1830
THOMAS H. DONNELLY
sorbed molecules a t the surface rather than the random orientation which we have assumed. The second effect can produce an increase or fall of AE with x, and hence cannot explain the generally observed fall of AE. The effect of surface heterogeneity, however, is always such as to produce a decrease of A E with increasing x. We have assumed the "graphite" surface to be composed of only two groups of sites, for simplicity, comprising 90 and 10% of the surface with values of AE/kT = 1.8 and -2.8, respectively. The adsorption on each group of sites was calculated using equation 25 and the total adsorption then used to give mean values of AE/kT. These are presented in Table I and show a reasonable fit with the experimental values. A similar analysis of the results of Kiselev and Platova for the graphitized carbon black showed 52% of the sites favoring the aromatic hydrocarbon, AE/kT = 2.6, the rest with AE/LT = - 3.2. The isotherm is not very sensitive to the choice of energies and hence these values must be considered rather approximate. Adsorption from binary non-electrolyte solutions has been used to characterize in a qualitative way the surface of carbon blacklo and other pigments." It is possible that such problems might be treated quantitatively as described above. (10) C. W. Sweitzer, L. J. Venuto and R. K.Estebu, Paint, Oil and Chem. Reo., 116, 22 (1952). (11) L. Dintenfoss, Chemistry and Industry, 560 (1957).
Vol. 64
It is known that the whole surface of the Spheron-6 is probably oxidized, thus presenting a fairly homogeneous surface. The artificial graphite is certainly less oxidized (see the analyses of Blackburn, Kipling and Tester) and the 10% of sites with AE/kT = - 2.8 may be associated with bare graphite sites. Observations12 that graphon (a practically pure graphite surface) adsorbs cyclohexane vapor more strongly than benzene would seem to be in accord with the above assignment. Such an adsorption of benzene and other aromatic hydrocarbons in preference to non-aromatic hydrocarbons of about the same molecular weight is also to be found for hydrated silica gel as adsorbentg where there is a well-established donoracceptor interaction between the acidic surface hydroxyls and the 7-electrons of the aromatic nucleus. Both hydroxyl and carbonyl groups' are present on the surface of the oxidized carbon black, and various possibilities exist for obtaining a donor-acceptor interaction between the surface and the aromatic nucleus. A fuller discussion of this problem has been given e1~ewhere.l~ Acknowledgments.-We acknowledge with gratitude the support of the Paint Research Institute of the Federation of Paint and Varnish Production Clubs which has made this work possible. (12) R. N. Smith, C. Pierce and H. Cordes, J . A m . Chem. Soc., 73, 5595 (1950). (la) G . Delmaa and D . Patterson, O f i c . D i g . Federation Parnt & Vatnish Production Clubs, 31, 1129 (1959).
THE STUDY OF LIMITED MOLECULAR WEIGHT DISTRIBUTIONS BY THE USE OF EQUILIBRIUM ULTRACENTRIFUGATION1 BY THOMAS H. DONNELLY Swift & Company Research Laboratories, Chicago, Illinois Received January
1960
I n the investigation of heterodisperse systems, the technique of equilibrium ultracentrifugation offers the advantage of ability to evaluate more than one moment, of known form, of the weight distribution function. Unique specification of such a distribution function from these moments alone is possible only in the case of paucidisperse systems, such as might be obtained from fractionated small polymers. The present work recasts the equations of Goldberg and of Johnson, Kraus and Scatchard in a form more suitable for use with such paucidisperse systems, incorporating a simple correction adequate as a first-order approximation of pressure effects. The measurable moments of the distribution are defined in terms of memurements at infinite dilution. Calrulations of the averages related to these moments are illustrated by application to a sample of linoleic acid polymers. The precision necessary in measuring these averages is made apparent by a consideration of the mathematics by which such a distribution function may be evaluated from its moments.
Introduction The advantages of the use of short columns for rapid attainment of ultracentrifugal equilibrium, as pointed out by Van Holde and Baldwin,2 have served to refocus attention on the capabilities of this technique for molecular weight studies. In simple, relatively ideal systems, attainment of equilibrium in the ultracentrifuge has the advantage of simpler calculations than those associ(1) Presented in part before the Diviaion of Polymer Chemistry at the 136th meeting of the American Chemical Society, Atlantic City, New Jersey, September, 1959. (2) K. E. Van Holde and R . L. Baldwin, THISJOURNAL,64, 734 (19SS).
ated with the Archibald p r i n ~ i p l e ~for , ~ rapid measurement of molecular weights. With complex systems, the simple relationship of the molecular weight averages measured by equilibrium ultracentrifugation t o the moments of mass of the molecular weight distribution will often outweigh any advantage gained by the use of the Archibald principle.6 (3) W. J. Archibald. ibid., 51, 1204 (1947). (4) S. M. Klainer and C. Kegeles. ibtd., 09, 952 (1955). (5) The types of averages evaluated by equilibrium ultracentrifueation have been described b y Lansing and Kraemer.6 The relationship of these averages to the momenta of mass of the distribution is given, for example, b y Goldberg.' The possibility of evaluating more than
Dec., 1960
STUDYOF LIMITEDMOLECULAR WEIGHTDISTRIBUTIONS
Equilibrium ultracentrifugation has been employed as a method for studying distributions since Rinde’s’O studies of particle size distributions in gold sols. Methods for the calculation of distributions directly from the technique are given by Rindello Wales, Adler and Van Holde,” and Herdan.12 These authors have often found that the distributions obtained are unsatisfactory, especially since they often indicate that certain size ranges are present in negative amounts. This is probably due to many general causes, one of which is the failure of the technique to afford sufficiently precise measurements, while another is the use of continuous rather than discrete distributions. The present study is concerned with discrete distributions, especially those obtained in systems of very small polymers. For such systems, it is seen that compressibility corrections are of the order of the reciprocal molecular weight, while variations of partial specific volume with molecular weight may be fala more important. Theory The thermodynamic description of the equilibrium state attained during ultracentrifugation has been given by Pedersen.18 A more detailed treatment applicable to multi-component systems is given by Goldberg’ in an extension of the classical thermodynamics of solutions in gravitational fields as given by Guggenheim.14 A further refinement has been added by Young, Kraus and Johnson,16 who have considered the effects of activity coefficients and compressibilities in simple systems. The studies of E h ~ j i t aand ~ ~ Cheng and Schachman1’ suggest methods of incorporating compressibility corrections, although these studies are not concerned with equilibrium measurements. The basic equation of the equilibrium ultracentrifuge, essentially as given by Pedersen,la is
Further evaluation of this equation may be made by considering the equation dVi =
(”5) bT P , N j dT + ( $ ) T , N j
one average molecular weight by t h e Archibald principle has been established by ICrIander a n d Foster,s while t h e form of such averages has been investigated b y Yphantia.9 (6) W. D. Lansing a n d E. 0. Kraemer, J. A m . Chem. Soc., 67, 1369
(1935).
