reaction kinetics in dilute solutions of chain molecules carrying randomly spaced reactive and...

J. POLYMER SCI.: PART C NO. 31, PP. 177-192 (1970)

REACTION KINETICS IN DILUTE SOLUTIONS OF CHAIN MOLECULES CARRYING RANDOMLY SPACED REACTIVE AND CATALYTIC CHAIN SUBSTITUENTS. 1. ESTIMATION OF RING CLOSURE PROBABILITIES FROM KINETIC DATA AND COMPUTER SIMULATION*

NEIL GOODMAN and HERBERT MORAWETZ

Polymer Research Institute, Polytechnic Institute of Brooklyn, Brooklyn, New York I1201

SYNOPSIS

Acrylamide was copolymerized with small proportions of monomers containing a p-nitrophenyl ester group and a small amount of a monomer containing a pyridine residue. The hydrolysis of the ester groups was studied under conditions where the rate of the process was controlled by interaction with pyridine residues attached to the same chain. Variables studied included the mode of attachment of the ester groups to the chain backbone, the concentration of the catalytic pyridine residues in the chain and the fraction of the pyridine present in the unprotonated, catalytically active form. Kinetic data obtained at different pH values gave the same results if the extent of reaction was expressed as a function of the product of the experimental time and the fraction of pyridine residues in the basic form, This proves that the intramolecular process is not controlled by the rate at which cyclic conformations are formed but by their probability. The reaction deviated from first-order kinetics because the reactive groups had varying spacings from the catalytic groups. Experimental data were in good agreement with the kinetics derived by computer simulation if the probability of group interaction was made inversely proportional to the square of their separation along the chain molecule. Initial apparent first-order rate constants were about 40% of the value predicted on the basis of a simple theory based on the average extension of the polymer coils.

INTRODUCTION

The reactivity of substituents attached to the backbone of polymer chain molecules may be substantially increased by the presence of a suitable neighboring group carried by the chain molecule.( la) Typical examples of such effects are the enhancement of the solvolysis of phenyl ester groups by neighboring ionized carboxyls,(2,3) of amide or anilide groups by un-ionized carboxyls,(4,5) and of acetate groups by neighboring hydroxyls.(6-8) A different problem arises when the interacting groups are not located on neighboring residues of the polymer chain. In this case, the rate of the process will reflect the

*Dedicated to Professor Herman F. Mark in honor of his 75th birthday.

177 01970 by John Wiley & Sons, Inc.

178 GOODMAN AND MORAWETZ

probability of chain conformations which bring the two interacting groups into juxtaposition and it will, therefore, be related to the chain flexibility.

In the present investigation we have studied the kinetic behavior of dilute solutions of acrylamide copolymers with small proportions of the reactive monomers I or I1 and the catalytic monomer 111. The solvolysis of the nitrophenyl ester groups is catalyzed by pyridine residues(9,lO) and the dilution

CH3 I

I co 1

cy= c

0

0 I

I

C%=CH 1 co I

1

6 I

(a) n = I (b) n = 5

I1

NO2

CH,=CH I co I NH

I

I11 of the polymer was such that the catalytic effect was significant only if the pyridine group was carried by the same chain as a given ester group. Since the ester groups were located at different spacings from the catalytic groups, it was possible to interpret the data in terms of the dependence of the ring closure probability on the size of the ring formed.

THEORETICAL CONSIDERATIONS

According to the well-known treatment of W. Kuhn,(ll) the probability distribution function of the end-to-end distance h of freely jointed chains containing a large number Z of infinitely thin links with a length b is given by

F(h)dh = ( 2 7 ~ Z b 2 / 3 ) - ~ / ~ exp { -3h2/2Zbz 1 47rhZdh (1)

Kuhn also pointed out that F(h)/47rh2 has the dimensions of concentration and that it approaches a limiting value for h2 Q Zb2. This limiting concentration [F(h)/4nh2] h = O c:flmay be viewed as the average concentration of the second chain end in the vicinity of the first chain end. Since the distribution function F(h) leads to Zb2 = v12), where UzZ) is the mean-square chain end displacement, one obtains when expressing c'&in units of moles per liter

