KANGYASG
562
But, as we have seen, r is equivalent to In (X/XO) or
Since the pua term is negligible and for all practical purposes constant, we have (74) where pLp0 is the chemical potential when phase separation occurs. We also observe
R,
In S
-
(75)
= s
b7
It can be deduced using eq. 71 and replacing the summation by an integration that Z -V, is a linear function S Br where A and B are non-zero of r of the form A coefficients. Thus
+
s=-
B
A
+ Bi-
From eq. 7 2 me have s*
=
(!3 2 IcT
9
3
r
(77)
TTol. 67
As expected, the average value and the maximum value are quite different. Reich has discussed his model extensively from a mass action viewpoint. We shall not restate his argument. It has been our purpose to illustrate, using Reich’s model, the results which are to be obtained when Reich’s model is analyzed from the two-phase viewpoint and from the constant temperature and pressure ensemble viewpoint. Conclusions.-The general statistical approach contains within it the various models used for the study of micelles and the relationship between these models. The nature of the approximations involved is laid bare. The appropriate interaction terms necessary for improving existing theory are displayed in the general distribution equations for iv,. The basic assumptions of incompressibility and additivity of volumes and the assumption of separability of partition functions used to separate solvent and solute contributions to r are of course inexact. However, they enable us to simplify the ensemble sum and do lead to results in accord with existing theoretical treatments. The separability of solvent and solute contributions can be eliminated easily in the formulation, leading to equations which represent an improvement over existing treatments. Acknowledgment.-The author wishes to express sincere appreciation and gratitude to Dr. Louis M7’itten and Dr. John Gryder for many helpful and stimulating discussions and to the Chemical Directorate of the Air Force ORce of Scientific Research for their support of this work.
FREE RADICAL REACTION INITIATED BY IONIZING RADIATIONS. 111. PARAFFIN REACTIVITIES I N HYDROGEN ATOM ABSTRACTION REACTIONS BY RANG YANG Radiation Laboratory, Continental Oil Company, Ponca City, Oklahoma Received J u l y $7, 196.9
+
+
Rate constants, k = BT1I2exp( - e / R T ) , for these reactions H 31 + Hz R have been determined by investigating the temperature dependence of hydrogen yields in the yradiolysis of paraffin-propylene systems. Assuming E = 2.2 kcal./mole for the hydrogen atom addition reaction with propylene, the following E values are estimated: ethane, 8.6; propane, 7.0; n-butane, 6.3; isobutane, 4.7. The LCBO treatment of an assumption that the energy required for isolating two electrons in a bond to be broken from the rest of the c-electron system SCH?. Here N C H plays the decisive role in determining paraffin reactivities leads to the expreseion: E = E is the number of additional CH bonds formed by the carbon atom that forms the CH bond to be broken. TWO constants, 4 and 7,have these meanings: E is the activation energy for the hydrogen atom abstraction reaction involving diatomic CH molecules; and 7 represents the major structural contribution to the reactivity and comes from the fact that the bond to be broken undergoes etabilizing interactions with neighboring CH bonds. Examination of experimental data in the light of the above equation indicates that each such interaction contributes 2.0 kcal./mole to the activation energy, thus reducing the paraffin reactivity about 30-fold a t room temperature.
+
Introduction Structurally similar compounds often exhibit markedly different reactivities in a series of similar radical reactions.1 I n the case of radical addition reactions with carbon-carbon double bonds, this difference is attributable to a difference in the energy, Eloc, required for localizing a n-electron a t the reaction center2-6; thus it is demonstrated that’ (1) See, for example, M. Srwaro, J . Phys. Chem., 61, 40 (1957). (2) G . W. Wheland, J . A m . Chem. Soc., 64, 900 (1942). (3) C. A. Coulson, J . Chem. Soc., 1435 (1955).
aEloc - b (E-1) where E is the activation energy for the hydrogen atom addition reactions in gas phase, and two constants, a and b, are expressible in terms of parameters characterizing potential changes involved in these reactions. Attempts halve been made to extend this successful localization energy concept to the case of abstraction E
=
(4) J. H. Rinks and M. Saw-arc, J . Chem. Phys., 30, 1494 (1959). ( 5 ) S. Sato and R. J. Cvetanovic, J . A m . Chem. Snc., 81, 3223 (1969). (6) K. R. Jennings and R. J. Cvetanovic, J . Chem. Phys., 36, 1233 (1961). (7) K. Y a w , J . A m . Chem Soc., 34, 3795 (1962).
