So far the energetic span model takes concentrations into account only for calculating the TOF.3, 4 Herein, we extend the model in the following directions:
to include the reactant and product concentrations in the degree of TOF control of the energies and
to determine a degree of TOF control of the reactant and product concentrations, i.e., a quantitative prediction of the influence of the various concentrations on the TOF.
The degrees of TOF control of the reactant and product concentrations can provide explicit predictions regarding the most favorable conditions to be applied in the experiment, e.g., if it helps using a reactant in high concentration, or if a product kinetically inhibits the reaction and should, therefore, be continuously extracted from the reaction mixture.
Modified Expression for the Degree of TOF Control of the Energies
Until now, the degrees of TOF control XTOF [eqs. (3a) and (3b)] were calculated using the Mi,j values as defined by the solely energy based representation of eq. (1), which means the XTOF values did not change when the concentrations of products and reactants were included in the prediction of the TOF. To include the reactant and product concentrations in the degree of TOF control of the energies, we redefine the degree of TOF control to be calculated from the M′i,j terms as defined in eq. (6), where concentrations are taken into account. This leads directly to eqs. (8a) and (8b):
The concentrations will influence the degree of TOF control significantly, when either several intermediates or transition states are important (i.e., more than one intermediate or more than one TS have an XTOF > 0.00), or when extreme concentrations are involved, e.g., when the solvent acts as reactant or only traces of a reactant are left. Applying this new definition to the hydroamination leads to the XTOF values shown in Table 4. Here, again a concentration of 0.1 mol L−1 was chosen for ammonia, ethylene, and ethylamine.
Table 4. XTOF Values for the Hydroamination Taking Concentrations into Account
Comparing the XTOF values calculated with concentration effect (Table 4) to those calculated without concentration effect (Table 3), one can see that under the chosen conditions the influence of I5c is lower than estimated before, whereas the influence of I2b and I5b is increased. Additionally, I1 now has a significant influence, which is even greater than that of I2b and I5b. From this example, we can see the importance of taking concentrations into account also for the XTOF values.
Degree of TOF Control of the Concentrations
Equations (8a) and (8b) reflect the influence of each energy state on the TOF at given values for the concentrations of reactants and products. Yet, it is also desirable to understand the influence of the concentrations themselves on the TOF, as concentrations are easily adjustable parameters within an experimental setup. Using again Campbell's concept,10, 11 but this time applied to the concentrations, we can express the degree of TOF control of the concentrations as shown in eq 9.
Replacing herein Δ′ and M′, according to eq. (6), we have to distinguish between reactant and product concentrations, as derived in the final eqs. (10a) and (10b):
The terms and are all the M′i,j terms that contain the kth reactant or product, respectively. These equations can be neatly interpreted within the picture of the thermodynamic driving force Δ′ and the kinetic resistance M′. The first fraction of equation (10a) is the change in the catalytic potential caused by a change in the reactant concentration. The numerator of this term is always positive, whereas the denominator is the catalytic potential itself, which has to be positive as well so that the reaction can proceed forward. Furthermore, the denominator cannot be greater than the numerator, so this term is always ≥1. The second fraction of eq. (10a) accounts for the influence of the reactant concentration on the kinetic resistance for the reaction flux, and it is always <1. Therefore, the difference of the two terms results in a positive XTOF,. This means that raising a reactant concentration always raises the TOF, which is consistent with expectations. In eq. (10b), both parts are negatively signed, so XTOF, is always negative, meaning that raising a product concentration lowers the TOF.
Applying this concept to the hydramination example leads to the interesting case of NH3 and C2H4 acting not only as reactants for the catalytic cycle but also as reactants and products during the formation and decomposition of the spectator species I2b and I5c. For instance, while going from I2b to I2 ammonia is released as a “product”, Table 5 shows the degrees of TOF control of the reactants and products at the different steps.
The product C2H5NH2 has an XTOF value of 0.00, so the reaction does not suffer from product inhibition. As a rule of thumb, one can say that only those reactants and products that enter or leave the reaction between the TDI and the TDTS are influential.3 However, it gets more complicated when more than one intermediate or more than one transition state have a significant XTOF.
As NH3 and C2H4 act both as reactant and product, their overall influence can only be seen from the sum of their XTOF values. Summing up all the XTOF values for NH3 gives 0.88, whereas C2H4 gives −0.35. This means that raising the concentration of C2H4 would actually lower the TOF. This is consistent with the chemical intuition, as an increased C2H4 concentration would promote the formation of the spectator species I5c and, therefore, the reaction is slowed down. As Table 6 shows, raising the concentration of C2H4 to 0.2 mol L−1 indeed leads to a lowering of the TOF to 0.70 h−1 (entry B). Furthermore, it can be seen that the degrees of TOF control themselves also change with the concentration. A further increase of the C2H4 concentration would lead to an even stronger decrease of the TOF as its degree of TOF control has changed from −0.35 to −0.75. On the other hand, an increase of the NH3 concentration increases the TOF as NH3 has a positive XTOF. Raising its concentration to 0.2 mol L−1 leads to a TOF of 1.9 h−1 (entry C). One can also see that an increased NH3 concentration lowers the inhibition effect of C2H4, as the XTOF value of C2H4 is changed from −0.35 to −0.21. This, too, is in line with chemical intuition, as NH3 competes with C2H4 to drive back the formation of I5c, while at the same time fostering the formation of I5, which continues the catalytic cycle.
Table 6. XTOF and TOF at Different Concentrations
|Entry||c(NH3) (mol L−1)||c(C2H4) (mol L−1)||∑X||∑X||TOF (h−1)|
Optimizing the concentrations leads to a sevenfold increased TOF of 7.1 h−1 at an NH3 concentration of 2.0 mol L−1 and a C2H4 concentration of 0.2 mol L−1 (entry D). Increasing the C2H4 concentration further very soon leads to a negative XTOF again and lowers the TOF (entry E). An additional increase of the NH3 concentration on the other hand would still increase the TOF (entry F). However, this effect has become smaller with the increased NH3 concentration as indicated by the XTOF value for NH3 of 0.37 (entry D) and 0.25 (entry F) compared to 0.88 (entry A) and 0.96 (entry B) at lower concentrations. So, at some point one has to decide if the handling of highly concentrated NH3 is worth the gain in TOF. However, it is clear that ammonia should be used in excess to ethylene in this reaction system.