PETER J. DUNLOP
994
Vol. 61
through the data. Two different apparent activation energies can be calculated from the slopes of these lines: 41.5 kcal./mole for the lower temperatures and 18.4 kcal./mole for the higher temperatures. A comparison of the X-ray patterns of samples 50% decomposed a t each temperature reveals some differences which may partially explain the sharp change in rate constant and apparent activation energy. The 85" pattern shows no lines of crystalline Na4P207,while the patterns of 95, 100 and 105" do show Na4Pz07lines, their intensities increasing TABLE I1 with temperature. The three higher temperaRATE CONSTANTS FOR THE THERMAL DECOMPOSITION OF tures, 110, 115 and 120°, produced materials SODIUM TRIPHOSPHATE HEXAHYDRATE whose X-ray diffract,ion patterns contain the difTemp., k, aec.-1 Temp., k, seo. - 1 OC . x 105 OC. x 10' fraction lines of Na3HP2O7.H20. The change in 85 1.01 110 11.6 the reaction rate a t approximately 110" may pos95 4.11 115 14.5 sibly be due to a change in the rate of diffusion of 100 0.30 120 21.3 the released water through the crystal. However, 105 21.3 the fact that the lines of crystalline N ~ ~ H P ~ O T H ~ O Examination of Table I1 reveals that the rate are found in the X-ray patterns of the products obconstants did not increase continuously with in- tained a t 110' and above, and that they are missing creasing temperature, and that a definite break oc- in the patterns of the products obtained a t the curred between 105 and 110". A plot of the loga- lower temperatures, may indicate that the crystalrithm of the reaction rate versus 1/T, Fig. 3, shows lization of Na3HP207.H20may be an important that two separate straight lines can be drawn factor. Figure 2 is a plot of the logarithm of the concentration of the hexahydrate versus time; this relation should produce a straight line for firstorder reactions. Reasonable straight lines are obtained with the data considering possible errors in the X-ray analysis, and the general difficulty of conducting precise rate studies for solid state reactions. The rate constants for the reaction at each temperature, calculated from the slopes of the lines, are presented in Table 11.
A STUDY OF INTERACTING FLOWS IX DIFFUSION OF THE SYSTEM RAFFINOSE-KCl-H,O A T 85 O BY PETER J. DUNLOP C'ontyibulionf r o m the Department of Chemistry, University of Wisconsin, dIadison, Wisconsin Received April 17, 1967
Onsager and Lainin have proposed equations, which are more general than Fick's first law, to describe the phenomenon of interacting flows in liquid systems of more than two components. This paper re orts experiments which establjsh the existence of the phenomenon of interacting flows in the three-component system raknose-KCl-HpO. The four diffusion coefficients which appear in certain experimental flow equations, based on the theoretical ones of Onsager, have been evaluated for this system for two sets of mean solute concentrations; the concentration dependence of two of these coefficients is digcussed. Because the system rafKnose-KCl-H*O is somewhat similar to a system in which a protein diffuses a! its,isoelectrlc point in a buffer, a set of initial conditions, other than those normally used, is suggested for studying the diffusion of proteins by the method of free diffusion. For three-component s stems consisting of a strong electrolyte and a non-electrolyte in an un-ionized solvent, a method is proposed for obtaining t i e correct diffusion coefficient to use in the classical Svedberg equation for computing molecular weights. However because proteins and related molecules are usually weak electrolytes, extreme caution should be exercised in applying the same method to obtain diffusion coefficients of molecules of large molecular weight.
Data, obtained in this Laboratory, have been reported for diffusion in three-component systems consisting of two electrolytes in water.' I n these systems it was found that the solute flows exhibited interaction ; hence flow equations,lV2 which are more general than Fick's first law,3were proposed and series solutions for the resulting solute concentration distributions in three-component systems were derived for the case of free diffusion. A later paper4 gave exact solutions to the differential equations for the solute concentration distributions. Several methods for evaluating the four (1) P. J. Dunlop and L. J. Gosting,
J. A m . Cham. Soc., 77, 5238
(1955). (2) R. L. Baldwin, P. J. Dun1011 and L. J. Gosting, ibid., 77, 5235
(1055). (3) A. Bick, Pogg. A n n . , 94, 59 (1855). (4) H. Fujita and L. J. Gosting, J. Am. Chem. Soc., 78, 1090 (1933.