March, 1963
PARAFFIN REACTIVITIES IS HYDROGEK ATOMABSTRACTIOK REACTIOKS
563
reactions involving a-electron ~ y s t e m s . ~Here the corresponding localization energy may be defined as the energy required for isolating two electrons in a bond to be broken from the rest of the a-electron systems. The present paper describes an experimental and theoretical test of (E-1) in which E l o c is defined as above. The experimental work consists essentially of accurate determinations of activation energy differences in a series of simple gas-phase reactions
H+RI-%-H~+R
(1) where A I denotes ethane, propane, butane, or isobutane. The main reason for this choice is that solvent effects are absent, and there can be little steric hindrance; also, although somewhat controversial,* the required information is available for the corresponding D atom reaction^.^ I n the case of the H atom reactions, reliable data obtained directly from temperature dependence of relative rates seem to be absent from the literature. The experimental method has been described in detail in part Il0and part 117 of the present series (hereafter denoted as part I or part II).” It employs hydrogen atoms produced in the yradiolysis of NI. These atoms undergo competitive reactions 1and 2 ks
H -I- C 3 H 6 + C3H7 (2) propylene being added initially. The decrease in hydrogen yield resulting from reaction 2 is then determined a t very low conversion of &I. TJnder this condition, [CgHG] call be considered as constant throughout the radiolysis and the following rate equation resultslO
Here ro and r, denote the hydrogen yields in the absence and presence of propylene, and r 2 is the hydrogen yield from processes in which the thermalized H atoms do not participate.l2 I n this method, hydrogen atoms are produced a t a controlled rate and the reaction temperature can be varied widely (50-250’). This is essential for obtaining accurate activation energy data. Disadvantages of the present method are, in general, that the radiolysis mechanism is quite complicated and that other non-radical prlocesses are conceivable by which olefins may reduce the hydrogen yield? I n spit,e of these, relative rates estimated by the present method agree well with that obtained by photochemical method.7*” I n addition, temperature dependence of inhibition effect strongly suggests that the above radical mechanism is correct? I n the case of substitution reactions involving 0-electron systems, the localizatioii energy concept has been treated theoretically. l 3 There, E l o c was calculated by using a mobile a-electron model14which regards hydro(8) For example, see S. W. Benson. “The Foundations of Chemical Kinetics,” McGraw-Hill Book Go., New York, N. Y., 1960, p. 293, footnote g.
(9) B. de B. Darwent and R. Roberts, Discussions Faraday Sac., 14, 65 (1953). . (10) K. Yang, J . Am. Chem. Soc., 84, 719 (1962). (11) See the following references also: (a) R. A. Back, J . Phg/s. Chem., 64, 124 (1960); (b) T. J. Hardwick, ibid., 66,101 (1961): (0) 66, 291 (1962). (12) L. M. Dorfman, ibid., 62, 29 (1958).. (13) K. Fukui, H. Xato, and T. Yoneeawa, Bull. Chem. Sac. Japan, 33, 1201 (1960). (141 H. Yoshisumi, Trane. Faradoy Sac., 63, 123 (19.57).
k,/k,,
for the reactions H
+M
CsH?.
km
H, -I- R and H
+ C3Hs +k. .
