NOTES
1440
Vol. 6.5
NOTES DEAD-END PORE VOLUME AS DISTRIBUTED SOURCES AND SINKS BY WALTER ROSE, H. C. TUNG
states, transport of matter will occur variously to and from the dead-end pore regions. The simplified theory of Goodknight, et al., apparently leads to the correct (quantitative) College of Enninemino, University of Illinois, Urbana, Illinois description of the influence of continuously distributed dead-end porosity, when limited to those AND CLAUDE NEWMAN cases where the medium is taken to be homogeneous Department ,of Applied Mathematics, Uniuersity of Colorado and isotropic. I n effect they apply equation 1 to Boulder, Colorado subregions of the porous medium (Le., to macroReceived October 80, 1080 scopic differential volume elements) in the way of Let ‘us state the mass (or energy) continuity implicitly saying that the pore space can be parcondition to describe transport phenomena in titioned into a continuous (central) domain, and a material bodies (e.g., diffusion, porous media continuous cul de sac (peripheral) domain. It is compressible fluid flow, heat transfer, electrical then assumed that Darcy’s law describes transport in the central domain region as long as the interconduction, etc.) as action between the fluid potential in the two condp/dt div @ = pu (1) tiguous domains is taken into explicit account. where In this Laboratory we are seeking more general divait? = div (PB) = a(pvl)/ax a(pu,)/ay.+ a(pu,)~& solutions of the diff usivity equations derived from M is the mass (or energy) flux vector per unit area equation 1, examining for example the analytic p is the maas (or energy) per unit volume @ is the velocity vector (with components u,, vy and us) methods discussed by S n e d d ~ n in , ~ order to treat is the density of distribution of internal sources and the unsteady-states of heat transfer in solid media sinks M a a function of time ( t ) and the space co- having randomly distributed internal sources arid ordinates (5,y and 2 ) . sinks (e.g., distributed radioactive crystals of Equation 1. is that given in the development of varying strengths). We suspect that in the most potential theory’ to apply to the unsteady-states general case, a microscopically inhomogeneous which ensue following an initial non-uniform dis- distribution of dead-end porosity ~ 1 1 be 1 found in tribution of :mass (or energy) throughout a space the macroscopic differential volume elements region of interest. Since the t,hree dependent to which the diffusivity equation is to be applied. variables are all indicated to be functions of the I n consequence, the function u ( x , ~ , z , will ~ ) have space coordinates, and of time, the solution of such a coniplicated representation that general equation 1 requires that a t least two other inter- methods of analysis will not be immediately specirelationships between the dependent variables be fiable. known. An analog representation of the situation we wish For exampk, if me limit attention to the subject to consider is obtained by constructing a network matter of the recent paper by Goodknight, et aLj2 of interconnected resistors (scaled to model local we describe compressible fluid flow in porous media conditions of diffusivity), with capacitors a t thc systems by combining equation 1 with: (a) the mesh points (scaled to model the local divergence relationship ‘between fluid density and pressure of mass or energy capacity of the system). Other as given by some appropriate equation of state; capacitors connected variously to one or more of (b) the relationship between the mass flux vector the mesh points through high resistance elements, and the fluid pressure (e.g., by invoking Darcy’s then. represent the dead-end pores. X simple Law). measurement of the resistance-capacitance timeI n addition, of course, the function, u ( ~ , y , z , t ) , constant of the network, therefore, serves to must be specified. The resultant diffusivity equa- characterize the transients of equation 1 for partion then is integrable in principle (e.g., if u is ticular systems with particular initial and boundary zero, the classic Laplacian solutions will be avail- conditions. a.ble if the transport is steady, and Fourier solutions In our initial work we have made use of a 160 can be adopted for the unsteady-states), upon mesh-point network (10 nodes wide by 16 elements specification (of the relevant initial and boundary between line source and sink region^).^ The exconditions of the problem. perimental results summarized in Table I thus show Goodknight, et al., have shown a physical the marked degree to which the transients [as reason to consider solutions of equation 1 for cases measured in ternis of the network time-constant ] where (r is non-zero. That is, they postulate the depend on: (a) number of dead-end pores; (b) possibility of the prevalence of dead-end pore location of dead-end pores; (c) point of measurespace in porous media systems of practical interest, (3) I. N. Sneddon, “Elements of Partial Differential Equations,” so much isolated (via high resistance paths) from Book Co., New York, N. Y.. 1957, p. 274 and 299 ff. the “central” transport paths that during unsteady- McGraw-Hill (4) 960-node network analyzer, Department of Applied Mathe-
+
+
(1) 0. D. Kellog, “Foundations of Potential Theory.” Dover Publications (reissiie of the original 1929 edition!, 1953, p. 45 ff. (2) R. C. Goodknight. Pi. A. Xlikoff and I. Fatt. J. Phus. Chem., 64, 1162 11960).
matics, University of Colorado. The network analyzer can be used to construct an arbitrary N-dimensional interconnection of resistors. with capacitors located as desired at the interconnecting points. Active components are supplied by a BEAC analog computer.
NOTES
August, 1961 lnent of the dependent variable; (d) time constant of the individual dead-elid pores. TABLE I
N o . of
deadrnd pores
0 1
Location
...
Network time constant in relative aec. (ca. Dcnd-end pore component 60% greater _.-__ than real Ca pa ci tance Resistance time in (farads) (ohm-) sec.)
...
...
