Necesidades de Formación en el Ecuador

It is vitally important to understand what a free translation means, and to be aware of the important consequences of motion through the critical configuration being a free translation. The force acting on a system is

force¼ dðPEÞ

dr ð4:19Þ

If the PE is constant, then dðPEÞ=dris zero, and the force is zero and there is no acceleration in the motion of the reaction unit as it passes through the critical configuration, i.e. through the region of the surface where the PE is constant. If there is no acceleration, then the reaction unit must pass over the barrier at a constant rate. This rate can be calculated from kinetic theory.

If the PE is not constant, then dðPEÞ=dris non-zero, and the rate of change of configuration over that region of the PE surface will vary depending on the value of dðPEÞ=dr. This happens at all points other than the critical configuration.

The concentration of activated complexes is thetotalnumber of units (X----Y----Z) per unit volume which are in the critical configuration, no matter what their past history is.

The rate of reaction is the total number of units (X----Y----Z) per unit volume which pass through the critical configuration per unit timefrom reactant valley to product valley.

Worked Problem 4.9

Question. Figure 4.27 gives a potential energy contour diagram. What can be said about configurations along the line AB? Identify the activated complex and activated intermediate in the diagram.

Answer. Along AB, rAB¼rBC. The activated complex lies at the saddle-point lying on AB. The PE contour diagram is symmetrical about the critical configura-tion. The activated intermediate lies in the well of the elliptical contours and at configurations where rBC>rAB. There is a further activated complex as marked along the line PQ.

4.3.3 An outline of arguments involved in the

configuration is constant. This rate can be calculated from kinetic theory. The concentration of activated complexes must also be calculated. This is done using equilibrium statistical mechanics, and by assuming that activated complexes are in equilibrium with the reactants, which is fully justified ifE0>kT.

The statistical mechanical contribution to transition state theory uses partition functions. These are statistical mechanical quantities made up from translational, rotational, vibrational and electronic terms, though the electronic terms can normally be ignored if the reaction occurs in the ground state throughout.

Calculation of partition functions requires spectroscopic quantities for the rota-tional and vibrarota-tional partition functions. The quantities required are moments of inertia, rotational symmetry numbers and fundamental vibration frequencies for all normal modes of vibration. The translational terms require the mass of the molecule.

All terms depend on temperature. Calculation of partition functions is routine for species for which a detailed spectroscopic analysis has been made.

Two equations emerge:

the statistical mechanical expression for the rate constant for reaction;

the equilibrium constant for formation of the activated complex from reactants.

k¼kT h

Q^6¼

Q_XQ_YZexp E0

RT

ð4:20Þ K^6¼¼ Q^6¼

Q_XQ_YZexp E0

RT

ð4:21Þ

B

Q

P A

r_AB

r_BC

Figure 4.27 A potential energy contour diagram

is the transmission coefficient and allows for the possibility that not all units (X----Y----Z) in the critical configuration may actually react,kandhare the Boltzmann and Planck constants,E0is the critical energy and can be found from the PE surface,Q^6¼ is the partition function per unit volume for the activated complex with one term missingandQX andQYZare ordinary completepartition functions per unit volume for the reactants. LikeQ^6¼,K^6¼is a quantity with one term missing. This is explained in the brief derivation given below.

Equation (4.20) is the working equation of transition state theory in statistical mechanical form.

Brief derivation of transition state theory

Reaction occurs when the reaction unit attains the critical configuration, becomes an activated complex, and passes on to product. The critical configuration occurs at the maximum on the potential energy profile and extends over a length along the reaction coordinate. The activated complex spends a time in the critical configuration, and from this the rate of passage through the critical configuration from left to right is given as

v¼

where the magnitudes of bothand are unspecified.

If N^6¼ is the total number of activated complexes, then the number which pass through the critical configuration per unit time per unit volume isN^6¼v=V, and so the rate of reaction can be given as

N^6¼v

V ¼c^6¼v

ð4:22Þ

whereis the transmission coefficient which accounts for the possibility that not all activated complexes reach product configuration, and c^6¼ is the concentration of activated complexes expressed as anumberof activated complexes per unit volume.

vcan be found from kinetic theory, and isv¼kT=2^6¼1=2

, giving rate of reaction¼c^6¼

kT 2^6¼ 1=2

ð4:23Þ where ^6¼ is the reduced mass for the change of configuration, calculable from the potential energy surface, but generally left unspecified.

The rate of reaction can also be given askcXcYZso that the calculated rate constant is given by

k¼ c^6¼ c_Xc_YZ

1

kT 2^6¼ 1=2

ð4:24Þ

TRANSITION STATE THEORY 133

There is no easy way to calculatec^6¼=c_Xc_YZexcept by assuming equilibrium between reactants and activated complexes when equilibrium statistical mechanics can be used to calculatec^6¼=cXcYZ:

K^6¼ ¼ c^6¼ c_Xc_YZ

¼ Q^6¼

Q_XQ_YZexp U0

RT

ð4:25Þ where U₀ isU at absolute zero for the process of forming activated complexes from reactants, and is equal to E0, the activation energy per mole found from the potential energy surface. The Q are complete partition functions per unit volume.

Substitution into the equation forkgives

k¼ Q^6¼ QXQYZ

1

kT 2^6¼ 1=2

exp E0

RT

ð4:26Þ

This equation contains the awkward terms and ^6¼ for which an assignment of magnitude would not be an easy task, and which conventionally is not made.

There is, however, a very convenient way out of this problem. The activated complex has been shown to have a free translation along the reaction coordinate over the distance. Statistical mechanics can furnish an expression for this quantity: the partition function for a free translation over a distance is 2^6¼kT1=2

h , and

this contains the awkward termsand^6¼. If this term is factorized out of the overall total molecular partition function for the activated complex Q^6¼, then Q^6¼ can be written as

2^6¼kT 1=2

h Q^6¼ ð4:27Þ

whereQ^6¼ is the partition function with the term for the free translation factorized out; the * summarizes the fact that this is a partition function with one term struck out.

This now results in an expression where cancellation of the awkward terms occurs:

k¼ Q^6¼ Q_XQ_YZ

2^6¼kT 1=2

h

kT 2^6¼ 1=2

exp E0

RT

ð4:28Þ

¼kT h

Q^6¼ QXQYZ

exp E0

RT

ð4:20Þ

which is the fundamental equation of transition state theory in statistical mechanical form.

Calculation of the partition functions for reactants is straightforward, but the partition function for theactivated complexneeds explanation. The activated complex has been shown to have the unique feature of a free translation along the reaction coordinate over the distance occupied by the activated complex. The statistical mechanical quantity for this free translation has already beenfactorized outfrom the total partition function for the activated complex in the derivation. This has been done simply because doing so allows cancellation of some awkward terms in the derivation of the rate constant equation. This is why the symbol has * appeared along with the symbol 6¼, this latter indicating that the process is one of forming the activated complex, often very loosely termed activation. Q^6¼ is now a partition function per unit volume for the activated complex but with one crucial term missing from it, i.e. the term for the free translation. This is more fully explained in the section below.

4.3.4 Use of the statistical mechanical form of transition state theory