Dec., 1954
DIFFUSION AND MEASUREMENT OF HETEROGENEITY IN SEDIMENTATION
becomes particularly clear in the following comparison of these two spectra (Table 111),not only concerns the ring structure but also the binding forces prevailing in the ring system. Experimental The infrared data were obtained with a model 12C PerkinElmer spectrometer. The prism used was NaCl. The cell windows, too, consisted of NaCl with a mm. path length. The l.,3,5-triazine2 was purified by distillation over sodium metal, followed by slow sublimation a t 35’ bath temperature, these operations being carried out under exclusion of air moisture. For the Raman measurements the solvent (CCld or CBHB,respectively) was distilled onto the sublimed sample
1081
of triazine until a solution saturated a t room temperature was obtained. Then this solution was filtered through a frit into the Raman tube. The measurements were carried out in a customary apparatus, for 1.5, 4.5 and 8 hours in Cc1, solution and for 6 hours in C&& solution, uRing Hg 4358 as tho artivating light, source. Activation with Hg 4047 A. led to decomposition of the substance.
8.
Acknowledgment.-Two of the authors (C. G. and A. K.) are indebted to the Mathieson Chemical Corporation for their generous support of this work. Furthermore we wish to thank Mr. J. A. Curtis, Mathieson Chemical Corporation, Research Department, Niagara Falls, N. Y., for assistance in the infrared measurements.
BOUNDARY SPREADING IN SEDIMENTATION VELOCITY EXPERIMENTS. 111. EFFECTS OF DIFFUSION ON THE R.IEASURER;IENT OF HETEROGENEITY WHEN CONCENTRATION DEPENDENCE IS ABSENT BYROBEBTL. BALDWIN Contribution fronz the Department of Biochemiftry, University of Oxford, and the Department of Chemistry, University of Wzsconszn, Madison, Wisconsin Received M a y 20, 1964
The method of Baldwin 5nd Williams for finding g(s), a substance’s distribution of sedimentation coefficient, is based on extrapolation to infinite time. In this article the reliability of the extrapolahon procedure is studied with the aid of the analytic expression for g*(S) (the quantity used in extrapahtion) that results when g(s) is taken to be Gaussian. Secondly, a method is presented for obtaining the moments of the boundary gradient curve direct1 from the continuity equation, without having to solve the differential equation for the shape of the boundary, and it is stown that moments obtained in this way confirm Faxen’s solution of the differential equation. Finally, higher order terms are derived for the relation between p (the standard deviation of g(s)) and the standard deviation, u,of the boundary gradient curve.
Introduction The first article’ of this series considered how the width of a sedimenting boundary could be related to the average diffusion coefficient (D) and the heterogeneity in sedimentation coefficient ( s ) of the sedimenting substance, for the case in which s and D do not depend on concentration (c). It was found that the contributions to the boundary width from diffusion and from heterogeneity in s depend on different powers of the time, so that it is possible to obtain the distribution of sedimentation coefficient, g(s), by extrapolation to infinite time, in the same way that mobility distributions can be obtained.2 Costing3made a thorough theoretioal study of the extrapolation to infinite time and found the correct function of time t o use in order to obtain a linear extrapolation as infinite time is approached. The problem of obtaining g(s) under these canditions (ie., no dependence of s and D on c ) thus becomes one of reaching this range of time where Gosting’s limiting law holds. However, there is a basic limitation on the time for which an ultracentrifugal experiment may be continued : the experiment must stop before the boundary reaches the bottom of the cell. The definition of s (s = (dz/ d t ) / d z ) may be rearranged to show that the final
value of su2t is limited4 by cell and rotor design; consequently the length of an experiment can be increased only by lowering the speed of rotation and this decreases the resolution that can be obtained. This situation poses two important problems in finding g(s) by extrapolation to infinite time. First, how can one recognize for a given system whether or not the heterogeneity in s is sufficiently resolved that Gosting’s limiting law will hold in the range of time accessible to experiment? Second, if one is outside this range, what other method could be used to find g(s)? The second problem, although very interesting, is also very difficult and will not be considered here. I n order t o study the first problem, an analytic expression for the quantity used in extrapolation (g*(S), the “apparent distribution” of s) has been obtained for the case that g(s) is Gaussian. With this expression, the extrapolation to infinite time can be carried out with calculated values of g*(S)6and comparison of the extrapolated with the true values of g(s) rhade to show by how much the experiment departs from limiting law conditions. The success of the extrapolation is related to the resolution of the various sedimenting species obtained by the end of the experiment. The degree of resolution is oharacterized here by the ratio of the contributions to u2 (the second moment about
(1) J. W. Williams, R. L. Baldwin, W. M. Saunders and P. G. (4) It I t is the time elapsed at the end of the experiment, sw%f = Squire, .J. Am. Chevn. SOC.,74, 1642 (1962). In (zr/zo) < 0.2, where co and SI are the initial and final positions of (2) R. L. Baldwln, P. M. Laughton and R . A. Alberty, THISJOUR- the boundary. NAL, 66, 111 (1961). (5) T h s method was used’ to check on the reliability of mobility (3) 4,J, Goeting, J . A m , Chem, Hoc., 74, 1548 (1952). distributions obtained by extrapolation t o infinite time.