diffusion coefficients for three-component systems also were given in the above article^.'^^*^ A t the time of the electrolyte experiments it was believed that three-component systems consisting of an electrolyte and a non-electrolyte in water, or two non-electrolytes in water, would not exhibit measurable flow interaction. By reporting diffusion data for the system raffinose-KC1-HzO this paper shows that measurable interaction of flows can be observed in the first case. More recent exp e r i m e n t ~have ~ shown also that a system of two non-electrolytes in water also requires use of the generalized flow equations. . The system studied in this paper bears some similarity to a system in which a protcin diffuses a t its iso-electric point in a buffer. It is hoped that ( 5 ) P. J. Dunloi~,unl)ublislied data.
July, 1957
INTERACTING FLOWS IN DIFFUSION OF THE RAFFINOSE-KCL-WATER SYSTEM
the measurements presented here will be helpful in formulating a suitable method to obtain diffusion coefficients of proteins for use in the classical Svedberg equation.6 Most workers, when measuring diffusion coefficients of proteins by the method of free diffusion, have dialyzed before diffusion to equalize the chemical potentials of the dialyzable components of the two solutions on either side of the initially sharp boundary. However in studies of free diffusion one normally measures concentrations or refractive indices, not chemical potentials, and furthermore it is difficult to be quite certain that the dialysis is complete. Hence it seems reasonable to suggest that a different set of initial conditions be employed for measuring the diffusion coefficients of proteins by the method of free diffusion, especially as dialysis can, a t best, only minimize the effects of the phenomenon of interacting flows; it cannot make the cross-term diffusion coefficients (see section on flow equations) equal to zero. This worker therefore suggests that, when using the method of free diffusion, a more fundamental method of measuring the diffusion coefficients of proteins and related molecules may be to utilize experiments in which the buffer concentration is accurately known on either side of the initial boundary. The results may then be treated in a similar or analogous fashion to that given in the following sections of this paper. However because proteins are weak electrolytes, this approach will require further theoretical and experimental developments, but it is felt that departure from the traditional approach warrants study. Flow Equations For many years it was believed that Fick's first law was adequate to describe the solute flows for diffusion in liquid systems. However both Lamm7 and Onsagerg have proposed more general flow equations which allow for the fact that a flow of one component might cause a flow of the other components in a given system. Evidence confirming this phenomenon recently has been reported1 and another set of generalized flow equations, based on those of Onsager, has been givena2 For a system of three components these equations are
995
Experimental The Gouy diffusiometer used to measure the reduced height-area ratios, PA,and the differe?tial refractive increments, Ri,has been adequately descrlbed p r e v i o u ~ l y , ~ ~ ~ ~ as have also the necessary experimental conditions and methods used to obtain the BA,the Ri and the graphs of the fringe deviations, hl, versus the reduced fringe numbers f(0. The above experimental quantities, used later to evaluate the four diffusion coefficients, have been defined in a previous publication.' It is suggested that the reader have that article available for reference. A single quartz Tiselius cell, 9 cm. in height, was used i,n all experiments. The cell dimension, a , along the optlo axis was 2.5103 cm. and the optical lever arm, b, of the Gouy diffusiometer was 306.86 cm. All diffusion measurements were made with the 5460.7 1.mercur line isolated from a G.E. A-H4 lamp with a Wratten 77A Hter. Materials.