Chapter 6. Reactivity in Thermalised Systems

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6. Reactivity in Thermalized Systems



requirement that

K=



k1

= e − ΔH RT

k2



(6.2)



which can only be true either at the temperature of absolute zero or if the reaction involves

no entropy change. The above equations ignore the steric factors, present in eq. (5.16),

which must involve an entropy term. Since the equilibrium constant is Kϭexp (Ϫ⌬G0/RT),

the collision theory must be modified to include an entropy of activation, as shown in

eq. (5.19). Consequently, the collision theory equation should have the form

2

k = dAB



8 RT



e − ΔS



‡



RT



e − ΔH



‡



RT



(6.3)



where the entropy of activation assimilates the steric factor. This critique by Eyring and

co-workers emphasises the fact that it is the free energy of activation, and not the heat of

activation, that determines the reaction rate.

6.1.1



Classical formulation



The TST, as Eyring’s theory is known, is a statistical-mechanical theory to calculate the rate

constants of chemical reactions. As a statistical theory it avoids the dynamics of collisions.

However, ultimately, TST addresses a dynamical problem: the proper definition of a transition state is essentially dynamic, because this state defines a condition of dynamical instability, with the movement on one side of the transition state having a different character

from the movement on the other side. The statistical mechanics aspect of the theory comes

from the assumption that thermal equilibrium is maintained all along the reaction coordinate. We will see how this assumption can be employed to simplify the dynamics problem.

Let us define a reaction coordinate s, joining reactants (s Ͻ0) to products (s Ͼ 0), and a

transition state with a fixed value of s, that is, with one degree of freedom less than a stable molecule (Figure 6.1). It is possible to derive TST from one fundamental assumption:

there is a quasi-equilibrium between the transition-state species (formed from the reactants) and the reactants themselves. The quasi-equilibrium state is characterised by the

equilibrium between the reactants and the transition-state species, and by the fact that the

concentration of these species does not vary with their disappearance to the products. This

quasi-equilibrium offers a method to calculate the concentration of transition-state species

using chemical equilibrium theories, and the dynamics problem is transformed into an

equilibrium problem, with a known solution.

Representing the transition-state species by ‡, the kinetic mechanism

K‡



⎯⎯⎯

→

A + B ←⎯⎯ ‡ ⎯ν → products

⎯

⎯

‡



6.II



leads to the bimolecular reaction rate

v = ν‡ [ ‡ ]



(6.4)



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Figure 6.1 Potential energy curve for a reaction going through a transition state ‡ contained in an

imaginary potential box with length ⌬s.



This expression indicates that the number of species transformed into products, per unit

time, is the product of their concentration and the frequency of their conversion. The frequency ‡ is the number of times, per unit time, that the transition-state species evolve

along the reaction coordinate in the direction of the products. This movement corresponds

to the conversion of an internal degree of freedom into a translational degree of freedom.

Therefore, the transition state has one degree of freedom less than a normal molecule,

because one of these is the reaction coordinate. The movement along this coordinate is

that of the relative displacement of two atoms in opposite directions. This frequency can

also be represented by the mean velocity of crossing the transition state, vq , over the

length of the transition state at the top of the barrier, ⌬s. Hence, the above equation can

be written as



v=



vq

Δs



[‡ ]



(6.5)



and the mean velocity can be obtained from the Maxwell–Boltzmann distribution of molecular velocities over one dimension

∞



vq



− 1 mvq kBT

2



∫ e

=

∫ e

0



∞



−∞



vq dv



− 1 mvq kBT

2



dv



=



k BT

2 m



(6.6)



where the limits of integration in the denominator are taken from –ϱ to ϱ to allow for the

fact that the complexes are moving in both directions, whereas in the numerator the limits

are zero to infinity because it is the mean velocity in the direction of the products that is

required. The mass m is the effective mass of the complex in this movement.