REACTION KINETICS. I 179

ctff = (lOOO/N) (3/2n<h2>r12

where fl is Avogadro’s number. In principle, it is possible to estimate czff from kinetic data involving a

reaction between two groups attached at the two chain ends. If the characteristic second-order rate constant for a bimolecular reaction between these groups is k,, then the first-order rate constant characterizing the intramolecular process will be k , = k2czff and

k, = c/<h2>3/2

C = (100OlcJfl) (3/2srl2 (3)

Let us now consider the behavior of real chain molecules. In 0 solvents, where effects of long-range interactions of chain segments on the extension of the molecular chain are eliminated, (h2) is still proportional to 2, but the chain stiffness is increased by a factor which reflects bond angle restrictions, the distribution of the internal angles of rotation and correlations between these angles of rotation around adjoining bonds. As a result #z2)/Zb2 assumes values ranging generally, for vinyl polymers, from 7 to 10.(12) We should also note, that this limiting value of (h2)/Zb2 is attained rather slowly. This principle is illustrated by calculations for polymethylene chains, where (h2)/Zb2 reaches only about 80% of its limiting value for a chain of 25 bonds.( 13) The probability of ring closure may also not be expected to be given precisely by Kuhn’s statistical theory for relatively short chains. For instance, estimates of ring closure probabilities by direct enumeration of conformations with very small end-to-end displacements in short polydimethylsiloxane chains led to significantly higher values as compared to those predicted by eq. (2) on the basis of the (h2)/Zb2 ratio characteristic of this polymer,(l4,15) In addition, it should be pointed out that the Kuhn statistics of chain conformation neglect changes in conformational energy accompanying changes in the end-to-end displacement of the chain molecule. It would be expected, for instance, that cycloalkanes contain a higher proportion of gauche bonds than the corresponding open chains. This factor should make a positive contribution to the free energy of activation for ring closure and thus reduce the cyclization rate below the values derived from theories in whch this factor is neglected.

When a chain molecule is placed in a medium in which the excluded volume effect does not vanish, the ring closure probability will be sharply reduced. Various attempts to estimate the magnitude of this effect have been made by generating, on a computer, chains on various lattices, rejecting those in which a lattice point was occupied more than once.(16-18) The results indicate that the chain closure probability falls off approximately as Z-2 if the excluded volume effect is operative, instead of being proportional to P 3 I 2 as predicted by Kuhn.

Ideally, ring closure probabilities should be investigated on molecularly homogeneous chain molecules carrying two interacting groups at the chain ends. The most extensive study of this type was carried out by Stoll and RouvC(19) on the homologous series of a, w-hydroxyalkanecarboxylic acids up to


HO(CH,),,COOH. Although the detailed interpretation of their data is incorrect(20) their results show clearly that rings of intermediate size (8-1 3 atoms) form with extreme difficulty because of the crowding of hydrogens in the interior of the rings. A statistical treatment of ring closure probabilities, in which effects of such steric interference is not taken into account, is, therefore, applicable only to rings containing more than 20 atoms, Such restrictions may be even more severe if rings are formed from branched chains, such as polymers of vinyl compounds. Clearly, the synthesis of molecularly homogeneous molecules with interacting groups at the chain ends and with chain lengths sufficient to test predictions of chain closure probabilities would encounter prohibitive difficulty.

It is for this reason that we decided to approach the problem indirectly, by an analysis of the kinetic pattern obtained with chain molecules carrying a small number of randomly distributed reactive (R) and catalytic (C) groups (Fig. 1). In such a system the rate constants characterizing the various reactive groups will vary within wide limits, depending on the distribution of the catalytic residues around a given reactive group.