-The potassium chloride11 had been recrystallized from conductance water, drained centrifugally and dried in vacuo a t 80" after which the crystals were fused in a latinum crucible and broken up with an agate mortar. affinose12 was heated in vacuo at 80" to remove the water of crystallization and, because of its slightly hygroscopic nature, it was always dried to constant weight before use. Solutions.-All solutions were prepared by weight using doubly-distilled water, saturated with air, as solvent. The weight fraction of each solute, corrected to vacuum, was converted to g./lOO ml. of solution, ci, by means of.densities measured in 3 0 4 . Pyrex pycnometers. Also using a method previously densities, d, were calculated using 0.997075 g./ml. for the density of water a t 25" and the following apparent specific volumes, 4, in ml./g.IaJ4 In most cases measured and calculated density values did not +KCI = 0.3587 O.OO9ec1/$ 0 . 0 0 0 ~ ~ (3) Graff. = 0.6091 (4) show the good agreement previously obtained for other three-component systems .I911
fk
+
+
Results Table I summarizes the experimental data for the diffusion experiments performed a t the mean solute concentrations Qae. = 0.75 and EKci = 3.73, while Table I1 gives similar data for the mean SolUte concentrations Gaff. = 0.75 and EKCl = 0.75. All concentrations are expressed as grams of solute per 100 ml. of solution. Lines 2-9 give the solute concentrations of the two solutions A (upper) and B (lower) used ill the diffusion experiments together with measured and calculated values of the solution densities. The mean solute concentrations, G, and the concentration differences, Aci, are given in lines 10-13. Concentration increments are considered positive if the concentration of a given solute is greater in the lower solution than in the corresponding upper solution. The JI = -Dii ( b c i / d z ) t - DIZ( b C Z / ~~ ) t (1) total number of Gouy fringes, J , obtained in each Jz = - Dai ( b C I / ~ Z )-~ Dzz (a c z / b ~ ) t (2) experiment is given in line 14. Using these values where JI and Jz are the solute flows in grams em. -2 of J and the solute concentration differences, set.-', (bcl/bx)t and (bc2lbx;)t are the first deriva- values of the differential refractive increments, Ri, tives of the solute concentrations expressed on the were calculated by the method of O ' D o ~ i n e l l ~ ~ volume scale, Dll and Dzzare the main diffusion co- and are given a t the bottom of each table. With efiients and D12 and Dzl are the cross-term diffu- the aid of the R i and the concentration differsion coefiients. The x coordinate is defined as being fixed relative to the cell and increasing in the ences, values of J were computed and are given (9) L. J. Gosting, E. M . Hanson, 0.Kegeles and M. S. Morris, downward direction. The above equations are Sci. Instruments, 20, 209 (1949). used in this article to describe the solute flows in Reu. (IO) P. J. Dunlop and L. J. Gosting, J . A m . Chem. Soc., 7 6 , 5073 the system raffinose-KC1-H20 and the four dif- (1953). (11) D . F. Akeley and L. J. Gosting, ibid., 76, 5685 (1953). fusion coefficients have been evaluated for two sets (12) Obtained from Pfanstiehl Chemical Co., Waukegan, Illinois. of mean solute concentrations. ( 6 ) T. Svedberg and K. 0. Pedersen, "The Ultracentrifuge," Oxford University Press, 1940, p. 5. (7) 0.Lamm, Arkiu K e m i , Mineral.. God., 18B,No. 3 (1944); Tms JOURNAL, 51, 1063 (1947). ( 8 ) L. Onsager. Ann. N. Y. Acad. Sci., 46, 241 (1945).
(13) B. B. Owen and S. R. Brinkley, Ann. iV. Y. Acad. X c i . , 61, 753 (1949). (14) P. J. Dunlog, T H IJOURNAL. ~ 60, 1464 (1956). (15) See I. J. O'Donnell and L. J. Gosting to be prosentod in :L
sylnposium at the 1957 meeting of the Electrochemical Society in Washington, D.C .
PETER J. DUNLOP
996
Vol. 61
for comparison in line 15. The agreement is seen to be not unsatisfactory. Then with the aid of the Ri and the concentration differences, values of ai, solute fractions on the basis of refractive index, were computed and are listed in lines 16 and 17. Values of the reduced height-area ratios and the maximum values of the fringe deviation graphs, Q-, obtained from each experiment are given in lines 18 and 19.