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6. Reactivity in Thermalized Systems



Using the quasi-equilibrium approximation,



[‡ ]

[A ][B]



(6.7)



k BT 1 ‡

K [ A ][ B]

2 m Δs



(6.8)



K‡ =



the reaction rate becomes

v=



The calculation of the rate constant is now focussed on the calculation of the quasiequilibrium constant. Statistical mechanics relate the equilibrium constant with the structures and energies of the reactants and products. These relations are developed in detail in

Appendix II. It is shown that the equilibrium constant for the system

K‡



⎯⎯⎯

→

A + B ←⎯⎯ ‡

⎯



(6.III)



can be written in terms of the molar partition functions, per unit volume, of A, B and ‡,

K‡ =



Q‡

QA QB



e − ΔE0



RT



(6.9)



The partition functions are a measure of the states that are thermally accessible to the molecule at a given temperature. In the equation above, the energetic factor

‡

A

B

ΔE0 = E0 − E0 − E0



(6.10)



is the difference between the zero-point energies (ZPEs) of the transition state and the

reactants, per mole, at Tϭ0 K. It is the amount of energy that the reactants must acquire at

0 K to reach the transition state.

As discussed above, one of the vibrational degrees of freedom of ‡ in mechanism (6.III)

became a translation along the reaction coordinate in mechanism (6.II). This can be factored out of the transition state in mechanism (6.II), Q‡, and related to the partition functions of the normal molecule, Q‡,

Q‡ = Qtrans,q Q ‡



(6.11)



As will be seen below, the translational partition function along the reaction coordinate has

the form



Qtrans,q



(2

=



m k BT )



1



2



Δs



h



(6.12)



and the rate constant can be written, using eq. (6.8)

k BT ‡

K

h

k T Q ‡ ΔE0

= B

e

h QA QB



k=



RT



(6.13)



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It is quite remarkable that the combination of the two terms involving the properties of

the transition state, namely the velocity of crossing the transition state and the translational

partition function along the reaction coordinate, gives the quantity kBT/h, which is the same

for reactants and reactions of all types. It represents the frequency with which any transition state becomes a product at a given temperature and its value is about 6ϫ1012 secϪ1

at 300 K. Additionally, the length ⌬s representing the transition state at the top of the barrier, is eliminated in this procedure, and there is no obvious restriction on its magnitude.

The most interesting achievement of TST is that the calculation of the partition functions

of reactants and transition state leads to the reaction rate, provided that the energy term is

known.

The constant K‡ is exactly analogous to any other equilibrium constant, and hence

should be related to ⌬G‡, ⌬H‡ and ⌬S‡, the standard free energy, enthalpy and entropy

changes, respectively, accompanying the formation of the transition state from the reactants by means of the familiar thermodynamic relationships. In this manner, eq. (6.13) can

also be written in the form

G

kBT − ΔRT

k T ΔS − ΔH

e

= B e R e RT

h

h

‡



k=



‡



‡



(6.14)



that satisfies the requirement that the reaction rate is determined by the free energy of activation. This is an improvement over the simple collision theory in that it gives a precise

and simple significance to the frequency factor. It must be emphasised that ⌬G‡, ⌬H‡ and

⌬S‡ refer to the reactants and transition state in their standard states, although the conventional zero superscript is omitted.

6.1.2



Partition functions



Statistical mechanics offers well-established methods to obtain the partition functions of

simple systems. In view of their relevance to the calculation of the pre-exponential factor

of eq. (6.13), they are briefly reviewed below for translational, rotational and vibrational

partition functions. The electronic partition function is assumed here to be identical to the

degeneracy of the electronic ground state, Qelecϭ gelec.

