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Steven D. Gammon Western Washington University Bellingham, WA 98225
Using Computer Graphics to Demonstrate the Origin and Applications of the “Reacting Bond Rules”
W
David R. Tyler* and David R. Herrick Department of Chemistry, University of Oregon, Eugene, OR 97403; *
[email protected]
The “reacting bond rules” (1) (also known colloquially as “Thornton’s rules” [2] ) describe how transition-state structures and energies vary as a function of changes in the energies of selected reaction parameters. Students in physical organic chemistry courses often have a difficult time understanding the origin of these rules; visualizing the application of the rules can be even more difficult. Fortunately, however, computer programs such as Mathematica1 have excellent 2-D and 3-D plotting capabilities that students can use to visualize and manipulate reaction surfaces, which in turn facilitates students’ understanding of the rules. This paper describes how students can use these programs, given a general introduction to the rules, to aid in their comprehension and application of the rules.
CR3Y + X
−
CR3(X)(Y)
(reactants)
(1,1)
C –Y Bond Order
(0,1)
−
Analysis of Reaction Coordinates A useful way to analyze reactivity is to plot the energy of a system as a function of the parameters that change during the course of the reaction. A plot of this kind for a typical SN2 reaction (CR3Y + X– → CR3X + Y–) is shown in Figure 1.2 In this figure, the x-axis is the carbon-entering group bond order, the y-axis is the carbon-leaving group bond order, and the z-axis is the energy of the system. In principle, each relevant molecular parameter that changes during the reaction can be assigned an axis, leading to a multi-dimensional plot. For obvious reasons, however, it is not particularly helpful to deal with multi-dimensional plots, and most systems are therefore analyzed in terms of only two (or rarely three [3] ) reaction parameters. The reaction pathway of the SN2 reaction is shown by the darkened arching curve in Figure 1; note that the highest energy point along the reaction pathway is the transition state (denoted with a ‡ in the fig-
(0,0)
C –X Bond Order
(1,0)
CR3X + Y −
+ − − CR3 + X + Y
(products)
Figure 2. More O’Ferrall–Jencks diagram for the SN2 reaction in Figure 1. The reaction coordinate is indicated by the curve running from reactants to products.
ure). The projection of the reaction pathway onto the x-y plane is called the reaction coordinate. A plot of the reaction pathway projected onto the x-y plane for the SN2 reaction in Figure 1 is shown in Figure 2. These two-dimensional diagrams are called More O’Ferrall–Jencks diagrams, after the two investigators who first popularized their use (4). (More O’Ferrall–Jencks diagrams are particularly useful for delineating the limiting
Energy
Figure 1. Energy surface for an SN2 substitution reaction.
CR3Y + X − (reactants)
nd Bo Y r C– rde O
(0,0,0)
CR3+ + X − + Y −
CR3(X)(Y)− (1,1,0)
(0,1,0)
Rea
ction
Coo
C –X Bond Order
rdina
te (1,0,0)
CR3X + Y − (products)
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Journal of Chemical Education • Vol. 79 No. 11 November 2002 • JChemEd.chem.wisc.edu
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Energy
'
Energy
C B
reactants A
products
Reaction Coordinate
Reaction Coordinate
Figure 3. The reaction coordinate diagram for the SN2 reaction in Figures 1 and 2.
mechanisms of a reaction and for showing the continuum of mechanisms between these limits. For example, a reaction coordinate that runs from the (0,1) reactant corner to (0,0) then to the product corner at (1,0) describes an SN1 pathway.) To simplify these pictures, it is traditional to plot the energy of the system as a function of the reaction coordinate. Thus, the reaction coordinate in Figure 2 becomes the x-axis and the energy of the reaction system is plotted on the y-axis. The resulting familiar “reaction coordinate diagram” for the SN2 reaction of Figures 1 and 2 is shown in Figure 3. More O’Ferrall–Jencks diagrams have found extensive use in the study and analysis of reaction mechanisms, particularly when used in conjunction with the reacting bond rules (5). In deriving the reacting bond rules, Thornton used an inverted parabola to represent the energy surface near the transition state and then he considered the effect on the transition state of raising or lowering the energy of the reactants or products (6). The energy changes were assumed to be caused by linear perturbations in the energy. As an example, the effect of one such change in shown in Figure 4. The parabola labeled A represents the initial, “unperturbed” reaction coordinate and the line (B) is the linear perturbation in energy (perhaps caused by changing a substituent on the carbon atom, and so forth).