carbons as composed of carbon skeletons only, thus neglecting the influence of CH bonds. Obviously, such an approach is incapable of treating abstraction reactions. In the present paper, E l o c is estimated by using the linear combination of bond orbitals (LCBO) method.16-17 The result, which explains the experimental data quite well, indicates that the major structural factor governing paraffin reactivities in abstraction reactions is the stabilization of the bond to be broken by neighboring CH bonds. Experimental Phillips research grade hydrocarbons were used. ill1 gases were degassed and distilled except for ethane, which was subject to additional bromine treatment in a high-pressure steel vessel.18 The concentration of added propylene did not exceed 3%. Other experimental details have been described previously.7)*0
Results The rate constant ratios, km/ks, were obtained by plotting rs against (TO - rs)[blI/ [SIas shown in (E-2). Kinetic data were similar to the ones given in part I and part 11,and good straight lines resulted in all cases. Figure 1 summarizes temperature dependence of the resulting ratios. From least squares treatment, we obtained these relations log k(ethane)/k,
=
(-0.01
f
0.18)
-
(6,380 h 370)/2.3RT (E-3) log k(propane)/k,
=
(0.15
=I=
0.05) -
(4,800 h 100)/2.3RT
(E-4)lg
(15) G. G. Hall, Prac. Rau. Soc. (London). A206, 541 (1951). (16) R. D. Brown, J . Chem. SOC.,2615 (1953). (17) For a recent review, see R . Daudel. R. Lefebvre, and C. Moser, “Quantum Chemistry,” Interscience Publishers, Inc., New York, N. Y . , 1959. (18) K. Yang and P. L. Gant, J . Phiis. Chem., 65, 1861 (1961). (19) This relation is taken from part 1.10
KANGYANG
5 64 log k(butane)/k, = (-0.18 i 0.14)
-
(4,070 f 250)/2.3RT log k(isobutane)/k,
=
(-0.44 (2,470
f: 0.13)
f
(E-5)
=
?ob
(E-6)
/ RT)
(E-7)
we computed the k , values given in Table I. I n estimating (E-7), k, at 31" is taken to be 4.8 X 10" cc./ mole-sec.20and the activation energy to be 2.2 kcal./ mole. As indicated in part 11,published E values range from 1.321to 4.29; hence the present choice introduces average deviations of about 1.5 in the data listed in Table I. With this in mind, it is of interest to compare the present results with published data. TABLE 1 BATECONSTANTS, k , ( CC./MOLE-SEC.),
+
1cm
FOR THE
REACTIOSS
+
H h I + Hz R lcm = BT112exp( - E / R T ) M
log B
Ethane Propane Butane Isobutane
12.0 12.2 11.8 11.6
c
x
e
Ethane 7.0
Propane 5 2
Butane 5.1
M
50"
Isobutane 4 3
Except for the isobutane reaction, E values seem to be too low. Bensons suggested that their data may contain contributions from hot atom reactions. This could be partially responsible for the present discrepancy. The uncertainties in the present absolute E values do not appreciably weaken the forthcoming discussion, because the major importance is the activation energy differences whose standard deviations do not exceed 0.4. I n discussing radiolysis mechanism, ra and r2 values are important. These are given in Table I1 in terms of G(H2) (molecules/100 e.v.). The yo values were reproducible within f3'%; for the present purpose of relative rate determinations, variations of TO with temperature may be regarded as quite small. Because of uncertainties in the intercept, deviations in r2 values (20) A. B. Callear and J. C. Robbs, Trans. Faraday Sac., 61, 638 (1965). (21) M. D. Sheer and R. Klein, 1. Phgs. Chem., 66, 375 (1961). (22) M. R. Berlie and D. J. LeRoy, Discusszons Faraday Sac., 14, 60 (1953).
1600
Ethane 8.8 Propane 6.4 Butane 5.4 Isobutane 5.4 a For dosimetry, see reference is molecular yield.
240'
8.9 6.5 5.5 5.4 18.
60-240'
9.3 6.7 5.8 5.5 T O is total yield
3.3 1.6 1.8 1.8 while
TZ
are relatively high (-10%). Within this limit, r2 values did not depend on temperature. Discussion To examine the data in Table I in light of the localization energy concept, we make this assumption: (E-1) is applicable in the case of abstraction reactions also. In estimating Eloo,Brown's LCBO treatmenP of paraffins may be used. Although his approach bas been described carefully, the following derivation of an expression for Elooin a form suitable for reactivity discussion is quite helpful in understanding our arguments below. As an example, we consider the ethane reaction
lo-'
8.6 7.0 6.3 4.7
Berlie and LeRoy2*investigated the ethane reaction using hydrogen atoms generated by Wood's method and exp( - 6,40O/RT). reported that lc(ethane) = 1011.01T1'2 In view of the present result, their E value seems to be too low. With presently available information alone, it is difficult to ascertain the exact source of the discrepancy. When isotope effects are neglected, Damvent and Roberts' datag on the D atom reactions can be compared also. For this purpose, it is desirable to recalculate their results to give e = 2.2 for the hydrogen atom addition reaction with propylene. When this is done, the following data result R.l
r2b
_-&---.
