Vol. 3, No. 5, September-October 1970
NOTES 699
Notes Derivation of the Ring Closure Probability from the Distribution of Reaction Products when Reagents of the Type X(CH2),Y Undergo Simultaneous Cyclization and Polycondensation
monomer by reaction with the functional groups at the ends of polymer chains. We have, therefore, reevaluated the relation between Y and X using the following kinetic scheme i; 1
HERBERT MORAWETZ* AND NEILGOODMAN
M+L
Polymer Reseurcli liistiiute, Polytecliriic Iirstitute of Brooklyii, Brooklyii. New York 11201. Receiced Juiie 19, 1970
More than 30 years ago, Stoll, et al.,2 carried out a n extensive study of the condensation of a,w-hydroxyalkanecarboxylic acids. The functional groups of these reagents may undergo both intramolecular and intermolecular condensation and the authors suggested a procedure for calculating the rate constant for the cyclization reaction from the yield of the monomeric lactone when the reaction was allowed t o go t o completion. The results of these calculations have been quoted3 as providing quantitative data to illustrate the difficulty with which rings of medium size (i.e., 8-13 atoms) are formed, due to bond angle distortion, partially eclipsed groups in 1,2-vicinal positions and trans-annular strain. In their derivation, Stoll, et a/.,* considered the processes h.1
k2
M+M-+-P (3) k2
M+P+P
k2
P+P+P
The monomer disappears according t o -d(M),’dt
=
kuC(M)
+ 2k2(M)’ + ks(M)(P) = ks(M)[C
d(P)/dt
=
k?[(M)’
- (P)’]
where M , L, and P represent monomer, monomeric lactone, and terminal monomer residue in the condensation polymer, respectively. They obtained then for the weight ratio Y of monomeric lactone to polymer2a
(5)
where we are neglecting the possible formation of polymeric rings. Introducing dimensionless parametersm=(M)/(MO),p=(P)/(Mo), wehave dpjdm
(1) k?
(4)
Similarly, the accumulation and subsequent decay of (P) is governed by
M+L
M+M+P
+ 2(M) + ( 0 1
= (p? -
+ 2m + p )
m2),/m(X
(6)
The ratio of the yields of the monomeric lactone and the polymer, respectively, is then obtained by integrating their rates of formation over all time, Le.
or, substituting for dr from eq 4 where X = C/(Mo), C is the “cyclization constant” k l l k 2 , representing the reagent concentration at which intramolecular and intermolecular reactions are equally probable, and (Mo) is the initial monomer concentration. The procedure of Stoll, et al., is in error o n two counts. First, it disregards the fact that a single monomer disappears in the cyclization, while two monomers are used up in their intermolecular condensation. Second, it neglects the disappearance of
* To whom correspondence
should be addressed. ( I ) This work is abstracted from a Ph.D. thesis to be submitted by Neil Goodman to the Graduate School of the Polytechnic Institute of Brooklyn in June 1971. Financial support by Grant GM-05811 of the National Institutes of Health is gratefully acknowledged. (2) (a) M. Stoll, A . Rouve, and G. Stoll-Comte, Helr. Chim. Acta, 17, 1289 (1934); (b) M. Stoll and A. Rouve, ibid., 18, 1087 (1935). (3) P. J. Flory, “Principles of Polymer Chemistry,” Cornell University Press, Ithaca, N. Y . , 1953, pp 96-97. (4) (a) V. Prelog, Pure A p p l . Chem., 6, 545 (1963); (b) A. C. Cope, M. M . Martin, and M. A. McKervey, Quart. Rer., Chem. Soc., 20, 119 (1966); (c) J. Sicher in “Progress in Stereochemistry,” Vol. 111, P. B. D. de la Mare and W. Klyne, Ed., Butterworths, Washington, D. C., 1962, pp 202-263; (d) J. D. Dunitz in “Perspectives in Structural Chemistry,” Vol. 11, J. D. Dunitz and J . A . Ibers, Ed., Wiley, New York, N. Y., 1968, pp 1-70.
or
y/(y
+ 1)
=
X
lo+ (2m
p
+ X)-ldm
(9)
Numerical solution of eq 6 and 9 for X values of 0.01, 0.1, and 1.0 gave values for Y of 0.0124, 0.113, and 0.988, respectively. In Figure 1 we compare the relation of X and Y obtained by our procedure (curve 2) with the results of Stoll, et a/. (curve 1). It may be seen that the cyclization constants estimated by Stoll, et al., from experimental Y values are too low by a factor which varies from 5 to 2.5 as Y increases from 0.01 t o 2. For long flexible chains, the displacement h of the chain ends from one another is given by a probability distribution function; W(h)dh
=
( 3 / 2 ~ ( / 1 ? ) ) ~exp( -3h2/2(h*))4~h*dh (10)
(5) W. Kuhn, Kolloid-Z. Z . Pol.im., 68, 2 (1934).
