KINETIC PROPERTIES AT HIGH PRESSURE
3.2.1 Molecular theory of reaction rate constants
The effect of pressure on the reaction rate constant can be interpreted by both the collision-, and the transition state or activated complex theories. However, it has generally been found that the role of pressure can be evaluated more clearly by the transition state approach [3].
The development of transition state theory goes back to Eyring [4], and Evans and Polanyi [5]. The concept assumes that the energy of a bimolecular reaction starting from the initial reactants A and B
a . A + b - B >X # > c . R + d . S (3.2-1)
proceeds to a state of maximum energy, the so-called transition state, and then decreases as the products R and S separate (Fig. 3.2-1). At the peak, the reactants have been brought to the degree of closeness and distortion such that a small distortion in an appropriate direction will send the system in the direction of products. This critical configuration is called the activated complex, X # .
This complex is in equilibrium with the reactants, A and B. The equilibrium product for the bimolecular reaction under consideration is defined by:
K # =
[A] a .[B] b (3.2-2)
X # activated complex
A, B
initial reactants ~ R, S
products Reaction coordinate Figure 3.2-1. Reaction profile.
*) References at the end of section 3.3
where [A] and [B] denote the concentrations of the reactants.
The quantity K # is related to the rate constant, k, through the equation k B . T #
k = x " . K
h (3.2-3)
in which k8 is the Boltzmann's constant, T stands for the absolute temperature, and h is Planck's constant. The product h / k ~ . T is the lifetime of the complex in the transition state. The transmission coefficient, Z, is a factor which defines the probability that the activated complex will decompose into the products, R and S, rather than to the original species, A and B.
Glasstone, Laidler and Eyring considered that the probability- or transmission coefficient, Z, is close to unity, and is independent of temperature and pressure [6].
We transform eqn. 3.2-3 by logarithm, to In k = In K # + In T + const.
and differentiate. At constant temperature we obtain:
(3.2-4)
O l n k
@ T = ~ OlnK# I OP T (3.2-5)
The equilibrium product is related to the standard change of free energy, AG #, when the transition state is formed from the reactants.
OlnK # 1 OAG # Av #
ap R . T c3p R . T (3.2-6)
It should be noted that k and K # must be based on the same concentration scale. If we select the mole-fraction scale, x, we obtain:
c31nk x
Op
Av #
= - ~ (3.2-7)
R . T T
where
#-(a vA +b.vB)
AV # = V x (3.2-8)
Here, Av # is the activation volume. It is the excess of the partial molar volume of the transition state over the partial molar volume of the initial species, at the composition of the
Eqn. 3.2-7 is comparable to the well-known Arrhenius law, Olnk/OT = - Ea/RT, which describes the dependence of the rate constant on the temperature by means of the energy of activation, E,. For reactions occurring in the liquid phase it is the practice to use molar concentrations instead of mole fractions. This implies some complications. The equilibrium product is now defined by molar concentrations. The quasi-thermodynamic development leads to:
K # = K#x 9 vs(a+b) (3.2-9)
in which vs is the molar volume of the pure solvent. The derivative (eqn. 3.2-5) assumes the form:
O(R. T. lnkc) I
~P c=O
- ( A v # f ~ - a . . . .
= + 0 b) R T K S (3.2-10)
where (Av#) ~176 is the activation volume when the solution is nearly infinitely dilute, and
s l lnvsl
@ r
(3.2-11)
denotes the compressibility coefficient of the solvent.
Eqn. 3.2-10 does reduce to eqn. 3.2-12 Olnk[ = Av #
Op Ir ~ r (3.2-12)
for first-order reactions when a = 1 and b = 0. Nevertheless, it is common practice to evaluate the dependence of the rate constant on the pressure by using eqn. 3.2-12 also when molar concentrations are used to express the composition of a solution. This simplification is justified by the fact that the compressibility coefficient of liquids is only small and is in the
range of 1 - 4 cm3/mol.
3.2.2 Activation volume
The activation volume is a composite function to which different effects contribute. It can be split into two major terms:
AVR # the change in volume of the reacting molecules of each partial reaction;
Av~ the change in volume of solvent, arising from changes in electrostatic forces.