10.5" 10.4" 25.5" 21.5"
At source 10 -6 2 x 104 At sink 10-6 2 x 104 Uniformly dis- 10-6 10 2 x 105 tributed 10 Uniformly dis- 10 -6 15.7' 2 x 105 tributed 40 15.9" Uniformly dis- 2 . 5 x 10-7 s x 106 tributed a 311Jasiircd nt sink. b Alcnsiircd in middle of nctmork. 1
In all cases m have held the total capacitance of the dead-end pores as a constant fraction of the
total capacitaiice of the central network domain region ; likewise, the tinie-c,onstant of the individual dead-end pores was held fixed (at 0.2 second). The central network itself was entirely isotropic and homogeneous, being made up of a rectangular array of 0.5 megohm resistors with 0.1 microfarad capacitors a t each mesh point. These values arbitrarily were chosen and scaled to be meaningful with reference to the experiment of Goodknight, et aL6; however, it is to be expected that, had a less uniform network form been chosen, an even greater influence on the transients (from deadend porosity) mould have been in evidence. It is clear that the analog network model offers t,lie opportunity to investigate problems of much greater complexity than those envisioned by Goodknight, et al. For example, media inhomogeneity a.nd anisotropy can be scaled without increased difficulty, and micro- as well as macroscopically irregular dead-end pore configurations can be represented. As for the latter, if each mesh point is thought of as the macroscopic differential volume clement to which equation 1 applies, the data of Table I refer to a discrete rather than a continuous distribution of dead-end pores in the macroscopic sense. On the other hand, if the entire netmork of many contiguous mesh points is taken as the macroscopic differential volume element of interest, t,he dat,a of Table I reflect on the influence of a given microscopic distribution of dead-end pores. cm.2, ( 5 ) If in the prototype system, permeability is taken as f i u i d compressibility is taken as per dyne per om.*, porosity is tsken as n . 2 5 fluid viscosity is taken as 10-2 poise, the length to cross-section31 a m i ratio is taken as unity, and the T'E/VI and H IiaramPters of Goodknight. et nl.. are talien as 0.625 and 10-1 reciprocal arconds. respidrely, the scale ratio between time in the electrical analog s p t r m and in the prototype system n-ill be 2 X 10-2.
ESTlnlATZON OF DISSOCIATIOX CONSTANTS OF ELECTROLYTES BY HENRYE. WIRTH Department of Chemrstry, Suracuse Unioersity, Syracuae, N . Y . Received Februarg 1. 1961
I n order to evaluate the dissociation constant,
K,' of a symmetrical electrolyte
1441 C$Y2*-
K =
(1)
I-ff
from conductivity measurements, lcuoss2 defined by the expression ff=
= _ -A
A
- S(Aj
hO(l
h°F(Z)
dV/CyC/Aa)
cy
(2)
where Z = S(n)dG(hO)-*/z and F ( Z ) is the continued fraction
(3)
P(Z) = 1
(4)
- Z(1 - Z( 1
-Z(1
- ..
.)-1/E)-'/z)-'/Y
The activity coefficient y.t is evaluated from the limiting law log Y t =
-S
w 6 c
(5)
Substitution in equation 1 gives the relation
so that Ao and K can be obtained from the intercept and slope of a plot of F ( Z ) / A versus cAy_t2/ F(Z). In the corresponding method developed by Shedlovsky,a a is defined by
where
This leads to the relation
from which A0 and R are evaluated from a plot of l/AS(Z) vs. chg&S(Z). The Shedlovsky method has been stated4 to give better results than the Fuoss method when I< is greater than lo+, but as shown in Fig. 1-3 both methods fail a t relatively low concentrations. The nature of the deviation in the case of the Shedlovsky treatment is such that the linear portion of the curve can be extended to higher concentrations by using the extended law for activity coefficients logy* =
- &*)&/(l+ .4'4/LYc)
(10)
with an appropriate choice of ('a," the distance of closest approach of the ions. Curves labeled S'(2) in Fig. 1-3 were obtained in this way. This is essentially the same procedure used by Owen and Gurryj to interpret the data on ZnS04. They made the product cuyh equal to the experimentally determined stoichiometric activity coefficient. Usually the Fuoss method cannot be modified in this way without introducing negative values for ((U.',
The use of the Fuoss and Shedlovsky methods has been simplified by the publication of tables (1) The notations used in this article are those employed by H. S. Harned and B. B. Owen, "Physical Chemiatry of Electrolytic Solutions," Reinhold Publ. Corp., New York, N. Y., 3rd Ed., 1958. (2) R. hf. Fuoes, J . Am. Cham. Soc.. 67, 488 (1935). (3) T.Shedlovsky, J . Fronklin Inst., 226, 739 (1938). (4) Harned and Owen, ref. 1, p. 290. (5) B. B. Owen and R. W. Gurry, J . Am. Chcm. Soc., 60, 3074 (1938).
NOTES
1442
Vol. 65
0.012
0.050
h
1 9 uz
0.011
- 15
e 0.040
IN
&.
,-. 0
I..
0
SG 0.030 kl
-51
d
kl
0.009
0.020 L
0
I
I
0.005
0.010
0.015
0.008 0
0.10
0.30
0.20
Fig. 1.--Estimation of A0 and K for ZnSOd in water a t 25“. The ordinate scale has been shifted 0.005 unit for W ( 2 )and 0.001 unit for S ( 2 ) to separate the functions. 0.014
i
+
Fig. 3.-Estimate of 110 and K in 82% dioxane-water solution a t 25’. The ordinate scale has been shifted 0.005 unit for W ( 2 )and 0.01 unit for S ( 2 ) . a!=
A --- A Ao(l - 2 )
Aow(z)
(11)
where W ( Z ) = 1 - 2. This substitution partially corrects for the fact that the conductivity of ions a t finite concentrations is less than that calculated by the limiting law. On substitution in equation 1, the relation
0.013 7
4
s
-
‘s,
d
4
S’d 81 0.012
,--.
53.4 kl
0.01 1
I
0
u.050
I
0.010
Fig.Z.---Estirniltton of A0 and K for HCl in 70% diovanemater solution at 25”. The ordinate scale has been shifted 0.0005 unit for ll’(Z) and 0.001 unit for S(2).
for F(2)I and S ( Z ) 6 for values of 2 between 0 and 0.209. For larger values of 2 the calculation of the functions is tedious. h simpler function can be developed if, in equation 2, A/An is substituted for the CY in the denominator of the defining term for 01 giving 16) H. M. Daggett, %bid.. 73 4977 (1951).
is obtained. The activity coefficient y h is obtained from either equation 5 or 10. The three functions F ( Z ) , S’(2)and W ( 2 )were used to estimate A0and K for three examples where the experimental values of A were available for both dilute and moderately concentrated solutions. The examples chosen were ZnS04 in water5 and HCI in 70% and 8270 dioxane-water mixtures? and cover a wide range of dielectric constant of the solvent. The curves are given in Fig. 1-3, and the results are given in Table I. The parameter “a” was chosen to give a straight line over the largest possible concentration range, and should be regarded as an arbitrary constant compensating for some of the errors introduced by using limiting laws at finite concentration. I t was pcssible to find a value of “a” that mould give a linear plot up to the point where l~hS”(2) or l V ( Z ) / h went through a maximum. This was usually close to the point where the calculated value of 01 started increasing with increasing concentration. In the case of HC1 in 82y0 dioxane where it ma‘ necessary to use equation 10 with TT’(2) t o giw :I linear plot, the value of lru” used is close to that estimated by other methods, whereas the valw (7) B. R. Owen and G. W. Waters, ibid.. 60, 2371 (1938).