ate the four diffusion coefficients. Here their method three, which involves use of the DA and the is modified for use reduced second moments, aZm, with the system raffinose-KC1-HzO. This modification is necessary because the Dzm which are normally obtained from the reduced height-area ratios and the deviation graphs' cannot, for the system in question, be evaluated with sufficient accuracy. The errors arise from the fact that the slopes [dQ/df(f)] in equations 47 and 48 of TABLE I" reference 1are not accurately measurable in the reDArA FROM EXPERIMENTS IN WHICH RAFFINOSE AND KC1 gion f({) = 1. Such inaccuracies greatly affectthe DIFFUSED SIMULTANEOUSLY I N WATERAT 25" two integrals in equation 47 (see ref. 1) Hence the Gaff. = 0.75 CKCI = 3.73 following method for evaluating the four diffusion 1 Exp. no. 1 3 5 7 coefficients was employed. 2 (Cmff)Bb 1 ,50040 0.74728 1.5000s 0.7527s Experiments were designed in such a way that 3 (CKCI~B 3.72740 4.4734r 3.7072; 4.47364 most of the deviation graphs were in the vicinity of 1.02610 1.02771 1.02597 1.02777 4 (dexp.)B zero. This can be done by choosing appropriate 5 (doslo.)B 1.02618 1.02778 1.02606 1.02780 0.00000 0.74684 0.00000 0.74730 6 (Craff.)Ab values of Acl and Acz, the concentration increments 7 (CKCI)A 3.72771 2.9819t 3.74808 2.9824: of raffinose and KC1 across the initial boundary. 8 (dexp.)A ... 1.01858 ... 1.OB63 Due to the fact that these two solutes have diffu9 !&alo.)A 1.02029 1.01864 1.02041 1.01865 sion coefficients which differ greatly, Qmax changes 0.75020 0.7470s 0.75004 0.75008 10 craff. 3.72758 3.72761 3.7276s 3.72804 11 dKCIC very rapidly with a1in the vicinity of a1 = 0 and 1.50040 0.00044 1.50008 0.00530 12 Acraff. also al = 1. In these two regions the slopes are 13 ACKCI -0,0003; 1.49121 1.49151 -0.04078 nearly constant; in the following we assume a 14 J(expt.1 100.48 88.69 98.07 89.01 linear relationship between Q,,, and a1 in these two 15 J(oa1cd.) 100.49 88.69 98.06 89.00 16 araff. 1.0002 0.0003 1.0247 0.0040 regions. Then by performing two experiments in 0.9997 -0.0247 0.9960 17 ~ K C I -0,0002 each region one may extrapolate to find the two 18 D A x lo6 0.43971 1.8344 0.42857 1.8191 values of a1for which Qm,, is zero. Hence we have 19 nmx x 104 26.0 -9.5 -4.0 -2.5 two sets of initial conditions for which Gaussian Rrarf. = 1.4572 X 10-8;RKCI = 1.2931-X 10-1 Units in Tables I and 11: concentrations, c,. g./lOO boundaries would have been obtained experimenml.; densities, d , g./m!.; reduced height-area, ratios., DA, tally. cm.2/sec. The subscripts B and A denote, respectively, At the given mean solute concentrations, 2l and the bottom and top solutions used in the diffusion experiments. The mean concentration CKCI = 3.727 in these CZ, 1 / 6 is~linear in a1 (see equation 56A). Usfour experiments corresponds to 0.5 mole/l. ing the data for % and a1in Tables I and I1 the method of least squares was employed to obtain TABLE I1 values of I A , SA and LA in equations 56A-59A. DATAFROM EXPERIMENTS IN WHICHRAFFINOSE A N D KC1 The values of these quantities are given in Table DIFFUSED SIMULTANEOUSLY I N WATERAT 25" I11 for the two sets of mean solute concentrations Cratr = 0.75 CKCI = 0.75 studied. In both cases the mean deviations from the least squared lines was less than 0.1%. Then 1 Exp. no. 2 4 6 8 2 (Cmff.)Ba 1.4978s 0.7495; 1.50011 0.7513s by plotting 3,, versus alone obtains two values of 3 (CKCI)B 0.74540 1 ,26762 0.7409r 1.26777 a1 for which the boundaries would be Gaussiane 4 (deap.)B 1.00763 1.00802 1.00764 1.00802 thus I?+ (see equations 50A and 51A) is zero in on; 5 (doal.)B 1.00768 1,00802 1.00766 1.00803 case and l?- is zero in the other. These plots are 6 (Craff.)A" 0.00OOo 0.75001 0.0000~ 0.74878 7 8
9 10 11 12 13
14 15 16 17 18
19
(CKCI)A
0.22374 0.75027 0.22389 1.00144 1.00145 1,00180 1.00145 1.00183 1.00144 crerf. 0.74892 0.74980 0,75000 0.7500s CKCIb 0.74551 0.74588 0.74561 0.74581 Acraff. 1.4978s -0.0004s 1.50011 0.00260 ACKCI - 0,00016 1.0438s -0.00932 1.0438a J(exp.) 100.85 04.73 100.47 64.95 J(ca1c.) 100.87 64.74 100.45 64.94 ararf. 1.0001 -0.0005 1.0057 0.0027 aKCl -0.0001 1.0005 -0.0057 0.9973 PAX 10' 0.4327s 1.8354 0.4307s 1.8240 nmax X 104 6.0 -7.0 0.5 2.8 Rlarf. = 1.4650 X 10-8 RKCI= 1.3497 X 10-8 (dexp.)A
0.74564
...
