Temperature and Pressure Effects
5.3 Transition State Theory
The transition state theory forms a theoretical basis on which more practical equations (such as Arrhenius’ law) can be based. Reaction kinetics in solution has some important aspects that are worth discussing. Reactions in a gas are due to isolated encounters between individual molecules, but this is not possible in solution. The reason is that reactant molecules interact continuously with solvent molecules. If a reaction has taken place, the products will tend to diffuse away, but because of the surrounding solvent molecules, this will take some time, and perhaps in the meantime something can happen to the products or reactants. This is called the ‘‘cage effect.’’ A typical consequence is that encounters last longer in solution (say 1011s) than in the gas phase (typically about 1013s).
1/T (K−1) ln Keq
y = 400.0x–0.9
0.3 0.35 0.4 0.45
0.0031 0.00315 0.0032 0.00325 0.0033 0.00335
FIGURE 5.2 Van’t Hoff plot for the mutarotation of glucose: a-glucose ! b-glucose between 308C and 458C.
The equation shown is the regression equation. Dataset in Appendix 5.1, Table A.5.2.
Transition state theory is well suited for reactions in solutions and is not concerned with rates of encounters (like in gas reactions) but considers thermodynamic and statistical mechanics principles to predict how many combinations of reactants will be present in the so-called transition state. This is a type of high-energy state in which molecules can be present in an unstable but activated condition, in which they will undergo some molecular change to yield products. Consider the reactants A, B that are transformed into products P, Q via a transition state ABz, as follows:
Aþ B Ð ABz! P þ Q (5:3)
Figure 5.3A shows schematically how the potential energy of a pair of molecules changes with the reaction coordinate, i.e., the path along the potential energy curve. The reaction coordinate indicates the state of the molecules in the transition from reactants to products. At this point it is perhaps useful to remark that the reaction coordinate does not refer to a state of the macroscopic system, it only refers to the behavior of a pair of molecules. It therefore does not make sense to plot free energy as a function of the reaction coordinate, as is sometimes done (remember that thermodynamic parameters such as
A + B
P + Q
P+Q A + B
AB‡ AB‡
∆G‡
G
∆G‡
(C) (B)
∆G > 0
∆G < 0 G
Reactants A, B
AB‡: transition state
Products P, Q
Reaction coordinate (A)
Potential energy
FIGURE 5.3 Schematic presentation of the potential energy of reactants A, B, transition state ABz, and products P, Q along the reaction coordinate (A). Standard free energies Gfor various possible states of reactants, products, and intermediates, where the standard free energy differenceDGbetween reactants and products is either negative (B), or positive (C). The activation standard free energiesDGzare always positive. Note that there is no label on the x-axis in the case of the diagrams in B and C.
free energy refer to huge ensembles of molecules); it should be potential energy. Plotting free energy would violate the property that it cannot show a maximum as the reaction progresses, as shown in Figure 3.16. In other words, the reaction coordinate should not be confused with the extent of reaction j discussed in Chapter 3. An alternative way of expressing energy profiles is to plot the standard free energy referring to 1 mol of the states in which molecules or complexes are in, but then without a label on the x-axis so as to avoid the suggestion that free energy is plotted as a function of reaction progress.
Figure 5.3B and C is such alternative ways of expressing energy profiles. Note that DG can be positive: it implies that the equilibrium constant for that particular change is smaller than unity. It can of course also be negative, implying that the equilibrium constant is higher than unity. In contrast, a reaction can never have a positiveDrG, or rather a positive dG=dj; if it does the reaction will run in the reverse way. This is extensively discussed before and the reader is referred to Chapter 3 for more details. The standard free energies for activation DGzare always positive, meaning that the equilib-rium constant that describes the position of the equilibequilib-rium is always much smaller than unity, expressing that the number of molecules in the activated state is very low compared to the molecules in the nonactivated state.