(7) R. J. Goldberg, THISJOURNAL,67, 194 (1953). (8) 8. R. Erlanderand J. F. Foster, J. Polu. Sci.,37, 103 (1959). (9) D. A. Yphantia, Tm8 JOURNAL, 63, 1742 (1959). (10) H. Rinde, Dissertation, Uppsala, 1928. (11) M. Wales, F. T. Adler a n d K. E. Van Holde, T H IJOURNAL, ~ SS, 145 (1951). (12) G. Herdan, Research, 5, Suppl. 35 (1950). (13) H. 0. Pedersen, in “The Ultracentrifuge,” Svedberg a n d Pedersen. Clareiidon P r c w Oxford, 1940, P. 48 ff. Reprinted by Johnson Reprint Corp., New York, 1959. (141 E. A. Guggenheim, “Thermodynamics.” North Holland Publishing Co., Amsterdam, 1957. pp. 403 E. (15) T. F. Young. K. A. Kraus a n d J. 8. Johnson, J . Chem. Phyn., 22, 87R (1954). (16) H. Fujita, J. A m . Chem. Sac., 18, 3598 (1956). (17) P. Y.Cheng and H. K. Schachman. ibid. 77, 1498 (1955).
dP
f
At constant temperature, eq. 2 is reduced to the same variables as those of eq. 1 by using the definition of the isothermal compressibility. p i = - - ( - )1 dVi Vi bp
T,Ni
=--(-) 1 bP bBi Bi
T,Nj
(3)
Combining the condition of hydrostatic equilibrium with the definition of the ultracentrifugal potential gives d P = -pd+ =
p o 2 ~dz
4)
Also
which has been used by Young, Kraus and JohnsonX5in a previous study of simple compressible systems. Using these substitutions, eq. 2 becomes, at constant temperature d In 8i = -pi,& z dx 3.
A completely general description of the mole fraction distribution in a solution column in which ultracentrifugal equilibrium is established may be obtained by combining eq. 1 and 6 with the definition n
E N,dl, 2=@
P-
5 N,M,r,
i=@
where the symbols are as used by Pedewen except for the representation of the partial specific volume by gi.
1831
NOM0
-
+
n
N,2lll i=l
NoM0ij,
+ 5 N,M,ii,
-
a=1
Because of the ambiguity18 regarding the terms, ( b lnf’ilb Nj), in any case, little is gained by writing such an equation except in the simplest cases. It would seem to be more appropriate to investigate the behavior of ideal systems and to presume that ideality may be approached by real systems. Further, in order to simplify eq. 7, it would seem most appropriate to assume that ideality is obtained at infinite dilution. In order t o simplify (18) The theoretical form of theae terms has been given b y Krigbaum a n d Flory.19 The complexity of these results, however, is such t h a t no straightforward simplification results from their me. A more realistic approach is the use of systems with ideal behavior, as shown by Mandelkern, Williams and Weissberg,a although Fujita and his co-workers*l.z2 have had some auccess with systems which are somewhat more non-ideal. (19) W. R. Krigbaum and P. J. Flory, J . A m . Chem. Soc.. 7S, 1775 (1953). (20) L. Mandelkern, L. C. Williams and J. G. Weiasberg, THIS JOURNAL, 61.271 (19571. (21) H. Fujita. ibid., 63, 1326 (1959). (22) H. Fujita, A. M. Linklater and J. W. Williams. J. A m . Chcm. Soc., 82, 379 (1960).
THOMAS H. DONNELLY
I832
this and subsequent treatment, it will be convenient to introduce a variable, u = (z2 - Xm2)/ ( 5 b 2 - Zm'), where XI, is the value of the outer radius of rotation of the column, while 2 m is that for the inner radius. Equation 1 may now be written as dN, = N , M , (1
- 8,p)A C I U
( 8)
where .4 is defined by the equation
d In & =
components or those in which all solute species are equally compressible by eq. 17 or its equivalent, this equation immediately suggests that, as should be expected, an appropriate approximation of compressibility effects might be obtained by considering all solute species to have compressibilities equal to the solvent, or pi = Po = P. Under these conditions, eq. 17 becomes dNi = NiWiA du
- p i p R T A du
(19)
Such a system may be related to one a t laboratory conditions through the usual conservation of mass condition, i.e.
Under these same conditions, eq. 6 becomes and eq. 7 is merely
Vol. 64
(10)
p = l/go.
This is just the analog of eq. 4 of Ginsburg, Appel and S~hachman,~' and Ci") is similar to their e. eq. 11 ci(u) is t'he same value as Ci"), but x: is now exmay be directly integrated to give pressed as the corresponding value of u. In the system considered in eq. 19, )@ ' ic is given by i& = Oo(") - po RTAu (12) where the superscript (0) denotes that the value is taken a t u = 0, or x: = xm. Equation 10 thus if we define Y by the expression becomes, for the species i = 1 to n
- Po R T A du If it is now assumed that Po is a do0 =
(11)
Y =
When eq. 12 is valid, and be integrated to give 8, =
Pi
is constant, this may
ii,(')(l - /~~~~ORTAU,)BW'BO
if l/fi0(O) is written as
poo.
