Reversible Multistep Synthesis with Equilibrium Properties Based on a

Article pubs.acs.org/JPCB

Reversible Multistep Synthesis with Equilibrium Properties Based on a Selection-Oriented Process with a Repetitive Sequence of Steps Sagi Eppel* and Moshe Portnoy School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel S Supporting Information *

ABSTRACT: Reversible processes and reactions exhibit equilibrium and error correction properties that allow them to surpass the limits of irreversible systems. However, such processes are currently limited to completely reversible one-pot transformations. In this work, we analyze and demonstrate a new system of reversible multistep syntheses that enable the introduction of reversibility and equilibrium properties, previously reserved for single-step processes, into step-by-step synthesis transformations. The system uses a repetitive sequence of steps such that the same sequence of reactions is performed on the material again and again in a loop. The final step in each loop transforms unequal fractions of all products back to the starting material. We show that such a system is reversible, even if some steps in the synthesis are irreversible. We mathematically analyze and experimentally demonstrate the properties of such systems and show that they include many features unique to reversible and equilibrium systems. This approach can enable new methods for controlling the distribution of the products of chemical transformations.

1. INTRODUCTION The construction of structures on the molecular level is usually done by one of two general methods. The first method involves the synthesis of a structure by using an irreversible sequence of mostly accurate reactions that leads to a target structure.1−3 This method is used in the majority of syntheses of organic molecules. However, it is limited by its lack of reversibility and error correction, which often results in reduced overall yield.1,4 The second method is based on using reversible equilibrium reactions, which by themselves could be inaccurate and could lead to a large number of products. Yet the reversibility of the reaction can cause the majority of the products to be transformed back into the starting materials while leaving only specific stable structures. Consequently, reversible system can selectively increase the fraction of some of the products concurrent with the reduction of the fraction of the others.5,6 As a result, reversible processes exhibit many interesting properties, such as equilibrium and error correction, that allow them to surpass the limits of irreversible ones.4−21 However, reversible systems are limited by the need for completely reversible reactions and the use of one-pot, one-step processes. Multistep processes are, on the other hand, mostly irreversible due to a nonrepetitive sequence of steps and the use of onedirectional reactions. In this work, we analyze and demonstrate a new selection-oriented synthesis system that enables the introduction of reversibility and equilibrium properties, which were previously reserved for one-step processes, into multistep synthesis transformations. The system uses a repetitive sequence of steps such that the same sequence is performed on the material again and again in a loop. The reactions used for each of the steps can be irreversible and of low selectivity, yielding a distribution of products. However, by having a © 2014 American Chemical Society

selection/backward step that transforms unequal fractions of all products back to the starting material, as the final step in the loop, together with repetitively performing the sequence of steps, the system selectively increases the fraction of some products relative to others and achieves equilibrium properties similar to those of reversible systems.22,23 This work will explore such a system from a theoretical and experimental perspective.

2. SYSTEM DESCRIPTION There are two main problems in introducing reversibility into stepwise synthesis processes. One problem involves the irreversibility of multistep processes: in a one-pot process, the steady set of conditions allows the reactions to recur continually, potentially enabling reversibility and equilibrium properties. In multistep processes, however, every step uses specific components and reaction conditions; therefore, a reaction that occurs at a specific step cannot recur in the next step. Hence, the irreversibility of the sequence results in the irreversibility of the entire process. This problem can be solved by using a repetitive sequence of steps. By repeating the same sequence of reaction conditions again and again in a loop, every process that occurs in one step of the cycle can recur in the next cycle when the step is repeated and the same conditions are reapplied. A second problem is that the multistep syntheses can include irreversible steps. However, the irreversibility of some steps in the sequence does not necessarily imply the irreversibility of the entire sequence. If the final step of the Received: May 26, 2014 Revised: July 22, 2014 Published: July 23, 2014 9733

dx.doi.org/10.1021/jp5051645 | J. Phys. Chem. B 2014, 118, 9733−9744

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Figure 1. System 1 diagram. Arrows represent fractional transformations from the starting component to the target product. The fraction of material transformed is given next to the arrow. Yn and Xn represent the fractions of products Y and X after the forward step in cycle n. Ynb, Xnb, and Zn represent the fractions of components Y, X, and Z after the backward transformation of cycle n. fXn and fYn represent the fractions of component Z that are transformed into products X and Y, respectively, in the forward step of cycle n, bXn and bYn represent the fractions of products X and Y that are transformed back into component Z in the backward step of cycle n.