NOTES
August, 1961 TABLE I COMPARISON OF EQUlTIONS FOR ESTIhL4TISG DISSOCIATION
COXSTANTS F(Z)
Function
S(Z)
S’(Z)
W(Z)
ZndOl in water A0
K (1.
x
132.P 5.25
103
(B.,
....
132.85 132.55 4.95 5.07 .... 5.40
132.90
5.04 0
HCl in 707, dioxane
K 11.
.... .... ....
93.27 93.52 6.76 7.48 0.867 0
57.9 2.0%
.,..
....
. . ..
58.55 1.95 7.31
!J3.18
.io
x
7.F
103
(H.,
,...
HC1 in 827, dioxane ‘10
x 104 (1. (A,) K
58.52 1.99 6.30
required with S’(2)was somewhat larger. The same observation has been made in other cases examined in testing these functions. Use of the functions S ’ ( 2 ) or W ( 2 ) permits the use of conductivity data obtained at higher concentrations in estimating dissociation constants. W(2)is to be preferred since it gives reasonable answers with less calculation. -___-
CRYOSCOPIC DETERMINATION OF XOLECULAR WEIGHTS I N AQUEOUS PERCHLORIC: ACID BY MICHAELXRDONAND AMOSLINENBERG Department of Inorganzc and Analytical Chemistry, The Hebrew Unmuerszty, Jerusalem, Israel Received January 17, 1961
The determination of molecular weights of small polynuclear ionic species, in aqueous solution, from freezing point data, is a difficult task, not only because the mean activity coefficients of the solutes usually are unknown, but mainly because the contribution of the counterions of such ionic species to the depression of the freezing point is often much greater than the contribution of the species investigated, thus making this method highly inaccurate. This difficulty may be overcome by the use, as a solvent, of a eutectic solution of a strong electrolyte in water. If a foreign electrolyte is dissolved in a solvent of this kind, only those ions which do not exist in the pure solvent (the “foreign ions”) will depress the freezing point of the solution.23 One could investigate the molecular weight of a polynuclear catioii by iising a coiinter-ion which already exists in the solwilt, thereby eliminating its effect on the freezing point. By this method, a much higher prwicioii in molecular 1% eight deterininatioiib is Another advantage of this nirthod iq the fact that the variation of the activity coefficient of the solute xvith concentration is small and linear in such a This makes possible a correct extrapolatioi~of K fto infinite diliition. (1) G. I’arissakis and G . Schwarienba~:l~, Helu. Chim. Acta, 4042, 41 (1968); 4426, 41 (1968). ( 2 ) H. J. Muller, Ann. Chim., I l l 18, 143 (1937).
1443
The systems hitherto used were mainly eutectic solutions of salts in water2 (I X , with moderate optic axial angle, and strong dispersion of the optic axes, ~ V ( TZ) > 2Vz (!), The mean refractive index is around 1.8, in agreement with the value 1.80 calculated from Lorentx-Lorenz molar refractivity of the end-member chlorides with linearly additive molar volumes. RbzPuClS.--This compound melts incongruently a t 560" with the peritectic composition 0.42 PuC13. The composition is chosen as being the simplest consistmt nith the data. Crystals are transparent, brownish or greenish yellow t o nearly colorlees by txansmitted light, and optically anisotropic Rith low birefringence. The mean refrao-
9
---
-
1
j
'
400 -
CPCl
I 0.2
* e
I
--i
* e
I
0.4 0.6 0.8 PILCli PuCb mole fraction. Fig. 2.-Phase diagram of the binary system PuC13-CnCl; (1) PuC13 melting point, 769 f 2'; (2) eutectic point, 011 a t PuCI3 mole fraction 0.70; (3) CsPuZCl7meltin point, 616 f 3'. (4) eutectic point, 504' at PuC& mole fraction 0.47; (5) ks3PuC1G melting point, 825 ?Z 3"; (6) Cs3PuCl~ polymorphic transformation, 410' (cooling); ( 7 ) eutwtic point, 592' a t PuC13 mole fraction 0.10; (8) CsCl melting point, 645 2 " ; (9) CsCl polymorphic transformation, 465" (cooling).
CsPuzCly.-This compound melts congruently a t 616 * 3". Crystals of this phase are transparent, pale blue in thinnest sections through greenish blue to deep green in thicker sections and not noticeably pleochroic. They are optically biaxial negative with large optic axial angle and moderate birefringence. Acicular crystals exhibit positive elongation and parallel extinction suggest-
August, 1961
SOTES
ing orthorhombic symmetry. The mean refractive index is 1.85 f 0.03, in agreement with the value 1.85 calculated as above. The differences in the optical properties suggest that this compound may not be isomorphic with the isoformular rubidium compound. Cs3PuC16.-This compound melts congruently a t 825 =t 3" and. on cooling, undergoes polymorphic transformation a t 410". Crystals of the roomtemperature modification are green in bulk by reflected light but, immersed in liquid of similar refractive index, are transparent and virtually colorless by transmitted light. They are optically anisotropic and exhibit very fine, complex, polysynthetic twnning. The mean refractive index is around 1.7, in agreement with the value 1.73 calculated as above. Discussion It is interesting to note the similarity between PuC13 and UCljb6in the formation of double salts with alkali chlorides. KO compounds occur in the binary systems involving LiCl or NaC1. There exist iLIPuzC17-type compounds, where hI = Rb or Cs, hl2PuCl5-type compounds where 31 = K or Rb, and R13PuC16-typecompounds where bI = K, Rb or Cs. Acknowledgments.-We are indebted to A. TIT. Morgan and J IT. Anderson for the plutonium metal, and to C . F. AIetz, W. H. Ashley, G. R. Waterbury, R. T . Phelps, C. T. Apel, &I. H. Corker, D. C. Croley, J. A. Mariner, 0. R. Simi, C. H. Ward, K.IT. Wilson and A. Zerwekh for the chemical and spectrochemical analyses.
polyphosphate, previously reported5 a t 25", evaluated a t 65'.