The translational partition function is conveniently derived from the energy levels accessible to a particle of mass m moving in a potential energy box of length l. According to the

Schrödinger equation for this system, the allowed energy levels are



En =



n2 h 2

8ml 2



(6.15)



where n is the quantum number associated with the energy levels. The one-dimensional

partition function is, then,

∞

⎛ − n2 h2 ⎞

Qtrans = ∑ exp ⎜

2

⎝ 8ml kBT ⎟

⎠

n=1



(6.16)



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6. Reactivity in Thermalized Systems



According to the definition of the partition function as a measure of the number of thermally accessible states at a given temperature, there are very large numbers of translational

states accessible for molecular systems at typical temperatures because the energetic separation between the translational states is much smaller than the thermal energy, 3/2 kBT

from eq. (2.43). Therefore, the summation can be replaced by appropriate integration, with

the integral having a standard form

∞

⎛ − n2 h 2 ⎞

Qtrans = ∫ exp ⎜

dn

0

⎝ 8ml 2 kBT ⎟

⎠



1 ⎛ 8 ml 2 kBT ⎞

= ⎜

⎟

2⎝

h2

⎠



1



2



(2

=



m k BT ) 2 l

1



(6.17)



h



for each dimension. The translational energy for the three dimensions is

E trans = E trans, x + E trans, y + E trans, z



(6.18)



and, consequently, the translational partition function for the three dimensions is

Qtrans = Qtrans, x Qtrans, y Qtrans, z



(6.19)



or

Qtrans



⎛ 2 mkBT ⎞

=⎜

⎝ h2 ⎟

⎠



3



2



V



(6.20)



where Vϭl 3.

The vibrational partition function of a harmonic oscillator has equally spaced energy

levels (eq. (1.5)). Representing the separation between the energy levels by , the vibrational partition function can be written as a geometric progression:

Qvib = 1 + e −

=



1

1 − e−



kBT



kBT



+ e −2

=



kBT



+



= 1 + e−



kBT



)



kBT 2



+

(6.21)



1

1− e



(



+ e−



− h ν kBT



where we used the relations

S = 1 + x + x2 +

xS = x + x 2 + x 3 +

1

S=

1− x



(6.22)



The linear, rigid rotor has energy levels given by



EJ =



J ( J + 1) ⎛ h ⎞

⎜ ⎟

2I ⎝ 2 ⎠



2



(6.23)



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where Iϭ miri2 is the moment of inertia (ri is the perpendicular distance between atom i

of mass mi and the rotation axis), and Jϭ0, 1, 2, . . . is the rotational quantum number. The

linear, rigid rotor has degenerate energy levels, and the rotational degeneracy is 2Jϩ1. The

corresponding rotational partition function is

∞

⎡ J ( J + 1) ⎛ h ⎞ 2 ⎤

Qrot = ∑ (2 J + 1) exp ⎢ −

⎜ ⎟ ⎥

J =0

⎢

⎣ 2 IkBT ⎝ 2 ⎠ ⎥

⎦



(6.24)



In general, at room temperature, the difference between the rotational energy levels is

much smaller than kBT. Under these conditions, many rotational levels are occupied and it

is reasonable to replace the summation by the integral

⎡ J ( J + 1) ⎛ h ⎞ 2 ⎤

2 J + 1) exp ⎢ −

(

⎜ ⎟ ⎥ dJ

0

⎢

⎥

⎣ 2 IkBT ⎝ 2 ⎠ ⎦

2

8 I k BT

=

h2



Qrot = ∫



∞



(6.25)



The H2 molecule is one of the rare exceptions to the conditions formulated above,

because its small mass, hence the small moment of inertia, leads to relatively large energetic separations between the energy levels. Given the H–H bond length lHHϭ0.741 Å,

we have I ϭ 4.5ϫ10Ϫ48 kg m2 and ⌬EJ ϭ 3 kJ molϪ1, whereas at room temperature, 3/2RT ϭ

3.7 kJ molϪ1.