cycle transforms some fractions of all of the products back to the starting materials, then the system can be considered to be reversible even if some of the preceding steps are irreversible. Such a sequence could be viewed as a reversible process in which the forward and backward reactions are separated and performed at different steps (Figure 1). Few types of systems fulfill the above conditions. We will show in section 3 that all such systems exhibit the same general behavior and can be described by the same equations. Three systems based on such reversible multistep processes are shown in Figures 1−3 and are described below. 2.1. System 1. System 1, illustrated in Figure 1, includes two steps per cycle: a forward step and a backward step. The forward step involves the transformation of starting material Z into one of two products X and Y, with the fractions of component Z that are transformed into products X and Y given by fX and fY respectively. The second step involves the backward process in which fractions of products X and Y are transformed back into component Z. The fractions of products X and Y that are transformed back into Z are given by bX and bY, respectively (Figure 1). This process could be considered to be a multistep version of the three-component equilibrium shown in Figure 4a. 2.2. System 2. System 2, shown in Figure 2, involves two components (X and Y) and one step per cycle. In each step/ cycle, the fraction tX of component X is transformed into product Y, while in the same step the fraction tY of component Y is transformed into product X. This could be considered to be the multistep version of the two-component equilibrium shown in Figure 4b. While the cycle is represented by a single step, this scheme can portray much more complex systems since each step in the scheme could by itself be composed of a sequence of reactions. For example, if we take the backward and forward steps in system 1 (Figure 1) as corresponding to one unified step, then the system becomes identical to system 2. 2.3. System 3. System 3, shown in Figure 3, involves a combination of the steps from systems 1 and 2. The system now involves three steps per cycle: (a) the forward step, which

Figure 2. System 2 diagram. Arrows represent fractional transformations from the starting component to the target product. The fraction of material transformed is given next to the arrow. Yn and Xn represent the fractions of components Y and X in cycle n. tXn gives the fraction of component X that is transformed into products Y in cycle n. tYn gives the fraction of component Y that is transformed into product X in cycle n.

involves the conversion of structure Z into one of two products X and Y with the transformation fractions of fX and fY, respectively (Figure 3); (b) the transformation step, in which the fraction tX of product X is transformed into component Y while at the same time the fraction tY of component Y is transformed into component X; and (c) the backward step, in which fractions bX and bY of components X and Y, respectively, are transformed back into component Z. This is a multistep equivalent to the three-component equilibrium shown in Figure 4c. If the first two steps are considered to be a single unified step, then this system is identical to system 1, whereas if all three steps are described as a single unified step, then this system is equivalent to system 2.

3. MATHEMATICAL BEHAVIOR A mathematical description for the behavior of systems 1−3 above, including the exact equations for the fractions of each component in each step, was developed based on the equations in Figures 1−3 and is given in Table 1. If we write the fraction of component X in cycles n and n + 1 as Xn and Xn+1 9734



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Figure 3. System 3 diagram. Arrows represent fractional transformations from the starting component to the target product. The fraction of material transformed is given next to the arrow. Yn and Xn represent the fractions of products Y and X in cycle n. Ynt and Xnt represent the fractions of products Y and X after the transformation step of cycle n. Ynb, Xnb, and Zn represent the fractions of products Y, X, and Z after the backward step of cycle n. fXn and fYn represent the fractions of component Z that are transformed into products Y and X, respectively, in the forward step of cycle n. tXn represents the fraction of component X that is transformed into product Y in the transformation step of cycle n. tYn gives the fraction of component Y that is transformed into product X in the transformation step of cycle n. bXn and bYn represent the fractions of components Y and X, respectively, that are transformed back into Z in the backward step of cycle n.

graphs 5d−f. If the transformation rates remain constant through the cycles, then the result is a monotonous increase/ decrease in the fractions of components X and Y (Figure 5d,e). In graph 5f, the transformation constants were chosen to be different for even and odd cycles, with X transformed into Y in odd cycles and Y transformed into X in even cycles. The result is again a jigsaw pattern that matches the experimental results for this system (Figure 13). Two examples of system 3 are shown in Graphs 5g,h. These exhibit a close similarity to those of the experimental results for system 3 in Figure 12. Finally, we note that after a few steps all of the systems in Figure 5 approach equilibrium/steady-state conditions with the fractions of the components remaining almost unchanged after each cycle.22−26