P
200)/2.3RT
1012.aT1'2 exp( -2,200
TABLE I1 THE YIELDS( M O I , E C U L E S /E.v.) ~ ~ ~ OF HYDROGEN IN HYDROCARBON RADIOLYSIS AT VARIOUSTEMPERATURES'
-
where uncertainties are indicated in terms of standard deviations. By assuming
k,
Vol. 67
I
I
I
/
-4-C-H
I
+-C-C*
I
*€I
I 1
The eigenfunction of the reactant is represented by the , linear combination of six CH bond orbitals, Y C Hand one CC bond orbital, YCC. I n the localized structure, two electrons in the bond to be broken are isolated from the rest of the c-electron system. The energy difference between these two structures gives E l o o . For the matrix elements, we use this notation
f Yc"YcHt
acc = ~ Y C H H Y dCr ;Hp =
p'
=
f YCHHYCG dr; y =
dr
S = JYCHYCH dr
4(p - Sacc); and A c ~=
LYC
+
CYH
-
20ccc
(E-8)
When a pair of CH bonds do not originate from the same carbon, p is taken to be zero. The term ac CXH in Aoc denotes the energy of two electrons in the bond to be broken. By using the variation method and expanding16 the resulting secular equation in a power series in S , it can be shown that
+
Eloo=
ACY4-2S7
03-91
In obtaining (E-9), we neglect terms containing second or highqr powers of X. It is also assumed that ( f l ' / p ) 2 N 0, which seems reasonable in view of a value P ' / p N 0.2 obtained by investigating ionization potentials of paraffins by the LCBO method.16 An extension of the above argument leads to the generalization Eloo=
Aa
+NCH~Y
(~-10)23
Here NCHdenotes the number of CH bonds surrounding the bond to be broken. For example, it is unity in case of the propane reaction. From (E-1) and (E-IO) (23) An equation similar t o (E-10) has been successfully used by BromnlB in treating bond energy data.
PARAFFIN REACTIVITIES IN HYDROGEN ATOMABSTRACTION REACTIONS
March, 1963 e
=:
5
+ NCHV
565
(E-11)
where ( = aAa - b, and q = aXr. The present E l o c j, different from bond strength, D,in two respects. First, the term ac CYH in El,, may contain a significant contribution from the interaction of two electrons in the bond to be broken, while this is not so in the case of 13. Secondly, structural differences between alkyl radicals in isolated and bound states affect D but not E l o c . For these reasons it is quite possible that Eloc< D. If these differences are assumed to make a constant contribution to e in a series of similar reactions, then Elo,!and D may be used interchangeably. I n this case, (E-1) implies a well known and successful expression2*--Z8
+
E
=
amD - Pm
(E-12)
where a,, and pm are constants. I n the present paper, however, it is not necessary to use (E-12). As shown below, a correlation of activation energy data is achieved without using bond energy data. This is worth emphasizing because, in discussing relative reactivity, the use of (E-12) demands AD values on which available information often contains a large per cent of errors. According to (E-11), a straight line should result when e in Table I is plotted against NCH. Figure 2 confirms this prediction. I n Fig. 2 only those bonds having the highest reactivities are coiisidered, and the experimental data are assumed to correspond with the reaction involving such a bond. Actually, in the temperature ranges investigated here, some less reactive bonds are also likely to participate. I n view of the linearity shown in Fig. 1, however, uncertainties in e from such sources are not likely to exceed 0.4. From Fig. 2, we estimate the E value for the reaction
H
+ CH4 +H2 + CHs
(3) to be 10.3. Recent>ly Fenimore and Jonesz9 investigated this reaction in a flame and reported that IC(methane) = 2 X 11013 exp(-l1,500/RT). This corresponds to an e value of 10, in excellent agreement with the present estimation. From the data in Fig. 2, we obtain
4
= 4.7
17 =: 2.0
(E-13)
We now examine the physical significances of these two constants. I n the case of the reaction
H
+ i-C4Hio +H2 f ~-CIH,
(4) NCH = 0; hence e = 4. The same relation results when the present theory is applied to the reaction involving an isolated CH bond.