...
!daalo.)A
a The subscripts B and A denote, respectively, the bottom and top solutions used in the diffusion experiments. b T h e mean concentration CKCI = 0.745 in these four experiments corresponds to 0.1 mole/l.
Evaluation of the Four Diffusion Coefficients.-In the following sections the author, for brevity, will refer to certain equations in the recent paper of Fujita and Gosting4; such equation numbers will be followed by the letter A. These workers gave several methods which can he used to eralu-
TABLE I11 DATAFOR USE I N COMPUTING THE DIFFUSION COEFFICIENTS AT THE Two SETSOF MEANSOLUTE CONCENTRATIONS OF THE SYSTEM RAFFINOSE-KCl-HzO 5" = 25') c.g.5. units Erati.= 0.75" EKCl = 3.73
IA
CrSrf. =
233.42 243.21 LA 476.68 (w)r+- 0 0.0054 1.0214 ( d r - -0 u+ x 10-5 2.3216 u- x 10-6 0.55098 IDijl x 1010 0.78177 Izm x 106 1.822' Szm x 10' - 1.3624 Lzm x 106 0.4599 a Concentrations are expressed as grhms of ml. of solution. SA
0.75
= 0.75
&C1
233.49 247. Os 480.54 0.OOls 1.0062
2.3239 0.54728 0.78634 1.8299 1.3910 0.438s solute per 100
-
July, 1957
INTERACTING FLOWS IN DIFFUSION OF
THE
997
RAFFINOSE-KCL-WATER SYSTEM
\
t
Fig. 1.-Method used to obtain initial conditions which correspond to Gaussian boundaries when the flows interact. Here
the maximum values, Qrnsx., of the fringe deviation graphs are plotted against the refractive index fraction of raffinose, 011. When Q ~ = ~0 the . refractive index gradient curxe is Gaussian if there is no skewing due to concentration dependence of the diffusion coeEcients.
given in Fig. 1. Using these values of a1 two values of DA, which correspond t o l?+ = 0 and I?- = 0, respectively, were calculated from the least squared lines of 1 / 6 versus ~ al;equation 55A then permits values of a+ and u- to be obtained Thus we have the determinant lD{jl (equation 17*) from equation 64A' these quantities are given in Table 111. It is now possible to solve for I2m7 Szm and Lzm, the quantities defined by equations 60A-63A. For the two Gaussian boundaries in question I?+ = 0 in the one case and r- = 0 in the other. Hence in order to obtain Izm, S z m and L z m the two equations to be solved, corresponding to I?+ = 0 and r- = 0, are
-E
[e+
- (Rz/EI)GI [u+
+
(a111
- H - ( R I / R ~ ) P(az11 ] =0
- E - (R~/R~)GI + [u- - H - (RI/R2)Fl (QZ)II
iU-
(5)
(Lyl)II
0 (6)
are given by equations wherein E , F, and 20A-23A and the subscripts 1 and 11 refer to the values of a1and a2for which Gaussian boundaries would be obtained. After some manipulations and remembering equations 62A and 63A we obtain from equations 5 and 6 =
I* 4- ( R 1 / R e ) F 1 l D i i l
= (a1)11u+
al)II(PI)I
and
- (alhu-
- (al)I(az)II
] IDiil
(7)
PETER J. DUNLOP
998
Vol. 61
EXP. 5
LXC. 8
. .