(14)
PpoO
RTA
(22)
A system a t laboratory conditions may be related to one a t ultracentrifugal equilibrium by evaluating eq. 20 using eq. 21 before and after redistribut,ion of species. This process is equivalent to the equation
Equation 8 may then
be written as dN, = h'iM, { l - 8,(O)poO(l
- popo'RTA~)(BdPd) A du
(15)
by substituting eq. 12 and 14. Equation 15 may then be expanded by the binomial theorem to give dN, = N I X x11 - 8,(o)p~O(l - (Pi (
(1
- (Pi
22P0)
(poo RTAu)
- Po)pooRTAu+
+
where ( N i ) o is the original mole-fraction of the ith species and is independent of compression. While the integral on the left is easily evaluated, that on the right has no simple evaluation. For the purposes of this work, a satisfactory evaluation may be made by expanding the denominator in the usual series and dropping terms in Y 2 . Since Y is of the order of 0.02 or less in the systems dealt with here, this will usually be adequate. Using such a treatment, Ni'O) may be related to (Ni),by the expression
...)I
A d u (17)
if Wi is defined as
w,= Mi (1 - 8,(O)p0O)
(18)25
(24)
This may be used to evaluate the constant in the integrated form of eq. 19, giving the result that
While it may be useful to treat systems of few (23) This assumption will usually describe amply well the sort of system which is likely t o be encountered in t h e use of this procedure. Use of the technique of Van Holde a n d Baldwin2 make such an assumption even more realistic. When such a n assumption is not adequate, i t is possible t o resort t o a n equation of s t a t e such as t h e T a i t equation.24 The mathematics of this procedure are far more cumbersome t h a n those encountered here, a n d such a procedure apparently adds little t o the understanding of present d a y experiments. (24) See, for example. H. S. Harned a n d B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," Rheinhold Publ. Corp., N e v York, N. Y., 1958, p. 379 ff. (25) Wi is used here, rather t h a n t h e similar Li of Johnson, Kraus a n d Scatchard28 or t h e M I * of Yphantis,'since p is not usually equal to POD.
(26) J. S. Johnson, K. A . Kraus a n d G. Scatchard, THISJOURNAL, 68, 1034 (1954).
Refractive increments may be defined so that ( n - no) =
2=1
2 h7iM,
(26)
i=O
which, for constant and equal R i , would mean that the difference between the refractive index of the solution and that of the solvent is propor(27) -4. Ginsburg. P. Appel a n d H. K. Schachman, Arch. Biochem. Biophys., 66, 545 (1956).
Dec., 1960
STUDYOF LIMITEDMOLECULAR WEIGHTDISTRIBUTIONS
1833
tional t o the percentage by weight of solute. These Extrapolation of the reciprocal of M,, has Ri are related to the Ri of Van Holde and Baldwin2 usually been taken as valid, although there is some question as to the reliability of any such by the relationship procedure.20 When such a procedure is valid, R, = poBi (27) where the B, are the R, of Van Holde and Baldwin2 one may extrapolate the reciprocal of the quantity evaluated a t infinite dilution.28 If (n - no) is on the left of eq. 34 and use the function so obabbreviated as n3,as suggested by Van Holde and tained to evaluate the quantity on the right. In Baldwin12eq. 25 and 26 may be combined to give order that this procedure be strictly valid, the values of A should be constant in all experiments. ne = - xm2) This usually means that the value of (a2 R,(N,)oM,TT,eA Wlu should be constant. As a first-order correction, A 1+values of ( x b 2 - r m 2 ) r e f . / ( d In (nc)/du) may be 2 = 1 ((eAW; - 1)(1 + YeATG’1 AW extrapolated t o zero c o n c e n t r a t i ~ n . ~This ~ proce~ _ _ _ n dure will define (d In (nc)/du) as a function of u only. N,N, This function is then used to define the various t=O molecular weight averages as well as they can be (28) defined by the use of equilibrium ultracentrifugaEquation 28 and succeeding results will be con- tion. The evaluation of such molecular weight siderably simplified by defining Wi’ by the ex- averages is by an extension of the classical methods pression of Lansing and Kraemer6 and Wales.3O By such a procedure, it may be shown, for example, that W , W,‘= (29) Y
5
( )’
A)
)
II
and A’ by the expression A’ = A (1
+ 9)
Equation 28 thus becomes n
R, ( N l ) o M , W l ‘ e A W I u / ( e A ~ ~l 1) = -ZEl (31) Y,M, A‘
nc(u)
5
2=0
Equation 26 may be written as n
R,( N J ~ M ,
(n& =
2=1
(32)
5 NLM, 2=0
where (nc)ois the value of (n - no) measured before redistribution of species. Equation 32 may now be combined with eq. 31 t o give Limit .vO+ 1
(($$?)
Ri(Ni)oMiWi’V7i2 i=l
=
(35-4
n
n
RI(NI)oM,W,’eAWlu/(eAWI
Ri(Ni)oMiWi’m’i
- 1)
-4’ %2
i=l
(33)
n
R,(N,)oM, 2=1
By combining the derivative of eq. 33 with that equation, there results 11
R,(Ni)oil/liWi’WieAWiU/(eAwi
- 1)
i=l
A --
E
n
( 34) Ri(Ari)oMiWi’eAWiU/(eAW‘
- I)
i=l
Equation 34 defines a quantity which is proportional t o the quantity, M,,, ordinarily defined. (28) I n fact, the Bi are identical t o t h e Ri of Van Holde a n d Baldwin,t since their Ri are constants independent of concentration. It should be emphasized t h a t t h e Ri used in t h e present work are also taken a s independent of pressure.
and so forth. The quantities on the right in eq. 35 are relat.ed to the usual weight, z and z 1 molecular weights. As in the cases described, for example, by Wales,30 the accuracy of the higher averages produced by further differentiation is determined by t’he accuracy with which the derivatives of the function, (d In n,/du), may be evaluated a t the end-points.
+
(29) Here ( Z b P - zm2),,fis a reference value of ( x b 2 - zmz) a n d determines t h e value of A for the s e t of values to be extrapolated. I t should, therefore, be t h a t value of A which minimizes the values of AA, t h e variation of the value of A for each set of d a t a from this value. (30) M. Wales, THIS J O U R X A L63, , 235 (1948). (31) This equation may be evaluated b y a procedure equivalent t o t h a t of Johnson, Scatchard a n d Krausa2 involving the use of interference optics. It should also be emphasized t h a t studies such as the present should be interpreted in the light of second-order effects such as those which are pointed o u t b y these authors. (32) J . 9. Johnson, G. Scatchard a n d K. A . Kraus, THISJOURYAL. 63,787 (1959).