Figure 4. One-pot/single-step equilibrium processes equivalent to the multistep processes in systems 1 (a), 2 (b), and 3 (c).

respectively, then the change in the fraction of component X from one cycle to the next is given by the formula Xn+1 = knXn + an, where kn and an are constants which depend on the transformation rates of the systems in cycle n (kn and an are given in Table 1, rows 5−7). The fact that all three systems (Figures 1−3) behave according to this formula suggests that all cyclic multistep processes should have essentially similar behavior derived from this equation (Table 1). It also allows the development of an exact formula for the fraction of each component in each cycle (Table 1, row 2). Another interesting result of this description is that, if the transformation probabilities (fX, fY, bX, bY, tX, tY, Figures 1−3) remain unchanged in each cycle (kn = k, an = a), then after a number of cycles the system will reach an equilibrium (steady-state) condition in which the fraction of the component will no longer change by repeating the reaction cycle (Xn+1 = Xn). The fraction of component X in this equilibrium state is given in Table 1, row 3. 3.1. Theoretical Examples. A few examples for the behavior of systems 1−3 are shown in Figure 5; these were generated using the equations in Figures 1−3. The behavior of system 1 (Figure 1) in graphs 5a−c shows a distinct oscillating jigsaw pattern,16,17,22−27 which results from the periodic increase in the fraction of components X and Y in the forward step (when Z is transformed into X and Y, Figure 1) and decrease in the backward step (when X and Y are transformed back into Z). This jigsaw pattern is the main characteristic of system 1 and also appears in the experimental results for this system (Figure 11). Graphs 5a,b show system 1 when the transformation rates (fX, fY, bX, bY; Figure 1) remain constant through all cycles, while graph 5c demonstrates system 1 with forward transformation rate fX (from Z to X) that constantly decreases with progressive cycles. The theoretical behavior of system 2 (Figure 2) under various conditions is depicted in

4. EXPERIMENTAL DEMONSTRATION Given the cyclic nature of the systems described above, any experiment that will emulate these systems will have to use a large number of steps with minimal effort and without a considerable loss of products. The system we chose for this emulation is a polymer-supported system in which the product is attached to a cross-linked polystyrene polymer support, also known as Wang polystyrene. This approach reduces the product loss in the separation and purification stages as well as the efforts devoted to these stages.2,3 4.1. System Description. The experimental scheme used to demonstrate the cyclic systems is shown in Figure 6. The synthesis starts with component Z (Figure 6) with two OH groups per structure and is based on three reactions per cycle: (1) The first step involves one-directional HBTU-induced esterification of the OH groups using a mixture of the two carboxylic acid forms of 1Y (Yn and Ya, Figures 6 and 7).28,29 Such esterification is nonselective and will therefore yield a random distribution of three products (Figure 7): dialkyne (Ya,Ya), diazide (Yn, Yn) and alkyne−azide (Ya, Yn) in an estimated ratio of 25%:25%:50% respectively (for the first cycle and for equal fractions of YnH and YaH in the reaction mixture (Figure 7)). (2) The second step is an alkyne−azide cycloaddition (CuAAC) reaction, also known as the click reaction, which occurs between the azide on the Yn group and the alkyne on the Ya group (Figure 8a). The reaction is carried 9735


fraction of component X in step n

component X in equilibrium (Xeq = Xn+1 = Xn)

fraction of product Y

kn and an for system 1

kn and anfor system 1

kn and an for system 3

2

3

4

5

6

7

j=0

u=j+1

∑ aj ∏

an only if this expression remains constant through all cycles 1 − kn

n−1 j=0

b

(1 − b Y )

k = [(1 − tX)(1 − f X) − tY (1 − f X)](1 − b X ) + {1 − tY − tX}f X

k n = [(1 − tXn)(1 − f Xn) − tYn(1 − f Xn)](1 − b Xn) + {1 − tYn − tXn}f Xn (1 − b Yn)

a = (1 − f X)tY (1 − b X ) + f X − (1 − tY )(1 − b Y )f X

k = (1 − tX − tY ), a = tY

k = (1 − b X ) + f Xb X − b Y f X

a = bY fX

X eq

a = 1−k

X n = X 0k n + a∑ k n − j − 1

a n = (1 − f Xn)tYn(1 − b Xn) + f Xn − (1 − tYn)(1 − b Yn)f Xn

k n = (1 − tXn − tYn), a n = tYn

k n = (1 − b Xn) + f Xnb Xn − b Ynf Xn

a n = b Ynf Xn

Yn = 1 − X n

X eq =

i=0

Xn = X0 ∏ ki +

n−1

Xn is the fraction of product Y in step n

n−1

kn, an are given in rows 5−7. n−1

X n + 1 = kX n + a

equations for system with constant transformation rates for all cycles:a,c rn = r, r = k, a, fR, fO, bR, bO, tR, tO