For a chain without the excluded volume effect, we may treat the probability of interaction of a reactive group attached to the nth monomer residue with a catalytic group on thejth residue as equivalent to the probability of interaction of groups attached to the ends of a chain with lj-nI segments. The mean-square separation of the jth and nth residues 'hii) is related to the mean-square separation of the chain ends (h2) by (hii) = (h2) lj-nl/Z,( 12) and we must take into account that the statistical theory is applicable only to rings containing more than some critical number x' of monomer residues. We obtain then, in analogy with eq. (2), for the effective local catalyst concentration in the vicinity of the nth residue of a chain of 2 residues with a random distribution of attached catalytic substituents:

'eff = ( l O O O / f l ) (32/2~012>)3/2 X

(4) ) ( x=l-n X=%'

x = -.J x=Z-n c ipiXi-3/2 t c ipiXi-3/2 + ceff

Here x = j-n; pl = 1 or pi = 0, depending on whether the j th monomer residue does or does not carry a catalytic substituent; cLfi is a correction term accounting for the probability of interaction of groups separated by fewer than x' monomer residues. The first-order constant characterizing a reactive group attached to the nth monomer residue is then again k, = k2c,

We have noted above that Monte Carlo calculations of the number of nonintersecting chains generated on a computer for various types of lattices led to the estimate that the ring closure probability for chains with excluded volume fell off at .P2 rather than 2-312. Assuming then that the ring-closure exponent Q is a function of the solvating power of the medium, we may express the first-order rate constant characterizing a given reactive group attached to a polymer which also carries catalytic substituents as

REACTION KINETICS. I 1 8 1

FIG. 1. Schematic representation of terpolymer with randomly spaced reactive and catalytic side chain substitutents.

where C' is a constant which should be related to u and k' is the contribution to the rate constant from catalytic chain substituents separated by fewer than x' monomer residues from the reactive group.

With a random distribution of catalytic groups, eq. ( 5 ) will yield rate constants characterized by a distribution function W(k) and the course of the reaction will then be given by

y = J W ( ~ ) exp { -kt 1 dk (6)

where y is the fraction of reactive groups remaining at time t . It is shown in Appendix I that the initial apparent first-order rate constant kFd is equal to the average value of k, i.e.

It is, then, the main purpose of the present study to compare the observed course of the reaction with predictions based on the W(k) which results from various functional dependences of the ring-closure probability on the spacing of the interacting groups along the polymer chain backbone.

RESULTS

Reactions of Low Molecular Weight Analogs The hydrolysis of monomers IIa and IIb was studied in solutions containing

monomer 111 as the nucleophilic.catalyst in buffers of pH 5 , 6, and 7. The results yielded k, = 4.5 l./mole-min for IIa and k2 = 1.3 l./mole-min for IIb. The dependence of the reaction rate on the pH of the solution was consistent with pK = 5.7 (referred to as the kinetic pK) for the pyridine residue of monomer 111. A spectrophotometric determination based on the dependence of the optical density at 260 nm on the state of ionization of the pyridine residue led to pK = 5.9.


Hydrolysis of Copolymers without Catalytic Residues

Results for the hydrolysis of ester containing copolymers without catalytic residues are listed in Table 1. All these rates are very low compared to those obtained with the terpolymers to be discussed below. Only with the copolymer of monomer IIb, which carried the ester groups at a considerable distance from the chain backbone, was a very slight intermolecular catalysis observed in solutions containing a mixture of reactive and catalytic copolymers.

Hydrolysis of Terpolymers Containing Reactive and Catalytic Chain Substituents

Dependence on the Mode of Attachment to the Polymer Backbone. Ter- polymers containing the reactive monomer I were hydrolyzed at the same rate as copolymers carrying no catalytic residues. This shows that steric inhibition is prohibitive when the ester carbonyl is attached directly to a quaternary carbon of the chain backbone. This result is hardly surprising in view of the recent demonstration(21) that catalysis of the solvolysis of nitrophenyl esters by pyridine requires the formation of a labile acylpyridinium reaction intermediate. A similar steric inhibition has recently been demonstrated as preventing the rearrangement of poly@-aminoethyl methacrylate) according to

CH3 I

I co I O ~ H P H ,

(-C-CHz-), - although the corresponding reaction is analog, 0-aminoethyl pivalate.(22)

CH3 I

I co I NHC,&OH

(-C-CHz -),

extremely fast with the monomeric

The solvolyses of terpolymers containing the catalytic monomer 111 and the reactive monomers IIa or IIb exhibit a pronounced effect of intramolecular catalysis. In interpreting the magnitude of this effect, we have to take account of the second order rate constant k2 characterizing the corresponding intermolecular process and the mole fraction w of the catalytic residues in the terpolymer. The ratio of the initial apparent first-order rate constant k \ ~ t to k,w for the two types of terpolymers is given in Table 11. We may see that extension of the side chain beyond the length characterizing monomer IIa has little effect on the efficiency of the intramolecular catalysis.

