KINETIC PROPERTIES AT HIGH PRESSURE
3.2.3 Evaluation of the activation volume from experimental data
The rate, r, of a single homogeneous reaction
A k > B (3.2-18)
can be expressed by the equation:
dc A
r = ~ = k n (3.2-19)
dt "CA
In order to determine the volume of activation, first the rate constant, k, is calculated from the measured reaction rate, r, and the concentration, CA, through eqn. 3.2-19. The exponent n can be obtained from experiments at different concentrations. The value of k is then plotted on a logarithmic scale versus the pressure, and Av # (10 .6 cm3/mol) is evaluated from the slope, a, of the resulting straight line (Fig. 3.2-2). For this purpose a, (MPa-1), is multiplied by the gas constant, R (8.314 J mol 1 K-l), and the temperature, T (K).
O t,.)
~D
.r
Pressure, p
Figure 3.2-2. Determination of the activation volume of single reactions.
The expressions for the rates of multiple reactions are more complex and cannot be easily handled to determine the activation volume. Eqn. 3.2-7 is strictly valid for single reactions.
If a reaction is accompanied by side reactions, which are not accounted for in detail, deviations from the straight line may be experienced in the plot of lnk/p when the overall rate is considered. This difficulty is well known from the determination of the activation energy for multiple reactions.
3.2.3.2 Parallel reactions
The course of concentration during parallel reactions
A k, ) R (3.2-20)
A k2 )S
is shown in Fig. 3.2-3.
The ratio of the rate constants kl and k2 can be obtained from the ratio of the rates of formation of the products, R and S.
r 1 = de R / d t = kl'CnA~ (3.2-21)
r2 dcs /dt k 2 "C~ 2
If the reactions have the same kinetic order, n, the expression:
rl kl
- - = ~ (3.2-22)
r2 k2
results and, using eqn. 3.2-7, the difference of the activation volumes can be determined.
O l n k l / k 2 Av # - Av#2
ap R . T (3.2-23)
3.2.3.3 Reactions in series
As an example for reactions in series or consecutive reactions, the simple reaction scheme
A kl ) R k2 ~P (3.2-24)
should be discussed. The course of the concentrations of the initial A, product R, and consecutive product P, is shown in Fig. 3.2-4. The reduction in the concentration of the initial reactant, A, follows the relationship
dcA = k 1 "el 1 (3.2-25)
n =--~-
1 1-
i
0
m R S
~ . . . A
Time, t
Figure 3.2-3. Change of concentration with time during parallel reactions.
o
O
=o
,.d
0
~ A
P
Time, t
Figure 3.2-4. Change of concentration with time during reactions in series, kl ~ k2.
The dependence of kl on the pressure is given by eqn. 3.2-7 with the activation volume AVl # . The change of the concentration of the intermediate product, R, with time is
dcR = k 1 "C/~ 1 - k 2 .c~ 2 (3.2-26)
r2 =---~-
The influence of pressure is determined by Av~ and Av2 #, the activation volumes of the two reactions in series. The determination of Av~ and Av~ requires one to evaluate first kl and k2.
If n l and n2 are unity, r2/cA is plotted versus CA/Cn. Then kl is obtained from the intersection
of the resulting straight line and the ordinate, whereas k2 is its slope. Standard mathematical methods, such as linear- and multiple regression, or search techniques based on least-squares- methods to minimize the deviation of measured and calculated reaction rates, must be applied to determine the rate constants when nl and n2 are different from unity.
3.2.3.4 Chain reactions
An important example of a complex reaction is a chain reaction in which free radicals are involved. These reactions consist of three essential steps: Formation of free radicals or initiation; propagation; and termination.
Radicals, I ~ can be formed by unimolecular dissociation:
I k~ ; i 9 (3.2-27)
The radicals, I',.may react with a further component, M, such as an olefin to form secondary radicals, R'.
I ~ k~ >R 9 (3.2-28)
By successive addition of M to radicals R~ in the propagation reaction, chain radicals Ri+ 1 are formed
R i~ + M kp ), Ri+l~ (3.2-29)
the growth of which is terminated by combination.
9 9 k t
R i + Rj ) Pi+j (3.2-30)
In the termination reaction, stable products P are obtained.
