MODEL OF A POTENTIAL ENERGY SURFACE1 J. I. DYE Michigan State University, East Lansing, Michigan
IN TEACHING rhemiral kinetics, much use is made of the concepts of transition-state theory. A useful outline of the development of the theory is given by E y ~ i n g . Potential-energy ~ diagrams if correctly used, can contribute a great deal to the students' understanding of this theory by offering a "physical picure" of some of the details of atomic interactions. When a teacher or other lecturer discusses the theories of reaction rates, he is almost certain to make use of a graph of the potential energy versus the "reaction coordinate." Such a diagram, illustrated in Figure 1, is very helpful; but students are often confused regarding the meaning of the term "reaction coordinate." This paramet,er has the general nature of a distance or combination of distances between atoms during reaction along the patahof lowest energy. Its meaning becomes clearer when the student acquires an understanding of three-dimensional potential-energy diagrams. I t is assumed t,hat for any collection of atoms in the ground state, the total energy is constant and the potential energy is a function of the relative positions of the nuclei. When only two atoms are involved, the energy is a function of the internuclear distance only and may be plotted in tmo dimensions. A potentialenergy graph for the HBr molecule is shown in Figure 2. This graph is actually a photograph showing one side of the model described in this paper. The potential energy for a system of three atoms is more difficult to visualize. I n fact, since there are three independent internuclear distances, the energy LPresentrsl :IS p a t of the Symposium on Use and Abuse of Models in Tme'ling before the Division of Chernic:l Eduo~ttionat the 130th Mcrting of the American Chemical Socieby, Atlantic City, September, 1956. ' EYRING, H., Chern. Revs., 17, 65-77 (1935).
VOLUME 34, NO. 5, MAY, 1957
ACTIVATED
>
COMPLEX
a -
L PRODUCTS
a R E A C T I O N ' COORDINATE Figure 1.
General Potential Energy Curat f o r e Remction
function is a four-dimensiona! "surface." Quantum mechanics allows one to shorv that, a t least when the valenre electrons of the center atom are in s states, the most favorable configuration is a linear arrangement of the three atoms. If we assume that something close to
this linear arrangement exista during most "successful" collisions, only two independent distances remain. This assumption is the first step in choosing the "reaction coordinate." A three-dimensional surface can be used to represent the variat.ion of potential energy with internuclear distances for such a linear configuration. The energy of a system of three atoms, X-Y-Z, cannot yet be calculated from quantum mechanics alone. Instead, an avproximate method such as the energy surface are not correct; but nevertheless a useful qualitative picture is obtained. Using the Heitler-London treatment, the energy of the system can be written as E=A+B+C*
.\/'A[(.
- 0)'
+
(a
- 7)'
+ (8 - ?)'I
(1)
in which A, B, and Care lLcoulomhic"energy terms and or, 0, and 7 represent "exchange" energies. The " semi-empirical" approach consists in evaluating all six terms from the potential-energy curves for the three diatomic systems, X-Y, X-Z, and Y-Z. These curves are represented by the Morse function using spectroscopic inf~rmation.~In addition, the coulombic energies A, B, and C are assumed to constitute a fixed percentage (l0%-209,) of the total energy of the respective diatomic system. Equation (1) allows E to be calculated as a function of the two independent distances, rxl. and rrz. (For this linear system, TXZ = TXY rye) Such calculations allow one to describe the chauges in energy which occur during the reaction,
+
with ruz becoming smaller and rxy becoming larger as the reaction proceeds. The calculations described above were carried out for the reaction H-H
+ Br
-
H
+ H-Br
by 27 students in a class studying kinetics. This reaction is postulated to be an important step in the photochemical reaction of hydrogen and bromine.' Each student made calculations of the potentialenergy curves for several fixed distances of a pair of atoms. Data listed by Herzbei-9 vere used assuming 14% coulombic energy. I t was possible for the class to prepare a rather detailed energy contour graph for the reaction without undue work for any individual. The contour diagram has the same appearance as shown in a top view of the model with constant energy lines drawn a t 10 kcal. intervals (Figure 3). CONSTRUCTION OF THE MODEL
Using this contour graph glued to a sheet of plywood, a clay mold was prepared which was used to make the final laster model. S t r i ~ sof 0.014-in. thick steel a E ~H.,AND ~ ~ M. POLANYI, ~ ~ 2. , physik. Chem., BIZ, 279311 H., Chem. Revs., 10,10&123 (1932). (1931); EYRING, HERZBERG, G., "MolecularSpectra,and Molecular Structure," 2nd ed., D. Van Nostrand Co., Ino., New York, 1950, p. 101. See for example: FROST,A. A., AND R. G. PEARSON, "Kinetics and--Meohanism," John Wiley & Sons, Inc., Now York, 1953, p. 224. V.ef.3, Table 39, pp. 501-81.
T I Y Y ~3.E P h o t o g l a p h Showing the Point on the Model Corresp~nrlsng t o the Activated Complex
"shim-stock" were cut to widths of 1 in., 2 in., etc., and were fitted to the appropriate contour lines with small finishing nails. Most of the space between the steel strips was filled with plaster of Paris which served to hold the 'strips firmly in place. Modeling clay was then used to finish the 18- X 18- X ll- in. high mold. The mold was greased, and a mixture of plaster of Paris and expanded mica was used to form the model. After the mold was removed, the model was mounted in a plywood box and finished with a commercial finishing plaster and paint. Lines of constant potential energy were drawn on the surface using a jig which rested on the floor. These lines were spaced a t 10 kcal. intervals (1 in. on the model), and were painted on the surface with the aid of masking tape. Figure 3 shows a top view of the finished model and Figure 2, a side view. The model illustrates quite effectively to a c1a.w the meaning of the reaction coordinate (the path of lowest energy), the activated complex, and potential-energy curves for diatomic systems. ANALOGY T O CHEMICAL REACTION
Figure 3 illustrates photographs which can be made into slides t o familiarize the students with the meaning of potential-energy surfaces. The three halls in a line form a scale model for the Br-H-H system in the activated complex. The black ball on the model surface shows the point on the surface corresponding to the activated complex. If a frictionless particle could be caused to slide on a perfect surface of this type, the motion of the particle would describe the atomic positions and velocities in an actual reaction. (To be an exact representation, the angle between the coordinate axes would have to be 45.4' for this system.) In practice, a ball rolling on the surface is quite useful in examining some of the details of the reaction, particularly the inierconversion of translational and vibrational energies. It is difficult for a large class to observe the motion of a ball on the surface, and even with small groups of students, the velocities involved are too great to permit easy study of the path. For these reasons, 16-mm. slow-motion moving pictures were made of a ball rolling on the surface. Such pictures, exposed a t 64 frames per JOURNAL OF CHEMICAL EDUCATION
Flour? 5.
second, and shown a t 16 frames per second, permit ready observation by the rlass. Figures 4 and 5 are photographs of the model with paths drawn on the photographs reprrsenting a "successful" and an "unsuccessful" collision, respectively. The paths mere sketched with the aid of the motion pictures. When the students have a thorough grasp of the ideas involved in this reaction, it is not diflicult t o
VOLUME 34, NO. 5, MAY, 1957
Tri>cr of r Bnil o n
ths. St:r!.rr Collision
tor a n
Uns~iricsri;xl
extend the discussion to more complicated reaction types, for vhich the potential energy cannot be graphed. ACKNOWLEDGMENT
The author wishes t o express his thanks t o all of the students who participated in the potential-energy calculations, and t o F. E. Hood and R. Wheaton for assist,ance in finishing and photographing the model'.
217