+
+
H CH +H2 C (5) Reactions 4 and 5 thus should have about the same activation energy. I n such a reaction involving three (24) A. F. Trotman-Dickenson, Discussions Faraday ’soc., 10, 111 (1951). (25) A. F. Trotman-Dickenson, “Gas Kinetics,” Butterworths Scientific Publications, Ltd., London, 1955, p. 199. (26) E. Warhurst, Quart. Rev., 5, 44 (1931). (27) R. E. Dodd, J . Chem. Phys., 26, 1353 (1959). (28) K. Otozai, Sei. Papew Osaka Uneu., 20 (1951): Bull. Cham. hoc. Japan, 24, 218, 257, 262 (1951): also see ref. 8, p. 317. (29) C. P. Fenimore and C. W. Jones, J . Phys. Chem., 68, 2200 (1962).
ISOBUTANE
-
I
2
3
s
Fig. 2.-Activation energy, e, for the hydrogen atom abstraction reactions as a function of the number, N C H ,of additional CH bonds formed by the carbon atom that forms CH bond to be broken.
atoms, Eyring and his co-workers’ semi-empirical method30 can be used to estimate E. Hir~chfelder~l has shown that, when the interactions between two outer atoms are neglected, this method yields a simple expression for e e
D z
-[2 - 3n
2
- (3(1 - 2n))l”J
(E-14)
where D is the CH bond strength in an isolated CH bond, and n signifies the ratio of coulombic enagy to total bindifig energy and is usually taken to be0.14. Here, D is different from D(C-H) in diatomic molecules, because the effective Hamiltonian in 4 is not the same as that needed in treating the CH molecules. For the present purpose, it is more appropriate to use D(C-H) (= 94 k ~ a l . / m o l e ~in~isobutanea2; ) then E becomes 5.2, in good agreement with the 5 value of 4.7 f 1.5. Unfortunately, E estimated from (E-14) seems to be not better than h 5 , and the above agreement only indicates that the present ( value is not inconsistent with the transition state theory. The physical significance of q is equally clear-cut. It is the major structural factor governing the activation energy dzerences (hence the relative reactivity also) (30) S. Glasstone, K. J. Laidler. and H. Eyring, “The Theory of Rbte Processes,” McGrrtw-Hill Book Co., Inc., New York, N. Y.. 1941. (31) J. Hirschfelder, J. Chem. Phys., 9, 645 (1941). (32) We express our appreciation for the referee who brought this p i n t to our attention. (33) T. L. Cottrell, “The Strength of Chemical Bonds,” Academic Press, Inc., New York, N. Y., 1954, p. 272.
566
T. A. OROFIKOAND F. WENGER
and comes from a stabilizing interaction of neighboring CH bonds with the bond to be broken. Equation (E-12) indicates that each such interaction contributes 2.0 to the activation energy, thus reducing the paraffin reactivity about 30-fold at room temperature.
TTol. 67
Acknowledgment.-We express our appreciation to Dr. F. H. Dickey, Dr. L. 0. Morgan, and Mr. C. L. Hassell for their valuable discussion and to Mr. P. L. Gant and Mr. J. D. Reedy for helping with the experimental work.
DILUTE SOLUTIOS PROPERTIES OF BRANCHED POLYMERS. POLYSTYRESE TRIFUNCTIONAL STAR MOLECULES' BY T. A. OROFINOAND FRANZ WENGER~ Mellon Institute, Pittsburgh, Pennsylvania Received July 50, 1961 The results of intrinsic viscosity, second virial coefficient and related thermodynamic measurements on samples of polystyrene trifunctional star molecules are reported. These branched polymers were synthesized by means of a coupling reaction between essentially monodisperse polystyryllithium and 1,2,4-tris-(chloromethy1)-benzene. The homogeneity of the materials obtained with regard to molecular weight and functionality was established by fractionation, sedimentation, and absolute molecular weight determinations. Intrinsic viscosities and second virial coefficients in both poor and good solvent media were found to be less for these materials than for linear polystyrenes of comparable molecular weights. The Huggins viscosity constant k' in good solvents was unaffected by branching; in poor solvent media, an augmentation of IC' with branching was noted. The results of these and other dilute solution studies are discussed from the point of view of Current theoretical interpretations. An attempt is made to extend them in applicability to more general branched polymer systems.