Character of the Kinetic Pattern. As pointed out in the section on theoretical considerations, the reactive groups attached to a terpolymer will have different reactivities depending on the spacing of a given reactive group from the various catalytic substituents along the polymer chain. Since those reactive groups which have the most favorable spacing from the catalytic functions will tend to disappear fastest, the apparent first order rate constant will decrease with the progress of the reaction.


TABLE I

Hydrolysis of Copolymers without Catalytic Residues (2SoC, pH 5) ~~ -

Reactive monomer in copolymer I IIa IIb

[ t l l& , dllg 0.63 0.89 0.76 Rate constant, no

Rate constant in

catalyst, min-1 4.8 x 10-5 7.4 x 10-4 2.1 x 10-5

copolymer, min-1 a 4.8 x 10-5 7.4 x 10-4 2.7 x 10-5 presence of catalytic

a The catalytic copolymer contained 1.2 mole % monomer 111 and had [q] 25 = H20 1.00 dl/g. It was used a t a concentration of 1.0 g/l.

TABLE I1

Dependence of Intramolecular Catalysis in Terpolymers on the Length of the Side Chain to Which the Reactive Group is Attached (25OC, pH 5)

Terpolymer Terpolymer A B

Mole fraction catalytic monomer ( w ) 0.012 0.010

Mole fraction monomer IIa 0.010 0

Mole fraction monomer IIb 0 0.010

[7I1-2H5,0, dl/g 1.65 1.57 Degree of polymerization 4,900 4,300 k, , 1 ./mole-min 4.5 1.3 kinit

1

kpii/k2w 2.7 2.9 7.1 7.1 emax

eff

0.148 0.038

It is easily seen that the dispersion of the rate constants given by the probability distribution function W(k), will tend to narrow down with an increase of the concentration of the catalytic groups in the polymer chain. In the limit of a copolymer containing a small number of reactive residues and catalytic groups on all the other monomer residues, all the reactive groups would be in an equivalent environment and first-order kinetics would be expected. The effect of the mole fraction o of catalytic monomer units on the appearance of a first-order plot is illustrated in Figure 2. For easier comparison of the kinetic curves we have multiplied the experimental time f by a. i t may be seen that the initial slope of these curves is very similar, i.e., the initial reaction rate is proportional to the mole fraction of catalytic monomer in the terpolymer. It can also be seen that deviation from first-order kinetics decreases with an increase in w .

It should be noted that for a terpolymer of finite molecular weight some


Y

I I -1.0) I I 1

0 0.5 1.0 2.0 3.0

FIG. 2. Hydrolysis of acrylamide terpolymer with 1.0 mole-% of reactive monomer 1Ia and m y @ amounts of catalytic monomer 111 at pH 5.0. The product of the experimental time, is in each case multiplied by w to facilitate a comparison of the relative deviation from first-order kinetics.

reactive groups will be attached to chains which carry no catalytic chain substituent. Such groups will exhibit no appreciable reaction under our experimental conditions. It is shown in Appendix I1 that the fraction of such unreactive groups should be

w t (min )

f = (1 tP,w)-2 (8) where % is the number-average degree of polymerization, and the polymer is assumed to have a normal distribution of chain lengths, The prediction of eq. (8) was in satisfactory agreement with an experiment on a terpolymer with w = 2.0 X in which a fraction f = 0.55 of the ester groups were unreactive. According to eq. (8), < = 1700 which is in reasonable conformity with the viscosity-average degree of polymerization Fv = 1500.