1463 iq
Experimental Materials.-Tetramethylammonium polyphosphates and imidophosphates in 99.9+ and 97+% puiitl , Iebpectively, were prepared as previously d e s c r ~ b e d . ~ The water used for solution make-up was distilled and freshly boiled to remove dissolved COz. Other chemicals were C.P. grade. Procedures.-The aciditv constant determination at a constant temperature and ionic strength WBR carried out ab previously described.3 A Leeds and Sorthiop pH meter with glass and calomd electrodes T T ~ Sutilized, and aa. calibrated a t each temperature with huffer solutions having a pH of 4, 7 and 10 Calibration oi the pH meter nith one buffer solution gave readings ~ i t thr h other two buffers that agreed to within 10.01 pH unit, indicating lineaiity of the pH scale. Temperature B A S controlled to 10.1" using a heater in combination with a heat-sensing Thermotrol unit, manufactured by Hallikainen Instruments, Berkeley, California. In the experiments below room temperature, the solutions were placed in an ice-acetone bath, a i t h the heater supplying enough heat to maintain the desired temperature. During titration the solutions \\err maintained untie1 :i nitrogen atmosphere. The magnesium coniplr\ing I)\ p\ iophosphate and tripolyphosphate vas measiirrd ti\ the same procedui e described by Lambert and W a t t t ~ sexcvpt ,~ that the measurements \\-ere made a t 25 and 65'. The p H mrasurements in the presence of excess magnesium nere made within a few minutes to aboid precipitntion of niagnefiirim phoqphates
Results and Discussion Acidity Constants.-Stepwise titration curves with definite breaks for the weakest two hydrogens were obtained with all of the investigated acids. The other hydrogens were so strong that no inflection points ~ e r eobserved. The acid-base titration dhta, obtained at a constant temperature and total ionic strength, were fit to a least-squares program of an IB3I 704 computer, as previously (5) C J Baiton R J Shell A B milkerson and W R. Grimes. described.3 The resultant acid dissociation conORNL-2548 or E M Leiin and H F RZcMurdie, "Phase Diagrams for Ceramists Part 11, %m Ceram Soc , 1959. stants with the statistical 95y0 confidence limits (6) J J Katz and E Rabinowitch 'The Chemistry of Uranium are listed in Table I. Part I , N N E S Di\ V I I I , Vol 5, RIcGraw-Hill, New York, N Y.. pu'o attempt was made to extrapolate the acid 1951, p 480 dissociation constants to infinite dilution since they were only determined a t fewer than four ionic strengths. For the polyphosphoric acids (H0)zOP METAL CORIPLEXISG BY PHOSPHORUS CORIPOUKDS. T'. TEMPERATURE [ ~ H ] ~ - P O ( O € € ) 2with n varying from 0-60, D E P E S D E S C E OF ACIDITY AND MAGSESIUM COMPLEXISG CONSTA4NTS no significant temperature dependence of the dissociation of the weakest two hydrogens was obBY RIYADR. 1 ~ 4 x 1 served. with the apparent, molal AH for dissociation RrseaTch Department, Inorganir Chemicals Dzuzston, Monsanto Chemibeing between -1 and 0 kcal. The apparent A H cal Company, St. Louts 66, Missourz for the dissociation of the stronger hydrogens lies Received April 3 , 1961 between - 2 and 0 kcal. As was previously3 found The acidity c ~ n s t a n t s l -of ~ polyphosphoric and at 25', the ratio of the dissociation constants of the imidophosphoric acids hare been reported at 25'. two weakest hydrogens approaches the value of However, the evaluation of the temperature de- four7 as the chain length of the phosphate chain appendence of metal complexing at pH values where proaches infinity. two hydrogen forms of a ligand coexist4requires the The acid dissociation constants of imidodiphosavailability of acid dissociation constants in the phoric and diimidotriphosphoric acids show sigsame temperature range. nificant temperature dependence, as was observed I n the present study the aciditj constants of poly- with their calcium complexes. Thus, the apparent phosphoric and imidophosphoric acids are presented molal AH for the dissociation of the weakcst hydroa t several temperatures and ionic strengths. Rlag- gen ( K J at :in ionic strength of 0.1 is -6.4 and nesiuni complexing by pyrophosphate and tri- - 4.5 kcal. for imidodiphosphorir and diimidotriphosphoric acids, respectively, where "
"
(1) J. I. Watters, E. D. Loughran a n d S. M. Lambert, J . A m . Chem. SOC.,78, 4855 (1956). ( 2 ) S. M. Lambert a n d J. I. Watters. ibid.,79, 4262 (1957). (3) R. R. Irani and C. F. Calli@,J . Phys. Chem., 6S, 934 (1961). (4) R. R. Irani and C. F. Callis, ibid., 64, 1398 (1960).
(5) S. ILl. Lambert and J. I. T a t t e r s , J. A m . Chem. Soc., 7 9 , 5608 (1957). (6) R. R. Irani a n d C. F. Callis, J . Phgs. Chem., SS,296 (1961). (7) S. W. Benson, J . A m . CAem. Soc., 80, 5151 (1958).
VOI. 63.5
XOTES
L
FOR
n
0
Ionic strength
Temp., O C .
0
0.1
10
.1
25
.I 1.0 0.1 .2 .3 .1 .2 .3 .1 1.o 0.1
05
Ib
0 10 25
.1 1.o 0.1 .2 .3 .1 .2 .3 0.1 1.o
50
05
4
14
58
1.o
25 37 50 25 37 50 25 37 50
pKa
9.08 f 0.16 8 . 9 7 f .13 8.95 rt .10 S . 7 4 f .07 8 . 9 4 f .OS 8 . 9 3 z t .09 8.88rt .09 s . 9 i r t .os 8 . 9 0 f .os 8.88rt .09 8 . 9 2 f .09 8 . 7 2 f .07 8 . 5 1 f .12 8 . 7 0 f .07 8 . 6 5 f .05 8 . 5 6 f .10 8 . 5 0 3 ~.OS 8.51 f .OS 8 . 5 2 f .08 8 . 5 5 f .OS 8 . 4 8 f .OS 8 . 4 8 f .OS 8 . 4 8 f .06 8 . 3 9 f .OG 8 . 1 3 f .14 8 . 0 2 f .2 8 . 0 0 f .15 8 . 0 S f .I0 8 . 0 2 f .13 8 . 1 5 f .10 8 . 1 7 f .04 8.033Z .OS 8 . 0 3 f .08
.1
37
Y = OXYGEN
PKi
1.0 1.o 1.o 1.0 1.0 1.0 1.0 1 .o
FORX 0
0.1 .3 .I .3 .1 .3
25 37 50
lb
25
.I
37 .I 50 .1 Est,imnted, large errnrs may exist.
