A. TJ. TOBOLSKY, R. H. GOBRAN, R. BOHME,AXD R. SCHAFFHAUSER
2336
Vol. 67
actly using Liouville’s solution to the equation V2u = eU.’ This has been done by Dresner and Kraus,6bwho found the results 1/yrt2 = 1
+ tR4/(192+ 2452)
(17a)
where ER
=
KR
(17b)
and K~ = 2e/a]/kTeR (17c) The value of the electrical potential corresponding to these results is
4*
=
(kT/xie) In [l
+ (tR2- 52)/8]2
5 R (17d) = 0 r 2 R (17e) where [ = KT, r is the radius vector (0 5 r 5 R), and XI r
I$*
is the valence of the ions of opposite charge to the fixed charge cr (counter-ions). When the electrolyte concentration is not vanishingly small, y& can be estimated variationally by use of the trial function
4
=
(kT/xle) In [l
4=0
+ b(5R2 -
$2)]e
r 5R T
(18a)
2 R (18b)
where b and c are as yet undetermined constants. If we introduce the notation PI for the ratio of nl, the number of counter-ions in V , to 2lrRLfal/e, the number of monovalent “fixed” ions in V , and the notation pz for the ratio of n2, the number of co-ions in V , to 27rRLI al/e, then a straightforward calculation yields the result
Here use has been made of the coiiditioii of charge neutrality in the form 1 pz = PI. The coefficient of p2 in eq. 19 is the value of In (l/yi2) = In (l/ylyz) defined by eq. 16b that corresponds to the trial function (18). Shown in Fig. 1 are values of the reciprocal mean ~ with the trial activity coefficient l / y ~ tcalculated function 18, the constants b and c having been chosen (by trial and error) to minimize the value of H given in (19). The 02 = 0 values are the same as those given by (17a), since for Pz = 0, the values b = ’/*and c = 2, which make (17) and (18) identical, will minimize H . The ease with which these estimates have been obtained demonstrates the power of the variational method ; on the other hand, a disadvantage of the present calculation is that it gives no clue to the magnitude of the error involved. Nevertheless, it is the author’s feeling that the precipitous drop in l/yi2 with increasing Pz near p2 = 0 is realistically described because the trial function 18 is exact a t pz = 0 and because it can accommodate itself through two independent constants to changes in p2. The capillary system discussed in this section has been used by Dresner and KrausGbas a model of a suggested salt filter composed of a porous bed of ion-exchange active particles. Their studies of the thermodynamic equilibria of such salt filters in contact with electrolyte solutions mere based on the 0 2 = 0 mean activity coefficient yi given by (17). Since in their work pz was always 50.01, the curves of Fig. 1 indicate that their results do not stand in need of correction.
+
HETEROGENEITY INDEX DURING DEAD-END POLYMERIZATION BY A. V. TOBOLSKY, R. H. GOBRAK, R. BOHME,ASD R. SCHAFFHAUSER Frick Chemical Laboratory, Princeton University, Princeton, N e w Jersey Received April 89, 1963 Expressions are developed for determining the cumulative number and weight average degrees of polymerization throughout a free radical polymerization.
Introduction It can be shown1 that in the absence of retardation of the termination reaction, radical-initiated polymerization may cease short of complete conversion even when the initiator does not undergo wasteful side reactions. This phenomenon, termed (‘dead-end polymerization,” is due to the fact that the initiator may be depleted before the polymerization has gone to completion. The theory of this type of polymerization has been checked2 quantitatively for the polymerization of styrene using 2,2’-azobisisobutyronitrile (azo-1) as the initiator. It was found that in the absence of the (1) A. V. Tobolsky, J. Am. Chem. Soc., 80, 5927 (1958). (2) A. V. Tobolsky, C. E. Rogers, and R. D. Briokman, ibid., 82, 1277 (1960).
