured rate of the reaction

is related to the Gibbs free energy of forming the transition state from the reagents (∆G^‡) as:

(3.4)

where h is Planck’s constant (= 6.62 × 10⁻³⁴ Js⁻¹).

The Gibbs free energy can be split into enthal-pic (∆H^‡) and entropic (∆S^‡) contributions as

∆G^‡ = ∆H^‡ − T∆S^‡, so:

(3.5)

and the values of ∆H^‡and ∆S^‡ can be estimated from the activation energy and frequency factor of an Arrhenius plot as:

‡

H Ea

∆ ≈ and (3.6)

Note that ∆G^‡, ∆H^‡and ∆S^‡ all refer to the for-mation of the transition state from the reagents and not the overall changes in Gibbs free energy, enthalpy, and entropy of the reaction. They are useful parameters as knowing them allows some assessment of the nature of the unobserved in-termediate state and thus the pathway of the reaction.

Example: Formation of a Transition State During α-lactalbumin Denaturation

Anema and McKenna (1996) used their measurements of the temperature depen-dence of whey protein denaturation rates (see Fig. 3.6) to calculate the free energy, enthalpy, and entropy changes associated with forming the transition state from native α-lactalbumin. For both the high and low temperature mechanisms, the Gibbs free energy to form the transition state was about the same 105–110 kJ·mol⁻¹. However for the low-temperature mechanism, ∆H^‡ = 192 kJ·mol⁻¹ and ∆S^‡ = 0.24 kJ·mol⁻¹ K⁻¹ while for the high temperature mecha-nism, ∆H^‡ = 54.5 kJ·mol⁻¹ and

Fig. 3.6 Arrhenius plot showing the effect of tem-perature on the rate of α-lactalbumin degradation in heated reconstituted whole milk. There are two mechanisms responsible for the degradation, each with a different activation energy (slope of the re-gression lines shown). (Adapted with permission from Anema and Mckenna (1996). Copyright 1996 American Chemical Society).

3.5 Effect of Temperature on Reaction Rate

∆S^‡ = −0.14 kJ·mol⁻¹ K⁻¹. That is to say at low temperatures there was a bigger increase in enthalpy to form the transition state than at high temperatures suggesting more chemical bonds needed to be bro-ken. Similarly, the change in entropy was positive for transition state formation by the low temperature mechanism, suggest-ing the transition was more disordered than the native protein while the entropy change was negative for the high tempera-ture mechanism, suggesting that the transi-tion state was more ordered than the native proteins. They argued that chain unfolding should have a high ∆H^‡ as many bonds are broken and a positive ∆S^‡ as the unfolded product is more disordered. An aggregation reaction on the other hand has a lower ∆H^‡ as fewer bonds are broken and a negative

∆S^‡ the product is more ordered. There-fore, that while the protein is denatured at both low and high temperatures, unfolding represents the rate-limiting step at low tem-peratures and aggregation at high tempera-tures.