ROBERTL. BALDWIN
1082
Vol. 58
the mean of the concentration gradient curve) from solution of the differential equation holds are that diffusion and from heterogeneity in s. Although s and I) be constant and that the concentration at the resulting conclusions are strictly valid only for the meniscus be zero for all times later than t = 0.l6 the case of a Gaussian g(s), they are quite useful in After substituting (3) and (4)into (2), the resultassessing the extrapolations used to find g(s) for the ing equations can be replaced by a series of known systems which so far have been reported.6-1n integrals of the form K 1 pe---y2-K z dy, where Theory y = K s ( s - S ) , by expanding the necessary Representation of g*(S) when g(s) is Gaussian. functions of s. as Taylor's series about (s - 8). -The definition' of g*(S) is The final expression for g*(S), in which terms contributing less than 0.2% to g*(S) have been g*(S) = bC/co w2t(x"xo") bX dropped, is where x is distance from the center of rotation, xo e - ( s -2) z/2p*z) 1 e2b 3 esb . . (5) g*(S) = p*24+ marks the position of the meniscus, w is angular speed of rotation, t is time and ( d C / C o ) / ( b z is ) the where gradient of the total concentration at x , divided by b = (w2t)(S - g ) / ( l plw%) (5a) the initial total concentration. The symbol S has been used in g*(S) to indicate that S is a variable p*" = p" f ~D/w'xxo~ (5b) derived from x by the relation In carrying out the extrapolation to infinite time = xoeSwzl (14 with calculated values of g*(S), it is not necessary and, unlike s, is not a property of a solute species." to consider what happens to the analytic expression In order to find (dC/'Cn)/(bX) , the boundary-spread- for g*(S) in the range of time beyond that accessible ing equation3must be integrated. to experiment ( i e . , sw2t > 0.29. The factors which determine the extrapolated value of g(s), for a given s, are the values of g*(S) for sw2t < 0.2, the way in which these values change with time, where ( b c / c ~ ) / b xiss the concentration gradient pro- and the manner of carrying out the extrapolation. duced by species of sedimentation coefficient s, The actual value of Q ( S ) is given by the distribution divided by this species' initial concentration. In function-in this case the Gaussian function-used this case, g(s) is given by the Gaussian function in obtaining the analytic expression for g*(S). Thus it is allowable to drop terms from the exg(s) = e-(8-i)z/2Pz/p(2~)'/2 (3) pression (5) for g*(S) which arb negligible during an where experiment even though these terms might become s = Jm 8 g(s) ds (34 large in the range of time corresponding to sw2t >> 1. A Method for Finding Moments and its Use in Faxen's12 solution of the differential equation for the Confinnation of Faxen's Series.-It is possible the ~ltracentrifuge'~ may be wed to give (bc/co)/ to confirm Faxen's solution of the differential equation by an unusual approach which is given here because Faxen's equation is used repeatedly in this article and because the approach gives promise of being widely applicable in the problems . . . ) 2 ( 1 / 2 - (9/16)a . , . ) + 2"(3/8 of sedimentation, diffusion and electrophoresis. (45/64)a . . ) z3(5/16 . . . ) z4(35/128 + . . , ) + . , , } This approach makes use of a method for finding moments of the boundary gradient curve of ( d c l b x ) (4) vs. x,when the equation describing this curve is not where known. The case considered by Faxen is that of sedimentation and diffusion of a single solute from a completely sharp initial boundary in a sector cell z = (xoesw'l - x)/x@w*t (4b) and changing field, where s and D are constants, the This series converges rapidly in the range of time concentration at xo is zero for t > 0, and the area accessible to experiment, since a will not exceed ( K x ) and field strength (w%) are proportional to about 4 X 10-4. The conditions under which this the distance from the center of rotation. (6) L. E. Miller and F. A. Hamm, THISJOURNAL, 67, 110 (1963). In this case, the n'th moment of the boundary (7) J. R. Cann, J . A m . Chem. Soe., 76, 4213 (1953). gradient curve is defined by (8) R. L. Baldwin, Bra. J . Ezp. Path., S4, 217 (1953).