The application of Kuhn’s statistical theory or more refined theories of chain configuration to the interpretation of reaction kinetics involving the intramolecular interaction of chain substituents implies that the rate of conformational transitions of the polymer chain is not rate limiting in such a process. In other words, it is assumed that an extremely large number of collisions of the interacting groups is required for the reaction, so that these groups will diffuse towards each other and separate many times before the reaction is accomplished. Only under these conditions will it be legitimate to use the probability of cyclic conformations to predict reaction rates. In our system, this assumption is very easily tested, since we may change the reactivity of the catalytic groups by changing the pH. If the rate of conformational transitions of the chain backbone is not rate limiting, the reaction rate should be proportional to the fraction 01 of pyridine residues in their basic form. This prediction may be formulated by the postulate that the progress of the reaction should be a unique function of tar, independent of pH. Figure 3 demonstrates that the experimental data are, indeed, in accord with this prediction. Data obtained at pH 4 and at pH 5 (which correspond, on the basis of the kinetic pK of the catalytic nucleophile, to ar = 0.02 and ar = 0.2) all lie on a single curve on this plot.


Computer Simulation

To assess the magnitude of the “ring closure exponent” governing the dependence of the interaction of a catalytic and a reactive group on their separation along the polymer chain backbone, kinetic patterns obtained by computer simulation were compared with experimental data. For this purpose samples of 100 chains with 1000 monomer units and with a random distribution of catalytic groups for various values of the probability o that any given monomer carries a catalytic group were generated on a computer. The k values were obtained according to eq. ( 5 ) by neglecting k‘ and using a = 3/2 or a = 2. Figures 4 and 5 show histograms illustrating the distributions of the ratio of the rate constant k to its average value OC) as a function of either the ring closure exponent a or the density of catalytic groups w.

In Figure 6 we compare the deviation from first-order kinetics observed with a terpolymer with the kinetic data derived from the computed distribution functions W(k) by using a = 3/2 or a = 2 and adjusting C‘ in eq. (5) so as to make the initial rate correspond to the experimental value. It may be seen that the computed points corresponding to a = 2 are in very good agreement with experiment, while a = 3/2 leads to a much smaller deviation from first order kinetics than observed experimentally.

Relation between the RingClosure Probability and the Overall Chain Dimensions

According to eq. ( 2 ) , czff is proportional to (h2)-3/2, i.e., the probability of ring closure involving the ends of a chain without excluded volume may be thought of as inversely proportional to the volume occupied by the molecular coil. In good solvent media the molecular coil tends to expand and it is then of interest to inquire whether the ring closure probability is related in some simple manner to this expansion. The mean-square end-to-end displacement (h2) of flexible chain molecules in strong solvent media approaches asymptotically proportionality to 21.2 according to Flory(23) and proportionality to Z4l3 according to Kurata et al.(24) This leads to (h2)-3/2 proportional to Z-’.8 and Z-2, respectively, which is the range of the ring closure exponents for chains with excluded volume predicted on the basis of computer simulation,( 16-18) It may, therefore, be conjectured that ctff u12)3/2 is independent of the solvent power of the medium.

According to Flory’s theory(23) the intrinsic viscosity [q] is related to the molecular weight M of flexible chain polymers by

<h2>j/’- = 1771 M / @ ( 9 )

where is a quantity which is almost constant in a wide variety of polymer-solvent systems. Over a limited molecular weight range, the relation of [q] and M may generally be represented by the Mark-Houwink relation [q]=KM7, where K and y are constants characterizing a given polymer-solvent system. This relation leads to ( ( h ; , ) / ( / ~ 2 ) ) - ~ / ’ - = ( l j -- nVZ)’+Y. For relatively short chain molecules, the chain expansion has been shown to be independent of the solvent power of the medium(25,26) and we iriay take this into account by


000

I O -

0 8 -

Y

0 6 -

04 -

0 2 -

Computed Distribution of Rote Constants

( o_ =3/2 1 r--i I I

W =O 048

- I I 1 ; I I

A p H 5 O a = 0 2

b

1

FIG. 3. Hydrolysis of acrylamide terpolymer with 1.0 mole-% of reactive monomer IIa and catalytic monomer 111 (w = 0.012) at different pH values. The experimental time is multiplied by a: the fraction of g o u p s in the catalytically active form, at each pH value.

0 12

k w (-1 <k>

O o 4 L 0

l4G. 4. Computer gcncrated distribution of k / a>demonst ra t ing the effect of a variation in the density w of catalytic groups at a constant ring closure exponent.


