The tworeacting functional groups can either be on the same monomer molecule, as inamino caproic acid, NH2ðCH2Þ5COOH, or on two separate molecules, as inthe reaction between ethylene gly
Trang 1Monomer molecules consisting of at least two functional groups canundergo step-growth polymerization In order to keep mathematics tractable,this chapter will focus on polymerization of bifunctional monomers The tworeacting functional groups can either be on the same monomer molecule, as inamino caproic acid, NH2ðCH2Þ5COOH, or on two separate molecules, as inthe reaction between ethylene glycol, OHðCH2Þ2OH, and adipic acid,COOHðCH2Þ4COOH If they are located on the same monomer molecule,represented schematically as ARB, the concentrations of the functional groupsremain equimolar throughout the course of the reaction, which can be schema-tically represented as follows [1–7]:
nðARBÞ ! ARB½ARBn2ARB ð3:1:1ÞHere, R represents an alkyl or aryl group to which the two functional groups Aand B are attached In case the functional groups are located on two differentmonomers, ARA and BR0B, an analysis similar to the one for ARB
Trang 2polymerization can be conducted As pointed out in Chapter 1, the overallreaction represented by Eq (3.1.1) consists of several elementary reactions,which can be represented as follows:
If two small molecular species, A and B, react as
A þ B ! Products ð3:1:3Þ
it is evident that the reaction will proceed only after a molecule of A diffusesclose to a molecule of B from the bulk Thus, the overall reaction between A and
B consists of two consecutive steps: (1) the diffusion of molecules from the bulk
of the mixture to within close proximity of each other and (2) the chemicalinteraction leading to product formation This is represented schematically as
A þ B ! Diffusion ½A B ! Chemical
reaction
ABðProductÞ ð3:1:4Þ
However, polymer molecules are very long and generally exist in a highly coiledstate in the reaction mass with the functional groups situated at the chain ends.Therefore, in addition to the ‘‘bulk’’ molecular diffusion of Pmand Pn, the chainends must diffuse close to each other (called segmental motion) before thechemical reaction can occur This can be represented schematically as
Pmþ Pn !Bulk
diffusion
½PmPn !
Segmental diffusion
Trang 33.2 ESTERIFICATION OF HOMOLOGOUS SERIES
AND THE EQUAL REACTIVITY HYPOTHESIS
[1,4,5]
The following esterification reactions of monobasic and dibasic acids of logous series illustrate the effect of molecular size on the rate constants:HðCH2ÞnCOOH þ C2H5OH!HCl HðCH2ÞnCOOC2H5þ H2O
re¼d½COOH
dt ¼ kA½COOH½Hþ ð3:2:2Þwhere reis the rate of esterification and [ ] represents molar concentrations Theconcentration of ethanol does not enter into Eq (3.2.2) because it is present in thereaction mass in large excess
In Eq (3.2.2) [COOH] represents the total concentration of the carboxylicacid groups in the reaction mass at any time, whether present in the form of amonobasic of dibasic acid; this is usually determined by titration ½Hþ is theconcentration of protons liberated by the hydrochloric acid Use of the rateequation in the form shown in Eq (3.2.2), together with experiments onmonobasic and dibasic acids having different n, makes it possible to isolate theeffect of the size of the molecule on kA
The rate constants for various values of n are tabulated inTable 3.1 Twoimportant conclusions can be drawn from the experimental results:
1 The reactivity of larger molecules does not depend on the size of themolecule for n> 8 [2,7]
2 For larger molecules, the rate constant is independent of whether thereare one, two, or more carboxylic acid groups per molecule
Similar conclusions have also been obtained on the saponification of estersand etherification reactions [4,5] If, in the chemical reaction step of Eq (3.1.5),
Trang 4the reactivity of a COOH group with an OH group is assumed to beindependent of n, these observations imply that the rate of diffusion of largemolecules is not affected by the value of n However, we know intuitively that thelarger the molecule, the slower is its rate of diffusion Consequently, it is expectedthat, as n increases, the diffusional rate should decrease, implying that kA mustdecrease with increasing n, a conclusion in apparent contradiction with theobserved behavior.