Tromsdorff-Korrish effe~t,~.4 the actual amount of conversion agreed quite well with the predicted value. The effect of chain transfer to monomer was neglected in the above study since only the amount of conversion was considered. In the present treatment, the effect of chain transfer to monomer is considered and expressions are_developed for calculating the cumulative values of Pn and P, (the number and weight average degrees of polymerization) a t any time during the polymerization. Theoretical I n a free radical polymerization where the growing radicals do not undergo transfer reactions with solvent (3) E. Tromsdorff, H. Kohle, and P. Lagolly, Illakromol. Chem., 1, 169 (1948). (4) R. G. W. Sorrish and R. R. Smith, Nature, 150, 336 (1942).
HETEROGESEITY INDEX DURIKG DEAD-END POLYMERIZATION
Yov., 1963
2337
or initiator and thermal polymerization is negligible, the number average degree of polymerization (pn)of the polymer former a t any instant is given by the relation6
I n expression 1, [MI is the monomer concentration, kt is the rate constant for the termination reaction, k , is the rate constant for the propagation reaction, and C T is~the transfer constant to monomer. The rate of polymerization (neglecting thermal polymerization) is given by
where f is the initiator efficiency, k d is the rate constant for catalyst decomposition, and [cat] is the catalyst concentration. Substituting this expression for R, in eq. 1, one obtains
(3) The cumulative I‘, at any time (t) during the polymerization is equal to the ratio of the total amount of polymer formed to the total number of chains present. Thus
Pn (cumulative)
=
D l l o - [MI, CT, [ [ M I 0 - [Mlzl
j[[catlo .- [catltl
+
Oo /
CONVERSION
Fig. 1.-Calculated weight average degree of polymerization during a styrene polymerization at 60” for varying concentrations of initiator (1, 0.0009 mole/l.; 2, 0.0016 mole/l.; 3, 0.0030 mole/l.) including (solid lines) and omitting (broken lines) transfer to monomer.
(4)
nI
5.5
where subscripts 0 and t refer to time zero and t, respect ively . The ratio of Pw,to p n i for the polymer which is formed in a very narrow range of conversion in vinyl polymerization can be obtained from the coupling distribution6
5.0
l---l-M
4.5
4.0
m’
where p is a parameter which specifies the actual distribution encountered and whose value lies between zero and unity. illthough p may change slightly as the polymerization proceeds, its value is quite close to unity if the number average degree of polymerization is a t all appreciable. Taking p equal to unity as an approximation, one obtains
‘0 x
3.5
Ia ‘ 3.0
2.5
for the polymer formed a t any instant.’ The cumulative weight average degree of polymerization is the product of the instantaneous fractional (5) A. T’. Tobolsky and R. B. Mesrobian, “Organic Peroxides,” Intersoience Publishers, Inc., New York, N. Y., 1954, pp. 138, 139. (6) A. V. Tobolsky, “Properties and Structure of Polymers,” John Wiley and Sone, Inc., New York, N. Y., 1960, Appendix ti’. - (7) For - the case where transfer is the only chain termination reaction,
P, i = 2Pni. Therefore in expression 6, etc., the factor 1.5 may be replaced by a weighted factor
2.0
5 1 -.
0
IO
’
20 O/o
I
30
I 40
I 50
I 60
CONVERSION.
Fig. 2.--Calculated number average degree of polymerization during a styrene polymerization a t 60’ for varying concentraions of initiator (1, 0.0009 mole/l.; 2, 0.0016 mole/l.; 3, 0.0030 mole/l.) including (solid lines) and omitting (broken lines) transfer to monomer.
A. V. TOBOLSKY, R. H. GOBRAN, R. BOHME,AND R. SCHAFFHAUSER
2338
dt
+ C'Tm
Vol. 67
dt
or p,(cumulative)
= JLMiu [MI
+d[Ml ([MI, - [MI)
Substituting the appropriate expressions for the rate of change in monomer and catalyst (found in ref. 1) and rearranging, one obtains an expression which can be integrated by increments I
0
10
20 'Io
40
30
50
60
CONVERSION.