0.12

Computed Distribution of Rote Constants

( w =0.012 1

FIG. 5 . Computer generated distribution of k/<k>demonstrating the effect of varying the ring closure exponent at a constant density w of catalytic groups.

- 0.4 -

log Y

- 0.6 -

-0.8 -

w = 0.012

0 expt l .

A Computed, a = 2 x Computed,&= 1.5

10 20 30 40 5 0 a t

FIG. 6. Hydrolysis of acrylamide terpolymer with 1.0 mole-% of reactive monomer Ila and catalytic monomer 111. The experimental results (0) are compared with computed kinetic points for two values of the ring-closure exponent.


specifying that ctff is independent of the nature of the solvent for l j - nl = XI.

We may then generalize eq. (4) to x=+' x=Z-n

c,, = c'za c l P i X P -I- c x=l-n x=-xr (

C' = (3/2nyI2 (1000*//"~]M) (XI)"-312

a = l + r

Consider now c$: the maximum possible value of ceff corresponding to a copolymer with a small number of reactive residues and with a catalytic group on all other monomer units (i.e., pi = 1 for all j values). On replacing the sums by integrals and neglecting Ckfir, we have

x = n - I x=Z-n

J x a d x t f x-WX) (11) ( x=xI x=xl

(12)

p a x =clza eff

which yields for reactive groups well separated from either chain end

cgaX = (3/2n)3/2 (~OOO@/RM: )

where Mo =M/Z is the molecular weight of the monomer residue. If a terpolymer contains only a fraction o of residues carrying catalytic

substituents, the initial probability that a reactive group picked at random will find a catalytic substituent at any given spacing x will be w and the initial apparent first-order rate constant kp t should be crRmok2. In the last line of Table I1 is the c r + value calculated by using Q, = 2.2 X lo2', the values for the Mark-Houwink constants K and y given by Scholtan(27) and xr = 10. We may see that it is about 2.5 times as high as the experimental values for kpt/k2w. The discrepancy may be due to several factors: (a) the value of c ~ F is dominated by the probability for the formation of relatively small rings for which the theory is least satisfactory; (b) the choice of XI may be too low if steric hindrance of ring closure is substantially more effective in vinyl polymer chains than in unbranched paraffins; (c) the difference in conformational energy of extended, and circular conformations may render ring closure less probable than predicted on the basis of a model in which this difference is neglected; (d) reactive groups attached close to the end of the polymer chain would, on the average, be subject only to half as many intramolecular collisions with catalytic substituents than groups attached at a great distance from either chain end.

DISCUSSION Two aspects of our result deserve further comment. We found unambiguous evidence that the rate of conformational transitions

of the polymer chain is not rate determining for the intramolecular process which we investigated. This result is analogous to the well-known fact that bimolecular reactions proceed at rates independent of the viscosity of the medium, provided the process is characterized by an appreciable activation


energy. This is so, since an increasing viscosity slows equally the diffusion of the reagents towards each other and their separation. Obviously, reaction rates do become sensitive to viscosity if only very few collisions of the interacting species are required and it would, therefore, be of interest to supplement the present study with one in which the interaction of the species attached to the polymer would be characterized by a low activation energy. Such a study might employ, for instance, chain molecules with fluorescing and quenching chain substituents.

Two previous studies concerned with the rates of interaction of groups attached at the ends of macromolecules were concerned with the question whether the diffusion of these groups towards each other is rate limiting. Wang and Davidson,(28) who studied the cyclization of a species of deoxyribose nucleic acid with “cohesive” chain ends, concluded that the process was not diffusion-controlled. Koenig and Banderet(29) followed the condensation of chains containing amino end groups with chains carrying acid chloride endgroups and tried to find kinetic evidence for the cyclization of the species

HZN ---- NHCO ----- COCl

They report that this process is diffusion-controlled, but their interpretation of the experimental data may be considered somewhat uncertain.(30)

The high ring-closure exponent which is indicated by this study should be considered in conjunction with the solvent power of the medium for the polymer under investigation. If we consider the exponent y in the Mark- Houwink relation [s] = K M y between the intrinsic viscosity [q] and the molecular weight M as a guide, then the value of 7 = 0.8 reported for the polyacrylamide-water system(27) indicates extremely strong polymer-solvent interaction. Even so, our procedure in which ring closure for a chain of x’ units is treated as independent of the nature of the solvent while the ring closure exponent has a constant value, dependent on the solvent, for all larger rings, is clearly only a rough approximation. In principle, a should vary continuously towards a limiting value for large rings. It is also of great interest to see, whether in 0 solvents, in which the excluded volume vanishes as judged by either osmotic behavior or by chain expansion, the ring closure exponent will have the value of 3 f2 expected from theory. This problem will be the subject of a future communication.