J-7 [i
(
+ .]
-
+
+ + +. +
+
+ +
(9) A. G . Ogston and E. F. Woods, Trans. Faraday SOC.,60, 635 (1954). (10) (a) J. W. Williams, W. M. Saunders and J. Circirelli. THIS JOURNAL, 68, 774 (1954); (b) J. W. Williams and W. M. Saunders, ibid., 68, 854 (1964). (11) Thus S is used in o*(S) becsuse this is a quantity which is defined (for a given time and experiment) by a position in the cell. On the other hand, 8 is used in d e ) because, for a given system, this is a property of the solute species with sedimentation coefficient 8 . (12) H. Faxen, Arkiu Mal. Astron. Fysik, I l B , No. 3 (1929). (13) 0. Lamm, ibid., 118, No. 2 (1929). (14) This equation was obtained by rearranging equation 32 of the article by Gosting,# who repeated Faxen's solution and carried additional terms of the aeries.
s,"
Po =
dx
X%
6"E
Jct X" dc =
Ct
(6)
dx
(15) If this aeries were expanded instead about the maximum of the concentration gradient curve, the magnitude of the terms in brackets would be reduced: consequently the choice of m 0 3 w a ~as an origin suggeats a greater skewness of this curve than in fact exists. The position in the boundary corresponding t6 zoeawzt is the square root of. the second moment of the concentratiofi gradient curve1)B and this position does not coincide with the center of a Gaussian curve. (15a) R. J. Goldberg, T H IJOURNAL, ~ k , 194 (1958).
u
Dec., 1954
DIFFUSIONAND
RIEASUREMENT OF
where ct, the concentration in the homogeneous solution beyond the boundary, is given byls ct = Coe-28wPt
The n'th moment is a function only of time, in a given experiment, while the concentration is a function of the two variables x and t. In differentiating (6), the equation of Leibnitz17 for the differentiation of a definite integral is used; 2 is treated as a function of c and t , since, within the boundary region, any given set of values of c and t uniquely determines x.
HETEROGENEITY I N SEDIMENTATION
1083
where the coefficients 01 = A p =B
y =
+ A'a + A"a2 + A"'a3 + . . . + B'a + B"a* + * . .
c + C'a + C"a2
+
9
.
.
(13%)
(1%) (13c)
...
are to be determined. Next the moments defined by (6) are found in terms of the coefficients of (13). For example
+ y22 + + + . . . I d2 = a + ya + 4 3 ~ 2+ ) ... 623
..