b
6 . 1 i f 0.15 6 . 0 3 f .18 6 . 1 2 f .10 5 . 9 S f .07 6 . 1 3 3 ~ .07 6 . 1 2 5 .08 6 . 0 8 f .09 6 . 1 3 f .07 (i.06rfi .07 6 . 0 4 f .08 6 . 1 6 f .08 6 . 0 1 f .OS 5 . 7 f .1 5 8 4 f .05 5 . 7 5 f .14 5 . 6 9 f .ll 5 . 7 7 f .07 5 . 7 7 i .07 5 . 7 8 3 ~ .07 5 . 9 0 f .07 5 . 8 0 f .OS 5 . 8 4 f .07 5 . 8 8 f .07 5 . 8 0 3 ~ .05 5 . 9 8 3 ~ .18 5 . 8 3 & .15 5.81 f .20 6 . 4 8 i .07 6 . 4 8 f .OS 6.50f .OS 7 . 2 2 f .04 7 . 2 8 f .08 7 . 2 8 4 ~ .OS
PKZ
2 . 5 f0.1 2.3 f . I 2.0 f . 1 1 . 9 5 3 ~ .04 1 . 9 5 f .05 1 . 9 7 f .OF 1 . 9 1 f .oc, I . 9 S I .of; 1 . 9 2 f .08 2.123~.Oi 2 . 1 2 f .0; 2 , 1 7 + .05 2.3 f .1 2.31 f .04 2 . 1 3 k .10 2 . 0 4 k .09 1 . 8 9 f .OG 1 . 9 5 3 ~.06 1 . 9 S f .oo 2 . 1 2 f .07 1 . 9 5 f .OG 2 . 6 2 f .05 2 . 1 5 3 ~ .05 2 . 1 0 f .08 2 . 1 9 f .10 2 . 2 2 f .07 2 . 2 2 i .12 2 . 9 2 3 ~ .09 2 . 6 4 f .05 2 . 5 2 f .13
..... .....
2.3 2.2 2 1.7 I .(I1 I .7 1.7
I !I I .:I 1.2
1.3 1.2 2.2 2.2 2 1.2 1.7 1.7 1.7 1 .7 1.7 1.7 1. 7 1.7 2 .I 1.3 1.3 2 2 2 .. ..
..
= NH
7.32 f 0.12 2.66 f 0.09 1 0 . 2 2 f 0.09 7 . 0 5 3 ~.I2 2.81 f .06 9 . 7 7 f .I2 7 . 1 6 f .04 2 . 6 0 + .OS 9 . 7 9 3 ~ .03 6 . 9 9 f .05 2 . 6 8 f .07 9 , 5 2 5 .05 6 . 9 0 f .06 2 . 8 1 f .07 9 . 4 1 3 ~ .06 6 . S 8 f .06 2 . 8 3 f .07 9 . 3 2 f .03 6 . 6 1 f .09 3 . 0 3 f .10 9 . 8 4 f .OS 6 . 8 0 f .OS 3 . 2 4 f .OS 9 . 5 0 f .OS 7 . 0 2 f .08 3.59* .09 9 . 2 8 f .OS p K j is estimated to be 1 f 0.5 at the various temperatures.
d In K , dT ICn = (H+) ( H ~ - I L - ( ~ - ” + ” ) (H,L -b -))
pKda
1.5 2 1.8 1.8
1.8 1.8 2 2.2 2.4
weakest hydrogens (&), the vahieq of AH at an ionic strength of 0.1 are -3.2 and +3.2 kcal. for imidodiphosphoric and diimidotriphosphoric ncids (2) respectively. Magnesium Comp1exing.-The nephelomet,ric and parentheses indirate concentration. The difference hetwecn A H I , and the thermodynamic value, titrations, previously utilized* in evaluating calcium complexing constants were not found as AH‘n, is suitable for magnesium complexing because of the d In f absence of a well defined magnesium precipitate. AH’n AH,, = R T 2 - - (3) dt Therefore, the pH-lowering technique was used. wheref is the o~cti~7ity cocfficient8ratio of the species The acid-base titrations in the presence of magin equation 2 . For the dissociation of the second nesium still showed stepwise complex formation. A H n = RT2--
-
August, 1961
NOTES
In Table I1 the pH values are given after adding 1/2 and 3/2 equivalents of hydrogen ion per mole of ligand in the presence of magnesium. Utilizing the appropriate acidity constants, the, formation constants for the following equilibria, previously proposed bv 1,xmbert and Wattersls were evahinted a t 25 arid Go, using the techniques previously described.
to a need for polarizability and molar volume data for substituted ammonium salts in particular. We report here the results of refractive index and density measurements for the systems tetra-nbutylammonium picrate (Bu4NPi) in nitrobenzene and in chlorobenzene, tetra-n-butylammonium iodide (BaNI) in o-dichlorobenzene and in water, and sodium tetraphenylboride, tetra-n-propylammonium iodide (Pr4NI), tetraethylammonium bromide and tetramethylammonium chloride in water.