THOMAS H. DONNELLY
1834
The problem of evaluability of higher molecular weight averages is therefore directly related t o the problem of evaluation of the derivatives of a function from the function itself. This is the inverse of the problem of evaluating the Taylor’s series for (d In (nc)/du) in the interval, u = 0 to u = 1. It is apparent from eq. 33 that the Taylor’s series is usually an infinite series in u whose derivatives do not vanish. Thus, it cannot be accurately represented by a finite polynomial. Further, representation of (d In nc/du) by any polynomial other than this exact Taylor’s series will lead t o incorrect evaluation of the higher molecular weight averages. Since it is usually possible to represent this function by a polynomial other than exact, such interpretation of equilibrium ultracentrifugation data will not usually determine uniquely the exact molecular weight distribution in polydisperse systems. However, other methods of molecular weight determination are even less attractive than equilibrium ultracentrifugation as regards accurate determination of this distribution. It should be apparent that only fractionating procedures have this possibility, and that present fractionation techniques leave much to be desired. Therefore, the failure of equilibrium ultracentrifugation to provide accurate distribution functions should not be surprising. Despite its present inability to produce eyact molecular distribution functions, the technique of equilibrium ultracentrifugation is valuable in such studies since it is virtually the only technique which provides an indication and a measure of polydispersity in a single experiment. While it is true that vapor phase chromatography is a superior technique for volatile, low-molecular weight samples, and that liquid phase chromatography and electrophoresis are superior for certain medium to high molecular weight samples, neither of these techniques is as generally applicable as equilibrium ultracentrifugation. This is especially true when general applicability is judged on a molecular weight basis. Thus, it is of interest to determine how well a molecular weight distribution function may be characterized by the technique of equilibrium ultracentrifugation. It is, of course, obvious that a system composed of one, two, or three components of known partial specific volume and molecular weight may be directly specified by equilibrium ultracentrifugation to within the limits of experimental error. This may be raised to four components in light of the suggestion by WaleP that derivatives corresponding to the (z 1) molecular weight average can be roughly evaluated. As Goldberg’ has pointed out, further information regarding the shape of molecular weight distribution curves is often available, especially if the system consists of polymeric species condensing according to some statistical rule. By virtue of the fact that it measures the first three moments of a molecular weight distribution, equilibrium ultracentrifugation may be used to verify that a system follows such a rule. The worst case is that cited by Williams, Van Holde, Baldwin and Fujita33 in which the distrihu-
+
Vol. 64
tion is specified by giving its measurable moments. To picture such a distribution, one may use the Chebychev i n e q ~ a l i t y . ~This ~ states that, if one has a distribution whose mean is M , and whose - l)’l2,then standard deviation is M,(M,/M, the fraction of the distribution lying within the range M,(1 f k (Me/Mw- 1 ) ’ / 2 ) is greater than or equal to (1 - l / k 2 ) . If a molecular weight distribution is specified in terms of weight fractions, then it is such a distribution, as is pointed cut by Williams, et aLa3 Experimental This procedure was applied to samples isolated from the materials formed during the thermal polymerization of “winterized” cottonseed oil. The preparation and isolation of the sample was based on the procedure of Chang and Kummerow35 and will be described elsewhere. Essentially it consisted of heating the oil under controlled condit,ions, and fractionation of the product by saponification, acidiF,cation, urea-adduct formation, and selective extraction with solvent mixtures. A portion of this material was used for analysis of the methyl esters by gas-liquid partition chromatography. The methylation was carried out by refluxing wit,h absolute methanol. A 50-microliter sample of the methyl esters was then analyzed using a column of Reoplex 400 on Celite 545. The Reoplex 400 was kindly supplied by the Geigy Chemical Company. This column was held at 220,; in a Perkin-Elmer Model 154-B “Vapor Fractometer, and helium was used as the carrier gas. A similar determination was carried out using methyl oleate as a reference standard. The sample for ultracentrifugal analysis was dissolved in absolute methanol. The centrifuge runs were made using a Spinco Model E ultracentrifuge, in conjunction with an analytical An-D rotor and two plain window analytical cells, Spinco No. 1190. ( I t was necessary to use two cells rather than one double sector cell due to the high speeds used.) One-tenth ml. of Fluorochemical FC-43,8eproduced by Minnesota Mining and Manufacturing Co., was placed in each cell using a small volume hypodermic syringe.” A 0.1-ml. ortion of the sample dissolved in methanol was then addef to one cell, while the same amount of methanol n-as placed in the ot,her cell. The two cells were then placed opposite each other in the rotor, and the run carried out in accordance with the procedure of Van Holde and Ba1dwin.a Other concentrations of sample were studied similarly. The runs were followed using the schlieren pattern as observed with a phase-plate.s8 The attainment of equilibrium was verified by failure of this pattern to change within one hour after apparent attainment. The resulting pattern was photographed, and calculations were made from enlarged tracings of the photographs. These calculations were done using a tabular scheme similar to that employed by S~hachman.~Q The values of d(n,)/dz were obtained by measurement of the difference between the schlieren pattern of the solvent and that of the solution. (33) J. W. Williams, K. E. Van Holde, R . L. Baldwin a n d H. Fujita, Chem. Reus.. 68, 715 (19581. (34) See, for example, H. Cramer, “Mathematical Methods of Statistics.” Princeton University Press, Princeton, N.J., 1948,p. 182-183. (35) S. R. Chang a n d F. A . Kummerow, J . A m . Oil Cnem. Soc.. 80, 403 (1953). (36) This material has been found t o have a satisfactorily high degree of insolubility in water, alcohols, ketones, aldehydes, tetrahydrofuran a n d ethylenediamine. I n fact, studies of solutions in these solvents are limited, when using this technique, primarily b y t h e solvent resistance of t h e components of boundary-forming cells. T h e material, however, is n o t satisfactory for use with hexane a n d benzene, and also cannot b e used with the halogenated hydrocarbons. (37) Although t h e volumes used here could be measured with a syringe such as a B-D Yale, I/i-cc. capacity, i t has been found to be more generally satisfactory to use one such 8s t h e Hamilton 100 microliter syringe. The Hamilton syringe is far superior in measuring, reproducibly, amounts of less t h a n 0.1 ml. (38) R. T r a u t m a n a n d V. W. Burns, Biochim. e t Biophys. Acta, 14, 26 (1954). (39) H.K. Schachman, “Methods in Enzymology,” I V , Academic Press, New York. N.Y.,1957, p. 32 ff.