X n + 1 = k nX n + a n

ku

equations for general systema,b

All of the constants in the equations are described in Figures 1−3 and Sections 2 and 3. General equations for cyclic selection-oriented synthesis (left column) in which the transformation probabilities depend on the cycle (n). cEquation for the specific case of selection-oriented synthesis in which the transformation probabilities (fX, fY, bX, bY, tX, and tY; Figures 1−3) remain constant during different cycles (right column). The development of these equations is based on the equations in Figures 1−3 and is described in the Appendix.

a

change in the fraction of component X from cycle n to cycle n + 1

1

description

Table 1. General Equations for the Behavior of Systems 1−3a−c

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Figure 5. Graphs of the behavior of systems 1−3; the graphs were generated using the equations in Figures 1−3. The transformation rates for each case are given in each panel.

out using a copper catalyst and in the presence of 1-hexyne.30 The reaction between the azide (Yn) and alkyne (Ya) of the alkyne−azide esterification product ((Ya, Yn), Figure 8a) is intramolecular and is therefore likely to occur first to form ring structure X (Figure 8a).31 The azide group on the diazide ((Yn, Yn), Figure 8b) will then react with 1-hexyne in the solution, forming the Yc structure (Figure 8b). The moieties of the dialkyne esterification product ((Ya, Ya), Figure 7) will remain unreacted. 1-Hexyne is added in order to prevent the dialkyne (Ya, Ya) and diazide (Yn, Yn) components from reacting with each other and being polymerized on the resin.32 1-Hexyne reacts with the azide groups of Yn and blocks them from reacting with the dialkyne (Figure 8b). Since both the esterification and click reactions are irreversible and result in product distribution, they are equivalent to the forward step in system 1. (3) The third step of the cycle is the selection/ backward step, in which a fraction of all structures of X and Y are selectively transformed back into the initial structure Z (Figure 6). This is done via a transesterification reaction14 with sodium methoxide, as shown in Figures 6 and 9. Structure X has two ester attachment points and therefore requires two sequential transesterification reactions in order to dissociate (Figure 9a). Thus, due to a chelating effect, the transformation of component X to component Z is slower and less favorable.33,34 At the same time, the 2Y groups of the Y component have one ester attachment point; therefore, their transesterification to yield structure Z occurs much more rapidly (Figure 9b). This means that the backward trans-

formation rate of product Y to Z will be faster than that of product X. After the transesterification step, the exposed OH groups of Z (Figure 6) have again been formed in the system, and the reaction cycle (steps 1−3) can be repeated. 4.2. Tracking Component Fractions Using NMR. The identification and evaluation of the fraction of each component in Figure 6 were performed based on the integration of their respective signals in the 1H NMR spectra. In order to identify typical peaks of each component, structures X, Y, and Z (Figure 10) were first synthesized separately as pure compounds, and their typical NMR peaks were identified (listed in the Supporting Information). A few peaks were found to be distinct for specific components. These were used to evaluate the product quantity. To make the analyses more objective, the integrations of all peaks in all steps were performed in a precisely applied chemical shift range, which remained constant for all steps, removing the possibility of manipulating the peak size by adjusting the integration range. The large number of peaks and components in the system, combined with the large overall number of steps, cause an increase in the noise and, as a result, limited NMR accuracy in determining the component fractions. An estimation of the measurement accuracy based on deviations from the expected values suggests that the sum of errors in all component fractions (X + Y + Z) is no more than 1 to 2% for the first two cycles and no more than 5% for the third cycle (captions of Figures 11−13). To further verify the existence of each component, after the first click reaction, mass spectrometry was performed, providing clear and accurate 9737