1465
TABLE I1 COMPLEXING OF MAGNESIUMBY PYROPHOSPHATE AND Experimental TRIPOLYPHOSPHATE ANIONS Salts.-These were from current laboratory stock and, "a" is the number of equivalents of H + per mole of ligand, with the exception of sodium tetraphenylboride and tetra-nand L is the polyphosphate ligand. Total ionic strength propylammonium iodide, were considered of sufficient purity was adjusted to 1.0 with tetramethylammonium bromide. to use without further treatment. The sodium tetraphenylboride was recrystallized from water, and the tetraNeg. log of formation n-propylammonium iodide was recrystallized from ethanol. constant of Ligand I emp., Solvents.-The solvents were purified by recrystalliza"C. "a" pH MgL MgzL MgHI, tion (nitrobenzene), distillation (water), and in the case of .,_.j U . 5 C.30 5 . 4 2 2 . 3 3 .. [' y ro 1111os 111, a tr ,'' chlorobenzene and o-dichlorobenzene, passed through an 1.55.11 .. .. 3.05 alumina column and then distilled. li5 1.5 5.00 .. .. 4.13 Apparatus.-Densities were determined with a reproTripolyphosphateb 25 0.5 6.10 5.81 2.13 .. ducibility of 0.02% using a Lipkin pycnometer, the volume .. .. 3.36 1 . 5 4.52 of the stems having been calibrated with mercury a t 25", 65 0.5 5.97 5.76 2.12 .. and the volume of the bulb with water a t 25". Solutions 1.5 4.60 .. .. 3 . 4 0 were equilibrated in an oil thermostat set at 25.0'. RefracTotal magnesium concentration = 9.930 X M . tive indices were obtained with a reproducibility of 0.01% Total pyrophosphate concentration = 2.955 X lo-' M . using a Bausch and Lomb Abbe-3L refractometer. The 6 Total magnesium concentration = 9.980 X M . light source waB an ordinary 40 watt incandescent lamp. Total tripolyphosphate concentration = 2.553 X ill. The refractometer was connected to a circulating 25" water thermostat. Mg f L MgL (4) Results 2Mg L MggL (5) The solution densities were plotted us. molar Mg HL MgHL (6) concentration of solute. All of the plots were r l
0
+ +
*
where L is the polyphosphate anion. At 65' and pH values over 7 a precipitate formed rapidly in solutions containing pyrophosphate and an excess of magnesium, so that no values for MgP2072-, and Mg,P207are reported a t that temperature. The results a t 25' agree very well with those previously reported. The formation constants for the magnesium complexes a t 65' are very close to those a t 25') and the AH for complex formation is 0 zt 1 kcal. Previous extensive work8 with calcium complexing by pyrophosphate and tripolyphosphate showed the AH values to be less than 4 kcal. and attributed complex formation to positive entropy changes. Obviously, in magnesium complexing by polyphosphat'es the same interpretation applies. These results suggest that a relationship may exist between the heats of dissociation of a hydrogen and a metal ion from a ligand. This is not surprising since the two processes are somewhat similar. Acknowledgment.-The author thanks Mr. William W. Illorgenthaler for making some of the measurements. POLARIZABILITIES AXD MOLAR VOLUMES
OF A NUMBER O F SALTS I N SEVERAL SOLVENTS AT 25'1 BY W. R. GILBERSON AND J. L. STEW ART^ Depurtment of Chemistry, University of South Carolina, Columbia, South Carolina Received April 7, 1961
Recent interest3+ in obtaining dipole moments of ion pairs from dielectric measurements has led
linear within the concentration range used (0.005 to 0.05 M ) . From the relation d = do
+ (ill, - & ~ ) C / l O O O
(1)
where d is the solution density, do that of pure solvent, M o the formula weight for the solute, V,' the partial molar volume of solute a t infinite dilution, and C the molar concentration of solute, the values of Vi shown in Table I were obtained using the slopes of the d versus C plots. The specific refractions, R, of the salt solutions were determined from the relation
where rllz is the refractive index of solution, q1 that of pure solvent, C1the molar concentration of solvent; and CIois the molar concentration of pure solvent. Plots of specific refraction 2's. salt concentration were linear in all cases. From the slopes of these plots, the solute polarizabilities, at, werc calculated from the relation CYZ
= 3000 X s l o p e / 4 ~ N
These results are listed in Table I. Discussion A comparison of the molar volumes of BulNPi and B a N I in the two solvents used for each is interesting. I n nitrobenzene, the picrate has an (I) This work has been supported in part by contract with the Office of Ordnance Research, U. 9.Army. (2) N. S. F. Summer Research Participant, 1960. (3) E. A. Richardson and K. H. Stern, J. Am. Chem. Soc.. 89, 1296 (1960).
(4) M. Davies and G. Williams, Trans. Faraday Soc., 66, 1610 (1960). (5) W. R. Gilkerson and K. K. Srivsstsva, J . Phys. Chem.. 65, 272 (1961).
1466
T'ol. 65
SOTES
MOLARVI)LUXES Salt
TABLE I POLARIZABILITIES AT 25"
AND
-
v,"
Solvent
cc.
BurNPi Nitrobenzene BulNPi Chiorobenzene Bu~NI o-Dichlorobenzene BQNI Water Na(CeH5),B Water PsTI Water Et4NBr Water Me4NC1 Water
407 402 302 316 277 25 1 175 107
4, cc.
10%'
1024 a + ,cc.
56.7 55.6 39.6 38.6 44.7 30.5 21.2 11.9
31.2 0.2 23.1 16.4 8.4
ion pair dissociation constant6 equal to 0.14 on the At M , the salt would be almost completely dissociated into free ions. I n chlorobenzene17 the dissociation constant is of the order so that the salt would be almost completely associated in this solvent. As can be seen in Table I, there is an expansion of 5 cc. going from the ion pair in chlorobenzene to the free ions in nitrobenzene. Similarly, in water, Bu4NI should be completely dissociated, while in o-dichlorobenzene, it is almost completely associated. The expansion in this case is 14 cc. Using tabulated values of the refractivities of the halides,g a table of ion refractions or polarizabilities can be set up, as seen in Table I, column five. The observed polarizability for Bu4NPi compares favorably with that estimated from bond cc. The small difrefractions (ref. 5)) 51 X ference in calculated and observed values will not require any modification of the charge-charge distance calculated from dipole m o r n e n t ~ . ~
c scale.
(6) E. Hirsch and R. 31. Fuoss, J . A m . Chem. Soc., 83, 1018 (1960). (7) P. H Flaherly a n d K. H. Stern, zbzd.. 80, 1034 (1958). (8) C. P Smyth, "Dielectric Behalior and Structure," JlcGrawHill Book c'o., New York, A'. Y., 1955, p. 407.
ductance data, one needs to evaluate ho,the limiting equivalent conductance for the salt. This has only been possible for salts having dissociation constants in the range of lo4 or greater. For most salts, this means that one is restricted to solvents of dielectric 8 or more. The temperature coefficient of K in such solvents is not very large.2+4-6 This results in some uncertainty in fitting eq. 1 to the data. For most liquids having dielectric constants in the range 5 4 , the ET product is almost independent of temperature. Further, it has been shown recently (ref. 3) that ho can be obtained for salts such as tetra-n-butylammonium picrate in 50 mole yo o-dichlorobenzene-benzene as solvent (E = 6.041 a t 25'). We thought that a study of the temperature dependency of K in solvents in this range of dielectric mould yield the best values of E, one could hope for, since t,he dielectric dependent factor in eq. 1 would remain practically constant. Consequently, we report here the electrical conductance of tetra-n-butylammonium picrate (Bu4NPi) in 50 mole % benzene-0-dichlorobenzene (DCB) a t 44.26 and 64.74' and in bromobenzene a t 25,35,44.58,55 and 65". Experimental Benzene and o-dichlorobenzene were prepared as before.s Bromobenzene was fractionated on a three-foot packed column, b.p. 156'. Before use, t,he solvent was passed through a 35 X 2 cm. column packed with rllcoa activated alumina, grade F-20. Analysis by vapor phase chromatography showed less than 0.1% impurity present. Based on retention times, this was apparent,ly chlorobenzene. The solvent conductivity was 1.8 X 10-10 mho/cm. a t 25'. The salt was prepared as previously.8 The density, viscosity and dielectric constants were obtained in t'he same manner as previously and the results appear in Table I. The conductance measurements were carried out with equipment already described.3 All solutions were made up by weight.