STUDY OF LIMITED MOLECULAR WEIGHTDISTIXIBUTIONS
Dec., 1960
1835
These were tabulated versus x, the radius of rotation a t which they were observed. Values of u and l/x(d(%)/dz) were also tabulated. The quantity
was determined, a t various selected values of u for each dilution, by interpolation of the l/z(d(%)/dz) data, and integration of the same data from u = 0, after determining (no)at u = 0. I t may be seen that (nC)cu)= (nc)(o) (zb) - Zm8)/2 l,lz(d(nc)/dz)du in analogy to the similar SOU equation by Ginsburg, Appel and Schachman.n I n the work here reported, the interpolation has been done graphically from a smoothed plot, and the integration has been done m t h an Ott-Type 22 planimeter using the same plot. The value of (no)a t u = 0 has been determined by using the relationship
+
I 8 12 16 20 ( n - no) u. Fig.1-An illustration of the method of extrapolation of data to infinite dilution to obtain the function from which the moments of the molecular weight distribution are evaluated.
+
+
100
200
0.5
which may be seen to be completely analogous to the comparable equation used by Ginsburg, Appel and Schachman.27 (nc)ois evaluated using the synthetic boundary cell, although this is run a t a speed such that the solution below the synthetic boundary is as compressed as the solution in the column in which equilibrium is established. Such a speed may be determined from ey. 12,20 and 21. For our conditions, it is approximately WB = 0.4 WE (36) if WB is the epeed used during the synthetic boundary run, while WE is used during the equilibrium run. Values of %/l/x(d n,/dz) have been multiplied by ( 5 b 2 - zmZ)ref/ (5152 - z,*) to provide the first-order correction described above.29 They are thus equal to (xb* - xm2),,f/2(d In n,/du). These values were then plotted us. (120) u and extrapolated, a t constant I L , to give ( 2 b 2 - zm2),,r/2(d In (%)/du).vo I. From this, values of (d In (nc)/du)No 1 were obtained as a function of 3. From these latter values, a table was drawn up and the functions (d In (nn)/dU)N, (ln((n,)@))/ n o ( o ) )=. ~I~ , o and (%(")/nc(o))du) tabulated us. u. No- 1 These were used to evaluate the moments defined here, and the value of A was determined using the value of ( 2 b 2 zm*)rei.Calculation of the appropriate distribution function is based on these moments.4o Throughout this work, it has been assumed that the errors involved were similar to those reported by Van Holde and Baldwin,* and that the results could be treated accordingly. Thus, we would expect that the error of measurement of the moment defined by eq. 35-b is about 1%. This has perhaps been verified by standardization with sucrose, for which we have measured, by Method I1 of Van Holde and Aaldwin,2 a value of 345 for the molecular weight. It should also be pointed out that runs using solvent only in both cells do not show measurably different schlieren patterns for similarly filled cells.
4
0.4
0.3
d 0.2
0.1
-
(jy
-
I
0
wi.
L 300
400
Fig. 2.-Crude representations of the spread of the molecular weight distribution based on the evaluation of the first two moments of the weight distribution as determined by these measurements, 1 using the Chebychev inequality, and [I using a distribution equivalent to the "most probable distribution in linear open chain molecules" as described by Flory.
up using the extrapolated data, and is used to evaluate the measurable moments of the molecular weight distribution. For the sample reported here, the averages defined by eq. 35-a (I) and 35-b (11) are, respectively,
1.284 - 1.000 Results and Discussion = 120 I = 1.133(2.08 X Characterization of the equilibrium state at(1.284)(0.238) - (0.211) = 155 tained during ultracentrifugation in terms of the I1 = (1.284 - 1.000)(2.08 X equations presented here, and the appropriate calculations, is shown in Tables I, I1 and 111. To compute the average based on eq. 35-c, we shall Table I presents the actual experimental data, a t resort to fitting the data by an analytic function points close to the chosen values of u,from the which is an infinite polynomial whose derivatives run of the most concentrated example. Table I1 constantly increase. The simplest such funcis calculated from Table I and is used for the extra- tion is exp { a bu/c - u 1. In the case at hand, the polation. Tables similar to Tables I and I1 are constants can be evaluated to give drawn up for each dilution of sample. Figure 1 - 1.556 4-3.4, illustrates the extrapolation of the data from Table I1 for the sample reported here. Table I11 is set This agrees with the values listed in second column of Table I11 to well within the experimental error, (40) With the help of Mr. Richard Clinite, of the Comptroller's Office, Swift h. Company, we have recently developed a program for as shown in Table IV. carrying out these calculations using the IBM 650 digital computer. Differentiating this function may be used to When such a (computer is not available, i t is advantageous to employ evaluate the average defined by eq. 35-12 (111). a desk calculator similar t o that recommended by Trautman.41 (41) R. Trautman, Biochim. Biophus. Acto, 28, 417 (1958). This derivative is
+
THOMAS H. DONNELLY
1836
give the moments is not actually a molecular weight distribution, but is a distribution over R i W i ' since ( N i) oLM i wi =
TABLE I EVALUATIUX OE' u
(1
AND
-
MENTAL
(T),
X,
cm.
mm.n
6.773 6.820 6.873 6.927 6.976
16 95 18.25 19.85 22.30 25.45
(Th'
-
Tm')
2
mm. cm.