As shown in Eq (3.1.5), there are two types of diffusional mechanismsassociated with the reaction of polymer molecules Although the rate of bulkdiffusion of two molecules decreases with n, the rate of the other step, calledsegmental diffusion, is independent of n The independence of n is due to the factthat there is some flexibility of rotation around any covalent bond in a polymermolecule (see Chapter 1), and there is restricted motion of a small sequence ofbonds near the ends, which constitutes segmental diffusion This brings thefunctional groups of two neighboring molecules near each other, regardless of thechain length of the entire molecule Thus, with increasing n, two polymermolecules diffuse slowly toward each other by bulk diffusion but stay togetherfor a longer time (the two effects canceling out), during which, segmentaldiffusion may bring the functional groups together for possible reaction.Based on the experimental results of Table 3.1, we can postulate a simplekinetic model for the study of step-growth polymerization in which all of the rateconstants are assumed to be independent of chain length This is referred to as theequal reactivity hypothesis The following section shows that this assumptionleads to a considerable simplification of the mathematical analysis However,there are several systems in which this hypothesis does not hold accurately, andthe analysis presented here must be accordingly modified [2,8–14]
TABLE3.1 Rate Constants for the Esterification of MonobasicChain length kA 104ð250CÞa kA 104ð250CÞa
Trang 53.3 KINETICS OF ARB POLYMERIZATION
USING EQUAL REACTIVITY HYPOTHESIS [2]
A chemical reaction can occur only when the reacting functional groups collidewith sufficient force that the activation energy for the reaction is available Therate of reaction, r, can thus be written as proportional to the product of thecollision frequency, omn, between Pmand Pnand the probability of reaction, Zmn(which accounts for the fraction of successful collisions), as follows:
where a is a constant of proportionality According to the equal reactivityhypothesis, Zmn is independent of m and n and is, say, equal to Z If thefunctional groups of the two molecules Pmand Pncan react in s distinct ways, theprobability of a reaction between Pm and Pn is given by sZ The collisionfrequency omnbetween two dissimilar molecules Pmand Pnin the forward step isproportional to ½Pm½Pn, whereas that for Pmand Pmis proportional to ½Pm2=2(the factor of one-half has been used to avoid counting collisions twice) Thus, if
kp is the rate constant associated with the reaction between functional groups,then under the equal reactivity hypothesis, km;n, the rate constant associated withmolecules Pmand Pn in the forward step, is given by
km ;n¼
r
½Pm½Pn¼ skp; m 6¼ n; m; n ¼ 1; 2; r
km;n¼ 2kp m 6¼ n; m; n ¼ 1; 2
kp m ¼ n; n ¼ 1; 2;
(
ð3:3:3aÞð3:3:3bÞThe various (distinct) elementary reactions in the forward step can now be written
and condensation product W; there is a bond scission in this process It may beobserved that Pn has n 1 equivalent chemical bonds where the reaction canoccur with equal likelihood It is thus seen that if kp0 is the reactivity of a bond
Trang 6with W, the reactivity of an oligomer Pnis ðn 1Þkp0 The mole balance equationsfor various molecular species in a constant-density batch reactor can now beeasily written Species P1 is depleted in the forward step when it reacts with anyother molecule in the reaction mass However, Pnðn 2Þ is formed in theforward step when a molecule Prðr < nÞ reacts with Pnr and is depleted byreaction with any other molecule In the reverse step Pn is depleted when any ofits chemical bonds are reacted and it is formed whenever a Pqðq > nÞ reacts at aspecified bond position For example, if we are focusing our attention on theformation of P4, a molecule having chain length greater than 4, say, P6, wouldlead to the formation of P4 if W reacts at the second or fourth position of P6 Themole balance relations are therefore given by the following:
d½P1
dt ¼ 2kp½P1f½P1 þ ½P2 þ g þ 2k0
p½Wf½P2 þ ½P3 þ g ð3:3:5aÞd½Pn
as shown through an example of the formation of P6 This occurs at a rate given
by ð2kp½P1½P5 þ 2kp½P2½P4 þ 2kp½P3½P3Þ The factor of the first two termsarises because km ;n is 2kp, whereas the factor of the last term, 2kp½P3=2, arisesbecause of the fact that two molecules of P3are consumed simulataneously when
P3reacts with P3 The first term in Eq (3.3.5b) for this is kp
P5 r¼1½Pr½Pnr, asshown
If the concentration of all the reactive molecules in the batch reactor isdefined as
Trang 7one can sum up the equations in Eq (3.3.5) for all n to give the following:
ðn 1Þ½Pn þ 2k0
p½WP1 n¼1
P1 i¼nþ1
½Pið3:3:7Þ
It is recognized that
P1
n¼1
P1 i¼nþ1
½Pi ¼ ½P2 þ ½P3 þ ½P4 þ þ ½P3 þ ½P4 þ þ ½P4 þ
¼ ½P2 þ 2½P3 þ 3½P4 þ
¼P1 n¼1
ðn 1Þ½Pn
ð3:3:8ÞTherefore, Eq (3.3.7) can be written as
dl0
dt ¼ kpl20þ k0
p½W P1 n¼1
ðn 1Þ½Pn ð3:3:9Þ
It may be observed thatP1
n¼1ðn 1Þ½Pn represents the total number of reactedbonds in the reaction mass It is thus seen that the infinite set of elementaryreactions in step-growth polymerization in Eq (3.1.2) can be representedkinetically by the following equivalent and simplified equation:
A þ B !
