## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 14. Some Special Topics: Reactions in Solutions

### 14.2. Transition State Theory

Transition state theory (TST), also known as “activated complex theory”, is an important follow-up from statistical mechanics, used throughout in chemistry and physics, which has been developed mainly by Eyring and coworkers. In this section we will illustrate this theory for chemical reactions in the gas phase.

**14.2.1 The Equilibrium Constant**

The first result we need is the equilibrium condition. Associating (as in __Chapter 5__) the macroscopic internal energy *U* with the statistical mechanics expression for the average energy ⟨*E*⟩, the macroscopic number of molecules *N* with the average number of molecules ⟨*N*⟩ and recalling that the grand canonical partition function reads *kT* ln *Ξ* = *PV*, we conclude from the expression for the grand canonical entropy

(14.12)

that ∂*S*/∂*N* = –*μ*/*T*. The equilibrium condition at constant *U* and *V* for a system of several components thus becomes d*S* = ∑* _{j}* (∂

*S*/∂

_{j}*N*) d

_{j}*N*= –∑

_{j}*(*

_{j}*μ*) d

_{j}/T*N*= 0, where the index

_{j}*j*denotes the various reactants and products. If we have a reaction

where R and P denote reactants and products, respectively, we obtain

Using the relation *μ _{j}* =

*kT*ln(

*N*/

_{j}*z*), obtained from

_{j}*μ*= ∂

_{j}*F*/∂

*N*with

_{j}*z*denoting the partition function of component

_{j}*j*, the result is

__(14.13)__

where *K _{c}*(

*T*) is the

*equilibrium constant*. This relationship is known as the

*law of mass action*. The reference level of energy for each factor in

__Eq. (14.13)__is the same. However, for the evaluation of the various partition functions it is more convenient to use the ground state of each species as a reference. Let us take the gas-phase chemical reaction AB + C ↔ A + BC as a simple example with

*K*(

*T*) =

*z*

_{A}

*z*

_{BC}/

*z*

_{AB}

*z*

_{C}. Shifting the reference level of all species to an arbitrary level and denoting the partition function with respect to the ground state for each species by

*z*′, we obtain

__(14.14)__

where in the last step the primes have been removed (using, from now on, as reference level for each of the species the ground state) and Δ*E* ≡ (*E*_{A} + *E*_{BC} − *E*_{AB} − *E*_{C}) represents the difference in ground-state energies of the reactants and products.

**14.2.2 Potential Energy Surfaces**

The second concept we need is the potential energy surface. We note that the potential energy of a system containing atoms or molecules can be written as a function of special combinations of the nuclear spatial coordinates, usually referred as *generalized* or *normal coordinates*. Generally, the pictorial representation of the potential energy hypersurface is difficult. The concept can be grasped from a simple example, for which we take the collinear reaction between three atoms A, B, and C. In __Figure 14.1__ a map is shown for this reaction with, as axes, the distances AB and BC, respectively. The map shows two valleys separated by a col, and in general, thermal fluctuation creates continuous attempts to pass from one valley to another. In order to calculate the rate constant for the chemical reaction A + BC ↔ AB + C, we must calculate all trajectories on the potential energy surface. With this in mind we need the concept of a *dividing surface*, defined as a surface which cannot be passed without passing a barrier. In __Figure 14.1__, for example, one of the dividing surfaces is given by *R*_{AB} = *R*_{BC}. When calculating the rate constant, we must take into account only those trajectories that do pass the dividing surface. An upper limit to the reaction rate *r*(*T*) is then given by

__(14.15)__

where *j*(*T*) represents the total amount of reactant systems crossing a dividing surface per unit volume and per unit time at a temperature *T* – that is, the flux. __Equation (14.15)__ is known as the *Wigner variational theorem*. All statistical methods for computing reaction rates are based on this principle. The calculation of all allowed trajectories to estimate the flux *j*(*T*) provides an enormous task if performed in a rigorous fashion, and therefore approximate methods are normally introduced. The variation theorem provides the opportunity to make an optimum choice for the dividing surface – that is, to select the surface that provides the lowest flux *j*(*T*). This may be achieved by using a surface depending on several parameters and choosing these parameters in such a way that a minimum *j*(*T*) is obtained.

