Mechanisms and Rate Laws
7.3 POLYMERIZATION REACTIONS
rco* = k&,foccH; =
chKd2{[1 +
f3~&&I,J(k&p - l}4k2
(7.2-2)
Furthermore, the rate of disappearance of CH4 is
(-~cIL,) = 2rczHs + rco, = klcMocCH, (7.2-3) which is also the limiting rate for either product, if the competing reaction is completely suppressed.
7.2.2 Computer Modeling of Complex Reaction Kinetics
v07O.v v0“OF
In the examples in Sections 7.1 and 7.2.1, explicit analytical expressions for rate laws are obtained from proposed mechanisms (except branched-chain mechanisms), with the aid of the SSH applied to reactive intermediates. In a particular case, a rate law obtained in this way can be used, if the Arrhenius parameters are known, to simulate or model the reaction in a specified reactor context. For example, it can be used to determine the concentration-(residence) time profiles for the various species in a BR or PFR, and hence the product distribution. It may be necessary to use a computer-implemented nu- merical procedure for integration of the resulting differential equations. The software package E-Z Solve can be used for this purpose.
It may not be possible to obtain an explicit rate law from a mechanism even with the aid of the SSH. This is particularly evident for complex systems with many elementary steps and reactive intermediates. In such cases, the numerical computer modeling pro- cedure is applied to the full set of differential equations, including those for the reactive intermediates; that is, it is not necessary to use the SSH, as it is in gaining the advantage of an analytical expression in an approximate solution. Computer modeling of a react- ing system in this way can provide insight into its behavior; for example, the effect of changing conditions (feed composition, T, etc.) can be studied. In modeling the effect of man-made chemicals on atmospheric chemistry, where reaction-coupling is impor- tant to the net effect, hundreds of reactions can be involved. In modeling the kinetics of ethane dehydrogenation to produce ethylene, the relatively simple mechanism given in Section 6.1.2 needs to be expanded considerably to account for the formation of a number of coproducts; even small amounts of these have significant economic conse- quences because of the large scale of the process. The simulation of systems such as these can be carried out with E-Z Solve or more specific-purpose software. For an ex- ample of the use of CHEMKIN, an important type of the latter, see Mims et al. (1994).
The inverse problem to simulation from a reaction mechanism is the determination of the reaction mechanism from observed kinetics. The process of building a mecha- nism is an interactive one, with successive changes followed by experimental testing of the model predictions. The purpose is to be able to explain why a reacting system behaves the way it does in order to control it better or to improve it (e.g., in reactor performance).
7.3 POLYMERIZATION REACTIONS
Because of the ubiquitous nature of polymers and plastics (synthetic rubbers, nylon, polyesters, polyethylene, etc.) in everyday life, we should consider the kinetics of their formation (the focus here is on kinetics; the significance of some features of kinetics in relation to polymer characteristics for reactor selection is treated in Chapter 18).
Polymerization, the reaction of monomer to produce polymer, may be self-polymeri- zation (e.g., ethylene monomer to produce polyethylene), or copolymerization (e.g.,
styrene monomer and butadiene monomer to produce SBR type of synthetic rubber).
These may both be classified broadly into chain-reaction polymerization and step- reaction (condensation) polymerization. We consider a simple model of each, by way of introduction to the subject, but the literature on polymerization and polymerization kinetics is very extensive (see, e.g., Billmeyer, 1984). Many polymerization reactions
are catalytic.
7.3.1 Chain-Reaction Polymerization
Chain-reaction mechanisms differ according to the nature of the reactive intermedi- ate in the propagation steps, such as free radicals, ions, or coordination compounds.
These give rise to radical-addition polymerization, ionic-addition (cationic or anionic) polymerization, etc. In Example 7-4 below, we use a simple model for radical-addition polymerization.
As for any chain reaction, radical-addition polymerization consists of three main types of steps: initiation, propagation, and termination. Initiation may be achieved by various methods: from the monomer thermally or photochemically, or by use of a free- radical initiator, a relatively unstable compound, such as a peroxide, that decomposes thermally to give free radicals (Example 7-4 below). The rate of initiation (rinit) can be determined experimentally by labeling the initiator radioactively or by use of a “scav- enger” to react with the radicals produced by the initiator; the rate is then the rate of consumption of the initiator. Propagation differs from previous consideration of linear chains in that there is no recycling of a chain carrier; polymers may grow by addition of monomer units in successive steps. Like initiation, termination may occur in vari- ous ways: combination of polymer radicals, disproportionation of polymer radicals, or radical transfer from polymer to monomer.
Suppose the chain-reaction mechanism for radical-addition polymerization of a monomer M (e.g., CH,CHCl), which involves an initiator I (e.g., benzoyl peroxide), at low concen- tration, is as follows (Hill, 1977, p. 124):
initiation:
propagation:
1%2R* (1)
R’ + M&P; (2)
P; + M%pI m
q+M%pf 0-9)
. . *
Y-l
+ M%P; (W. . .
termination: P’k + P; AP,+, k,e= 1,2,... (3)
in which it is assumed that rate constant k, is the same for all propagation steps, and k, is the same for all termination steps; Pk+e is the polymer product; and PF, r = 1,2, . . . , is a radical, the growing polymer chain.
(a)
By applying the stationary-state hypothesis (SSH) to each radical species (including R’), derive the rate law for the rate of disappearance of monomer, (-Q), for the mechanism above, in terms of the concentrations of I and M, andf, the efficiency of utilization of the R’ radicals;f is the fraction of R’ formed in (1) that results in initiating chains in (2).7.3 Polymerization Reactions 167 (b) Write the special cases for (-rM) in which (i)f is constant; (ii) f m CM; and (iii)
f a&.