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Figure 6. Scheme of the experimental cyclic reversible multistep synthesis system. This process was performed under two sets of different transesterification conditions, referred to as cases A and B. The synthesis performed on Wang polystyrene is based on a repeating cycle of three reactions: (1) esterification; (2) click, i.e., alkyne azide cycloaddition (CuAAC); and (3) transesterification. This cycle was repeated twice for case A and three times for case B, with each row in the figure representing one reaction cycle. The first two steps in the first cycle are common for cases A and B. Each of the Y, 1Y, 2Y, and A symbols represent a distribution of the moieties indicated for these groups in the upper boxes. The bottom-row arrows indicate the relationship between the reactions of the synthesis system and the relevant steps in system 1 (Figure 1). 1YH is the carboxylic acid form of the 1Y group moieties. Since each structure has two branches, it can contain two different (or identical) components of X, Y, and Z. The fractions of X, Y, and Z components embedded under each product refer to the fraction of branches belonging to each group in this step (for cases A and B). The Y and Z groups occupy a single benzylic branch each. As a result, a single structure can contain both Y and Z components. Since each branch is counted separately, one X ring is counted as two X branches.

Figure 7. Esterification of (Z, Z) with equal fractions of YnH and YaH yields dialkyne (Ya, Ya), diazide (Yn, Yn), and alkyne−azide (Yn, Ya) products that are formed in an estimated ratio of 25%:25%:50%, respectively.

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Figure 8. Click (CuAAC) reactions. (a) The reaction of alkyne−azide (Ya, Yn) yielding X is intramolecular and occurs first. (b) Subsequently, the azide groups of the diazide (Yn, Yn) react with 1-hexyne to form the (Yc, Yc) product.

Figure 9. Transesterification (backward) reaction. The reaction was performed using sodium methoxide in dry THF at room temperature. (a) The X product has two ester connecting moieties. Accordingly, the chelating effect leads to slow/unfavorable dissociation. (b) The 2Y moiety of structure Y has one attachment point; therefore, its dissociation will be more favorable and will occur much faster. Group A is detailed in Figure 6

Figure 10. Connections between components X, Y, and Z in theoretical system 1 (Figure 1) and their experimental equivalents X, Y,and Z (Figure 6). Groups Yc, Ya, and Yn are all accounted for as the Y component.

The exposed OH groups of Z in the initial structure of the experimental system (Z, Figure 6) correspond to starting component Z in the theoretical system (Figures 1 and 10). After the esterification and click reactions (Figure 6), the OH groups of component Z are transformed nonselectively to one of a few products shown in Figure 6. Structure X (Figures 6 and 10) is distinct among these products both by its probability to be formed (Figures 6 and 8) and by its lower probability to be transformed back into component Z in the transesterification (backward) step (Figures 6 and 9). It has therefore been defined as the component in the experimental system which corresponds to component X in the theoretical system (Figure 1). All the other products of the esterification + click steps (Y,

peaks for all of the expected products of the system (Supporting Information).

5. MATCHING THE EXPERIMENTAL AND THEORETICAL SYSTEMS 5.1. Matching Components of the Experimental and Theoretical Systems. To see how the experimental scheme described above (Figure 6) matches theoretical system 1 described in Figure 1, we must first match the components in the experimental system (X, Y, and Z in Figure 6) to those used in the theoretical system (X, Y, and Z in Figure 1). These relationships are shown in Figure 10 and explained below. 9739



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Figure 11. Experimental behavior of system 1 in terms of component fractions. The same results are presented twice: once as a transformation scheme (upper panel, same as in Figure 1) and once as a graph (lower panel), both sharing the same horizontal (x) axis. The reaction and step names on the horizontal axis are identical to those given in Figures 6 (reactions) and 1 (steps). Panels A and B are for experimental cases A and B, respectively. Whenever the experimental results deviate from the predicted theoretical values by 2% or more, the expected value is marked below the measured result in gray. (These deviations were used to estimate the measurement accuracy.)

Figure 6), that do not involve ring (X) formation, have an almost identical probability to be formed and an almost identical probability to dissociate in the transesterification step (Figure 9b). We will therefore categorize them in the second group of products that we shall name Y; this is equivalent to component Y in theoretical system 1 (Figures 1 and 10). The fraction of each component is measured by the fraction of branches it occupies. Components Y and Z occupy a single benzylic branch of the structure while X occupies two, thus a single structure can contain both Y and Z groups (or two identical groups). The ring structure of the X components occupies two branches and therefore contributes two X components. 5.2. Matching Steps in System 1 to the Reactions in the Experimental System. The connection between the reaction in Figure 6 and the steps in system 1 (Figure 1) is explained below. The synthesis starts with structure Z as the only component (Figure 6). The esterification reaction followed by the click reaction (Figure 6) transforms all Z components to either Y or X components. Both the click and esterification reactions are irreversible or completely onedirectional and give a distribution of products. Therefore, the combination of the click and esterification reactions (Figure 6) is equivalent to the forward step in system 1 (which transforms Z to a distribution of X and Y components [Figure 1]). The transesterification reaction (Figure 6) selectively transforms some of the X and Y components back into the initial Z component. This step, therefore, is the equivalent of the backward transformation step in system 1 (Figure 1), wherein a fraction of each component X and Y is selectively transformed back into initial component Z. The selectivity in the backward step results from the chelating effect of X (Figure 9a), which makes the fraction of the X component that transforms back to