It will be shown later that the rate of the chain reaction mentioned above can be calculated from the expression:
dM k~[M][ill/2
r = ~ = k p . = kbr[M][I] 1/2 (3.2-31)
dt
The dependence of the overall rate constant, kbr, on the pressure can be elucidated from eqn. 3.2-7
Olnkb__________~ r = _ A b_______Lr v # (3.2-32)
ap R.T
with
1 Av/# _ l A v t #
AVer = AV#p +-~ (3.2-33)
where Avp # , Av/#, and Avt # are the activation volumes of chain propagation, initiation, and termination reaction, respectively.
3.2.3.5 Heterogeneous catalytic reactions
A number of different steps are involved in heterogeneous catalytic processes. Among chemical reaction, adsorption, and desorption, transport processes may influence the overall rate. These steps are illustrated in Fig. 3.2-5.
1. Transport of reactants, A and B, from the bulk phase to the surface of the catalyst pellet.
2. Transport of reactants into the catalyst pores.
3. Adsorption of reactants, A and B, on the catalytic site, L:
A + L ~ > AL B + L ~ BL
4. Surface reaction between the adsorbed species:
A L + B L ~ R L + S L
(3.2-34) (3.2-35)
Figure 3.2-5. Steps involved in heterogeneous catalytic reactions.
5. Desorption of the products, R, S:
RL ~. >- R + L SL -- >-- S + L
6. Transport of the products back to the pellet surface.
7. Transport back to the bulk stream.
(3.2-36)
In the following discussion we will concentrate on the surface reaction, adsorption, and desorption. The complications induced by the transport phenomena will be ignored. In order to develop an expression for the overall rate, the surface concentrations, AL, BL, etc., are related to the concentrations of the reactants in the bulk phase by an "Equilibrium constant".
For example:
[AL]
XA = [AIL] (3.2-37)
Quite often it is found that one of the above-mentioned steps is much slower than the others and therefore controls the rate of the catalytic reaction. As an example, the rate expression for the bimolecular reaction
A + B _~ ~- R + S (3.2-38)
is shown below, when the surface reaction of the adsorbed species is rate controlling.
e . . . - . p s / X )
r = (3.2-39)
0 + KA "PA + KB "PB + KR "PR + KS "ps)2
Here, PA, PB, etc., are the partial pressures of the reactants, K = PR " P s is the equilibrium PA "PB
constant of the reaction, KA, Ks, etc., are the equilibrium constants of the individual steps, and k is the rate constant of the rate-controlling reaction in which the total concentration of active sites is incorporated.
The volume of activation of the rate-controlling step can be evaluated from the rate constant, k, using eqn. 3.2-7. First, k must be evaluated through eqn. 3.2-39 from the rate of the catalytic process measured at different total pressures. The standard-methods of least squares are applied again to fit values for k and the other parameters, KA, K~, etc.
3.2.3.6 Reactions influenced by mass transport
If the rate of a reaction is influenced by mass transport, the effect of the pressure both on the rate of the chemical reaction and on the rate of mass transport must be taken into account.
As an example, a heterogeneous catalytic reaction governed by the rate of diffusion within the pores of the catalyst is considered.
The overall rate of reaction, rov, can be described using the concept of the effectiveness factor, ~7, by the expression [13]
rov = k . c . ~l (3.2-40)
When the pore-diffusion is limiting, the effectiveness factor is 7"/= 1/q~, with the Thiele modulus, q~ = rK/3 (k/Dp) 1/2, rx being the radius of the catalyst particle, and Dp the coefficient of pore-diffusion. The overall rate of the process depends then on the reciprocal modulus, q~:
roveral l = k . c . dp-1 which means that:
roveral l - ( k . D ) 1/2 and roveral l - exp[-(Av # + A V # D ) . ( p - p 0 ) / 2 - R - T ] (3.2-41) Taking into consideration the fact that the activation volume of the diffusion process is small compared to that of the chemical reaction, it follow that:
,nrovl 'nr~ I - AV~ (3.2-42)
dp r 2 . R . T dp T R . T
with Avo#v = A v # / 2 . (3.2-43)
From eqn. 3.2-42 it can be concluded that the dependence of the overall rate of a heterogeneous catalytic reaction on the pressure is decreased when the pore-diffusion is rate limiting.