For a polyatomic molecule with moments of inertia Ia, Ib and Ic along the principal axis,

the rotational partition function is

Qrot =



6.1.3



12



⎛ 8 2 I a k BT ⎞

⎜

⎟

h2

⎝

⎠



12



⎛ 8 2 I b k BT ⎞

⎜

⎟

h2

⎝

⎠



12



⎛ 8 2 I c k BT ⎞

⎜

⎟

h2

⎝

⎠



12



(6.26)



Absolute rate calculations



Using eq. (6.13), together with eqs. (6.20), (6.21), (6.25) or (6.26), and knowing the ZPE

difference between the transition state and the reactants, it is now possible to calculate

“absolute” reaction rates. In practice, TST can only be applied if the structure of the transition state and its vibrational levels are also known, because the former is required for the

calculation of the rotational partition function of the transition state, and the latter enter the

transition-state vibrational partition function. These data and ⌬E0 can be obtained from ab

initio calculations or experimental information employed in the making of potential energy

surfaces (PESs). The rapid development of computers and software has made it possible

to carry out accurate ab initio calculations of transition-state properties for many tri-atomic

and some tetra-atomic systems in the gas phase. The best-known system is the atom

exchange in the H ϩ H2 system, and the properties of its linear transition state are shown

in Table 6.1.



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6. Reactivity in Thermalized Systems



Table 6.1

Data from the PES obtained by the double many-bodied expansion (DMBE) method for the

linear transition state {H–H–H}‡ [5]

⌬Vcl‡ (kJ molϪ1)



‡

lH1H2 = lH2 H3 (Å) νsym (cmϪ1)



40.4



0.9287



νasym (cmϪ1) νbend (cmϪ1) ⌬Vad‡ (kJ molϪ1)

1493ia



2067



899



38.2



a



this is an imaginary frequency



Using the data presented in Appendix III, on the properties of stable molecules, and in

Table 6.1, on the linear transition state of the H+H2 reaction, it is possible to calculate the

rate constant for this reaction as the product of the partition function ratios

k=



k BT

h



⎛ Q‡ ⎞ ⎛ Q‡ ⎞ ⎛

⎞

Q‡ V

− ΔE0

⎜

⎟ ⎜

⎟ ⎜

⎟ e

QH QH2 ⎠ ⎝ QH QH2 ⎠ ⎝ QH V QH2 V ⎠

⎝

vib

rot

trans



RT



(6.27)



that can be calculated separately. The ratio of the translational partition functions at 440 K is

⎛ mH + mH2 ⎞

⎛

⎞

Q‡ V

=⎜

⎟

⎜

⎟

⎝ QH V QH2 V ⎠ trans ⎝ mH mH2 ⎠



32



⎛ h2 ⎞

⎜2 k T⎟

⎝

B ⎠



(



32



)(



= 2.683 × 10 40 kg3 2 3.900 × 10 −71 kg3 2 m 3



(6.28)



)



= 1.05 × 10 −30 m 3

and the ratio of the rotational partition function for the linear transition state is simply

given by the ratio of the moments of inertia



⎛ Q‡ ⎞

I‡

=

⎜

⎟ =

⎝ QH2 ⎠ rot I H2



(



‡

H1H 2



mH1 l



)



2



(



+ mH3 l



‡

H 2 H3



)



2



mH2 mH3

mH2 + mH3



(m

−



l



(



)



‡

H1 H1H 2



‡

− mH3 lH2 H3



mH1 + mH2 + mH3



lH2 .eq



)



2



(6.29)



2



= 6.28

Finally, the ratio of the vibrational partition functions is

⎛ h νH 2 c ⎞

1 − exp ⎜ −

⎝ k BT ⎟

⎠



⎛ Q‡ ⎞

⎜

⎟ =

2

⎝ QH2 ⎠ vib ⎡

⎛ h νsym c ⎞ ⎤ ⎡

⎛ h νbend c ⎞ ⎤

⎢1 − exp ⎜ −

⎥ ⎢1 − exp ⎜ −

⎥

k BT ⎟ ⎦

⎝

⎠

⎠

⎝ k BT ⎟ ⎦ ⎣

⎣

= 1.12



(6.30)



where we considered two degenerate bendings at the transition state, one in the plane, the

other in and out of the plane, and the symmetric stretching. The anti-symmetric stretching



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has become the reaction coordinate and has an imaginary value. The vibrational frequency

of H2, employed in the construction of its Morse curve in Appendix III, is vH2 ϭ 4400 cmϪ1.