kp
where AB represents a reacted bond The representation of an infinite series
of elementary reactions by only one elementary reaction [Eq (3.3.10)] involvingfunctional groups is a direct consequence of the equal reactivity hypothesis Thisleads to a considerable simplification of the mathematical analysis of polymer-ization reactors
Example 3.1: Consider the ARB step-growth polymerization in which monomer
P1 reacts with Pn (for any n) with a different rate constant, as follows:
P1þ Pn !k1
Pnþ1þ W; n ¼ 1; 2; 3 ðaÞ
Pmþ Pn !kp
Pnþmþ W; m; n ¼ 2; 3; ðbÞDerive the mole balance relations for the MWD of the polymer in a batch reactor
Trang 8Solution: s in Eq (3.3.2) is 2 because the polymer chains are linear Let us firstconsider the reaction of P1 In the reactions of P1 with P1, similar molecules areinvolved, and the reactivity would be 2k1=2 However, for the reaction of P1withany other molecule, the reactivity would be 2k1 Therefore,
P1 is formed:
d½P1
dt ¼ ðForward reaction of P1with P1Þ
ðForward reaction of P1with P2; P3etc:Þ
þ ðreverse reaction of W at chain ends to give P1Þ
½Pn
The rate of formation of P2is k1½P1=2, and P2is depleted whenever it reacts withany molecule in the forward step or its bond reacts with W in the reverse step:d½P2
dt ¼ ðForward reaction forming P2Þ ðForward reaction of P2with P1Þ
ðForward reaction of P2with P2with P2; P3; etc:Þ
ðReverse reaction of bonds of P2with WÞ
þ ðreverse reaction of W with P3; P4; etc: to give P2Þ
½Pi
Trang 9Similarly, the mole balance relaxation for species Pn is given by the following:d½Pn
dt ¼ ðForward reaction of P1; P2; etc: with Pn1; Pn2; etc:Þ
ðForward reaction of Pnwith P1Þ
ðForward reaction of Pnwith P2; P3; etc:Þ
ðReverse reaction of W with n 1 bonds of PnÞ
þ ðReverse reaction of W with Pnþ1; Pnþ2; etc: to give PnÞ
½PrThe zeroth moment of the MWD can be easily found as follows:
ðn 1Þ½Pn
STEP-GROWTH POLYMERIZATION OF ARB
we could either consider only the dimers, trimers, and all other homologs toconstitute molecules of the polymer, or, alternatively, include monomer molecules
as well Naturally, the results using the second approach would be lower than thatobtained from the first one In the following analysis, the monomer is included in
Trang 10the computation of the average molecular weight This is not a drawback because,for practically important situations, the concentration of P1 is usually negligible.