t
-10;
'
1
p4 = I.OOS7
I
'
I
,
I
,
,
0.5
I
30 r
I
0.5
0
7
i -. '.5;
$
8
'
/
/'
\
\'
A -
EXP. 9
' \
\
\-
al = 1.0001
10 t
\
I
EXP. 7
,
,
,
0.5
0
1
I
EXP.
a
a,=O.OOLT
1
0
1
0.5
fa),
b
I
1
1
I
Fig. 2.-Fringe deviation graphs for the system raffinose-KC1-HtO. The four graphs on the left are for the mean solute concentrations erraft.= 0.75 and h C 1 = 3.73; those on the right are for &fr. = 0.75 and BCI= 0.75. At a given value of f ( l ) crosses indicate the average of the experimental points (dots) obtained from ten different Gouy photographs of the same boundary. The dashed curve indicates the values of predicted by equations 10 and 12. L~rn= [ E
+ (Ra/Ei)G]jDijl
=
- (Ly2)IU- ] l ~ q(8) - [( orl)II(ol2)1 - (orI)I(Ly2)II (LyI)IIU+
TABLE IV DIFFUSIONCOEFFICIENTS FOR Two SETSOF SOLUTECONCENTRATIONS T = 25O, c.g.8. units R s f f . 0.75," tr*fr. = 0.75, CKCl = 3.73: ' EKCI = 0.75 Dll x 106 0.4309 0.4303 D12 x 106 0.0066 0.0022 D2l x 106 0.0327 0.0094 Dzz X 10' 1.815 1.828 Concentrations are expressed as grams of solute per 100 ml. of solution. Subscripts 1 and 2 denote raffinose and KCl, respectively. VALUESOF
THE
and Szmis given by equation 61A. The values calculated by means of these equations are given in Table 111. Equations 70A and 71A and the necessary quantities in Table I11 were then used to solve for values of Dll and DZ2and equations 62A and 63A employed, with the aid of the differential increments in Tables I and 11, to obtain D12 and Dzl. Values of the two sets of diffusion coefficients are given in Table 117. It should be hoted that the four diffusion coeffi- trations of both solutes are expressed on the same dents will have identical units only if the concen- concentration scale. However, if the units of the
July, 1957
INTERACTING FLOWS IN DIFFUSION OF
two solute concentrations differ, the units of the main diffusion coefficients will remain unchanged but the units and numerical values of the crosst>ermdiffusion coefficients will change. Deviation Graphs.-Since (a-/a+) is not close t o unity but approximately 0.25 for this system, the series expression previously given for the fringe deviations (equation 72A) is not applicable to the experiments reported in this paper and so expressions analogous to equations 13 and 23 of reference 11 must be used. These equations now hecornel6
o
r1 =
=
r2
o
=
U+/L
r+F(b, r1) - r+2 G(b, r l ) + . . . . . . =
L/U+
r- F(l, ra) - r-2 G(b, r2) + . . . . . .
(9) (10)
(11) (12)
THE
RAFFINOSE-KCL-WATER SYSTEM
999
tion of the thermodynamic quantities needed to verify the above-mentioned relations. It is hoped that, in the near future, experimentalists will provide such data since the testing of the Onsager reciprocal relations is one of the main problems in the field of diffusion today.
Discussion I n order to see how well the computed diffusion coefficients in Table IV reproduce the measured values of DA, equation 55A was employed to cal~ ~ ~ ~in. Table V. Equations culate the ( B A ) values 50A and 51A and the values of al and a2 in Tables I and I1 were used to calculate the r+ and r-, while the a+ and u- were obtained by use of equations 30A and 31A. The agreement between ~ is seen to be calculated and experimental d ) values quite good.