EXPERIMENTAL

Monomers

Acrylamide (Eastman Kodak) was recrystallized from chloroform. p-Nitro- phenyl methacrylate (1) (mp 94-96OC) was prepared by adding methacrylyl chloride to an aqueous solution of p-nitrophenol. The product was recrystallized from ethyl acetate. To prepare the p-nitrophenyl ester of N-acrylylglycine (Ha) (mp 121-123”C), glycine was first acylated with acrylyl chloride under Schotten-Baumann conditions. The resulting N-acrylylglycine was then esterified by using bis(p-nitrophenyl) sulfite. The product, Ila, was recrystallized from ether-acetone. Compound llb, the p-nitrophenyl ester of N-acrylyl-6-amino-


caproic acid (mp 113-115°C) was prepared in an analogous fashion. The catalytic monomer (111), 4-N-picolyl acrylamide (mp 71-72"C), was prepared by adding acrylic anhydride to 4-picolylamine. The product was recrystallized from benzene.

Preparation and Characterization of Polymers

Polymerization of acrylamide with the reactive and catalytic monomers was carried out in deaerated solutions of spectroquality dioxane (Metro Scientific), at a temperature of 60°C. The reactions were initiated with 0.1 wt-% of azobisisobutyronitrile (based on the monomers). The polymer precipitated during the polymerization. The concentration of reactive and catalytic groups in the polymer was obtained after complete hydrolysis of the ester groups and after addition of pH 7 buffer by determination of the p-nitrophenol produced, using the molar extinction coefficient of 9.5 X lo3 at 320 nm.(31) The analysis for the catalytic pyridine groups was complicated because the polymer absorbed in the same region as the pyridine residues (260 nm). The extinction coefficient at pH 2 was found to be higher by 2.1 X lo3 l./mole-cm as compared to the value at pH 10. This differential extinction coefficient was used to find the concentration of pyridine residues in polymer solutions.

The molecular weight of the copolymers and terpolymers was characterized by measurements of the intrinsic viscosity in water at 25°C. The intrinsic viscosity was related to the molecular weight by [Q] = 6.31 X 10-5fl.80 for polyacrylamide .( 27)

Kinetic Measurements

For experiments involving samples of low molecular weight, stock solutions of the esters were prepared in dioxane, and 0.1 ml of the stock solution was mixed with 100 ml of the appropriate buffer solution at the start of the run. In studies involving copolymers and terpolymers, the samples were dissolved in 0.002M HCl, and the kinetic run was started by mixing 1 part of this stock solution with 9 parts of 0.05M buffers (biphthalate at pH 5, phosphate at pH 6 and 7). In both cases, the solutions were thermostated at 25.0"C and the progress of the ester hydrolysis was followed by recording the increase in absorbance at 320 nm. in a Beckman DU spectrophotometer as a function of time. The pH of the solution was checked on a Cambridge Model pH Meter at the end of the run.

APPENDIX I

Consider a system with a variety of reagents with an initial concentration ci characterized by first-order rate constants ki. If Zci= c,

-dcldt = kici

The apparent initial first-order rate constant is defined by

-dcldt=kPt ci

REACTION KINETICS. I 19 1

Comparing ( 13) and ( 14) yields

kpt = k,ci/ ci s. ( ki) (15)

where (ki) is the average rate constant characterizing the molecules in the system.