Figure 4. The effect on the transition state (‡) of a “parallel” linear perturbation that increases the energy of the product. Parabola A represents the initial state, line B is the linear energy perturbation, and C is the sum of A and B. Note the new transition state (‡’) is higher in energy and further to the right along the reaction coordinate (“more product-like”).
The sum of the original parabola and the line is a new parabola labeled C. Note that the transition state has moved to the right along the reaction coordinate and it is higher in energy. In mechanistic parlance, the transition state is said to occur “later”. Similar plots can be done for perturbations that stabilize the transition state by lowering the energy of the products or that raise or lower the energy of the reactants.3 In each case, the plots reveal how the transition state structure changes (“earlier” or “later”) and whether its energy increases or decreases. The results of these plots are summarized in the top half of Table 1. (A cautionary note when using this table: recall that the rate constant for a reaction is determined by the difference in energy between the reactants and transition state. Thus, an increase (decrease) in the absolute energy of the transition state does not necessarily lead to a decrease (increase) in the rate constant for the reaction because the perturbation might also raise the energy of the reactants.) The perturbations in the preceding paragraph are all caused by changes along (“parallel to”) the reaction coordi-
Table 1. The Effects of Parallel and Perpendicular Perturbations on a Transition State Direction of Perturbation to the Reaction Coordinate
Type of Perturbation
Change to Transition State
Absolute Energy of Transition State
Energy Gap between New Transition State and Reactants
Rate of Reaction
parallel
raise products
occurs later
higher
larger
slower
parallel
lower products
occurs earlier
lower
smaller
faster
parallel
raise reactants
occurs earlier
higher
smaller
faster
parallel
lower reactants
occurs later
lower
larger
slower
perpendicular
a
lower left edge
shifts left
lower
larger
slower
perpendicular
a
raise left edge
shifts right
higher
smaller
faster
a
raise right edge
shifts left
higher
larger
slower
a
lower right edge
shifts right
lower
smaller
faster
perpendicular perpendicular a
These analyses assume the reactants are on the left side of the saddle point in Figure 5.
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Figure 5. Illustration of a saddle point showing that the transition state lies at an energy maximum along the reaction pathway but at an energy minimum for a direction perpendicular to the reaction pathway. The reaction coordinate is represented by the solid curve and the perpendicular direction is represented by the dotted curve. A color version of this figure is printed on p 1283.
A
Energy
nate. Perturbations can also occur “perpendicular” to the reaction coordinate, and Thornton analyzed the effect these perturbations would have on the structure of the transition state as well. The concept of a change “perpendicular” to the reaction coordinate can be understood by reference to Figure 1. Note that the transition state occurs at a maximum along the reaction coordinate but at a minimum with respect to a coordinate that is perpendicular to the reaction coordinate. (Stated differently, the transition state is a saddle point, shown in Figure 5. Movement along the reaction coordinate away from the transition state will lead to a decrease in energy, but movement along the perpendicular coordinate will lead to an increase in energy, that is, the transition state sits in a well along the perpendicular coordinate.) One example of Thornton’s analysis of this situation is depicted in Figure 6. Again, the unperturbed, initial energy well is labeled A and the sum of this well and the linear perturbation, B, is labeled C. Lowering the energy of the left side of the parabola shifts the transition state to the left and lowers its energy. Other perturbations are possible, and they are summarized in the bottom half of Table 1.
C
The Reacting Bond Rules (Thornton’s Rules) Based on the results in Table 1, Thornton formulated two rules describing how transition states move in response to parallel or perpendicular perturbations (6). Restated in modern nomenclature, the rules are the following: A perturbation that lowers (raises) the energy of a species along the reaction coordinate (that is, that lowers (raises) the energy of the reactant or product) will decrease (increase) the energy of the transition state and move it structurally in the direction away from the end that is lowered (toward the end that is raised). A perturbation that lowers (raises) the energy of a species perpendicular to the reaction coordinate will decrease (increase) the energy of the transition state and move it structurally in the direction toward the side that is lowered (away from the side that is raised).