Equations 10 and 12 converge rapidly when r+ is small in the first case and when I?- is small in the TABLE V other. COMPARISON OF EXPERIMENTAL AND CALCULATED VALUES Using equations 50A and 51A values of F+ and OF THE REDUCEDHEIGHTAREARATIOS I'- were calculated (see Table V) for all experiT = 25", c.g.s. units ments and deviation graphs calculated by means of Craff. = 0.75"; CKCI = 3.73 equations 10 and 12 and the tabulated values of Expt. no. 1 3 5 7 F({, rz) and G({, T ~ given ) by Akeley and Gosting.l' ai 1,0002 0,0003 1.0247 0.0040 The calculated and experimental values are given in a? -0.0002 0.9997 -0.0247 0.9960 Fig: 2. Agreement between calculated and exrt 0.979, -0.0050 1.0038 -0.001r perimental values of fi is seen to be excellent except I1 0.020~ i.oo50 -0.003~ 1.001~ ( 9 A ) c a l o d . x io5 0.44008 1.8341 0.42928 1.8196 in experiments 1 and 3 ; some of this difference is ( D A ) w ~X ~105 , 0.43977 1.8434 0.42961 1.819s due undoubtedly to the fact that Qma, in the reCrrff. = 0.75; CKCI = 0.75 gions f@) = 0 and f([) = 1 is not perfectly linear in al. The experimental values of D are believed Exp. no. 2 4 6 8 to be accurate to f 2.0 X a1 1.0001 -0.0005 1.0057 0.0027 That the flows interact in this system may be aa -0,0001 1.0005 -0.0057 0.9973 r+ 0,993s -0.0022 0.999s 0.000~ seen by inspection of some of the deviation graphs r0,0061 1.0022 0,OOOa 0,9991 in Fig. 2. In experiments 1 and 2 the refractive 1.8360 0.43071 (DA)oalc. x lo5 0.43302 1.823, solute fractions of KC1, a2,were essentially zero and (DA)exp. x io5 0.4327s 1.8354 0.43078 1.8241 hence the fringe deviations would have been zero a All concentration units are in grams of solute per 100 (corresponding to Gaussian boundaries) had the ml. of solution. flows been independent. Similarly inspection of the graphs for experiments 3 and 4 in which araff. In all the experiments reported in this paper the mas zero shows that in each case a flow of raffinose mean concentration of raffinose, El, was held constant while & was varied. Equation 2 indicates was caused by the flow of KC1. that as Cz tends to zero Jz tends to zero and hence Reciprocal Relations.-Previous papers*," have pointed out that the four diffusion coefficients in Dzl must go to zero with &, a t constant A, since the flow equations for three-component systems (bcl/bx)tmay remain finite. DLzon the other hand are riot all independent. Recently discussions may remain constant or vary but need not tend to have led to the formulation'* of expressions, for zero with Fz. Experimentally it was found that Dzl direct use by experimentalists, Rrhich relate the did tend to zero with cz a t constant El; Fig. 3 shows diffusion coefficients to the thermodynamic prop- that this is the case. However the nature of the erties of the system in question. For a system of slope cannot be accurately determined because of three components a single relation exists between the difficulty in measuring accurately the four difthe four diffusion coefficients and the testing of fusion coefficients. The straight line in Fig. 3 inthis relation constitutes one test of @nsager's dicates that the slope is constant within the error reciprocal r e l a t i ~ n s . * J ~ ~ 'At ~ - ~the ~ present time, of measurement. D12 is much smaller than Dzl however, thermodynamic data for three-component and it is impossible to indicate its variation with systems are rather sparse and t,he accuracy of the KC1 concentration a t constant cl. It may be shown from equations 30A, 31A, 50A, those measurements available for dilute solutions and 55A that since Dpl + 0 as Ez ---f 0 a t con51A is not sufficiently great to permit accurate evalua(10) In the event t h a t both Dm and 0 2 1 are both zero, r+ = 011' = 012, ra = DII/DZP and 71 = Dre/D11. Equations 11 and 12 then reduce to equations 13 and 23 of referencr 11. (17) G.J. Hooyman, Physica, X X I I , 751 (1950). (18) R. L. Baldwin, P. J. Dunlop, L. J. Costing. G. Kegeles and J. G. Kirkwood, in preparation. (19) L. Onsager, PhU8. Reu., 37, 405 (1931). (20) L. Onsager, ibid.. 38, 2265 (1931). (21) L. Onsager and R. M. Fuoss, THISJOURNAL, 36, 2089 (1932).