The last parameter required for the calculation of the rate constant for the H+H2 reaction is the energetic factor ⌬E0. As mentioned before, it corresponds to the difference

between the ZPEs of the transition state and the reactants. In Table 6.1, the classical potential energy difference between transition state and the reactants is represented by ⌬V‡ and

the barrier corrected for the ZPE is represented by ⌬Vad‡. It is often designated as the vibrationally adiabatic barrier and corresponds to ⌬E0. The rate constant calculated at 440 K

with the universal frequency of TST, ⌬Vad‡ ϭ38.2 kJ molϪ1 and the ratio of the partition

functions calculated above is 1.2ϫ106 dm3 molϪ1 secϪ1, fairly close to the experimental

value of 5.7ϫ106 dm3 molϪ1 secϪ1.

6.1.4.



Statistical factors



An even better agreement can be obtained considering the role of statistical effects in the

transition state. The origin of such effects can be easily understood comparing the reactions

H + H 2 → Η ⋅⋅⋅⋅H ⋅⋅⋅⋅H ‡ → H 2 + H



(6.IIIa)



H + HD → Η ⋅⋅⋅⋅H ⋅⋅⋅⋅D → H 2 + D



(6.IIIb)



‡



The partition functions for the two reactions are similar. However, the first reaction may

result from two possible arrangements of the hydrogen molecule. This is clearer when we

number each identical atom of the molecule

H + H1 − H 2



(6.IVa)



H + H 2 − H1



(6.IVb)



In contrast, reaction (6.IIIb) can only occur from one transition-state arrangement,

because the other one leads to a different product, HDϩH. It is natural that reaction (6.IIIa)

is favoured by a statistical factor that is twice that of reaction (6.IIIb). The basis for the

introduction of such statistical factors relies on the nature of the TST: the reaction rate is

determined by the number of species that enter the transition state and do not return to the

reactants. A simple method to obtain the statistical factors is to label the identical atoms and

count the number of equivalent transition states. For example, the reaction

H + CH 4



⎯⎯

→

←⎯

⎯



H 2 + CH 3 (planar)



(6.V)



leads to the labelled species represented in Figure 6.2. The forward reaction leads to four

sets of products with different labels, thus f ϭ 4. The reverse reaction also leads to four

distinct sets of products, and thus r ϭ 4. The equilibrium constant written as the ratio of

forward and reverse reactions agrees with that of equilibrium thermodynamics

K eq =



kf [H 2 ][ CH 3 ]

=

=

kr

[H ][CH 4 ]



f



QH2 QCH3



r



QH QCH4



e − ΔE0



RT



(6.31)



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6. Reactivity in Thermalized Systems



H1H2+CH3H4H5



H1H3+CH2H4H5

statistical factor σ = 4



H1+CH2H3H4H5

H1H4+CH2H3H5



H1H5+CH2H3H4



H1+CH2H3H4H5



H1+CH3H2H4H5

1 2



statistical factor σ = 4



3 4 5



H H +CH H H



2



1 3 4 5



H +CH H H H



H2+CH3H1H4H5



Figure 6.2 Distinct species present in the reaction H+CH4→H2+CH3 (planar). The methanes

CH2H3H4H5 and CH3H2H4H5 as well as CH1H3H4H5 and CH3H1H4H5 are enantiomers.



H+

R2



R1

C



R1

R3



R2



R

H3



Figure 6.3 Addition of a hydrogen atom to an asymmetric methyl radical leading to two enantiomers. The statistical factor of the forward reaction is twice that of the reverse reaction.



