26.4 The Activated Complex Theory of Bimolecular Chemical Reaction Rates in Dilute Gases 1107
temperature, but is independent of the concentrations. The exponentαis called the
order with respect to substance Aand the exponentβis called theorder with respect
to substance B. The sum of the orders with respect to the different substances is called
theoverall order.Ifαandβboth equal unity, the reaction is said to be first order
with respect to substance A, first order with respect to substance B, and second order
overall. The ordersαandβare not necessarily equal to the stoichiometric coefficients
aandb.Potential Energy Surfaces
A reaction that takes place in a step that cannot be divided into simpler steps is called
anelementary reaction. An elementary reaction involving two molecules is called a
bimolecular reaction. Consider a bimolecular elementary gas-phase reaction between
a diatomic molecule and an atom:CD+F→C+DF (26.4-4)where the letters are abbreviations for the chemical symbols of the elements. We assume
that the Born–Oppenheimer Schrödinger equation for the electrons in the system has
been solved for various positions of the three nuclei. As with a bound molecule, the
Born–Oppenheimer energy acts as a potential energy for nuclear motion. As the reaction
occurs, the C and D nuclei move apart and the D and F nuclei move toward each other.
If the three nuclei remaincolinear(on the same line) the potential energy of the nuclei
is a function of two internuclear distances:V V(rCD,rDF) (26.4-5)This potential energy can be represented by a surface above a plane in which the
distancerCDis plotted on one axis and the distancerDFis plotted on the other axis.
The earliest calculation of a potential energy function for a three-atom reacting sys-
tem was a semi-empirical calculation for the collinear reaction of a hydrogen atom
with a hydrogen molecule.^4 This system has been repeatedly studied ever since.^5
Figure 26.1a shows schematically a view of a three-dimensional graph of the potential
energy function for the collinear conformation. Figure 26.1b shows the same informa-
tion in a different way by giving contours that represent positions of equal potential
energy.
There is a trough that runs from the point labeledaat the upper left of the diagram
(corresponding to reactants) along a roughly L-shaped path past the point labeledbto
the point labeledcat the lower right (corresponding to products). In a region nearbat
the lower left of the diagram the bottom of the trough rises to a maximum height so that
in this region the potential energy surface is shaped like a saddle or a mountain pass. A
curve is drawn along the bottom of the trough and over the saddle at pointbin Figure
26.1b, and the energy as a function of position along this curve is shown schematically
in Figure 26.1c. Motion along the curve in Figure 26.1b corresponds to motion along
the axis in Figure 26.1c and represents progress toward completion of the reaction, and
we call the distance along this curve thereaction coordinateζ.(^4) H. Eyring and M. Polanyi,Z. Phys. Chem.,B12, 279 (1931).
(^5) T. J. Park and J. C. Light,J. Chem. Phys., 91 , 974 (1989); see D. G. Truhlar and C. J. Horowitz,J. Chem.
Phys., 68 , 2466 (1978) for an accurate potential energy function.