CaC12 salting-out agent, the per cent acetone phase recovery was almost independent of the salt concentration in the region of maximum separation. The solubility of acetone in sucrose was very dependent on the temperature. Hence, it is important to control the temperature, pH, and concentration of sucrose for reproducible acetone recoveries.
Studies are currently under way in which solvent extraction of oxine chelates with acetone are successfully being performed using sucrose as the salting-out agent. Received for review October 12, 1972. Accepted March 16, 1973.
Error Analysis of Density Determination by a Unique Falling Body Method A.
s.Roy
Department of Chemical Engineering, University of the Negev, Beer-Sheva, lsrael
Sources of errors are discussed of the unique method of density measurement of a solid sample by comparing its fall time to that of a reference sample in two different liquids. Equations are derived for computing the quantitative effect of the various experimental parameters on the final density result, indicating that the accuracy of the method can be increased by using a reference body of density, size, and shape close to those of the tested body, and liquids of wide density difference, and high, approximately equal viscosities. The self-correcting features of the method make it insensitive to some lack of control on experimental conditions. Comparison with experimental results is presented.
where y is the ratio of the net gravitational forces acting on the body j in the two liquids,
A novel method has previously been reported for measuring density of an individual small, solid sample of high density and arbitrary shape ( I ) . In the study of isotopes and solid state imperfections and when the sample available for investigation is smaller than a millimeter in size, the method becomes invaluable for densities above 4 or 5. For obtaining accuracies of 0.1 to 0.3% a simple procedure with a limited attention to the choice of materials has been found adequate ( I ) . The present work deals with the error analysis of the method aimed to indicate the steps required for attaining higher accuracies. It will be shown that prudent choice of materials, specifically a standard body of density, size, and shape close to those of the tested body, and liquids of wide density difference and high, approximately equal viscosities, will permit the measurement of densities with additional one or more significant figures. The method consists of separate fall time measurements of two bodies between arbitrary marks on two cylinders. The two cylinders are each filled with a different Newtonian viscous liquid; one of density u and the other of 0' (gram/cm3). Two of the fall times, ti and t,' (sec), are taken for the tested body i in the two liquids, respectively, and similarly, t , and t,', for the reference b o d y j having a known density p , (gram/cm3) acting as a standard. The density p r of the tested body is obtained from the equation specifically derived for this method ( I ) :
where u, g, F , and D are, respectively, the fall velocity (cm/sec) of the body, the acceleration due to gravity (980 cm/sec2 at sea level), viscosity (poise), and the diameter or characteristic dimension of the body (cm), while k is a shape-orientation factor which is a numerical constant (unity for a sphere) typical of the geometry and particular fall orientation of the body. Applying Equation 4 twice to the same body i in the two different liquids of viscosities and p' (and densities c and u'), respectively, and then dividing one of these equations by the other, results in
PI
=
?@af - -a
rp - 1
(1) A. S. Roy,Ana/. Chern., 33, 1426 (1961)
(1)
and p is the gross fall time ratio,
(3) This results from 4 times use of Stokes' law, extended for bodies of arbitrary shape and constant fall orientation (2-4): (4)
(5) Similarly, with the reference body j P pf
-
P, p,
-
-a
t,
- -a/ t,'
Dividing Equation 5 by Equation 6 results in P,--a - fJJ
Pi
P,--a @ - fJJ
- P,
from which Equation 1is obtainable by rearrangement. The method requires thermostating of the liquid-filled cylinders in order to keep the liquid viscosities and densities constant and uniform during the fall runs. Care is also needed to prevent convection, air bubbles, or con(2) G. Barr, "A Monograph of Viscometry," Oxford University Press, 1931, Chap. VIII. (3) J. F. Heiss and J. Coul, Chern. Eng. Progr., 48. 132 (1952). (4) H. Lamb, "Hydrodynamics," 6th ed., Dover Publication, New York. N.Y., 1945. pp 597-605.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
1921
tamination in the liquid. These and other precautions are also, and much more so, required by the flotation and falling sphere (or drop), or any of the liquid substitution methods, where accuracies of 1 part in 100,OOO have been reported (5, 6). However, the present method, while being considerably less sensitive to these disturbances as will be shown below, is effective for measuring densities of small individual bodies which are too dense to yield to the flotation method (no transparent liquids of densities above 4 or 5 are available), and too nonspherical in shape to yield to the falling sphere method. This effectiveness stems from the specific nature of the present method, namely, using two, rather than one, liquids, and computing the sought density from fall time data of both the test body and a reference body. The density of p L of the test body is thus obtained indirectly, according to Equations 5-7 resulting in Equations 1-3. It is therefore of primary significance to investigate whether any sacrifice of accuracy might be incurred in consequence of the indirect feature of the method, to understand what governs accuracy and decide upon improvements necessary for controlling growth of errors. For this end, it is necessary to express analytically the total error differential dp, (computed based on Equations 1-3) as a function of the major experimental error differentials dp,, da, du’, and the dt-s. This is obtainable by differentiating Equation 1. The resulting equation will express propagation of errors, namely, the error dpL appearing in the computed pL arising from the differential errors (dp, etc.). From such an equation, conclusions can be drawn on the sensitivity of the computed density towards various sources of errors and what could be done in order to increase accuracy, namely, decrease dpL.