€2'
(14)
(The limits of the cell are taken to be infinitely far from the boundary; this corresponds to the as!dt $= -2ctsd (74 sumption used in deriving (4) and (11) that the and xw denotes the value of x a t which c becomes concentration a t xo is zero for t > 0 and that ct = ) ~ coe-280% .) By comparing (14) with (124 one sees equal to ct,. The partial derivative ( b ~ / b thas the physical significance of being the rate of move- that ment of a plane a t constant concentration. A = I (154 The continuity equation for a sector cell where13
A'+C=O A" C' 3E = 0
+ +
(15b) (15c)
and so forth. The significant characteristic of this method for finding moments is that equation 7 splits into a series of integrals, each of which may be evaluated separately. This is a result of the way in which The flow per unit area, J, is given byi3 (bxlbt),is derived from the flow equation and of the ac additivity of flows from separate processes such as J 7 -D - + CSW~X (10) bX sedimentation and diffusion (cf. equations 9 and for a species of zero net charge. Substituting (S), 10). I n order t o illustrate the usefulness of this (9) and (10) into (7), and rearranging with the aid method for other problems, consider the case of concentration-dependent diffusion in a rectangular of integration by parts gives cell. The approach shown in equations 7-10 *n = nswaMn + (n )(n- 2)Dp.-2 yields, for this case (11) dt may be used to transform this into a more tractable
for integral values of n. This recursion formula may be integrated in a straightforward manner for n = 0 and n = 2 and then, in successive steps, the positive even moments may be found. PO
= 1
1 pr/xo4e48wnt= 1 pe/xo6e6aw2t = 1
p2/xo%+8wzt =
+ 4a + 12a + 24aa
(1%) (12b) (12c) (12d)
where a is the quantity defined in equation 4a. In general P 2 n / x o ~ e 2 n a w 2= t
m = l m!(n
-
m)l
(n - l)! (2a)n--m -_ (In - l ) !
(12e)
The first five even moments are sufficient to coiifirm the coefficients of Faxen's solution which have been given in equation 4. First bc/bx is represented by a series of the form
ac--
dx
et
(xoeao't)( 2na)'h
e-Z'/20( a
+ @.z + yza + + + . J 623
This equation contains two important relations found by Gra14n1*: the first moment is stationary and the second moment is 2Dd, where DI is the integral diffusion coefficient. It is interesting to note that equation 16 is derived without use of Boltzmann's assumption that c is a function of the single variable xt-''~. The Second Moment About the Mean of the Entire Boundary Gradient Curve When Several Components Are Present.-An earlier derivation1 of the expression for u2 was based on several approximations, which could be checked only by numerical examples. It is possible now, with the knowledge that'5a
and hence that
€24
(13) (16) T. Svedberg and H. Rinde, J . Am. Chem. Soc.. 46, 2677 (1924). (17) C/.I. S. and E. S. Sokolnikoff, "Higher Matliematics for Enpineers and Physicists," AlcGraw-Hill Book C o . , Inc., Now l'ork, N. Y . , 1041, p, 167.
t)o derive this expression in a simple and general manner. (18) N. OralBn. Dissertation. upiJOd&, I'J4-1.
ROBERTL. BALDWIN
1084
Val. 58
native derivation for the relation2 between u2, D and the standard deviation of the mobility distribution. In estimating the magnitude of the higher terms .of' (253, it is possible to assign reasonable sample.values to p , q and T by considering the Since (bC)/(bx),the total 'concentration gradient a t properti.es of familiar distributions. For example, x is (dcilbx), these integrals may be replaced in q is zero for any symmetrical distribution and i 3p4 = r4 for the Gaussian distribution. Calculathe following way. tion for various cases shows that the higher order terms of (23) ( i e . , those not included in equation 24) will rarely be equal to 2% of u2. Discussion The Extrapolation of g*(S) to Infinite Time.Figure 1 shows the change in g*(S) with time when g(s) is Gaussian, for the case that p = 1 X 10-13 sec. and D = 5 X lo-' cm.2 set.-'. The effects of diffuwhere Co = Zcoi. The same assumptions that were sion are very considerable near the beginning of the made previously (si apd Di are constants, Ci = 0 experiment ( t = 1 X lo3sec.) and still quite impora t zo) are applied so that, according to (18) and tant at the end ( t = 5 X lo3sec.). Figure 2 shows, incomparison with the true distribution, the extra(12b) polated one obtained by placing a straight lihe through two values of g*(@ plotted against l/t, at t = 2 X lo3 and 5 X lo3sec. The extrapolated values of g(s) are low a t the center of the curve, he error in extrapolation is most serious, and high at the sides. Consequently the errw io Equations 18 and 4 may be used to give ui2 the area is small and deviations from unitye of 1 ui2 = xo2e2siw2t ai + ai2 + (21) Jmg(s)ds are likely to reflect difficulties with base( 2 Then expanding e-nslQ2tas a Taylor's series about lines. (The error in the extrapolated values, which e -n&t gives is caused by curvature of the plot of g*(S) us. l/t, xc,,je-mo*t =: Cpe-6w2t(l + (np&)2/2 could be reduced by choosing the two times to be ( n q ~ ~ t ) ~ (nru2tI4/24 /6 - 1 (22) 4 X lo3and 5 X lo3sec. instead; however, such a where p2, q3 and r4 are the second, third aiid fourth procedure would unduly magnify the effeots of the moments about the mean of the distribution of uncertaint\y in experimental data on the slope of the extrapolation.) sedimentation coefficient. After making the necessary substitutions, equaTABLE I tion 18 becomes CALCULATIONS FOR THE CASE WHDN g(s) IS CAUSSIANI The second moment about the mean is defined by
.)