TABLE I PHYSICAL PROPERTIES OF SOLVENTS
THE CONDUCTANCE OF TETRA-n-BVTYLSMMONIUM PICRATE IN 50 MOLE % ' BENZENE-0-DICHLOROBENZENE AND BROXOBENZENE S S A FUNCTION OF TEMPERATURE BY W. R. GILKERSON AND R. E. STAMM Department of Chemzstry, Unwerstty of South Carolzna, Columbza, South Corolzna Receaaed A p ~ z l7 , 1961
It hac1 been hopedlJ that a study of the temperature dependence of ion pair dissociation constants would reveal information both about ionsolvent interaction in terms of E, and the distance of closest approach, a. These two factors enter into the dissociation constant K for the equilibrium AB
+ B-
.4+
as shown in eq. l . 3 K
=
lOOlld (V,V -51 ~
/v-'v-'j
exp(-E./RT) eup(-eS/e a k T ) (1)
In order to obtain a good value for K from con(1) W.R Gilkernon. J . Chem P h y s , 25, 1199 (1956). (2) H. L Curry and W. R. Gilkerson, J A m . Chem. Soc , 79, 4021 (1957). (3) W.R. Gilkerson and R E. Stamm, zbad , 82, 5295 (1960).
Solvent
50 Mole % bensene-DCB 50 Mole % benzene-DCB 50 Mole % beneene-DCB Bromobenzene Bromobenzene Bromobenzene Bromobenzene Bromobenzene
TEmp., C.
Density, g./ml.
Viscosity, Dielectric centipoise constant
25
1.1110
0.839
6.04
44.26
1,0913
,665
5.68
64.74 25 35 44.58 55 65
1.074 1.4885 1.4750 1.4619 1.4477 1.4341
,559 1.068 0.940 ,834 ,740 ,668
5.34 5.37 5.24 5.12 4.99 4.87
Results The equivalent conductances as a function of concentration C, moles/l., are given in Table 11. In order to evaluate A,, in these solvents, nieasurements must be made in the concentration range M . At 25 and 3.5") such measurements are possible, with reasonable precision. At 45 and 65" , however, the conductance changes with time so that the results become unreliable. This may be due to solvent or salt decomposition. For the (4) K. H. Stern and A . E. Martell, ibid., 77, 1983 (1955). (5) J. T. Denison and J. B. Ramaey, ibid.. 77, 2615 (1955). (6) P: H. Flsherty and K. H: Stern. ibid.,80, 1034 (1958).
NOTES
August>,1961 TABLE I1 CONDUCTANCE OF BulKPi 50% Benzene-DCB -64.74'--
44.260-
7 -
I
104~
0.6420 0.8218 1.099 2.231 3.114
19.54 9.949 5.046 1.014 0.5089
lO*C
19.86 9.808 4.882 0.9980 0,4952
A
0.8245 1.050 1.386 2.766 3.751
Bromobenzene 7-350lodc
2.50
c
A
104C
9.940 4.891 2.483 0,9987 0,5001
r-25'---
10%
A
8.420 7.073 6.061 5.187 4.349 3.025 2.014 1.321 0.5534
1.520 1.647 1.765 1.897 2.057 2.417 2.881 3.449 4.902
0.1847 .2415 .3230 .4899 .6737
A
1.982 2.187 2.384 2.677 3.074 3.200 4.511 6.754
7.580 6.145 5.103 3.972 2.945 2.278 1.248 0.4634
-44.58"-
-65"-
104~
A
9.762 4.807 2.439 0.9810 .4913
0.2704 .3549 .4756 .7201 ,9927
104~
A
9.576 4.715 2.392 0.9621 .4818
0.3823 .5041 .6760 1.026 1.414
50 mole % benzene-DCB solvent, we can only observe values of Ao2R, the reciprocal slope of a Shedlovsky since measurements were confined to 44 and 65". In bromobenzene, low concentration data were obtained a t 25 and 35" and values of Ao, as well as K , appear in Table I11 for these two temperatures. TABLE I11 DERIVEDCONSTANTS FOR BurNPi Solvent
50 mole % benzeneDCB Bromobenzene
Temp., "C.
253 44.26 64.74 25 35 44.58 65
10'do*K
Ao
10'K
30
2.74
1467
35", there is little justification in discussing the significance of this value. It is probably best concluded that a study of the temperature coefficient of an ion pair dissociation constant will not allow much to be said concerning the values of the molecular parameters that determine the coefficient. T H E SYSTEMS T I N T S L U M PEKTACHLORIDE-FERRIC CHLORIDE AND NIOBIUM PENTACHLORIDE-FERRIC CHLORIDE BY CHARLESM. COOK,JR.,AND ROBERTB. HAND E. I. du Ponl de Yemours & Company. Pzgments Department, Walmzngton,DeEawarc Receaued Mag 5,1961
During a study of the properties of ferric chloride the solid-liquid equilibria of the system T a c k I:ezCl6and NbCI6--Fe2Cl6were reinvestigated. Experimental Ferric chloride (Fisher Scientific Company-purifiedsublimed) was purified by slow resublimation in a stream of oxygen-free chlorine. TaC15 was prepared by reaction of T a metal with oxygen-free chlorine. NbClj was prepared hv chlorination of NbzOj followed by passing the product with Cle over carbon a t 600" to remove SbOC13. Chlorides were stored and manipulated under inert atmospheres. Mixtures of FeC13 and TaC15 or NbClb, sealed, under ca. 0.1 atm. Clz, in a cylindrical Pyrex vessel of 25 mm. I) X 80 mm. L, were melted within an insulated cavity enclosed bv an externally heated steel jacket. During cooling a constant temperature difference was maintained between jacket and sample, the jacket power being regulated by a differential thermocouple input to a variable reluctance furnace controller. Sample temperatures, measured by a Pt-lO% Rh thermocouple in a Silicone oil-filled thermowell extending into the sample, were plotted as cooling curves on a Leeds & i'iorthrup recorder. St intervals this recorder was standardized by observing the indicated signal when a known voltage, from a Leeds & Northrup t y e K bridge, was impressed upon it. Agitation was provic&by a Burrell wristaction shaker to which sample and jacket mere attached.