The use of statistical factors requires extra care when the reaction introduces or removes

an asymmetric centre. For example, the addition of a hydrogen atom to an asymmetric

methyl radical leads to two enantiomers, because the H atom may bind to either side of the

symmetry plane (Figure 6.3). In the forward direction we have f ϭ2, but in the reverse

direction, rϭ1 and the equilibrium constant is multiplied by a factor of two

f



QCHR1 R2 R3



r



K eq =



QH QCR1 R2 R3



e − ΔE0



RT



(6.32)



The analyses of statistical factors may provide important information on the symmetry

of the transition state. For example, the isomerisation of cyclopropane to propene may proceed via a symmetrical or an asymmetrical transition state (Figure 6.4). In the first case,

f ϭ2, whereas in the second case f ϭ1. The symmetrical transition state corresponds to



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C2



D3h

H



H

C



110°

C



H



60°

C

C



H



‡



H



H



C



H

C



H

H



H



H



H



H



H



C

H



H



H



‡



H



H



C



H



‡



H



H



C

H



C2



H



C

C



H



C



C



C



H



C



H



H



C1



H



H



H

120° H



H

C



H



C



H



H



C



H



C



120°

C



C



Cs

H



H

H



H



‡



H



H

C



H



H



110°



H



H

H



H



C



C



H



H



60°

C

C



C



H



H

H

D3h



C2v



Figure 6.4 Symmetrical and asymmetrical transition states in the isomerisation of cyclopropane to

propene. The symmetrical transition state does not have any physical meaning because it requires

the splitting of the reaction path into two paths at the transition state.



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6. Reactivity in Thermalized Systems



the generation of two reaction paths from the same transition state. This cannot happen in

a PES, where one transition state can only connect one reactant’s valley to one product’s

valley. Consequently, the symmetrical transition state is unphysical.

The consideration of statistical factors in the TST is a simple and necessary refinement

of this theory. When the statistical factor is included, the classical rate constant of TST is

expressed as

kTST =



‡

k BT Q ‡

e − ΔVad

h QA QBC



RT



(6.33)



where explicit mention is made of the fact that the energetic factor is the vibrationally

adiabatic barrier, ⌬Vad‡. The inclusion of a statistical factor in the rate formulation is not

compatible with the inclusion of symmetry numbers in the rotational partition functions.

Indeed, such symmetry numbers were not included in our previous calculation of the

partition functions. Now that we know that ϭ2 for the H+H2 reaction, we can calculate

a classical rate of 2.4 ϫ 106 dm3 molϪ1 secϪ1 at 440 K. The calculated rate now

approaches the uncertainty of the experimental value, 5.7 ϫ 106 dm3 molϪ1 secϪ1, but

remains too slow.

6.1.5



Beyond the classical formulation



Further improvement of TST requires a critical analysis of its fundamental assumptions.

The development of this theory was based on the following approximations: (i) the distribution of energy in the reactants and transition state follows the Boltzmann distribution

law, (ii) every system that crosses the transition state becomes a product and (iii) the

movement along the reaction coordinate is adiabatic and can be described by classical

mechanics. Next, we analyse each one of these approximations to identify the conditions

in which they fail and see how the theory can be improved in terms of both generality and

accuracy.

The quasi-equilibrium assumption, that is the basis for the use of the Boltzmann distribution law, may lose its validity for rapid reactions. In such reactions, the most energetic

reactant molecules may disappear very rapidly and the concentration of species at the transition state may be lesser than that for a true equilibrium. In practice, even when

Ea /RT 1.3, as in the Cl+HBr →HClϩBr hydrogen atom abstraction, internal-state nonequilibrium effects are very small [6].

The assumption that the transition state is only crossed once by every reactive system

neglects several types of trajectories that the system may follow, as illustrated in Figure 6.5.

In fact, the classical TST assumes that the reactive systems can only follow trajectories of

type 1, illustrated in this figure. However, trajectories of types 2, 5 and 6 also cross the

transition state in the direction of the products, but do not correspond to reactive systems.

Furthermore, although trajectories of type 3 are reactive, they cross the transition state

twice in the direction of the products, but only contribute once to the actual rate. In general, a trajectory with just the energy necessary to cross the transition state will do so at a

region very close to the saddle-point, which is the point of minimum energy separating the