3.2.2.1 Terms contributing to AVR #
The main contribution to Av~ results from the formation of new bonds and the stretching of existing bonds in the transition process. Minor contributions are associated with the relaxation of parts of the molecules remote from the reacting centers.
In an unimolecular dissociation
AB--> A .... B---> A + B (3.2-13)
the covalent bond in the molecule AB is first stretched during the activation process. The activated complex will have a larger volume than the initial state, and therefore the activation volume is positive for this type of reaction.
A typical unimolecular reaction is the decomposition of organic peroxides for which always positive activation volumes of up to 15 cm3/mol have been observed. The decomposition of di(t-butyl)peroxide, an effective initiator for the high pressure polymerisation of ethylene, into two t-butoxyradicals, exhibits a positive activation volume of 13 cm3/mol (Table 3.2-1, a).
When new bonds are formed as in the association
A+B---> A .... B ---> AB (3.2-14)
the distance between A and B shrinks during the activation process, resulting in a negative activation volume.
Large negative values of Av # (-25 to -50 cm3/mol) have been found in the investigation of Diels-Alder reactions, such as the dimerization of cyclopentadiene (Table 3.2-1, b). The cyclic transition state has a compact structure similar to the reaction products. Av # is only a little smaller than the volume change between the initial- and the product-structures, indicating a late transition state.
When bond-formation and-breaking take place simultaneously
A + BC ---> A .... B .... C ---> A B + C (3.2-15)
the net change of volume is dominated by the contribution of the formation of new bonds.
Therefore, the activation volume is negative. Examples are rearrangement reactions, such as the Claisen rearrangement of allyl vinyl ether, where Av # is between 8 and -20 cm3/mol (Table 3.2-1, c). In these reactions, covalent O-C-bonds are stretched and broken and simultaneously new C-C-bonds are formed. Radical polymerization reactions, such as the polymerization of ethylene, also exhibit negative activation volumes o f - 1 0 to -25 cm3/mol (Table 3.2-1, d).
New bonds are formed by the addition of monomer molecules to radicals whereas the double bond of the monomer is stretched in the transition state, resulting in a single bond. By this mechanism the activation volume of the radical polymerization of ethylene, -25.5 cm3/mol, can be explained.
Table 3.2-1
Typical values of the activation volume of different reactions
Reaction Range Av # Example
[cm3/mol]
Av # [cm3/
mol]
a Unimolecular 0 to +15 dissociation
b Diels-Alder -25 to-50 reactions
c Rearrangement -8 to-20 reactions
d Radical polymeri- zations
-10 to -25
e Menshutkin -20 to -40 reactions
Rad. decomposition of di(t-butyl)per0xide (CH 3)3COOC(CH 3)3 -- 2 (CH 3)3CO "
Dimerisation of cyclopentadiene
Claisen rearrangement of allyl vinyl ether
/C\ CH 2 /C~" ... 'CH2~ --~--C~ H
Radical polymerization of ethylene
~ ~ H 2 + C2H4 r- ~ ~ H 2 Reaction between pyridine and ethyl iodide
+13
[7]
-31
[8]
-16 [9]
-25.5 [10,
11]
-30 1) -27 a) -24 3) [12]
1), acetone; 2), cyclohexanone; 3), nitrobenzene.
3.2.2.2 Terms contributing to Av~
In the dissociation o f a neutral molecule A B into two free ions, A + and B-,
AB ~ X # ~ A + + B- (3.2-16)
electrical charges develop during the transition from AB to X #. The ionic charges exert strong attractive forces on the permanent or induced dipoles of the solvent molecules, causing a contraction. The magnitude of this effect is larger than the increase of volume owing to the stretching of the bond between A and B, leading to a negative activation volume.
In the bimolecular association to non-ionic products
A + + B- --, X # ---~ AB (3.2-17)
the reduction in polarity results in an expansion of the solvent. Although there is an opposing contraction owing to bond-forming, the activation volume is positive. It must be pointed out that the predominance of Av # over AVR # is often observed but is not a strict rule.
The dependence of Av # on the solvent was investigated on Menshutkin reactions. In all solvents the reaction is strongly accelerated by pressure due to high activation volumes o f - 2 0 to -40 cm3/mol (Table 3.2-1, e).