7CO NOTES
0.5
Mucromolrcules
/”
oo21
0.0I
O.Oo5I 0 002 1
/ 1
0002
I
l
OD1
l
1
005
1
1
1
02
05
1
1
IO
20
1
expected to affect significantly ( h 2 ) for chains with as few as 24 atoms in the backbone, this factor would be expected to have a much greater influence on the probability of conformations with a very small end-toend displacement. It is, therefore, quite likely that even with relatively short chains, the cyclization constant is sensitive to the solvating power of the medium. Finally, we may note that any comparison of experimental data with theory assumes that the reaction was in fact carried to completion and that isolation of the monomeric lactone was quantitative. Any failure to achieve these ends could account for part of the discrepancy between the experimental results and the theoretical prediction.
X
Figure 1. Relation of X and Y (1) according to Stoll, et al., and (2) according to the present work. where ( h ? )is the mean value of h 2 . Kuhn has pointed out that W(h)/4rh2 yields in the limit of h +. 0 the “effective concentration” of one chain end in the neighborhood of the other. This is identical with the “cyclization constant” discussed above and we have then, with C in units of moles per liter
where is Avogadro’s number. Let us now see to what extent the formation of the largest rings investigated by Stoll and RouvC,*” Le., a ring with 24 atoms, conforms to the prediction of eq 11. According to Flory,6 a polymethylene chain with 24 atoms in the chain backbone would be characterized by (h2)’ = 1.75 x 10-7 cm, yielding C = 0.1. Stoll and RouvC’s experimental resultZh was Y = 1.7 at (Mo) = 0.0046. With our treatment, these data yield C = 0.0076. Ring formation is thus found to be an order of magnitude more difficult for a ring of this size than predicted by the simple theory outlined above for a cycloalkane with the same number of atoms in the ring. There may be several causes for this discrepancy. (1) Configurational statistics, leading to eq 10 and 11, apply only to chains with a large number of segments. A significant error may be involved in applying this result to a 24-membered chain.’ (2) Although both experimental and theoretical studies show that ring strain in cyclolakanes falls off rapidly with ring expansion beyond cyclodecane,s it seems obvious that even large cycloalkanes would have a larger proportion of gauche conformations than the corresponding open chain compound, so that ring formation would lead to an increase of the conformational energy. The difference between the heat of combustion of cyclolakanes and an equal number of methylene residues in open-chain paraffins seems to level off for rings containing 14 or more members. Using the calculations of Dunitz,’d we may estimate this limiting value of ring strain energy as 2 kcal/mol. (3) Although the excluded volume effect would not be (6) P. J. Flory, “Statistical Mechanics of Chain Molecules,” Wiley, New York, N. Y . , 1969, p 147. (7) See, e.g., J. B. Carmichael and J. Kinsinger, Can. J . Chem., 42,1966 (1964). (8) M. Bixon and S. Lifson, Tetrahedrorz, 23,769 (1967).
Apparent Molal Volume of Poly(acrylic acid) in Aqueous Solution
S. FRIEDMAN,‘ A. C A I L L AND ~,~~ H. D A O U S T ~ ~ Deparimeiit of Chemistry, Uiiicersitt de Moil frkal, Motitreal, Canada Receiced April 2 7, 1970
For weak electrolytes, the decrease of the apparent molal volume, $,., with decreasing concentration (Le., increasing dissociation) has been attributed to the electrostrictional effect. Values of dvcan be calculated from
&
=
+
(M?/wn)[(wl
Wr)/d12
-
Wl/dI]
(1)
where M 2 is the molecular weight of the solute, w1and w2 are the number of grams of solvent and solute, respectively, in a given volume of solution, dl and d12 are the densities of the solvent and the solution, respectively. & can also be calculated from
4,. = Vzo
+ AV
(2)
where V p is the molar volume of the pure solute in its liquid state and A V is the apparent change in volume or the apparent excess volume upon dissolution of 1 mol of the liquid solute in a given solvent. For a weak electrolyte, A V is the sum of two terms, namely, AV,, the volume change due to the mixing of solute with water at zero degree of dissociation ( a ) and AVd, the volume change due to the dissociation process. Equation 2 can be rewritten in the form =
4“
+ AV~I
(24
+
where #I“ (= V2” AVJ is the apparent molal volume of the undissociated electrolyte. For weak carboxylic acids AV,, is always negativeYwhile it can be predicted to become positive for poly(acry1ic acid) (PAA) at very high dilution when a begins to increase markedly as shown by Wall and G i L 4 More recently, Ise and Okubob have determined & as a function of concen(1) Department of Biochemistry, Faculty of Sciences, Lava1 University, Quebec, Canada. (2) (a) Department of Biophysics, Faculty of Medicine, Universite de Sherbrooke, Sherbrooke, Canda; (b) to whom correspondence should be addressed. (3) H. S. Harned and B. B. Owen, “The Physical Chemistry of Electrolytic Solutions,” 3rd ed, Reinhold Publishing Corp., New York, N. Y . , 1958. (4) F. T. Wall and S. J. Gill, J . Phys. Chem., 58, 740 (1954). (5) N. Ise and T. Okubo, J . A m w . Chenz. Soc., 90,4527 (1968).