the initial structure Z much lower than that of the Y component (bX < bY, Figure 1).33,34 After the backward step, the exposed OH groups (Z, Figure 6) are free to react again in the next cycle. Therefore, the process can be performed again and again with no (theoretical) limit on the number of cycles.

6. ANALYSES OF THE EXPERIMENTAL RESULTS The experiments of the system described in Figure 6 were performed for two cases, with different conditions in the backward step for each case (referred to as cases A and B). The results are shown in Figure 11. The graphs of the component fractions in Figure 11 exhibit a distinct oscillating jigsaw pattern,16,17,22−27 which results from the periodic increase in the fractions of components Y and X (Figure 10) in the forward step and their decrease in the backward step. Such a pattern exactly follows the expected behavior of system 1, as can clearly be seen by comparing the experimental results in Figure 11 and the theoretical results for system 1 in Figure 5a−c. As can be seen in Figure 11, the forward step transforms starting material Z into the X and Y products in an almost 50%/50% ratio. This is the maximal fraction of the X product that can be expected from the forward step in this method. The backward (transesterification) step clearly demonstrates the selective transformation of structures Y and X back into component Z, with the fraction of component Y that is transformed back into Z being more than twice that of component X (Figure 11). As a result, for case A, the fraction of component X after the backward step is more than 2 times the fraction of component Y (Y1/X1 = 34%/16%), which is well beyond the 1/1 maximal ratio of the forward step alone and represents a clear demonstration of the selectivity of such a system. Repeating the reaction cycle showed that the system is indeed cyclic, with a periodic increase in the Y and X product 9740



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However, if we consider the esterification and click reactions as two distinct steps, then we get a three-step process that fits the description of system 3 (Figures 3 and 12). In this process, the esterification reaction (Figure 6) corresponds to the forward step (transforming structure Z to structure Y [Figures 3 and 12]); the click reaction (Figure 6) corresponds to the transformation step of system 3 (transforming structure Y to structure X, [Figures 3 and 12]), and the transesterification reaction corresponds to the backward step (transforming components Y and X back to Z). The graph of experimental case A analyzed in terms of system 3 is depicted in Figure 12. This graph bears a close similarity to the theoretical examples of system 3 in Figure 5g,h. (The graph for case B is provided in the Supporting Information for brevity.) Similarly, the experimental scheme (Figure 6) can be described by system 2 (Figures 2 and 13). This can be done either by considering the three reactions in the cycle (Figure 6) as corresponding to a single transformation step of system 2 or, alternatively, considering the transesterification and esterification reactions as corresponding to one cycle (odd cycle) and the click reaction as corresponding to a second cycle (even cycle). This representation corresponds to system 2, in which the transformation rates (tX, tY, Figure 2) favor X in even cycles (after the click reaction, Figure 6) and Y in odd cycles (after the esterification reaction). The graphs of such a representation of cases A and B are shown in Figure 13 and bear considerable similarity to the theoretical graph of system 2 in Figure 5f, which was also generated using different transformation rates for odd and even cycles (tX(2n) < tY(2n), tX(2n+1) > tY(2n+1)).

