Physical Chemistry Third Edition

1108 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

Energy of

activated

complex

Activated

complex

Reaction coordinate, 

rCD

rCD/ bohr

rDF

/ bohr

rDF

1

1

2

3

23

(a)

(b)

(c)

0.75 eV

0.35 eV

0

1 eV

2 eV

1

2

3

4

5

9

9





c

b

a

Figure 26.1 Potential Energy Surface of H 3 for Linear Geometry.(a) Perspective

view of the surface (schematic). (b) Contours of constant potential energy for H 3. From

R. D. Levine and R. B. Bernstein,MolecularReactionDynamicsandChemicalReactivity,

Oxford University Press, New York, 1987, p. 124. (c) Potential energy along the reaction

coordinate.

Theactivation energyis the energy that must be added to the reactants to bring

them to the state at the saddle point, which is called thetransition stateor theactivated

complex, denoted by CDF‡. The height of the maximum in the curve of Figure 26.1c

above the energy of the reactants is approximately equal to the activation energy, since

except for zero-point vibrational energy it is the minimum excitation energy that is

required for the reactants to pass over the maximum to form products. Calculations

show that the activation energy for the hydrogen reaction is lowest for the collinear

conformation, and many calculations of three-atom reactions include only the collinear

geometry. A more complete discussion would take an average over the possible values

of the angle of approach, and this has been carried out for some reactions, including

the hydrogen reaction.

Trajectory Calculations

Once the potential energy surface is obtained, the reaction can be studied by solv-

ing for the nuclear motion along the reaction coordinate. Most nuclei other than

hydrogen are sufficiently massive that classical mechanics is thought to be an ade-

quate approximation, but quantum calculations are also carried out. Neither the classical

nor quantum equations can be solved in closed form, and the motions are numerically

simulated using computer programs. If classical mechanics is used, the calculation is

carried out for a number of different trajectories. The fraction of the trajectories that