** Figure 14.1** The potential energy surface for the reaction A + BC ↔ AB + C, showing a col between two valleys.

**14.2.3 The Activated Complex**

The simplest approach along the above lines is to take into account only the trajectory following the minimum energy path from one valley to the other valley. In the case of the chemical reaction A + BC ↔ AB + C, this means the path from the configuration represented by A + BC (point R in __Figure 14.1__) to the configuration represented by AB + C (point P in __Figure 14.1__). The coordinate along this path is called the *reaction coordinate*. The top of the col between the two valleys, which actually is a saddle point of the potential energy surface, is known as the *transition state*. A configuration in the neighborhood of the transition state is addressed as an *activated complex*. Transition state theory is based on a number of assumptions. The first assumption is that the optimum dividing surface passes through the transition state and is perpendicular to the reaction coordinate. The second assumption is that activated complexes are at all times in equilibrium with both the reactants and the products. For the forward reaction A + BC ↔ (ABC)^{‡}, where (ABC)^{‡} denotes the activated complex, this implies that the equilibrium constant *K*_{for} is given by

__(14.16)__

where, as usual, [X] denotes the concentration of species X. The equilibrium constant *K*_{rev} for the reverse reaction AB + C ↔ (ABC)^{‡} is

__(14.17)__

Since in __Eq. (14.16)__ and __(14.17)__ [(ABC)^{‡}]_{for} and [(ABC)^{‡}]_{rev} are equal, we have [(ABC)^{‡}]_{for} = [(ABC)^{‡}]_{rev} = ½[(ABC)^{‡}]. The combination of these two conditions together is called the *quasi-equilibrium assumption*. Finally, it is assumed that once the transition state is reached the reaction completes – that is, the reactants do not return to the nonreacted state.

Here, we consider the forward reaction in some detail. __Equation (14.14)__ shows that the equilibrium constant for the reaction A + BC ↔ (ABC)^{‡} can be written as

__(14.18)__

We also assume that the partition function of each species X can be (approximately) factorized as

where *z*_{ele}, *z*_{tra}, *z*_{vib}, and *z*_{rot} represent the electronic, translation, vibration, and rotation partition functions, respectively. The contribution of *z*_{ele} reduces to the degeneracy number of the ground state, usually 1, since we assume the reaction to proceed on a potential energy surface, which exists only by virtue of the adiabatic assumption. Therefore, no electronic transitions are allowed.

If we have sufficient information on the transition state – that is, we know the structure and relevant force constants – we can construct its partition function. The translation and rotation partition function do not provide a problem in principle, as they can be constructed as for normal molecules. However, the vibration partition function requires some care. The usual normal coordinate analysis of an *N*-atom molecule can be made, and from that we obtain 3*N *– 6 normal coordinates for a nonlinear molecule or 3*N *– 5 normal coordinates for a linear molecule. Of these vibration coordinates, all but one has a positive coefficient in the second-order terms in the potential energy expansion. The last, negative coefficient corresponds to an imaginary frequency for this vibration coordinate, and represents the reaction coordinate. This implies that, in the transition state, a small fluctuation in the reaction coordinate will lead to an unstable configuration with respect to this coordinate. It is customary [4]__ ^{3)}__ to treat this coordinate as a translation coordinate over a small length

*δ*at the top of the potential energy barrier with a partition function

*z*= (

*δ*/

*h*)(2π

*m*

^{‡}

*kT*)

^{1/2}. The complete partition function

*z*

_{(ABC)}

^{‡}for the activated complex is then

__(14.19)__

where *z*^{‡} represents the partition function for the remaining coordinates – that is, the true vibration coordinates, the translation and rotation coordinates. Further, *m*^{‡} denotes the mass of the activated complex and *k*, *T* and *h* have their usual meanings.

The concentration of the activated complexes due to the forward reaction is given by __Eq. (14.16)__, where the forward equilibrium constant is given by __Eq. (14.18)__. Since there are [(ABC)^{‡}]_{for} complexes per unit volume which populate the length *δ* and which are moving forward, the forward reaction rate *r*_{for} is *r*_{for} = [(ABC)^{‡}]_{for}/*τ*, where *τ* is the average time to traverse the length *δ*, given by *τ* = *δ */*v*_{ave}. Thus, we need the average velocity *v*_{ave} in one direction over the length *δ*. Using the same approximation of a free translatory motion again, we borrow from kinetic gas theory [5]

__(14.20)__

Combining __Eq. (14.16)__, __(14.18)__, __(14.19)__ and __(14.20)__ with *r*_{for} = [(ABC)^{‡}]*v*_{ave}/*δ*, we obtain

__(14.21)__

A similar expression is obtained for the reverse reaction, and it is easily verified that the forward and reverse reactions are in equilibrium. The forward rate constant *k*_{for} can thus be calculated given the relevant information. However, for a first-principles calculation many pieces of information are required: the energy barrier for the reaction; the structure; and the force constants associated with the dynamics of the reactants, products, and activated complex. Typically, this information is incompletely available. This is also true for other mechanisms, although of course for mechanisms other than gas reactions different arguments apply. The exponential dependence on the barrier energy is generally valid, however, and rationalizes the generally observed *Arrhenius-type behavior*.