THE TOTAL DIFFERENTIAL OF EQUATION 1 The differentiation operation of Equation 1 is quite lengthy. The final result is PI - c 7 PI dp, = -
P/
-
PJ -
-
c 7
fJ’
PI
0‘
6’ - u
~
dg, - -( P I -
0’)f
+
where t is the relative change in the gross time ratio 0,
This result of the total differentiation is very interesting. It shows that the inevitable experimental errors (dp,, do, etc.) yield quite unequal contributions toward the total error pi as determined by the respective coefficient functions that multiply each experimental error. Consequently (from Equation 8), the sensitivity of the computed pi toward the experimental errors depends greatly on the magnitudes of the density difference values p j - U, p j - u’, u’ - U , etc., and their functional relationships in the coefficients. This recognition gives us a powerful control on the magnitude of the total error dpi by allowing us to make a reasonable choice of materials having such densities that will minimize the coefficient functions. Prior to considering such a minimization, it is of significance to identify the origin of the experimental errors appearing in Equation 8. The error in the density of the standard body, dpJ, originates either from a discrepancy ( 5 ) N. Bauer, in “Technique of Organic Chemistry.” A Weissberger, Ed., Vol. 1, part 1, Interscience, New York, N.Y., 1959, pp 171-3. (6) F. H. Horn, Phys. Rev., 9 7 , 1521 (1955).
1922
between assigned density value p j (taken from the literature for the material of the standard body j ) and the actual density of the particular standard body j used in the experiment, or from an error in measuring p j (e.g., by a pycnometric method, using a large quantity of material j ) . The errors d a and da’, are errors in liquid density arising, again, from either a discrepancy between literature values and the actual magnitude of liquid densities, inaccuracies in measurements, or uncontrolled changes in the temperature of the liquid. The overall relative time error t is composed of several components (additive for small e-s), t
=
€Re
+ + €&
tt
(10)
The error t~~ results from lack of extreme laminarity; t W , from wall effect, and t t , from any inaccuracy of the timing technique. Other sources of errors may include changes in orientation of the falling body during the period of its fall between the measuring lines, any vibration of the cylinder due to any external interference, irreproducibility of the vertical position of the cylinder, changes in time of the liquid homogeneity, and other disturbances resulting from lack of care. Adequate experimental attention is, of course, necessary in this method as in any other, in order to eliminate such lack of control and also to reject data in which such interferences are observed to occur. It is now significant to assess the specific characteristics of the coefficient functions appearing in Equation 8, and how they control the magnitude of dp,. The difference pL - u appearing in the coefficient dp, expresses a measure of the net gravitational force acting on the test body i in the liquid of density u, and similarly, pL - u ’ , in the other liquid. By the same token, p, - u and p, - U’ for the standard body j . The coefficient thus expresses the product of ratios of net gravitational forces. For the particular case when either p, - a or pL - U’ equals zero, practically all the coefficients become zero. This represents the flotation method: the body floats in a liquid of a like density. This particular case cannot be realized for a density exceeding 4 or 5 , and the accuracy obtainable is identical to the accuracy of the density of the floating liquid. In this particular case, there is no significance to the reference body except for calibrating the liquid density. If p, - u or pc - U’ will not be zero but rather small values, this will decrease the values of the coefficient functions and thus will contribute to the decrease of dp,-namely, increase accuracy. This effect, too, cannot be utilized for densities above 5 . An opposite effect will occur if any of the differences appearing as denominators in Equation 8 become smallnamely, p, - u, p, - u’, or u’ - u. This will increase the coefficients and, consequently, increase the effect of the experimental errors, and thus decrease accuracy. In the extreme case, when any of the denominators become zero, the uncertainty dp, in p , will increase indefinitely, and pL cannot be evaluated a t all. The reason behind this is, that in this case the theoretical basis of the method is lost: no fall a t all of one of the bodies, or use of liquids which are not different with respect to the buoyant force they effect on a falling body.