+
THE
RATIOOF
THE EXTRAPOLATE€) TO
g(s) AS A
R"
(23)
where
S'
3 :
C s , c ~ ~ D CoiDi ~/~) z
p" =
(s, a
( 2 3 ~
1
- S')2~oiDI/CCO,D,
(23~)
i
It should be noted that 3, p , D,etc., all refer to the original solution and not to the homogeneous solution beyond the boundary after it has been diluted by sedimentation. The expression derived for u2 is u2
(pu22t)2(1
+ . . . } + 2Bt( 1 + Su2t + . .
,]
(24)
This same approach may be applied to the analogous problem in electrophoresis, to obtain an alter(19) See equations 1 and 2 of reference 1; the term in p 2 of equations 3 and 4 of this reference should be multiplied by esw21.
T X U E VALUEIS OF
FUNCTION OF R Q*G)/Q(i)
t/tf = 0 . 2 b
tlb
Extrap. c
0.995 I . 000 .SO8 .954 0.996 .696 .913 ,981 .601 .866 ,956 2 .523 .817 ,922 1 .398 .707 .828 a R is the ratio, at the end of the experiment, of the contributions to v 2 from heterogeneity in s aqd from diffusion. (dis the second moment about the, mean of the concentration gradient curve and R = p2u4xxotf/2D). These values are included to show by how much g*(S) changes with time during the experiment. The are calculated from g*(S)/ g(S) = (1 2D/p2u4xxot)-1x (see equation 5). For the purpose of calculating the change of x with time, in this equation,xowastakento be5,8cm.,S t o b e 4 X 1O-l3sec.,u2 tobe3.9 X lO+sec.-2andtrto be5 X lO3sec. "Thevalues of g*(S) were extrapolated to infinite time by placing a straight line through values of g*(S) plotted against l / t for the two times t / t t 0.4 and t / t t = 1. (By g (3) i s meant! 100 10 5 3
0.974
+
g*(9 = 31.1
The ratio, at the end of the experiment, of the boundary width produced by heterogeneity to that by diffusion is a convenient parameter with which to characterize the extrapolation. This ratio ( R )
Dee,, 1954
DIFFUSION AND MEASUREMENT OF HETEROGENEITY I N
SEDIMENTATION
1085
Fig. l.-An illustration of how y*(S) changes with time when g(s) is Gaussian. In this example, the standard deviasec., D = 5 X 10' cmd2sec.-l and up = 3.9 X lo7sec.-a tion of the distribution of 8, p , = 1 X
is given with sufficient accuracy for this purpose by p 2 ~ 4 $ x ~ t ~ /(cf. 2 D equation 24), where bf is the time at the end of the experiment. In Table I, the ratio of the extrapolated to the true value of g(8)max. (where the error in extrapolation is most serious) is given as a function of R. Table I should be a useful guide to the feasibility of obtaining g(6) by extrapolation in the case of any symmetrical distribution with only one maximum. In Table I1 the value of R has been calculated for the various determinations of g(s) reported in the literature6-10; p was estimated from the maximum height of the distribution by the formula for a Gaussian curve and, since t f has not usually been reported, it was assumed in all cases that swZtf = 0.15. Also given is the error in the extrapolated value of g(s)max. that would be expected were the distribution Gaussian and the extrapolation to infinite time carried out after the manner of Table
I.