Fig. 3.-Calculated variation of the heterogeneity index during a styrene polymerization at 60" for varying concentrations of initiator (1, 0.0009 mole/l.; 2, 0.0016 mole/l.; 3, 0.0030 mole/l.) omitting transfer t o monomer.
PW=
sox
1.5 dz ~'zhilhLtl/~[cat]gl/~ kp[WI]0(l -exp 2) - - 4 t / 2
+ c,,]
x
(11) where x = ([?(/I],- [M])/[MIo, and 1.5 may be replaced by y, equation in footnote 7, for a more accurate treatment. If the kinetic constants for a particular monomerinitiator system are k_nown a t a given temperature, the cumulative P n and P, at any time during the polymerization can be calculated from eq. 4 and 11. Results and Discussion Polymerization of Styrene.-Equations 4 and 11 were applied to the case of styrene initiated by azo-1, the kinetic constants used being listed in Table I. The variations of I', and pwwith degree of conversion were calculated for three catalyst concentrations a t a temperature of GOo and are shown in Fig. 1 and 2. The influence of chain transfer to monomer is shown by the calculation of I', and P , with and without the inclusion of chain transfer. The ratios of pwto P , (the heterogeneity index, H.I.) throughout the polyrnerization were determined as a function of per cent conversion and are shown in Fig. 3 and 4. RATE CONSTANTS FOR
'Io CONVERSION. Fig. $.--Calculated variation of the heterogeneity index during a styrene polymerization a t 60" for varying concentrations of initiator (1, 0.0009 mole/l.; 2, 0.0016 mole/l.; 3, 0.0030 mole/l.) including transfer to monomer.
increase of polymer (Wi) and the p,i of this fraction summed over all times P,(cumulative) where Wi =
-d[M1
=
andP,i
i
IViPwi
=
1.5Pni
[Mlo - [Ml P n i is defined by the differential form of eq. 4
(7)
TABLE I POLYMERIZATION OF STYRENEBY Azo-1 AT 60"
THE
Reference
f
0.60 a b k d = 9 . 7 X lO-B/seo. k P / k t ' I 2 = 0.0341 c d C T =~ 6 X lo-' [ M I 0 = 8.03 a J. P. Van Hook and A. V. Tobolsky, J . Polymer Sci., 33, 429 (1958). J. P. Van Hook and A. V. Tobolsky, J. Am. Chem. SOC., D. H. Johnson and A. V. Tobol80,779 (1958). See ref. 2. sky, J . Am. Chem. Soc., 74,938 (1952). =
The main interest in calculating Pn and P , is t o observe their variation as the polymerization proceeds. Initially, p , remains practically constant while pw increases. I n the later stages of polymerization, the
VOLTSUE CHANGES ON MIXINGOF ELECTROLYTE SOLUTIONS
Nov., 1963
rate of increase of Pn is accelerated while that of pw becomes relatively constant. From the figures representing Pn, P,, and H.I., it is evident that a small amount of chain transfer to monomer has an important effect on P , and Pvl but that the H.I. remains relatively unaffected. Although the general shapes of the Pn and iswus. per cent conversion curves are not altered by chain transfer, its inclusion lowers them considerably. This effect is most apparent, a t low catalyst concentrations. To observe the effect of tempera,ture on the variation of 2", and P,, a few calculations were made for polymerization of styrene a t 100'. While larger rates of increase in both Pn and P , were indicated, a limiting value for H.I. of 5.0 was found. These considerations are merely qualitative, however, since styrene undergoes appreciable thermal polymerization a t this temperaturelsthe effect of which was assumed to be negligible in these calculations. Polymerization of Isoprene.-The kinetic constants for the bulk polymerization of isoprene at various temperatures have been determined recently using the dead-end polymerization t h e ~ r y . Using ~ these kinetic constants (Table 11) and applying the expressions developed in this paper, the cumulative number and weight average degrees of polymerization were computed for isoprene a t 80' assuming no chgin iransfer to monomer. The calculated values for Pn, P,, and H.I. are listed in Table 111. (8) R. F. Boundy and R. F. Boyer, Ed., "Styrene, Its Polymers, Copolymers and Derivatives," Reinhold Publ. Corp., New York, N. Y , 1952, p. 218. (9) R. H. Gobran, M. B. Berenbaum, and A. V. Tobolsky, J. Polvmer Scl., 46,431 (1960).