It remains to be discussed how to connect the results to experimental data. Experimentally, it is often observed that, if the logarithm of the rate constant is plotted against the reciprocal temperature, a straight line is obtained. The gradient of this line is used to define the (*empirical*) *activation energy* *E*_{act} by

__(14.22)__

Substituting __Eq. (14.21)__, taking logarithms, and differentiating with respect to *T* yields

Using __Eq. (14.22)__ and the van't Hoff relation , where Δ*U*^{‡} is the change in energy, we obtain

This equation thus provides a link between the experimentally observed *E*_{act} and Δ*U*^{‡}. It should be noted that neither *E*_{act} nor Δ*U*^{‡} is identical to Δ*E*, which appears in __Eq. (14.18)__ because of the temperature dependence of the various partition functions. For the various partition functions, the temperature dependencies read

For the vibrations, however, the behavior is less simple. For *kT *<< *ħω*, *Z*_{vib} ≈ *T*^{0}, while for *kT *>> *ħω*, *Z*_{vib} ≈ *T*, where *n* is the number of vibrational modes of the molecule. At intermediate temperature, *Z*_{vib} ≈ *T ^{a}* with 0 ≤

*a*≤

*n*. For a restricted (intermediate) temperature range

*a*is approximately constant. In total, this yields

explaining the modified Arrhenius equation. As noted above, typically |*m*| < 2.

Finally, we note that formalism can also be interpreted thermodynamically, and to this purpose we write __Eq. (14.21)__ in terms of molar quantities

(14.23)

(14.24)

with Δ*S*^{‡}, Δ*H*^{‡}, Δ*U*^{‡} and Δ*V*^{‡} the molar entropy, enthalpy, energy, and volume of activation, respectively. In a gas reaction, the term *P*Δ*V*^{‡} may be put equal to *RT*Σ* _{j}ν_{j}*, where Σ

*is the stoichiometric sum for activated complex formation. For a unimolecular reaction the stoichiometric sum Σ*

_{j}ν_{j}*= 0 and Δ*

_{j}ν_{j}*H*

^{‡}= Δ

*U*

^{‡}=

*E*

_{act}−

*RT*, or

*E*

_{act}= Δ

*H*

^{‡}+

*RT*. Consequently,

*A*= e(

*kT*/

*h*)exp(Δ

*S*

^{‡}/

*R*). For a bimolecular reaction Σ

*= −1 and Δ*

_{j}ν_{j}*H*

^{‡}= Δ

*U*

^{‡}−

*RT*=

*E*

_{act}− 2

*RT*, or

*E*

_{act}= Δ

*H*

^{‡}+ 2

*RT*. Consequently,

*A*= e

^{2}(

*kT*/

*h*)exp(Δ

*S*

^{‡}/

*R*). In solution, the volume term is nearly always negligible, and therefore Δ

*H*

^{‡}≅ Δ

*U*

^{‡}. We see that the pre-exponential factor in the empirical expression

*k*

_{rea}=

*A*exp(−

*E*

_{act}/

*RT*) corresponds approximately to the entropy.

Here, our brief overview on gas-phase TST ends, and in the next sections we apply TST to reactions in liquids. It must be re-emphasized that, for a detailed calculation, a considerable amount of information is required.

Problem 14.1: The dissociation of I_{2}

Consider the reaction I_{2} ↔ 2I at 300 K. Note that the ground state for the I atom is ^{2}P_{2/3} (hence fourfold degenerate), while that for the I_{2} molecule is (hence nondegenerate). For I_{2} the rotational temperature *θ*_{rot} = 0.054 K, the vibrational temperature *θ*_{vib} = 308 K, the dissociation energy from the ground state is *D* = 1.5417 eV, and the molecular mass *m*_{I} = 127 g mol^{−1}.

**a)** Show that the translational and electronic partition function for the I atom are *z*_{tra} = *Λ*^{−3}*V* and *z*_{ele} = 4 exp[−½(*D*/*kT* + *θ*_{vib}/2*T*)], respectively.

**b)** Show that the translational, rotational, vibrational and electronic partition function for the I_{2} molecule are *z*_{tra} = 2^{−3/2}*Λ*^{−3}*V*, *z*_{rot} = *T*/2*θ*_{rot}, *z*_{vib} = [exp(*θ*_{vib}/2*T*) − exp(−*θ*_{vib}/2*T*)]^{−1}, and *z*_{ele} = 1, respectively.

**c)** Show that *K _{c}* = 32(π

*mkT*/

*h*

^{2})

^{3/2}(

*θ*

_{rot}/

*T*)[1 − exp(−

*θ*

_{vib}/

*T*)]exp(−

*D*/

*kT*), and that

*K*=

_{P}*kTK*.

_{c}**d)** Calculate the numerical value for *K _{P}* at 300 K.