r-
ca-90
stant El. Hence in the limit the measured reduced height-area ratios, under the above conditions, extrapolate to the differential diffusion coefficient of raffinose a t cl. In Fig. 4 both the DA, for those experiments for which ~ K C I= 0, and the DII have
1000 0.04
H. A. DROLL, B. P. BLOCK AND W. C. FERNELIUS
-
Vol. 61
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therefore that, in order to use the Svedberg equation for determining molecular weights of nonelectrolytes in such three-component systems, one must first extrapolate either the correct DA values, for which a s a l t = 0, or the Dll to zero concentration 0.02 of salt and then extrapolate the values obtained in this way to zero concentration of the non-electrolyte. Since proteins are weak electrolytes and since their net charge varies with the buffer and protein concentrations, a rigorous method for obtaining the correct value of the diffusion coefficient 0 I 2 3 4 of a protein for use in the Svedberg equation is not, c, as yet, available. But it is hoped that the above Fig. 3.-Plot of the cross-term diffusion coefficient, Dill methods may be helpful in solving the problems versus h t c ~at Craft. = 0.76. which lie ahead. L It is felt that extreme caution should be exercised when using the Svedberg equation to calculate molecular weights of proteins and related molecules. The diffusion coefficients used in these calculations usually have to be obtained from measurements with a system of a t least three com0.435 ponents and the difficulties in obtaining the correct values are great. Thus it is not a t all surprising that the numerous values in the literature for the diffusion coefficients of various proteins do not 0.430 agree very well. Furthermore, recent ~ o r k in ~ ~ 9 ~ ~ this and other laboratories, indicates that a pro0 I e 3 4 tein sample, such as BPA, may well change its Fig. 4.-Plot of one of the main diffusion coefficients, Dll, physical properties while stored in the solid state. and also (D&,o versus CICCI a t Crsff. = 0.75. Acknowledgments.-It is a pleasure to thank Professors L. J. Gosting and R. W. Baldwin and been plotted against the concentration of KC1 at also Mr. I. J. O'Donnell for helpful discussions constant raffinose concentration. As expected and suggestions during the course of this work from equation 13 these values are seen to extrap- and for their criticism of this manuscript. Finanolate, within the error of measurement, to the cial support to make this investigation possible value of the differential diffusion coefficient of was supplied by the U. S. Public Health Service raffinose a t 0.75 g./lOO mi24 At EKCl = 3.73 E-1426(C-2) and by the Rockefeller Foundation. g./100 ml. the difference between the DA for QKCI Grateful aclcnowledgment is hereby recorded. = 0 and Dll is seen to be approximately 3% so that (22) T. J. O'Donnell, t o be published. use of such DA values in the Svedberg equation (23) M . Halwer, G. C. Nutting and B. A. Brice, J . Am. Chem. Sac., would lead to considerable error. It would seem 73, 2786 (1951). -
C,=O.75
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STUDIES ON COORDINATION COMPOUNDS. XV. FORMATION CONSTANTS FOR CHLORIDE AND ACETYLACETONATE COMPLEXES-OF PALLADIUM(I1)I BY HENRY A. DROLL,B. P. BLOCKAND W. CONARD FERNELIUS Contribution from the College of Chemistry and Physics, The Pennsylvania State University, University Park, Pennsylvania Received April 86, 1067
The stepwise formation constants for the chloride complexes of palladium( 11) have been determined from spectrophotometric measurements made a t 21, 29.5 and 38". The reaction between palladium(I1) chloride and acetylacetone in aqueous solution has also been investigated, but from pH measurements a t 20, 30 and 40'. The logarithm of the over-all formation constant for PdC14-P, estimated to be 15.8 at 25O, is observed to be of the same order of magnitude as that for PtCld-2 and HgC4-2. The thermodynamic stability of bis-(acetylacetonat0)-palladium(I1) is greater than would be anticipated from considerations of the electronegativity and the second ionization potential of palladium.
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Introduction first transition series of the periodic table, a few The bulk of the studies of formation constants to non-transition-series elements, and silver(1). The date has been concerned with the cations of the Primary reason for the choice of cation appears be the desire t o have a simple species, the aquated (1) li'rolll a Portion of a thesis presented by Henry A. Drollin partial ion, as the reference cation. Because mostof the fulfillmcnt of the requirements for the degree of Doctor of Philosophy, secondelements do not JanllsrJr, ISM. Preceding paper R. M. Ieatt, w. C. ~ ~and ~ ~ and third-transition-series ~ l i ~ ~ B. P. Block, THISJOURNAL, 69, 235 (1955). readily form.Tsimple aquated cations, studies of