I
2 3 4 5 6' 7 S (Svedberg units). Fig. 2.-Comparison of g(s) (solid line) with the extrapolated values of g(s) (dashed line) when the constants in the expression for g*(S) (equation 5 ) are those given in Fig. 1 and the range of time used for extrapolation is 2 x 10s to 5 X l o 3 see. ( R = 2.8). 0
When there is more than one maximum in g*(S),
TABLE I1 the patio of bdundary spreading from heterogeneity RESOLVINGPOWEROF THE EXTRAPOLATION TO INFIVITEto that from diffusion is not a good parameter TIME,AS ESTIMATEDFOR VARIOUSDETERMINATIONS OF with which to characterize the extrapolation to Y(5)
U*(3
extl'ap.)
hlaterial
Ra
Q(S) true ( X 10') 0.92 4.3
P
8
( X 1 0 - 9 ( X 10-13)
?,-G lobulinl 1.9 0.7 7.0 PolyvinglpyrrotidoneE (No. 11) 1.8 .91 4.1 .3 1.4 Shiga toxin* 2.2 .93 5.7 .8 4.8 Dextran'0 9.4 .99 ( 3 . 8 ) .9 3.0 r-Glohulin' 0 4 3.% .3 6.6 In calculating R ( R = p204zz&/2D),u4 was taken to be 3.9 X 107 mo+-* in all cases exeept that of the Shiga toxin, where w* = 3.2 X 1 0 7 see-2. If g(s) were Gaussian and the extrapolation to infinite time carried out after the manner of Table I, this would be the ratio of the extrapolated to the true value of g(s)max. expected from the corresponding value of R.
infinite t h e . It would be possible to have a mixture of two compaents, each homogeneous, where the value of p for the system would be large because of the difference in s of the two components ; a high value df R in this case would not mean a good extrapolation to infinite time. Consequently, other ways of checking on the extrapolation are needed. It is probably safe to infer from Table I that if, at one of the mhxima, the value of g*(S) at t/tf = 0.2 is less than 2/3 its final value ( ~ J af A ~t/tf = l ) , then the extrapolation to infinite time will not give a good representation of g(s) because too much of the boundary Ecpread has been caused by diffusion. A comparable test is whether or not the same value of g(s)max. is obtained from the extrapolation of
1086
C. T. EWING, J. A. GRANDAND R. R. MILLER
Vol. 58
g*(S) vs. l/t as from the extrapolation of [g*(S)]-* us. l/t. In the limiting law range, both pro-
theoretical correctionz1 to g(S). However, extrapolation to infinite dilution of "apparent" values of cedures give the same intercept,20whereas, outside p ( i e . , values calculated from u2 without correction this range, the error in the value of g(S)nlax. ob- for the dependence of s on c) is not feasible, because tained is less when [g*(S)]-* is extrapolated vs. such "apparent" values would change with time l/t. (Equation 5 shows that, for a Gaussian g(s), within an experiment. Work is in progress on the the plot of [g*(S)]-2 vs. 1/zt gives the correct explicit correction of uz for the dependence of s value of g(S)max., regardless of the resolving power.)' on c; thus far only the case of a single solute has This article reveals two reasons for focussing been solved rigorously.22 interest on the calculation of p from measurements Acknowledgments.-A portion of this material of u2. First, since direct calculation rather than was taken from a thesis presented in June 1953 for extrapolation is used, p can be obtained from values the degree of Doctor of Philosophy at the Uniof u2 when the degree of resolution is too low for versity of Oxford. This work was done with the g(s) itself to be obtained from the extrapolation to guidance of Dr. A. G. Ogston, to whom I am infinite time. Secondly, comparison of the values indebted for helpful advice and encouragement. of p found from u2and from g(s) would be a valuable More recently, the detailed suggestions of Dr. L. J. general check on the method. Before this can be Gosting have brought about several improvements done, the effects of the dependence of s on c must in the development of the problem, Thanks are be taken into account. This is accomplished in also due to Dr. J. W. Williams for his continued finding g(s) by extrapolating1Oat'Ob g(8) to infinite and beneficial interest in this research. Finally, dilution (where g(S) is the curve obtained by extra- the author is indebted to the du Pont Co. for finanpolation of g*(S) to infinite time, without correc- cial support during the academic year 1953-1954. tion for the dependence of s on c ) or by applying a (21) R. L. Baldwin, J. Am. Chem. Soc., 7 6 , 402 (1954). The theory (20) The difference in the extrapolated value of
~
(
8
)
when ~ ~
g*(S) is plotted against 1/1 rather than against r~/zV (the variable
suggested by Gostingn) is less than 0.5% for the cases listed in Table I: this is less than the present experiniental uncertainty of measuring S*(S).