It is, however, possible that some molecules pass over an energy barrier when they have acquired sufficient kinetic energy. This becomes clear, at least qualitatively, from the Maxwell–Boltzmann distri-bution, showing the distribution of the average kinetic energy of molecules (Figure 5.4). An important point is that the shape of this distribution becomes wider at higher temperature, so that more molecules are able to overcome an energy barrier at higher temperature. Coming back to Figure 5.3A, when molecules start to interact, their potential energy increases, and a maximum is reached in the activated, or transition, state. It decreases again when products are formed. The population of molecules in the transition state is very small as compared to the number of reactant or product molecules.
It is assumed that there is some type of equilibrium between reactants and the activated complex (as shown in Equation 5.3), and also between products and activated complex (not shown in Equation 5.3).
However, if the activated complex is formed from reactants it is assumed that the activated complex must move on to form products; similarly, the activated complex formed from products must turn into reactants. At complete equilibrium, the forward rate and the reverse rate are equal, as discussed in the previous chapter, and then the concentration of activated complex formed from reactants equals that of the activated complex formed from products. If we next consider the condition that the equilibrium is
Velocity Number
frequency
T1
T2
FIGURE 5.4 Schematic picture of the Maxwell–Boltzmann distribution of the number frequency of molecules having a certain velocity at two temperatures, T1< T2.
disturbed, e.g., by removing products, the concentration of activated complex molecules formed from products will be reduced but not the concentration of activated complex formed from reactants. Clearly, this is not an equilibrium as usual, and therefore it is called quasiequilibrium. This is one of the postulates of transition state theory and many modifications have been proposed. We will not go into these details, but it is remarkable to note that the basic tenets of the original transition state theory are still valid.
The rate at which the equilibrium depicted in Equation 5.3 is established is assumed to be fast compared with the rate of conversion of ABzto P, Q, so the position of the equilibrium is not perturbed significantly. To be sure: ABzis not an intermediate that can be isolated readily; rather, it represents the configuration of molecules at the moment of collision. The interesting aspect of transition state theory is that it connects kinetics and thermodynamics. On the one hand, the quasiequilibrium between the transition state and the reactant molecules is considered as a thermodynamic equilibrium, so that we can postulate a dimensionless thermodynamic equilibrium constant Kzbased upon activities (cf. Equation 3.100):
As discussed in Chapter 3, aA¼ yA[A]=c, with yA the activity coefficient for the molar scale.
(A thermodynamic constant is dimensionless, as discussed in Chapter 3, so the concentrations are made dimensionless by dividing by the standard concentration c.) On the other hand, the formation of products out of activated complex is treated using the law of mass action (discussed in Chapter 4) and the rate is considered accordingly to be proportional to the concentration of activated complex
r/ [ABz]¼ kz[ABz] (5:5)
Thus, the formation of products is considered to be unimolecular characterized by a rate constant kz. Incidentally, one may wonder whether we should not use the activity of the activated complex rather than the concentration in Equation 5.5, but because of the postulate of the quasiequilibrium between reactants and activated complex it should be concentration indeed. As a reminder, this postulate is that all activated complex molecules, once formed out of reactants, should move to products, and therefore it is the number of molecules that is important, not their activity. Considerations based on statistical mechanics result in
kz¼kBT hP
(5:6)
so that, by combining all this with Equation 5.4, the rate of product formation r is r¼d[P] and T as usual the absolute temperature (K). kzhas dimension of frequency (s1). An interesting feature is the appearance of the activity coefficients of reactants and that of the activated complex in the rate equation.
This is very relevant for reactions in solutions, as we shall see in later chapters, much less so for reactions in the gas phase where the activity coefficients are usually close to 1. In solutions this would be true only for ideal solutions, which are in practice only very diluted solutions
Comparing Equation 5.7 with a‘‘normal’’ rate equation for a bimolecular reaction, such as the one in Equation 4.50, shows that for ideal systems the rate constant kidcan be expressed as
kid¼kBT
hP Kz(c)1 (5:8)
However, for nonideal solutions the relation becomes the observed bimolecular rate constant kobs
kobs¼kBT
As shown in Equation 3.113, the thermodynamic equilibrium constant relates to the standard Gibbs energy change, and consequently the enthalpy and entropy, of activation, as follows:
Kz¼ exp DGz
Combining Equation 5.8 with Equation 5.10 gives then as a general equation
kid¼kBT
A similar equation would be found for the nonideal case by combining Equation 5.9 with Equation 5.10.