It may be observed that during polymerization the total number of repeatunits at any time remains unchanged and is equal to the initial number ofmonomer molecules, N0 These repeat units, however, are now disturbed over Npolymer molecules at time t, so the average number of repeat units per molecule
is equal to N0=N This is defined as the number-average chain length, mn
(sometimes called the degree of polymerization), and is given by
where ½A0, ½B0, and ½A and ½B are the concentrations of the functional groups
A and B at times t ¼ 0 and t ¼ t, respectively It is convenient to work in terms ofthe (fractional) conversion of functional group A (or B), defined as
½W þ l0¼ ½W0þ ½P10 ð3:4:4Þwhere [W] and l0are the concentrations of condensation product and polymer atany instant of time We substitute [W] from this equation into Eq (3.3.9) toobtain
½Pn
ð3:4:5Þ
We further observe thatP1
n¼1n ½Pn is the first moment of the MWD and is equal
to the total number of repeat units, which means that the first moment, l10, is timeinvariant Therefore, Eq (3.4.5) becomes
dl0
dt ¼ kpl20þ k0
pf½W0þ ½P10 l0gðl10 l0Þ ð3:4:6Þ
Trang 11which can be integrated as follows:
q0¼2m2½P10þ m1 d
2m2½P10þ m1þ d ð3:4:9fÞThe number-average molecular weight can be easily obtained by multiplying mn
by the molecular weight of ARB (because the molecular weight of W is usuallysmall)
Example 3.2: Suppose NA0 moles of AR1A monomer are reacted with NB0moles of BR2B monomer to form the polymer Derive an expression for theaverage molecular weight of the polymer formed
Solution: We first observe that there are 2NA0moles of A functional groups and2NB0moles of B functional groups present at time t ¼ 0 Whenever a functionalgroup A (or B) reacts, the total number of molecules in the reaction massdecreases by 1 Let us, for the moment, assume that NB0is greater than NA0
In order to determine the molecular weight, we needed to determine the totalnumber of molecules at time t when the conversion of A functional groups is pA:
pA¼2NA0 2NA
2NA0The total number of moles of unreacted A functional groups at time t is equal to2NA0ð1 pAÞ
Trang 12The total number of moles of unreacted B functional groups at time t is equal toð2NB0 2NA0pAÞ
At any time t molecules of (A A), (A B), and (B B) types are present, and all ofthese are equally likely to occur If we know the total number of moles ofunreacted A and B functional groups, the total number of moles of polymer issimply half of this In other words, the total number of moles of polymer, N , attime t is equal to1
2f2NA0ð1 pAÞ þ 2NB0 2NA0pAg Similarly, the total number
of moles of polymer initially, N0, is equal to NA0þ NB0:
Observe that even when 100% conversion of A functional groups (i.e.,
pA¼ 1) is achieved, the average chain length mn has a limiting value ofð1 þ rÞ=ð1 rÞ instead of 1, as predicted by Eq (3.4.3) It is thus seen that
an equimolar ratio (i.e., r ¼ 1) is desirable for the formation of polymer of highmolecular weight
Example 3.3: The polyester PET, commonly used in the manufacture ofsynthetic fibers, is prepared through polymerization of bis-hydroxyethylterephthalate (BHET) During polymerization, several side reactions occur, but
if these are ignored, PET formation can be modeled by ARB kinetics asdiscussed Experiments have shown that
kp¼ 0:5 ðindependent of temperatureÞFor the initial monomer concentration, ½P10¼ 4:58 g mol=L, find the conver-sion, the average chain length, and the polydispersity index Q after 10 min ofpolymerization at 280C and 200C
Solution: At 280C
kp ¼ 4:0 104
exp 15; 0001:98ð273 þ 280Þ
¼ 4:49 102L=mol min
kp0 ¼ 2:25 102L=mol min
Trang 13Because t ¼ 0, BHET does not have any W,
l0=½P10 0:41
0:59 ¼ 0:172r2:91
; l0
½P10 ¼ 0:41 þ 0:172ð0:59Þð0:0545Þ ¼ 0:416Conversion ¼ 1 l
0
½P10¼ 0:584
mn¼ 2:40; Q ¼ 1 þ p ¼ 1:584
Trang 14dt ¼ 0 ¼ 2kp½P1el0eþ 2k0
p½We P1 i¼2
½Pied½Pne
dt ¼ 0 ¼ 2kp½Pnel0eþ 2k0
p
P
n1 r¼1
½Pre½Pnre
k0
p½Weðn 1Þ½Pneþ 2k0
p½We P1 inþ1
Trang 15where x and y are some parameters that are not dependent on chain length n Wehave already observed that the first moment l1is time invariant and is the same asthe initial value l10 Therefore, Eq (3.5.2) must satisfy the following relation:
ðn 1Þ½Pn ¼ kpl20eþ k0
p½Weðl10 l0eÞ ¼ 0
ð3:5:4Þ
This is a quadratic equation and can be easily solved In addition, we can also find