~
is derived for the case that diffusion is negligible; in order to apply it when diffusion is not negligible, the assumption is required that g(S), the curve found by extrapolation to infinite t h e , is identical with the curve that would be found if diffusion were negligible. (22) R. L. Baldwin, Biochem. J . , to be published.
.
VISCOSITY OF THE SODIUM-POTASSIUM SYSTEM BY C. T. EWING, J. A. GRANDAND R. R. MILLER Physical and Inorganic Brunch, Chemistry Division, Naval Research Laboratory, Washington 25, D. C . Received M a y 24, 1064
Viscosity coefficients in the low temperature range from 60' (or the m.p.) to 200' for sodium, potsssium and several alloys have recently been published. A nickel viscometer of the Ostwald type has been used to substantiate these measurements and to extend results to 700". For each metal and alloy, the extended coefficients also exhibited normal temperature variation which is adequately expressed by an equation set forth by Andrade. The coefficients a t higher temperatures were intended for engineering application and no attempt was made to duplicate the precision of the results measured in glass. A composite curve of isotherms representing both sets of measurements can be drawn from which the viscosity-temperature curve for any alloy in the sodium-potassium system can be derived.
Introduction Fluid and thermal properties of liquid metals should become increasingly important as further applications for their use as heat transfer agents become apparent. These properties are also important in basic theoretical studies because of the simplicity and ideality of their atomic structures. Viscosity coefficients to 200" for sodium, potassium and their alloys were measured by the present authors' in a modified Ostwald viscometer of glass. A larger capillary type viscometer of nickel has been used to extend these measurements to 700". In overlapping temperature ranges, the two 111dependent sets of measurements show good agreement. Viscosity coefficients for the pure metals and their alloys were found to vary with temperature in a continuous family of curves. Composition isotherms, therefore, show no apparent discontinuities a t any temperature. Coefficients for the liquid metals that appear in the literature were covered by references in the (1) C. T. E'rving, J. A. Grand and R. R. Rliller, J . A m . Chen. S%., 75, 1108 (1951).
previous article. The most reliable work was apparently that of Chiong2 who used an oscillating sphere method to measure coefficients for the pure metals to 350". The Naval Research Laboratory results presented for sodium show excellent agreement with Chiong's work having a maximum disagreement of less than 2% at 350". On the contrary, the values for potassium, though coinciding with those by Chiong a t 70°, diverge at higher temperatures and differ by as much as 14% at 350". Experimental Of the applicable viscosity methods, the capillary flow type was most readily adaptable to the conditions of measurement. The chemical activity of the alkali metals with moisture, oxygen, and container materials dictated the design of a nickel, closed-type viscometer. A description of this viscometer and the factors involved in its operation as a relative measuring tool by calibration with water will be described. Apparatus and E uipment.-The viscometer consisted essentially of two cylindrical 3-liter tanks which were connected by a long ca illary. The rate of liquid flow through the capillary was ogtained from observed weight change i n (2) Y. 8. C l h n g . f'voc. Roy. SOC.(London),A167, 204 (1036).