MAJOR REQUIREMENTS ON THE CHOICE OF MATERIALS The foregoing inspection of the coefficient functions leads to the conclusion that besides efforts necessary to decrease the experimental errors, no less significance should be placed on the choice of materials in order to
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
minimize the coefficients. Be the experimental procedure precise as it may, the resultant error, dpl, could still turn high if any of the difference terms appearing in the denominator of Equation 8 is small. On the other hand, proper choice of materials to minimize the coefficients will be rewarding. This can easily be done. There are many available liquids (at a very wide viscosity range) of densities around 1 and around 2 so that a value of about unity for U' - u can easily be realizable. For solid samples of densities above 3 (and up to and above 20), no denominator will be small. Another very important conclusion can be drawn from inspection of Equation 8. The selection of a reference body j to have a density, p j , close to that of the test body, p i , such that pi - p j be a small value, would minimize the coefficients of the experimental errors do and d d . As will be shown below, a small pi - p j constitutes also a major factor in decreasing t . As dp, has the smallest value [lO-5 for several materials (7)] of the various experimental errors, the choice of reference body of density close to that of the test sample should play a very significant role in increasing the accuracy of the method. For high-density samples, the coefficient of e becomes large. As the method is particularly important for high densities, an effort is justified to assess t and its components (Equation 10) and investigate how it can be minimized. ANALYSIS OF T H E RELATIVE TIME E R R O R t The Error € R e Due to Deviation from Laminarity. Equation 4 is valid only for extreme laminar flow, charac. terized by a very small Reynolds number, Re = D u ~ / pAs the state of extreme laminarity can only be approached as a limit, a correction is needed to be included in Equation 4 for actual fall runs. The correction causes the velocity, u, to be reduced ( t becomes larger) by factor of 1 (3/ 16)Re (8-121, which, for irregular shapes, incurs (by Equation 9) an error of
be quite accurately estimated with both magnitude and sign (by Equation ll),it can be used as a correction factor in order to eliminate altogether the error from lack of laminarity. For this end, the expression of (Equation 8) Api = € R e ( P l - u ) ( p i - u')/(u' - u) should be added as a correction to the pi obtained from Equation 1, and in this manner no error results due to deviation from complete laminarity . The Error c, Due to Wall Effect. Stokes' law is stated for an infinite fluid medium. In a cylinder of a finite diameter W and height H (cm), the right side of Equation 4 must be multiplied by a factor (2, 9-15) K,, smaller than unity, which, for laminar conditions of a sufficiently small Reynolds number, depends only on geometrical factors (2, 3, 9-15),
Ku =
1
(1
+ 2 4 D / W ) ( l + 1.7 D / H )
(12)
The magnitude of this factor is not negligible. For D = 0.1 cm, W = 7.5 cm, and H = 18 cm, the value of K, is 0.958. However, in the present method, it cancels out altogether in the ratios of Equations 5 and 6. This cancellation is exact as long as the body falls strictly along the axis of the cylinder (and the fall time is measured for the middle third portion of its height) for which Equation 12 is derived. For a case of small drifts of the fall paths from the axis of the cylinder (and the cylinder diameter is very large compared to the falling bodies), the error for an irregular shape would be
(13)
+
(11) For liquids of high viscosity, say of 10 poises, D of the order of 0.1 cm and u, of 0.1 cm/sec (Equation 4), the order of each ukDu term in Equation 11 is 10-4. (The value of k for compact bodies is close to unity.) However, because of the reciprocating signs of these terms, the value of € R e may come out to be considerably smaller. The diminishing of the value in the bracket can particularly be enhanced by using a standard body of density, size, and shape factor close to those of the test body, namely Di N Dj, pi E p j , ki k j , hence also (by Equation 4) vi N u j and vi' N u j ' . Even approximations of around 10% may reduce t R e by one or more orders of magnitude. Also, for increasing accuracy, liquids of high viscosity should be useful. This increases the values of the p's and decreases the values of u's appearing in Equation 11 and thus decreases t~~ considerably. An interesting point to note about € R e is that, as € R e can (7) A . Smakula, J. Kalnajs, and V. Sils, Phys. Rev., 99, 1747 (1955). (8) C. W. Oseen, Ark. Mat. Astron. Fys., 6, No. 29 (1910). (9) H. Lamb, "Hydrodynamics," 6th ed., Dover Publication, New York, N.Y., 1945, pp608-17. (10) C. E. Lapple, "Fluid 8 Particle Mechanics," University of Delaware, Newark, Del., 1951, pp 208-94. ( 1 1 ) H. Rouse, "Fluid Mechanics for Hydraulic Engineers," McGraw-Hill, New York, N.Y., 1938, pp 212-18. (12) C. N. Davies, in "Symp. Particle Size Analysis," Suppl. Trans. Inst. Chem. Eng., 25,25-39 (1947).