Vol. 64
u)
(37)
DATA
i=l 22,
crn.2
u
(1
-
mm.a/ cm.
u)
2.50 2.68 2.89 3.22 3.65
45.874 0.000 1.000 2.50 46.512 ,229 0,771 2.06 47.238 ,489 ,511 1.48 47.983 .756 ,244 0.79 48.665 1.000 ,000 0.00 u) 1 d(nJ du = 1.99 mm."-cm.
J (1 - ;(=)
(nJ0 = 18.48 mm."-cm. (no)(O) = 16.49 mm.a-cm. These numbers were measured on enlarged tracings using a viewing table and an accurate scale. Their actual magnitude is determined, of course, by the magnification factors, bar angle, etc., all of which have been held constant throughout these experiments. Use of different methods of measuring the refractive index gradient at x will, of course, lead to different magnitudes. Units are immaterial, since they cancel out in forming the quotients to be extrapolated. Q
where wi is the weight-fraction of the ith species. The RiWi' may be evaluated for each species and listed as in Table V. In so doing it is necessary to draw on all available information regarding the parameters characteristic of each molecular species which may be present. In the sample presented here, it was assumed that oxidative polymers are essentially absent, and that the variation of t j ( O ) with i may be predicted on the basis of parachor data.42p43 For this calculation, values of ( ~ i ) ' ' ~ have been assumed constant. Values of R i have been taken as constant based on computation from estimated molar refraction^.^^ T.4BLE
v
EVALUATION OF ALLOWED VALUESOF MOLECULAR WEIGHT AND RELATED FUNCTIOXS z
Jkf i
v,(o)a
(I
-
S ~ ( O ) ~ ~ O )w i
Wi'
b
1.107 0.125 34.7 34.6 1.076 .149 83.6 83.4 TABLE 11 1.067 .156 131 130.7 EVALUATION O F QUAKTITIESFOR EXTRAPOLATION 1.061 .I60 181 181 1.060 ,161 226 225 1.059 ,162 273 272 7 1.059 ,162 319 318 2.50 16.49 6.59 0.0 7.06 16.49 8 1.059 .162 364 363 0.25 2.68 17.41 6.50 6.96 17.66 9 1.059 .162 410 409 18.41 6.34 6.79 18.91 .5 2.90 10 1,059 .I62 454 453 .75 3.22 19.55 6.07 6.51 20.30 a Computed from the equation 1.0 3.65 20.69 5.69 6.09 21.69 694 ( i 1) 4-40 TABLE I11 694i 40 a f l EXTRAPOLaTED RESULTSAND CALCULATIONS THEREON derived on the basis of parachlor data, assuming ( y i ) ' / 4 is d In (no) nc(u) constant. is estimat,ed as 1/0.903 at 20°.46 Values of u (-;ir).vc=i (nc(0i))sfi-1 ( S ) N o = l (1 - &(O)poo) are computed a t 20°,and taken as independent 0.0 0.211 0.000 1,000 of temperature, using the density of methanol as 0.791.46 .25 ,217 ,059 1.061 b b for methanol is taken from Gibson4' as 1.25 X 10-'0 2 dyne a t 25'. f j o ( 0 ) is taken as 1/0.786 by interpola.5 221 ,120 1.128 Thus @RT/i&(o) is 2.43 g. A is 2.08 X lo-$ (g.)-l 75 ,229 ,182 1,200 for this particular run. 1.0 ,238 .250 1.284
1 2 3 4 5 6
280 561 841 1122 1402 1682 1963 2243 2524 2804
+ +
(
t
(K (s) du)ywl
Calculation of t8hebest distribution from the averages measured here is done by the most suitable available method. If the sample can be adequately represented by a finite distribution, a
= 1.133
TABLE IV FIT OF EXPERIMEKTAL RESULTS TO IKFINITE POLYNOMIAL d In (ne) u
0.0 .25 .5 .75 1.0
0.211 ,217 .221 .229
,238
exp
0.211 ,216 ,222 ,229 ,238
f'(0) = (0.0856)(f(O)) and f'(1) = (O.l69)(f(l)),or f'(0) = (0.0181)and f'(1) = (0.0402). Therefor?, (1.284)(0.0567 0.0402)- (0.0181 0.0446) rrI = (1.284)(0.238)- (0.211)
(-
(42) See, for example, G. W. Thompson, in A . Weissberger, "Physiof Organic Chemistry," Vol. I, Interscience Publ. Co., New York, N. Y., 1946, p. 202 ff. (43) This procedure has been checked approximately by using t h e second estimated molecular weight distribution given here t o predict t h e value of which should be observed for this sample. T h e agreement Kas satisfactory, b u t i t should be emphasized t h a t this value of G is primarily determined b y t h e paucimeric species in this particular sample. Thus, i t is not very different from F for t h e monomer in methanol, a n d t h e apparent check is not adequate. A further verification is given by comparison with t h e density d a t a of Chang a n d Kurnrnerowa3 f o r their analogous fractions. T h e values of Table V are t h u s seen t o be quite reasonable a n d far more reliable t h a n those obtained b y negkcting t h e dependence of ;icO) on i. (44) See, for example, K. Fajans, in A . Weisshrrper. "Physical 3Iethods of Orpanic Chemistry," Vol. 1. Interscience Piihl. Co., New T o r k , N. Y . , l Q l f i ,p. 672 ff. (45) K. S. hlarkley, "b'atty Acids," Interscience, New Tork, N. T., 1947, p. 216. (46) "International Critical Tables," Vol. 111, AlcGran.-Hill Book Co., New York, N. Y., 1933, p. 27. (47) R . E. Gibson, J. A m . Chem. Soc., 69, 1525 (1937).