B
'
Coordinate axis that is perpendicular to the reaction coordinate
Figure 6. The effect on the transition state (‡) of a linear perturbation that decreases the energy of an edge to the left on a More O’Ferrall diagram. Parabola A represents the initial state, line B is the linear energy perturbation, and C is the sum of A and B. Note the new transition state (‡’) is lower in energy and further to the left. CR3Y + X −
CR3(X)(Y)−
(reactants) (0,1)
(1,1)
As an example of applying the reacting bond rules, consider the effect of using a better leaving group on the transition state structure in an SN2 reaction. Using a better leaving group is equivalent to raising the energies of the species located along the (0,1)–(1,1) edge in Figure 2. (This statement is usually rationalized by noting that the bond to a better leaving group is weaker than a bond to a poorer leaving group. Hence, the species located along the (0,1)–(1,1) edge are higher in energy when Y is a better leaving group.) The rules predict therefore that the transition state will move toward this edge along the reaction coordinate and away from this edge perpendicular to the reaction coordinate. The net movement of the transition state is shown in Figure 7. Note that, in the new transition state, the bonding to the leaving group is approximately unchanged compared to the initial reaction, but there is less bonding to the entering group (again compared to the initial reaction, which involves a poorer leaving group). 1374
C –Y Bond Order
Applications of the Reacting Bond Rules
(0,0)
C –X Bond Order
+ − − CR3 + X + Y
(1,0)
CR3X + Y − (products)
Figure 7. A More O’Ferrall–Jencks diagram showing the effect on the location of the transition state of using a better leaving group (raising the energy of edge (0,1)–(1,1)).
Journal of Chemical Education • Vol. 79 No. 11 November 2002 • JChemEd.chem.wisc.edu
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A
(1,1)
.
C –Y Bond Order
(0,1)
(0,0)
C –X Bond Order
(0,1)
(1,0)
(1,1)
The ability of Mathematica (and similar programs) to display complicated functions graphically is a distinguishing feature of these programs, and this feature can be put to excellent instructional use by helping students visualize in three dimensions the transition-state movements predicted by the reacting bond rules. To demonstrate this usefulness, the threedimensional energy surface of the SN2 reaction in Figure 1 was constructed and plotted in Mathematica using the functions given by Dunn (7). The resulting plot is shown in Figure 8A. (The functions used by Dunn to construct an energy surface are provided in the supplemental material for this paper, as are the routines for entering the functions in Mathematica and then plotting Figure 8A. Although numerous functions for energy surfaces have been reported, the Dunn functions were chosen because they are relatively simple, yet they effectively illustrate the ability of Mathematica to graphically demonstrate the reacting bond rules and their applications.) The next step is to perturb the energy surface, and using Mathematica it is straightforward to manipulate the energy surface in ways that correspond to raising or lowering the energies of species along an edge of Figure 2 (or likewise changing the energy of one of the corners). This is done by simply adding or subtracting appropriate functions to the functions describing the energy surface
.
Using Mathematica to Visualize the Changes Predicted by the Reacting Bond Rules
C –Y Bond Order
B Figure 8. A 3-D plot from Mathematica showing the effect on the location of the transition state of raising the energy of edge (0,1)(1,1) in the More O’Ferrall–Jencks diagram in Figure 7. (A) shows the initial energy surface and (B) shows the surface after raising the edge. Note the energy of the (0,0)–(1,0) edge in the More O’Ferrall–Jencks diagram is held unchanged. In three dimensions, raising the energy of the (0,1)–(1,1) edge corresponds to adding the plane function z = cy (c = a constant) to the function of the energy surface in (A). The view in these plots is looking from the reactant corner to the transition state; the product corner is obscured by the energy surface in this view.