where C is a factor smaller than unity, probably of the order of 0.1, and the Ws, the shortest distance of the fall path from the cylinder wall. In terms of small deviations A W of the fall trajectories from the axis of the cylinder, Equation 13 can be given the form of
,.
tu
L
= 2.4*[k1DJ(AW,'
- AW,) - k,D,(AW,'
- AW,)l
(14) For W = 7.5 cm and D's of 0.1 cm, the value of t, will be less than 10-3 for A W deviations of 1 cm and less than for AWs of 0.1 cm. For equal AWs, c, would vanish, of course. Like € R e so also t, can be estimated if the AWs are recorded and then e, used for correction of p $ . Or else, this type of error can be avoided by just rejecting data of runs which show any visible drift from the axis of the cylinder. It is useful, therefore, to have vertical lines marked on the cylinder by which any deviation of the fall path from the axis can be detected and measured. The Timing Error t t . The timing error t t is the relative time error originating from inaccuracies in time readings. When a constant time error, dt, is involved in all fall measurements because of a systematic time deviation originating from the timing technique, then a systematic timing error t t is involved. In this case, after Equation 9,
It is desirable therefore to use one of the available precise (13) R. Ladenburg. Ann. Phys., 23, 447 (1907) (14) H. Faxen. Dissertation, Upsala (1921). (15) H. Faxen, Ann. Phys., 68, 89 (1922).
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1 1 , SEPTEMBER 1973
1923
timing techniques, e.g. photographic (16-18), by which d t can be made very small. For t values of 100 and d t of 0.01 (sec), the d t l t values will be of the order of 10-4 and et of the same order or smaller due to the reciprocating signs in Equation 15. This tendency of mutual cancellation will be enhanced if the t values are similar in magnitude. This, again, can be made to occur if the density, size, and shape factors of j are close to those of i, because in this case all fall velocities become approximately equal. A more quantitative treatment is as follows. From Equation 4 applied once to body i and once to body j and divided by each other, one obtains
(16)
In order to draw Equation 1 out of Equations 5 and 6, it is sufficient that p/p’ of Equation 5, related to the fall runs of body i, be equal to p/p’ of Equation 6, for body j. If one liquid in one cylinder be at one fixed temperature, and the other liquid in the other cylinder, a t another fiied temperature, so that p / p ’ is unchanged in time, Equation 8 is still true. It is only a matter of convenience, in general, to keep both cylinders in one thermostatic bath. It can also be shown that if the temperature inside the cylinder is not uniform, but is constant in time during the fall runs, Equations 5-7 still hold. Considering a vertical non-uniformity of temperature, let the upper part of length 11 (cm) of the cylinder have a temperature slightly (e.g. by 1 ‘C) higher than the lower part, 12. One may rewrite Equation 4 for a body i for the upper part of the cylinder:
(17)
and similarly, for the lower part of the cylinder:
where
!Jtl
-1 t,’
1 t,
1 p-a’ (”-Q ti’ p, - d
-
7= -
1)
(18)
K, (19)
Substituting ti’ on the right side of Equation 18 by Equation 19, gives
1 jl t,’ - t, 1.1‘ 1
p, p,
--d
-
p, u (-Qpi
-
- u’
t, = t,1
1)
(20)
-
It is apparent that when Q 1 (bodies of equal size and shape) and ( p i - p j l