l0e from the assumed form of the MWD in Eq (3.5.2) as follows:
l0e¼ xð1 þ y þ y2þ Þ ¼ x
Between Eqs (3.5.3) and (3.5.5), we get
l0e¼ l10ð1 yÞ ð3:5:6Þand the MWD in Eq (3.5.2) is given by
½Pie¼ P1 n¼1
½PiePn n¼1
Trang 16On substituting these in Eq (3.5.1), we have the following:
½P1 ¼ ½P10 ð3:6:2aÞ
½Pn0¼ 0; n ¼ 2; 3; ð3:6:2bÞ
Trang 17We guess the form of the MWD as
½Pn ¼ ½P10ð1 pÞ2
where p is the conversion of the functional groups defined in Eq (3.4.2) Wefurther observe [steps of the derivation are identical to those in Eqs (3.5.5) and(3.5.6)] the following:
The molecular-weight distribution of ARB polymerization was originallyderived by Flory using statistical arguments and is presented here for its historicalsignificance [17] A polymer molecule of chain length n has n 1 reacted A (orB) groups and one unreacted A (or B) Therefore, the probability of obtaining asequence of n 1 reacted and one unreacted A group in a polymer molecule ofsize n would be pn1ð1 pÞ and the number of molecules of size n, Nn, would begiven by the product of this probability and the total number of molecules present
in the reaction mass at that time; that is,
Nn¼ Npn1ð1 pÞ ð3:6:6ÞUsing Eq (3.4.3), we obtain
Nn
N0¼ ð1 pÞ2
which is identical to ½Pn=½P10, given by Eq (3.6.3)
If M1 is the molecular weight of the monomeric repeat unit, the weightfraction, Wn, of a molecule of size n would be given by
Trang 18FIGURE3.2 Number fraction distribution in ARB step-growth polymerization in batchreactors using pure monomer feed [Reprinted from P J Flory, Chem Rev., 39, vol 137(1946) with permission of American Chemical Society.]
FIGURE3.3 Weight fraction distribution in ARB step-growth polymerization in batchreactors using pure monomer feed [Reprinted from P J Flory, Chem Rev., 39, vol 137
Trang 19only shifts to higher and higher molecular weight but also broadens out Figures3.2 and 3.3 dictate that even though the concentrations of the low-molecular-weight homologs P1, P2, and so forth are always the highest, their weightfractions decrease significantly as p changes from 0.95 to 0.99.
It may be re-emphasized that in deriving Eq (3.6.3) it is essential to assumethat the feed to the batch reactor is a pure monomer If higher homologs arepresent in the feed, as would be encountered in any intermediate reactor in asequence of batch reactors, the molecular-weight distribution would be differentand the polydispersity index (PDI) of the polymer formed would not necessarily
be restricted to the limiting value of 2, as shown in Appendix 3.1 As a matter offact, one of the practical methods of achieving a PDI of more than 2 is to partiallyrecycle a portion of the product stream, as shown in Figure 3.4 [19–22].Polymerization is carried out in a tubular reactor, and a fraction F of the product
is mixed with the monomer feed The mole balance equations for tubular reactorsunder suitable variable transformations become identical to those for batchreactors [23]; these must be solved simultaneously, along with mole balanceequations for the mixer The solution of polymerization with mixing is involvedand has, therefore, been omitted in this book
Example 3.4: Prove the following summations:
ðm þ nÞPnPn ð1Þ
S2¼P1
n¼1
nk P1 m¼nþ1
Trang 20The general term of this series is ðm þ nÞkPmPn and, therefore, the sum S1 isgiven by
S1¼P1
n¼2
P1 n¼1
ðm þ nÞ2
PmPn
¼P1 n¼2
P1 n¼1
ðm2þ n2þ 2mnÞPmPn
¼P1 n¼1
Pm¼ 1kðP2þ P3þ P4þ Þ þ 2k
ðP3þ P4þ P5þ Þ þ 3kðP4þ P5þ Þ
¼ ½P2ð1kÞ þ ½P3ð1kþ 2kÞ þ
¼P1 n¼2
P
n1 m¼1
Trang 21Summation S3 is the kind which appears in redistribution reaction
Pn P1 j¼2
P
j1 i¼1
P
j1 i¼1
ðn þ iÞ2
Pj
¼ P1 n¼1
Pn P1 j¼1
P
j1 i¼1
ðn2þ 2ni þ i2Þ
Pj
¼ P1 n¼1
Pn
P1 j¼1
Example 3.5: Consider the polymerization of AA þ BC monomers where both
B and C react with A at different rates Determine the number-average molecularweight of the polymer
Solution: Let us say that at time t ¼ 0, the concentrations of A, B, and C are
½Ao, ½Bo, and ½Co, respectively, such that the total number of molecules, N0, perunit volume is
N0¼½A0
2 þ ½B0and
½B0 ¼ ½C0
However, as time progresses, these concentrations become different due todifferent reactivities, and the number average molecular weight cannot beobtained from pure kinetic analysis of functional groups The analysis presented
Trang 22is strictly true for batch reactors where probabilities can be equated to sions.