Results The TaC15-FezC16 system, presented in Fig. 1,
4.85 8.20
___....
13 18
.
1.53 1.01
___.-.
0.443
____I.-..__... b . b
0.902
Discussion In bromobenzene, the A0 values a t the lower temperatures are increasing faster with t.emperature than one would expect if the Walden product, A o ~ a were , a constant. Further, the value a t 25" 60 80 20 is much lower than is observed in high dielectric 0 MOLE % FeZCls. solvent^.^ This result is another indication that system TaC15-FeZCls: 0, this work; 0 , Aoqo decreases with decreasing solvent dielectric Fig. 1.-The Morozov. constant.8 The uncert,ainty in A0 is of the order of *1 equivalent conductance unit. We are thus left with an uncertainty of 14% in the value of K shows a single eutectic at 14.5 mole % Fe2C16and a t 25". This is the most unfavorable case. Since 203" in good agreement with Morozov,1 who lo' and 200". there is only a 40% change in K between 25 and cated the eutectic a t 13.9 mole % Freezing points of 215 and 307" were observed for 40
(7)
T.Shedlovsky, J . Franklin Inst., 226, 739 (1938).
(9) R . AI. Fuoan, P r o ? . S a t i . Amd S e i ) , 46, 807 (1958);
(1)
I. S.Morozov. Zhur. A7eovu. Khim., 1, 2792 (la.%)
100
1468
COMMUNICATIONS TO THE EDITOR
TaC152 and for FeC13, respectively. Schafer and Bayer3 record the melting point of FeC4 as 307.5" and point out that the apparent melting point of FeC&when measured in a CLdeficient atmosphere becomes depressed by contamination with FeC12. Their dat,a indicate that 7 mole yo FeCh in FeCh lowers the apparent melting point to 303", the FeC13m.p. reported by Moroxov. FeCL if present should depress the observed ferric chloride liquidus temperatures, and Morozov's values consistently lie below the liquidus calculated using d In X F ~ ~ C I ~ / dT-l = -AHr/R with AHr = 20.6 kcal./mole and represented by the dotted curves in the figures. 1 0 The NbC15-Fe2C16 system, presented in Fig. 2, shows a eutectic a t 9.5 mole yo FezCls and 191". Fig. 2.-The The NbC15has :m.p. 205".* During cooling of com(2) H. Schiifer and C. Pietruck. Z. anorg. Chem., 261, 174 (1951), report Tach m.p. 216.5'; NbCIs m.p. 201.7'. J. B. Ainscough, R. Holt and F. Trowse, J . Chem. Soc., 1034 (1957), report TaCls n1.p. 216.9'. NbCls n1.p. 203.4'. ( 3 ) 13. Schzifer and L. Bayer, Z . a n o ~ g .Chen., 271, 338 (1963).
20
Vol. 65
40
MOLE
60
%
80
I
100
FezCI,j.
system NbCla-FepCla: Morozov.
0, this work;
0,
positions containing > 0.1 a weak evolution of heat, which occurred reproducibly and which was independent of rate of cooling, was observed at 193" before crystallization of the eutectic a t 191".
COMMUNICATIONS TO THE EDITOR DETECTION OF STRUCTURAL DIFFERENCES IN POLYMERS IN A DENSITY GBADIENT ESTABLISHED BY ULTIRACENTRIFUGATION Sir: W7ehave developed a method of separating polymers based on differences in partial specific volume. We have applied the method to separating highly branched material from a previously described copolymer' and Lo the separation of atactic polystyrene from stereoregular polystyrene. It is seen easily that in the vicinity of a branch point of a polymer molecule, the "density" of the molecule will be slightly greater than in the linear portion of the molecule. Likewise, if the polymer consists of stereo regular sequences the volume that the molecule occupies in solution mill vary with the amount of :stereoregularity. We have adapted the density gradient method first introduced by Meselson, Stahl and Vinograd2 to synthetic polymers in the analytical ultracentrifuge. Essentially, a system of two solvents is used to set up a density gradient; one solvent is much more dense and has a higher molecular weight than the other solvent. The concentration of the more dense solvent and the speed of ultracentrifugation are chosen so 1,hat in the vicinity of the middle of the cell the qliaiitity (1 - 6 p ) equals zero. Here d is the partial specific volume of the polymer, and p is the density of the solution. If differences in partial specific volume of the various components of the bulk polymer do exist, then each component will collect in its own region of (1 - V p ) = 0. If (1) L. H. I'eebles. Jr.. J . A m . Chem. Soc., 80, 5603 (1958). (2) hl. Meselson, IF. 1%'. Stahl and J. Vinograd, Proceed. Nut. Acad. Scz., 43, 581 (1957).
care is taken to use very small density gradients within the cell, quite small differences in 8 can be measured. A 1 E.A. solution of sample 6,' dissolved in dimethylflormamide containing 135.6 g./l. of bromoform was spun in the ultracentrifuge for 150 hours at 33,450 rpm. During this time the material clearly separated into three discrete bands, each band being located at a different position in the region of the center of the cell. I n another experiment, 85 hours at 19,180 rpm., the branched material clearly separated into two distinct bands with a partial specific volume difference of 6.1 X lo-* ml./g. This difference is too small to be detected by standard means. Under these conditions, the linear polymer did not sediment. In order to see whether atactic polystyrene could be separated from stereoregular polystyrene, a mixture of 380 g./L of bromoform in benzene containing 0.32 g./l. of atactic polystyrene of 5 X lo6 molecular weight and 0.32 g./l. of stereoregular polystyrene of 20 X lo5 molecular weight was spun for 56 hours at 33,460 rpm. The latter material is largely stereoregular since only 2% of it is soluble in hot methyl ethyl ketone. The different molecular weights were deliberately chosen so that after separation into the component parts, the fractions could be identified since the breadth of the band depends in part upon the molecular weight. The molecular weights of these polymers are sufficiently high so that the molecular weight dependence of the partial specific volume is negligible. Again, the two materials separated into distinct bands. Comparison of these results with samples run under identical conditions except that the individual polymers were used instead of a mixture showed that the individual polymers collected a t the identical density value. The atactic polymer, however, apparently contained a small