0 297 u { - 1.556 + e u ] cal Methods
+
+
1
2.08 X 10
-3 = 308
The distribution of which these measurements
Dec., 1960
STUDYOF LIMITEDMOLECULAR WEIGHTDISTRIBUTIONS
matrix method such as is developed in the Appendix might be employed. The spread of values of I, I1 and I11 may be used to indicate whether or not this is a good approximation. In the present case, the great disparity between I1 and I11 indicates that the actual distribution has a high molecular weight “tail” which must be accounted for. This is done using the most appropriate statistical rule. When no such rule is appropriate, the spread of the distribution may be pictured using the Chebychev inequality34 applied as an equality. The value of the mean, p , I or 120, and the value of the standard deviation, u, I(II/I - l)’/z, 120(155/120 - 1)’12or 64.8, are applied t o those values of W ’ more than one standard deviation from the mean. One cannot, obviously, determine any properties of the distribution within ,u f u from this approach. Outside the interval ,u f u, or 55 < Wi‘ < 185, however, k, may be computed for each value of W,’ as (W,’ - 120)/64.8 and the distribution constructed by assuming the weight percentage of imer to be equal to l O O ( 1 - l/kI2) less the percentage already accounted for by i-mers whose values of W,’ are closer to the mean. Thus, for this sample, the W,’ values 83.4, 130.7 and 181 lie with the 1 u interval. The value of W,’ next closest to the mean is that for the monomer, 34.6. In this type of calculation, the minimum amount of material which could be in the range 34.6-205.4 is thus counted as monomer. It could, however, be spread over the four W,’ values, 34.6, 83.4, 130.7 and 181. Also, more material could be present in this range of W1‘. In addition, it may be recalled that the mean is often near the median and that one-half of the material is distributed on each side of the median. This might be used to improve the estimate of material present as monomer and dimer, for example. Subject to such inaccuracies, however, the distribution may be further delineated. The value of W,’ next furthest from the mean is 225, and the minimum weightpercentage of material within the range 15-225 is determined from the corresponding value of 12,. The incremeiit between this weight percentage and the previous one is taken as the weight per cent. of 5-mer. Such a process is continued to give the distribution in the second column of Table VI. The hazards of such a procedure are apparent if one redetermines the averages of eq. 35 from the values of Table VI, since the values of I and I1 thus vomputed do not reproduce the values from which Table VI is determined. This is presumably due to the failure of the procedure near the mean. In the present case, it is interesting to use the values of Table T’ to see how well the moments of the molecular weight distribution of the sample reported here agree with those predicted, for example, for a distribution of the simplest type ~ * most probable distribudescribed by F l ~ r y , the tion in linear, open-chain molecules. This is the distribution in which the mole fraction of i-mer is equal t o some constant, less than unity, times the mole fraction of (i - 1)mer. In Flory’s terms, this is (48) P J Flory, “Piinciples of Polymer Chemistry,” Cornell Universlty Press, Ithaca, h . Y , 1953, p 318 ff.
1837
TABLE VI DISTRIBUTION OVER ALLOWED VALUESOF Wit
CRUDE APPROXIMATIONB O F
W1’
z
WIa
Wtb
34.6 0.424 0 281 83.4 ... ,264 130.7 ... ,186 181 ... ,116 225 .195 ,0684 272 .199 .0385 7 318 .075 0211 8 363 ,036 ,0113 9 409 ,0211 ,00598 10 453 .0131 ,00312 Computed from Chebychev inequality described in text. Computed from most probable distribution in linear, open chain molecules as described in text. 1 2 3 4 5 6
- p)
(38)
w, = ip@-1) (1 - p )
(39)
A;, =
pCt-1)
(I
and w,is given as The moments defined by eq. 35-a, b, and c are seen to be
f i=l
i=l
respectively, for this system. Further, since
2
n
(Wi’/Wi) may be approximated ii’O.997 based on the data of Table V. I, the moment described by eq. 35-a, is thus wi
= 1,
wi
i= 1
(1
I =
- p)2
”.
2 ipC”-’, i=l
0.997
wit
(40)
From the expression, p may be evaluated as 0.470. When applied t o eq. 39, this gives the values in the third column of Table VI. The higher moments measured by eq. 35-b and c may be predicted from this value of p . They are thus given as I1 = 183 and I11 = 245 for this distribution. Thus, this sample is an example of the case which will often be met, ie., one in which a good picture of the distribution is not immediately available. At present, there seems to be no generally satisfactory method for such samples, and the development of such a procedure is beyond the scope of the present work. I n an attempt to verify that the moments measured for this sample are in accord with the actual molecular weight distribution, the Wi’s predicted from eq. 39 are used to compute a number-average molecular weight of 528. This may be compared with the value of 406 reported by Chang and Kummer0w3~for their equivalent sample. The distribution pictured by eq. 38 can be brought into agreement with this value only by the addition of more material on the low molecular weight side, especially in the range of monomer. A similar conclusion is reached using the technique of gas chromatography for monomer analysis. Such determinations were carried out on the methyl
THOMAS H. DONNELLY
1838
Vol. 64
may then be estimated from the manner in which the first three averages approach W m a x . These could then be used to compute moments of the distribution similar t o those defined by Goldberg.’ For any system, the equations
esters of the sample used in the preceding studies. Only one peak was observed, and it was found to correspond to methyl linoleate. A calibration run was carried out using an equal amount of a sample of pure methyl oleate. As shown in Table VII, only half as much material was detected when the unknown methyl esters were analyzed. This indicates that monomeric linoleic acid makes up about half of the unknown sample on which the ultracentrifugation studies were carried out.
WlWl’
WlWl’W,
TABLE VI1 Sample
Area of peak, om.’
Unknown Me oleate
39.0 f 1 . 4 77.2 f 2 . 8
=I
WmxWmsx’
WZWZ’WZ
(b)
WnWs’VYI
WmxWmsxfWmsx
MONOMER ANALYSIS BY GAS CHROMATOGRAPHY
(A-%a)
+ WZWZ’+ W S W ~+‘ . . . + + + .. .
=I
x
11 ( c )
etc., could then be written, and E might be estimated as in the text. This family of equations could then be solved by determinants. An interesting case which has a fairly general solution is one in which Wi = iW and Wi’ = iW‘. This will also illustrate the hazards and shortcomings of this procedure. In this case, eq. A-2 becomes
From these results, we may legitimately conclude that the distribution computed using eq. 38 is in error. The actual distribution is Drobablv
ment of I11 by this technique is in error by as I I I much as 50% since a distribution which will check 1 2 3 the values of 120 and 155 for I and 11, and will agree with the monomer analysis data, is not likely , , . to give a value of 305 for 111 if it is a statistically . . .