SOLUTION
r,. = 2fkdcI - kiCR.Chl = 0
t-i& = ?-[Step (2)]) = k$R.cM = 2fkdcI [from (4)]
cc
YP; = rinit - kpc~cp; - ktcp; C Cp; = 0
k = l
where the last term is from the rate of termination according to step (3). Similarly, rp; = kpCMCp; - kpcMcp; -
k = l
. . .
cc
rp: = kpcMcF’-, - k,cMcp: - k,cp: c cpk = o k = l
(4) (5) (6)
(7)
(8) From the summation of (6), (7), . . ., (8) with the assumption that k,cMcp: is relatively
small (since cp: is very small),
(9) which states that the rate of initiation is equal to the rate of termination. For the rate law, the rate of polymerization, the rate of disappearance of monomer, is
m (-TM) = rinit + kpCM C cq
k = l
= k,CMlF “q
k = l
= kpcM(riniJkt)1/2
= k,c,(2 f kdcIlkt)1’2 We write this finally as
(-rM) = k f 1’2C;‘2CM (7.3-1)
where k = k,(2k,lk,) 112 (7.3-2)
( i ) (-TM) = k’C;‘2CM (ii) (-)iL1) = k”c:‘2cz2 (iii) (-?-M) = k”‘cte?c&
Gfrinit a (-TM)1
[from (911 [from (511
(7.3-la) (7.3-lb) (7.3-lc)
7.3.2 Step-Change Polymerization
Consider the following mechanism for step-change polymerization of monomer M (PI) to P2, P,, . . .) P,, . . . . The mechanism corresponds to a complex series-parallel scheme:
series with respect to the growing polymer, and parallel with respect to M. Each step is a second-order elementary reaction, and the rate constant k (defined for each step)’ is the same for all steps.
M + MIP, M + P,+P,k
(1) (2)
M + P,-, +P,k (r - 1)
where r is the number of monomer units in the polymer. This mechanism differs from a chain-mechanism polymerization in that there are no initiation or termination steps.
Furthermore, the species P,, P,, etc. are product species and not reactive intermedi- ates. Therefore, we cannot apply the SSH to obtain a rate law for the disappearance of monomer (as in the previous section for equation 7.3-l), independent of cp,, cp,, etc.
From the mechanism above, the rate of disappearance of monomer, (- rM), is (-TM) = 2kcL + kcMcPz + . . . + kcMcp, + . . .
= kc,(2c, + 2 cP,)
r=2
(7.3-3)
The rates of appearance of dimer, trimer, etc. correspondingly are
+2 = kCM(CM - CP,)
(7.3-4)+3 = kCM(CPZ - CP3 > (7.3-5)
9 . .
+r = kcM(cp,-, - cpr), etc. (7.3-6)
These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncou- pled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case.
For the CSTR case, illustrated in Figure 7.2, suppose the feed concentration of monomer 1s cMo, the feed rate is q, and the reactor volume is V. Using the material- balance equation 2.3-4, we have, for the monomer:
cMoq -cMq+rMV = 0
‘The interpretation of k as a step rate constant (see equations 1.4-8 and 4.1-3) was used by Denbigh and Turner (197 1, p. 123). The interpretation of k as the species rate constant kM was used subsequently by Denbigh and Turner (1984, p. 125). Details of the consequences of the model, both here and in Chapter 18, differ according to which interpretation is made. In any case, we focus on the use of the model in a general sense, and not on the correctness of the interpretation of k.
7.3 Polymerization Reactions 169
L-J
Figure 7.2 Polymerization of monomer M in a CSTR at steady-stateor
(-d = (CM0 - c,)/(V/q) = (CM0 - C&/T where r is the space time.
Similarly, for the dimer, P,,
0 - cp2q + rplV = 0 or
and
Q* = cp2/r = kCM(CM - cp,)
Similarly, it follows that
CP2 = kc&Q - cp2)
+3 = bdCP* - CP,) . . +, = b&P,-, - CP,>
and, thus, on summing 7.3-8 to 7.3-10, we obtain
2 cp, = kCMyl7(CM - cp2 + cp2 - cp, + CPj + * . . - +ml + cP,d1 r=2
= kCM7(Ch$ - cp,) = kc&
since cp --f 0 as r -+ cc).
Substkution of 7.3-7 and -11 in 7.3-3 results in
CM0 - CM = kCh4T(2Ch4 + k&T) from which a cubic equation in cM arises:
CL + (2/kT)& + (l/k%2)c, - ch,&W = 0 Solution of equation 7.3-12 for cM leads to the solution for cp,, cp3, etc.:
From equation 7.3-8,
(7.3-7)
(from 7.3-4)
(7.3-8)
(7.3-9)
(7.3-10)
-CP,>
(7.3-11)
(7.3-lla)
(7.3-12)
CP, =
k&T
1 + kcMT (7.3-13)
Similarly, from 7.3-9 and -13,
CP, = kcM “P2 CM(kCMT)2 1 + kCMT = (1 + kCMT)2
Proceeding in this way, from 7.3-10, we obtain in general:
CP, =
CM( kc&-l
(1 + kCMT)‘-l = cM[l + (kcM+l]l-’
(7.3-14)
(7.3-15)
Thus, the product distribution (distribution of polymer species P,) leaving the CSTR can be calculated, if cMO, k, and T are known.
For a BR or a PFR in steady-state operation, corresponding differential equations can be established to obtain the product distribution (problem 7-15).