Introduction The interpretation of dilute solution data obtained on branched polymer systems to date has been complicated to varying degrees by structural and molecular weight heterogeneity present in the samples investigated. Although some general conclu~ions3~~ concerning the effect of branching have been deduced from the considerable quantity of experimental data accumulated, the lack of adequate sample characterization has in large part precluded attempts a t quantitative evaluation of the results. A general method for the controlled syntheeis of branched polymers has recently been outlined by one of usj5the initial application of which has led to the preparation of polystyrene trifunctional star moleculesbranched polymers formed by three linear polystyrene chains joined a t one end through a common junction. The skeletal structure of these materials involves only carbon-carbon linkages, coupling of the three constituent linear chains in each molecule being achieved through use of a suitable low molecular weight aromatic compound. The virtual chemical identity of the branched chains with linear polystyrene, together with their homogeneity in regard to functionality and molecular weight, make them ideally suited for physical measurements, the results of which in comparison with known properties of their linear counterparts be attributed unequivocally to the branched nature of the chains. (1) The results of this investigation were presented a t the 141st American Chemioal Society National Meeting, March, 1962, Washington, D. C. (2) Data obtained on earlier branched polymer samples have been reported in previous communications (a) F. Wenger, Discussion Contribution No. 187. Paper A41, International Symposium on Macromolecular Chemistry, July, 1961. Montreal, Canada-see J . Polymer Sci., 6'7, 481 (1962); (b) T. A. Orofino and F. Wenger, Division of Polymer Chemistry, Amerioan Chemical Society, Preprints 3, 274 (1962). These results have s h o e been supplemepted through mare extensive studies on better characterized materials, data for which are herein reported. (3) C. D. Thurmond and B. H. Zimm, J . Polymer Sci., 8, 477 (1952). (4) W. H. Stockmayer and M. Fixman, Ann. N . Y. Acad. Sa., 67, 334 (1953). ( 5 ) F. Wenger and S.-P. 8. Yen, presented a t the 141st $meriaan Chemical Society National Meeting, March, 1962, Washington, D. C. (to be published). See also, Dir ision of Polymei Chemistry, American Chemical Society, Preprints 3, 162 (1962).
The principal aim of the present investigation thus has been to determine the dilute solution properties of the model branched structures described and to compare these with the corresponding properties of chemically identical, linear polymers of the same molecular weights. The significance of this undertaking rests directly upon the degree to which three essentially independent aspects of the investigation can be pursued in an accurate and internally consistent manner. These are (1) the establishment of the physical constitutions of the branched samples selected for study, (2) the detailed investigation of the solution properties of these, and (3) the evaluation of the corresponding solution properties of their linear counterparts. In addition to the analyses of primary measurements utilized to establish the architectural structure of the particular materials selected for investigation, the data presented here include the detailed results of osmotic pressure, light scattering, and viscometric studies as functions of temperature in both poor and good solvent media. I n a separate study pursued by one of us6 the corresponding solution properties of a (reference) linear polystyrene sample have been extensively examined. The results of that investigation, coupled with other relevant studies carried out on linear polystyrenes in the course of the present project, provide the principal basis upon which the solution properties of the trifunctional star structures and their linear analogs of the same molecular weight are herein compared. Also, we have included in our presentation a summary of results obtained on a mixture of star and linear polystyrenes. These data provide a supplementary measure of the degree to which various dilute solution parameters are affected by branching of the kind here considered. The results of our investigation are discussed from the point of view of various prevalent theoretical develop ments appropriate to the analysis of branched polymer systems. (6) T. 8. Orofino, Division of Polymer Chemistry, American Chemical Society, Preprints, 2 , 161 (1961); T. A. Orofino and J. W. Mickey, Jr., J. Chem. Phus., 88, in press.