R A CONSTANTS' ~ FOR
2339
TABLE I1 POLYMERIZATION OF ISOPRENE RY
THE
Azo-1 AT 80' f = 0.60 k d = 1.0 X 10-4/sec. k,/kt'/Z = 1.1 x 10-2 M o = 10 See ref. 9.
HETEROGENEITY INDEX IN Fractional conv.
0.02 .06 .10 .14 .18 .22 .24 .26 .28
Cum.
TABLE I11 THE POLYMERIZATION OF ISOPRENE
Pw
Cum.
97.5 184.8 238.5 281.0 320.5 363.3 389.1 421.4 468.9
F,,
H.I.
58.4 60.1 62.2 69.3 75.5 82.3 86.4 91.2 96.7
1.67 3.07 3.83 4.05 4.25 4.41 4.50 4.62 4.75
The technique suggested in this study might well be used to prepare polymers of desired molecular weight and molecular weight distribution. However, one should bear in mind that at the onset of diffusion controlled termination2s4 these theoretical calculations become inaccurate; hence, a polymerization should be quenched before this effect becomes dominant. Acknowledgment:.-The partial support of the Army Research Office (Durham) and the Goodyear Tire and Rubber Company is gratefully acknowledged.
VOLUIKE CHANGES ON MIXING SOLUTIONS OF SODIUM CHLORIDE, HYDROCHLORIC ACID, SODIUM PERCHLORATE, AND PERCHLORIC ACID CONSTANT IONIC STRENGTH. A TEST OF YOUNG'S RULE1
Kr
BY HENRYE. WIRTH,RICHARD E. LINDSTROM, AND JOSEPH N. JOHNSON Department of Chemistry, Syracuse University, Syracuse 10,New York Received May 1, 1963 The volume changes on mixing two solutions of equal ionic strength were determined for the six possible combinations of the four electrolytes, NaC1, HC1, NaC104, and HClOI, at two concentrations, 1.000 and 4.1724 m. A dilatometer capable of measuring volume changes to ic 1 X ml. in a total volume of 100-300 ml. was used. It was found that the volume change on miking two heteroionic electrolytes (NaCl and HC1O4, or NaC104 and HC1) could be calculated from the volume changes observed in homoionic solutions. I n the most unfavorable case the mean apparent volume calculated by Young's rule is in error by 0.6 ml. (2%), and the corresponding density is in error by only 0.2%.
The mean apparent molal volume of a mixture of electrolytes (@I is defined by
V - 55.51D1' (1) m2 m3 where V is the volume of solution containing 1000 g. of water, D'I is the molar volume of pure water, and m2 and ms are the molalities of the two electrolytes. Young and Smith2 have shown that their mixt,ure rule cp=
+
(1) Preaented in part a t the 144th National Meeting of the American Chemical Society, Los Angeles, Calif., April, 1963. (2) T. F. Young and M. B. Smith, J . Phys. Chem., 68,716 (1954).
4,=
m2+2
m2
+ ma43 + m3
(2)
accurately represents data for KC1-NaCl13 KBrNaCllS and NaC104-HC1044 mixtures. I n eq. 2, 42 is the apparent molal volume of one of the electrolytes in a solution containing only water and this electrolyte a t an ionic strength pw corresponding to m2 m3, and 43 is the apparent molal volume of the other electrolyte in a binary solution a t this same ionic strength. A
+
(3) H. E. Wirth, J. Am. Chem. Soc., 69, 2549 (1937). (4) H. E. Wirth and F. N. Collier, Jr., ibid., 72, 5292 (1950).