fractions in the forward step and a selective decrease in the backward step (matching the system’s predicted behavior in Figure 5a−c). 6.1. Cyclic Behavior. Despite our initial expectation, the transformations rates did not remained constant through different cycles. Most notably, the probability of Z being transformed into product X in the forward step (fX, Figure 1) has decreased in the second and third cycles (Figure 11). This reduction can be explained by the fact that component Yc, that was formed after the first cycle (Figure 10), is unable to participate in the click reaction and therefore in the formation of ring structure X in the subsequent cycle (Figure 10, 5). Moreover, the backward transesterification reaction is highly sensitive to the conditions, resulting in limited control of the backward transformation rates. Despite these obstacles, experimental case A still shows an increase in the fraction of component X to 60% after the forward step in the second cycle (Figure 11), while the theoretical maximum for the fraction of X generated by the forward process alone is limited to 50%. This demonstrates the ability of the backward/forward repetitive synthesis sequence to increase the yield of specific components in a way that exceeds the limit of a purely onedirectional process. 6.2. Steady-State Conditions in Case B. Case B (Figure 11b) arrived surprisingly quickly at an equilibrium (steadystate) condition,19,20 where the component ratio remained constant after each cycle (Xn+1 = Xn). This is surprising for two reasons: (1) the backward transformation rate is much higher for component Y than for component X (bY > bX, Figure 1), suggesting that the ratio between the X component and the Y component (X/Y) should increase with each cycle, and (2) the transformation rates are not equal for each cycle, although this is the main condition for the equilibrium described in section 3. However, constant transformation rates are not the only way to reach equilibrium in such a system. In this case, the equilibrium can be explained by the fact that the rate of the formation of product X, in the forward step (fX, Figure 1), constantly decreases in the second and third cycles (Figure 11b), balancing the lower backward transformation rate (bX, Figure 1) of component X in the backward step (Figure 11b). This explanation is supported by the similarity of the graph in Figure 11b to the theoretical graph in Figure 5c, which was generated using the formation rate of X (fR) that decreases with progressive cycles (fXn > fX(n+1)).

8. SCOPE AND POTENTIAL APPLICATION OF REVERSIBLE MULTISTEP SYSTEMS In theory, multistep reversible systems could be applied in any nonselective stepwise process in which some of the products could be transformed back to their initial states. Since the majority of synthetic chemical processes used today are multistep,1−3,31 this implies a large number of potential applications, such as byproduct recycling and yield increase in nonselective chemical processes.35,36 From a scientific point of view, reversible multistep processes represent a new class of equilibrium systems with discrete behavior that is very different from the dynamics of continuous single-stage processes. Such systems are not limited by the number of steps per cycle or by the type of transformation used in each step. As a result, they can have a much wider range of possibilities compared to conventional equilibrium systems (which are limited to singlestage and thermodynamically reversible processes). Examples of reversible systems that could potentially be expanded to multistep processes reside in such fields as dynamic combinatorial chemistry and system chemistry as well as selfassembly.6,10,12−14,19,20,37 Finally, multistep reversible processes are not limited to chemical systems and could be applied to any system with a selection-oriented behavior. Examples of such systems are mesoscale and dynamic self-assembly processes in which various building blocks, such as magnets and even robots, can be arranged from random motions by the effect of selective interactions.19,20,38−41 Currently these systems are mostly limited to one-stage processes but have the potential to be extended into multistep processes.

7. DESCRIBING THE EXPERIMENTAL SCHEME USING THEORETICAL SYSTEMS 2 AND 3 As a result of the equivalence between the various cyclic multistep systems explained in sections 2 and 3, the experimental scheme in Figure 6 could also be described by systems 2 and 3 (Figures 2 and 3). For example, in the previous section we considered the combination of esterification and click reactions as a single process equivalent to the forward step of system 1 (Table 2). Table 2. Reactions (Figure 6) and Their Corresponding Steps According to Systems 1−3 (Figures 1−3) transesterification system 1 system 2 system 3

esterification

click

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Figure 12. Experimental behavior of system 3 in terms of component fractions for case A. (Case B is given in the Supporting Information.) The same results are presented twice: once as a transformation scheme (upper panel, same as Figure 3) and once as a graph (lower panel), both sharing the same horizontal (x) axis. The reactions and step names on the horizontal axis are equivalent to those given in Figures 6 (reactions) and 3 (steps). Whenever the experimental results deviate from the predicted theoretical values by 2% or more, the expected value is marked below the measured result in gray. (These deviations were used to estimate the measurement accuracy.)