APPENDIX I1

Consider chains with a normal distribution of the degree of polymerization P, which is to be expected for vinyl polymers in which the chain is terminated either by chain transfer or by disproportionation of the chain radicals(1b). The weight distribution function W(P) is then given by

WP) = ( 1 IF, P exp { -p/pn 1 (16)

where pn is the number-average degree of polymerization. If w is the probability that any monomer residue picked at random carries a catalytic group, the probability that a chain contains none of these groups is exp { -UP/. Since the distribution of reactive groups among chains of degree of polymerization P is WP), the fraction f of reactive groups attached to noncatalytic chains is

f = (I/P,)~ J P exp - P / F ~ 1 exp {-UP/ d~ (17) 0

Solving the integral leads to

f = (1 + PnW)--' (18)

This paper is based in part on a Ph.D. thesis to be submitted by Neil Goodman to the Graduate School of the Polytechnic Institute of Brooklyn in June 1971. The grant of a predoctoral fellowship No. 1-FOl-GM-41,173 to N. Goodman and of research grant MG-05811 by the National Institutes of Health provided financial support of this research and they are gratefully acknowledged.

REFERENCES

(1) H. Morawetz, Macromolecules in Solution, Wiley, New York, 1965, (a) pp. 422426;

(2) H. Morawetz and P. E. Zimmering, J. Phys. Chem., 58,753 (1954). ( 3 ) E. Gaetjens and H. Morawetz,J. Amer. Chem. SOC., 83,1738 (1961). (4) E. W. Westhead, Jr. and H. Morawetz, J. Amer. Chem. SOC., 80,237 (1958) . (5) G. Smets and A. M. Hesbain, J. Polym. Sci., 40,217 (1959). (6) L. M. Minsk, W. J. Priest, and W. 0. Kenyon,J. Amer. Chem. Soc.. 63,2715 (1941). (7) E. Nagai and N. Sagane, Kobunshi Kugaku, 12,195 (1955). ( 8 ) K. Fujii, J. Ukida, and M. Matsumoto,J. Polym. Sci. 8, 1,687 (1963). (9) T. C. Bruice and R. Lapinski,J. Amer. Chem. SOC., 80,2265 (1958).

(10) W. P. Jencks and J. Carriuolo,J. Amer. Chem. SOC., 82, 1778 (1960). (11) W. Kuhn, Kolloid-Z., 68, 2 (1934). (12) P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley, New York, 1969, pp.

(13) P. J. Flory and R. L. Jernigan,J. Chem. Phys., 42,3509 (1965).

(b) pp. 13-1 6.

40-42.


(14) J. B. Carmichael and J. Kinsinger, Can. J. Chem., 42,1966 (1964). (15) J. B. Carrnichael and R. Winger,J. Polym. Sci., Pt. A, 3,971 (1965). (16) F. T. Wall, L. A. Hiller, Jr., and W. F. Atchison,.!. Chem. Phys., 23, 2314 (1955). (17) B. J. Hiley and M. F. Sykes,J. Chem. Phys., 34,1531 (1961). (18) M E. Fisher,.!. Chem Phys., 45,1469 (1966). (19) M. Stoll and A. Rouve, Heh. Chim. Actu, 18,1087 (1935). (20) H. Morawetz and N. Goodman,Macromolecules, 3,699 (1970). (21) W. P. Jencks and M. Gilchrist,.!. Amer. Chem. Soc., 90,2622 (1968). (22) D. A. Smith, R. H. Cunningham and B. Coulton,J. Polym. Sci. A-I, 8,783 (1970). (23) P. J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y., 1953,

(24) M. Kurata, W. H. Stockrnayer, and A. Roig,.!. Chem. Phys., 33,151 (1960). (25) C. Rossi, A. Bianchi, and E. Bianchi,Mukromol. Chem, 41,31 (1960). (26) R. Okada, Y. Toyoshima, and H. Fujita,Mukromol. Chem., 59,137 (1965). (27) W. Scholtan, Makromol. Chem., 14, 169 (1954). (28) J. C. Wang and N. Davidson,J. Mol. Biol., 14,469 (1966). (29) R. Koenig and A. Banderet,.!. Chim. Phys., 65,2108 (1968). (30) H. Morawetz,Accfs. Chem. Res., 3,354 (1970). (31) W. Forbes, A. Ralph, and R. Gosinc, Can. J. Chem., 36,869 (1958).

Chap. XIV.

Received June 18, 1970

reaction kinetics in dilute solutions of chain molecules carrying randomly spaced reactive and...

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