1;
I:
IW,’ W/W’ . . ’ X w z = I/W’ .. ~ , , ’
;:
. I i .. j
,
2
Fvi I
~
:
’
~ (A-4)~
~
/
RADIATION CHEMISTRY OF AQUEOUS PERMANGANATE SOLUTIONS
Dec., 1960
2 X 2 inverse, which can be evaluated by eq. A-6 and A-7. The first few inverses are thus 2 -1
-1 1
3
-3 1
--5 2
-1 2
4 -1
3 2
I1
--3
-4
-7
2
-7 2
as may be verified by simple calculation. This scheme has been verified by calculation using the IBM 650 digital computer up to 10 X 10 matrices. Inspection of these inverses shows that they
1839
produce convergent values of Wi only as small differences between large numbers of opposing signs. Thus, the accuracy with which the W averages should be observed increases as the number of components present increases, and is of an order which is not easily obtained even when only a few components are present. For example, the relative concentrations in a three component system of this type can usually be specified only to two places when the measured averages are accurate to three places. Since similar considerations would hold for any system to be studied by equilibrium ultracentrifugation, it is seen that the unequivocal determination of molecular weight distributions by this technique is usually a highly imaginative undertaking. An essentially equivalent conclusion has been reached by Goldberg’ and by Williams, et
THE R,ADIATION CHEMISTRY OF AQUEOUS PERNIAKGAKATE SOLUTIONS’ BY MALCOLM DANIELS~ Chemistry Division, Argonne National Laboratory, Lemont, Illinois Recciued March 9,1960
The reduction of permanganate solutions, induced by Cogo y-rays, has been investigated in acid, neutral and alkaline solution. I n 0.8 N H B O , the major products are MnOz and 0 2 , and yields have been determined over a wide range of radiation intensity and permanganate concentration. A chain reaction has been found and the over-all rate of reduction can be represented by G(-MnO4-) = GO kl(Mn04;)’/o k2(MnO~-/I)’/~,where GO = 2.3 kl = 10 and k2 = 0.8 x 1011 (G is defined as molecules/100 e.v., concentration units are M , and intensity units are e.v./l./min.) An explanation is offered involving intermediate formation of 0 atoms and MnOE-. Yields of hydrogen gas (“molecular yield”) have been determined at various permanganate concentrations, and reactions of solutions saturated with H2 have been investigated. Explanation in terms of Mn06-- is suggested. At pH 5.6 colloidal Mn(1V) is produced but in 0.5 M OH-, manganate is the initial product. In both neutral and alkaline solution curves of product yield vs. dose exhibit decrease in slope when the product accumulates; this ifi attributed to re-oxidizing reactions of OH radical with the products.
+
+
Introduction Progress in the understanding of radiation chemistry of liquids has largely been based on detailed studies of chemical changes induced in reactive additives, usually termed “scavengers,” as exemplified by ferrous ion, ceric ion and formic acid in aqueous solution, and I2in organic liquids. The ability of permanganate ion to oxidize water (expressed by its high oxidation potential), coupled with its high solubility over a wide pH range, suggested investigation of its possible utility as such a scavenger. The possibility of clarifying the mechanisms of reduction of permanganate was a further factor. Previous work was sparse, limited in scope, and contradictory in n a t ~ r e . ~ -In~ the course of the (1) Based on work performed under the auspices of the U. 5. Atomic Energy Commission. Presented in part at the April 1958 ACS meeting held in San Francisco, California. (2) Chemistry Department, U. College of West Indies, Kingston. Jamaica. (3) K. Chamberlain, Phys. Reo., 26, 525 (1925). (4) G. PI. Clark and L. W. Pickett, J . Am. Chem. Soc., 62, 465 (1930). (5) G. W. Clark and W. S. Coe, J . Chem. Phya., 6, 97 (1837). JOURNAL, 42, 1229 (1938). (6) F. C. Lanning and S. C. Lind, THIB (7) L. Bloch-Frankenthal and G. Goldhaber, Bull. Res. Council of Is7ae2, 1, 117 (1951). (8) V. I. Veselovsky. Ts. 1. Zalkind, N. B. Miller, N. A.Aladzhalova, Symp. on Radiation Chem., Acad. Sei. U.S.S.R., Moscow, 1955. p. 36. (9) E. L. Alexander. Dissertation, Vsnderbilt Univ.. 1956.
present work results of more detailed investigations appeared,l0J1and will be discussed later in the appropriate places. Experimental Irradiation Arrangements.-Irradiations were carried out with sources of Coco y-rays whose activities ranged from 80 to 4500 curies. Dosimetry was carried out using the standard ferrous sulfate dosimeterlz [G(Fea+) = 15.61, together with the ferrous-cupric system’s for the higher doserates. The range of dose-rates used were from 1.2 X 1018 e.v./l./min. t o 3 0 X lOZ*e.v./l./min. Preparation of Solutions.--All solutions for irradiation were prepared with triply distilled water14 using analytical grade HzS04, NaOH and KMnO,. The sodium salt, iYaMn04.3H20,was “purified” grade. It was noted that concentrated solutions of NaMnO, showed considerably less tendency to decompose heterogeneously than those of the potassium salt. Degassing procedure has been previously described14 as have the cells used for the irradiation of degassed solutions14 and the syringes used for hydrogen saturated solutions.16 Cells and syringes were used in the “radiation-browned’’ condition rather than being furnacetreated a t 550°, for the latter condition seemed to cause heterogeneous thermal decomposition of the permanganate.18 (10) B. A. Gvoedev and V. N. Shubin, Treatise of 1st All-Union Conf. on Radiation Chem.. March 25-30th. 1957. (11) G. Simonoff, J . chim. phys.. 66, 547 (1958). (12) C. J. Hochanadel and J. Ghormley, J . Chem. Phys., 21, 1080 (1953). (13) E. J. Hart and P. D. Walsh, Rad. Res., 1, 342 (1954). (14) E. J. Hart, J . Am. Chem. Soc., 7 8 , 68 (1951). (15) E. J. Hart, 9. Gordon and D . A. Hutchinson, ibid.. 76, 6165 (1953)