conver-Let us define probabilities pA, pB, and pC for finding reacted A, B, and Cfunctional groups, respectively, at time t These can be taken as equal conversions
pAỬơB0
ơA0 đ pBợ pCỡand in terms of N0, one has
WtỬ đTotal number of molecules and of AA monomerỡ MAA
ợ fTotal number of molecules of BC monomer đơA0=2ỡMAA
Trang 23The number-average molecular weight mA is
step-In order to confirm the equal reactivity hypothesis, Flory originally studiedthe polymerization of adipic acid with decamethylene glycol in the absence of astrong acid [4,5] The course of the polymerization was followed by titrating thecarboxylic end group Flory assumed that the carboxylic groups act as a catalylst,and he represented the polymerisations as
Ok
COOH þ OH !COOH CO þ H2O ð3:7:1Þwith the rate of reactions given by
d½COOH
dt ¼ kp½COOH2½OH ð3:7:2Þ
In this experiment, conditions were maintained such that reaction (3.7.1)remained irreversible during the entire period of study If the hydroxyl and thecarboxylic acid groups are present in an equimolar ratio, Eq (3.7.2) can beintegrated to give
1
½COOH2¼ 2kpt þ const1 ð3:7:3Þ
If the initial concentration of the carboxylic acid group is ½COOH0, then itsconcentration at any time can be expressed as a function of the conversion, p, asfollows:
½COOH ¼ ð1 pÞ½COOH0 ð3:7:4Þ
Trang 24On substituting this into Eq (3.7.3), we obtain the following:
1ð1 pÞ2 ¼ 2kpt½COOH20þ const2 ð3:7:5Þ
A plot of 1ð1 pÞ2versus time should be linear; in Figure 3.5, it is found to be soafter 1=ð1 pÞ2
values of about 25.7 This means that this kinetic representation
is valid only after 80% conversion If the reaction is catalysed by a strong acid
FIGURE3.5 Catalyzed and uncatalyzed polymerization of ethylene glycol–adipic acid(DE-A) and ethylene glycol–caproic acid (DE-C)
Trang 25(e.g., toluene sulfonic acid), the reaction rate represented by Eq (3.7.2) then has
to be modified to
d½COOH2
dt ¼ kp*½Hþ½COOH½OH ð3:7:6Þwhere ½Hþ is the concentration of the acid Because the concentration ofhydrogen ions remains constant during polyesterification, ½Hþ can be absorbedwith k*p, and Eq (3.7.6) can be easily integrated If the concentrations offunctional groups COOH and OH are again equal, the following integratedform is obtained:
1
½COOH¼ 2kpt þ const1 ð3:7:7Þwhere kpis equal to k*p½Hþ and is a constant Equation (3.7.7) can be rewritten asfollows:
mn¼ 1
ð1 pÞ¼ kp½COOH0t þ const2 ð3:7:8ÞThe acid-catalyzed polyesterification of ethylene glycol and adipic acid has beenstudied by Flory The data are plotted inFigure 3.5 This figure also reveals that
Eq (3.7.8) holds after about 80% conversion More extensive experimental data[24–27] do not, however, confirm Flory’s conclusions It has been argued thatonly a limited amount of adipic acid dissociates in ethylene glycol (0.390 molesper mole ethylene glycol), and only this acid contributes to the catalysis in thepolymerization without strong acid Hence, instead of Eq (3.7.2), it has beenproposed that
d½COOH
dt ¼ kp½COOH½OH2 ð3:7:9ÞSimilarly, for acid-catalyzed polymerization, instead of Eq (3.7.6),
d½COOH
dt ¼ k00½COOH2 ð3:7:10Þhas been proposed If adipic acid and ethylene glycol are fed at a molar ratio of