The factor (c)1Dmis necessary to obtain the right units for rate constants. cis the concentration in the standard state, chosen as 1 mol dm3as discussed in Chapter 3, andDm is the molecularity (in this case (D)m ¼ 2, see Equation 5.3). The above derivation is based on a bimolecular reaction mechanism. For monomolecular reactions, the same reasoning can be followed, by assuming that the activated complex is formed because of frequent collisions with solvent molecules, related to the above mentioned cage effect.
This is called the Lindemann postulate.
Equation 5.11 has the correct units for a rate constant of any order because of the factor (c)1Dm, the concentration in the standard state to which the thermodynamic parameters are referred.
Coming back again to the discussion on thermodynamics in Chapter 3,DGz should be seen as the standard Gibbs free energy change which would occur if 1 mol of reactants is completely converted into 1 mol of transition state (at the specified temperature and standard state). In other words, it is the change in the value of free energy between these two states at the extreme ends of a possible process; it does not vary with the progress of the reaction as the free energy of a reaction does (which does decrease!).
Equation 5.11 is sometimes referred to as the Eyring equation, after one of the developers of the transition state theory. The importance of this equation is that it relates the effect of temperature on the reaction rate constant to fundamental terms of enthalpy and entropy changes. If, for instance, a high enthalpy of activation exists, this would make the reaction quite slow at moderate temperatures, but this may be compensated by a large activation entropy, whereby the reaction can still proceed at a measurable rate. A striking example of such a phenomenon is the unfolding of proteins, to be discussed in more detail in Chapter 10. This indeed requires a high activation enthalpy because of the high number of bonds being broken simultaneously upon unfolding but, at the same time, the entropy of the unfolded chain increases enormously. In other words, high activation enthalpies and entropies are characteristic for protein unfolding. On the other hand, bimolecular reactions usually have a negative activation entropy (entropy of the two reactants is lost because of bond rearrangements and bond formation). The energy released and needed in breaking and forming bonds results usually in a moderate activation enthalpy. Monomolecular reactions are usually characterized by a moderate activation entropy (either slightly negative or positive, depending on intramolecular changes, the exception being protein unfolding) and an activation enthalpy depending on the type of mechanism.
As it happens, most chemical reactions, though not all, will have a moderate activation entropy so that differences in rate constants between chemical reactions are for the most part determined by differences in activation enthalpy.
As an example, reaction rate constants were determined for the heat-induced deamidation in aqueous solutions of caseinate in the range between 1108C and 1458C. The resulting logarithmic plot (according to the natural logarithm of Equation 5.11) is shown in Figure 5.5.
Usually, the activation parameters are estimated via linear regression of the logarithmic plot as given in Figure 5.5. For statistical reasons to be discussed in Chapter 7, it is better to use nonlinear regression. An estimate of the activation enthalpy from these data via nonlinear regression isDHz¼ 92.0 13.6 kJ mol1 and DSz¼ 69.9 13.8 J mol1 K1 (95% confidence interval). The negative activation entropy is consistent with a bimolecular reaction of hydrolysis of amides.
The activation enthalpy and entropy are usually assumed to be independent of temperature, which in general is probably not true, but for the heat treatment of foods the temperature range is mostly not so large on the absolute temperature scale, so the approximation may then hold. A notable exception is, again, protein unfolding in an aqueous environment, because interaction with water comes into play.
Upon unfolding, hydrophobic groups are exposed and cause increased structuring of water. There is thus also a contribution of enthalpy and entropy changes of the solvent water that may oppose the positive enthalpy and entropy for protein unfolding. The difference in heat capacity between unfolded and folded (native) proteins is quite large, resulting in temperature dependency of (activation) enthalpy and entropy.
In general, when it comes to hydrophobic bonds, their temperature dependence is quite intricate. A very schematic impression is given in Figure 5.6, illustrating that hydrophobic bonds increase with tempera-ture (i.e., free energy becomes more negative), especially between 08C and 608C, but also at higher temperature.