Figure 13. Experimental behavior of system 2 in terms of component fractions for cases A and B, respectively. The same results are presented twice: once as a transformation scheme (upper panel, same as in Figure 2) and once as a graph (lower panel), both sharing the same horizontal (x) axis. The reaction and step names on the horizontal axis are equivalent to those given in Figures 6 (reactions) and 2 (steps). Panels A and B are for experimental cases A and B, respectively. Whenever the experimental results deviate from the predicted theoretical values by 2% or more, the expected value is marked below the measured result in gray (these deviations were used to estimate the measurement accuracy).

reserved for single-step, one-pot processes, into the field of multistep synthesis. We show that such reversible multistep syntheses exhibit many properties unique to reversible processes, such as equilibrium conditions and selective

9. CONCLUSIONS In this work, we have analyzed and demonstrated a new system of repetitive, selection-oriented synthesis that expands the notion of reversible and equilibrium processes, previously 9742



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n−1

n−1

increases/decreases in the amounts of products in nonselective reactions, all in the context of multistep processes that can include one-directional and irreversible reactions. The mathematical analysis of such systems suggests that, while there can be many variations of such processes, they all obey similar rules and can be described by the same mathematical series. The analysis also suggests that the use of such processes enables particular product fractions to be selectively increased or decreased, enabling new methods of controlling the product distribution of nonselective reactions and processes. The experimental demonstration of such a system, while affected by technical imperfection, clearly demonstrates all of its main features, and the results show a close resemblance to the predicted graphs generated by the theoretical equations describing the system. While the specific experimental demonstration described in this work is probably too complicated to have practical applications, applying this approach to other processes might lead to interesting and useful results.

Xn = X0 ∏ ki + i=0

n−1

∑ aj ∏ j=0

ku

u=j+1

where X0 is the initial fraction of component X. If we take a system in which the transformation probabilities are constant in each cycle (ki = k, ai = a), then this expression becomes Xn = X0kn + aΣj n=−0 1kn−j−1. The series Xn+1 = kXn + a converges to Xeq = (a/(1 − k)), which is the steady-state/equilibrium fraction of X. This expression can be derived by taking the equilibrium state (Xeq) as the condition in which X does not change over several cycles: Xeq = Xn+1 = Xn.

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ASSOCIATED CONTENT

S Supporting Information *

Synthesis and characterization data for the described compounds and the reversible multistep process. Detailed development of the equations in Table 1 (section 3). This material is available free of charge via the Internet at http:// pubs.acs.org.

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APPENDIX

Developing Mathematical Equations of Systems 1 and 2

AUTHOR INFORMATION

Corresponding Author

The development of the equations for systems 1 and 2, based on section 2 and Figures 1 and 2, is given below. This development was used to create the equations in Table 1 and section 3. More detailed equations are given in the Supporting Information.

*E-mail: [email protected]. Fax: +972 3640 9293. Tel: +972 3640 6517. Notes

The authors declare no competing financial interest.

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System 1

ACKNOWLEDGMENTS This research was supported by grant no. 955/10 from the Israel Science Foundation.

On the basis of Figure 1 and section 2, system 1, we can write the fraction of components X and Y after the forward step of cycle n + 1 as Xn+1 = Xnb + fXnZn and Yn+1 = Ynb + fYnZn, where Ynb, Xnb, and Zn are the fractions of structures Y, X, and Z after the backward step of cycle n (Figure 1) and fXn and fYn are the fractions of component Z that are transformed into X and Y in the forward step of the nth cycle. The fractions of components Y, X, and Z after the backward step of the nth cycle (Figure 1) are given by Xnb = Xn (1 − bXn), Ynb = Yn (1 − bYn), and Zn = XnbXn + YnbYn, where bXn and bYn are the fractions of the X and Y components that are transformed back into component Z in the backward step of cycle n. Combining the above equations and using Yn = 1 − Xn gives the fraction of structure X in cycle n + 1 as Xn+1 = knXn + an with an = bOnfXn and kn = (1 − bXn) + fXnbXn − bYnfXn.

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System 2

On the basis of Figure 2 and section 2, system 2, the fraction of structure X in step n + 1 can be expressed as Xn+1 = (1 − tXn)Xn + tYnYn, where Yn, Xn, and Xn+1 are the fractions of components Y and X in cycles n and n + 1. tYn is the fraction of Y that was transformed into component X in step n, and tXn is the fraction of component X that was transformed into component Y in step n. The fractions of all of the components sum to 1: Yn + Xn = 1. Using Yn = 1 − Xn, we get Xn+1 = knXn + an, where kn = (1 − tXn − tYn) and an = tYn. System 3

System 3 is described by a combination of the equations for systems 1 and 2. A detailed development of its equations is given in the Supporting Information. Series Behavior

The series Xn+1 = knXn + an, which describes the fractions of the components in systems 1−3, gives an equation for the fraction of component X in step n as 9743



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Reversible Multistep Synthesis with Equilibrium Properties Based on a

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