(0,0)
C –X Bond Order
(1,0)
Figure 9. (A) A Mathematica contour plot of the energy surface in Figure 8A. (B) A Mathematica contour plot of the energy surface in Figure 8B. The transition state is indicated by the dot along the reaction coordinate.
in Figure 8A. (The functions for raising/lowering an edge or a corner are also included in the Mathematica routines provided with the web material.) For example, the new energy surface obtained by raising the energy of species along edge (0,1)–(1,1) of Figure 2 is shown in Figure 8B. A comparison of the two plots in Figure 8 clearly shows that the transition state has moved. Mathematica can change the “viewpoint” from which one looks at the energy surface in Figure 8B, and by doing so it is possible to see that the transition state has moved toward the (0,0)–(0,1) edge, as predicted by the reacting bond rules. More useful in this regard, however, is Mathematica’s ability to make contour plots of three-dimensional surfaces. (A moment’s reflection reveals that these plots are More O’Ferrall–Jencks diagrams but with contours.) The contour plots of the energy surfaces in Figure 8 are shown in Figure 9. The transition-state structural change predicted by the reacting bond rules for a reaction with a better leaving group (see Figure 7) is readily apparent in the contour plots shown in Figure 9.
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A
(1,1)
.
C –Y Bond Order
(0,1)
.
Figure 10. Energy surface for an SN1 substitution reaction. The reaction pathway is indicated by the heavy curve and the two transition states by the dots. A color version of this figure is printed on p 1283.
(0,0)
B
C –X Bond Order
(0,1)
(1,0)
(1,1)
. .
The functions given by Dunn can also be used to construct the three-dimensional energy surface for an SN1 reaction (Figure 10; the contour plot is shown in Figure 11A). Consider the effect of raising the energy of the carbocation corner on the two transition states in Figure 10. Thornton’s rules predict that the transition states will move toward the corner (along the reaction coordinate) and away from the edge (perpendicular to the reaction coordinate). Indeed, as clearly shown by the contour plot in Figure 11B, the transition states do move as predicted by the rules. This movement can be convincingly demonstrated to a class by overlaying transparencies of the contour plots on an overhead projector or equivalently by using a computer to superimpose the two contour plots. What makes this demonstration so dramatically convincing for students is that they can follow the procedure each step of the way starting from the original three-dimensional surface, then raising the corner to get a new energy surface, and finally overlaying the two contour plots to see the shift in location of the transition states.
C –Y Bond Order
Other Examples
(0,0)
C –X Bond Order
(1,0)
Figure 11. (A) A Mathematica contour plot of the SN1 substitution energy surface in Figure 10. The filled circles indicate the transition states. (B) A Mathematica contour plot showing how the transition states in (A) move when the (0,0) corner of the corresponding More O’Ferrall–Jencks diagram is raised in energy.
Notes 1. Mathematica is available from Wolfram Research. See www.wolfram.com. 2. Figure 1 is based on a similar figure in ref. 2, page 216. 3. A plot showing the effect on the transition state of a linear perturbation that decreases the energy of the products is included in the supplemental material for this paper.W WSupplemental
Material
A plot showing the effect on the transition state of a linear perturbation that decreases the energy of the products, the functions used by Dunn to construct an energy surface, and the routines for entering the functions in Mathematica and then plotting are available in this issue of JCE Online.
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Literature Cited 1. Müller, P. Pure Appl. Chem. 1994, 66, 1077. 2. Lowry, T. H.; Richardson, K. S. Mechanism and Theory in Organic Chemistry, 3rd ed.; Harper and Row: New York, 1987; p 216 ff. 3. (a) Scudder, P. H. J. Org. Chem. 1990, 55, 4238. (b) Grunwald, E. J. Am. Chem. Soc. 1985, 107, 4715. 4. (a) More O’Ferrall, R. A. J. Chem. Soc. B 1970, 274. (b) Jencks, W. P. Chem. Rev. 1972, 72, 705. 5. (a) Jencks, W. P. Acc. Chem. Res. 1980, 13, 161. (b) Murdoch, J. R. J. Am. Chem. Soc. 1983, 105, 2660. 6. Thornton, E. R. J. Am. Chem. Soc. 1967, 89, 2915. 7. Dunn, B. M. Int. J. Chem. Kin. 1974, 6, 143.
Journal of Chemical Education • Vol. 79 No. 11 November 2002 • JChemEd.chem.wisc.edu
