Some Examples of the Numerical Solution of Nonlinear Integral Equations RICHARD S. NICHOLSON Chemistry Department, Michigan State University, East lansing, Mich.
b A numerical method is described for solving nonlinear integral equations of the convolution form. The method is applicable to several electroanalytical relaxation techniques for which only approximate linearized theories have been obtained in the past. The procedure is illustrated for three cases: ( 1 ) stationary electrode polarography with uncompensated ohmic potential losses; ( 2 ) stationary electrode polarography for the case of a rapid dimerization following charge transfer; and (3)the coulostatic diffusional relaxation method not restricted to small potential excursions. Results of the calculations for each case are discussed. of electrochemical relaxation techniques are described by boundary value problems containing nonlinrar boundary conditions. These conditions arise from the dependence of current (or concentration) on potential as expressed through the Eyring (or Sernst) equation. When potential is controlled, and double layer and ohmic potential losses are not considered, these problems in general take the form of linear integral equations with variable coefficients. Typical examples include stationary electrode polarography and chronoamperometry. When potential is not controlled, and,'or double layer and iR drop effects are considered, these boundary value problems take the form of nonlinear integral or integro-differential equations (10). Many of the more important relaxation methods are included in this last category-for example, voltostatic (14) and coulostatic ( 2 , 9) experiments. I n addition, the galvanoqtatic technique is of this class when account is taken of the electrical double layer ( 1 ) . However, boundary value problems of this last category usually have been solved only approximately, because of the mathematical difficulties associated with the solution of nonlinear equations (10). The most common theoretical approach has been linearization of the Eyring (or Nernstj equation by expansion as a Taylor series in which only first-order terms are retained (10, 1 4 ) . Solutions obtained in this way naturally are valid only for small potential variaNUMBER
tions-for example, in the case of faradaic impedance theories ( I S ) . In some instances these results have been modified by treating the effect of higher order terms in the series expansion as a perturbation of the linearized solution (8). Recently Reinmuth has reviewed these and several other approaches to the solution of nonlinear integro-differential equations (10). I n some cases-e.g., the second case discussed below-these methods are not applicable. Because of the importance of electrochemical techniques described by nonlinear integral equations, it appeared important to consider solutions to these problems without using restrictive simplifications. The only general approach appeared to be numerical. Although direct numerical solution of the partial differential equations comprising a boundary value problem is inconvenient, an integral equation approach can greatly reduce the complexity of the numerical calculations. On this basis numerical solutions usually can be obtained without loss of generality, provided the integral equations are dimensionless. Thus, converting integral equations to dimensionless form reveals functional relationships which may exist between variables (10). An example of these ideas is evident in a recent theoretical discussion of stationary electrode polarography (?'), where linear integral equations with variable coefficients are encountered. The numerical solution of an integral equation involves replacing the integrals by finite sums, and then solving the resulting system of equations. I n the case of nonlinear integral equations this procedure gives transcendental equations which usually cannot be solved analytically. However, numerical solution of such equations should be possible in every case. To illustrate this approach as applied to electrochemical theory, we have attempted the solution of three boundary value problems which take the form of nonlinear integral equations. I n each case the Nernst equation has been used as one of the boundary conditions. However, the method is believed to be general and should be applicable to problems 1%ith other nonlinear boundary conditions. The three
cases actually considered are : stationary electrode polarography with ohmic potential losses; stationary electrode polarography for the case of a rapid dimerization following charge transfer; and the coulostatic diffusional relaxation method not restricted t,o small potential excursions. Since none of these cases has been solved previously, the results of the numerical calculations are discussed briefly. For each case the solution of the transcendental equations was obtained by Sewton-Raphson iteration (12 ) . This method is particularly suited t'o the problems considered here, since convergence is assured if the initial guess is sufficiently close to the actual solution. I n addition, when the iterations converge, they always converge to a root of the equation. Iterations always were continued until successively calculated solutions differed by less t'han lop7. In all of the calculations an upper limit of 100 iterations was set in the machine programs, and this limit never was exceeded. All calculations were performed on the Michigan State University Cont'rol Data 3600 digital computer. STATIONARY ELECTRODE POLAROGRAPHY WITH UNCOMPENSATED OHMIC POTENTIAL LOSSES
The availability of modern threeelectrode potentiostats would appear to make the question of the effect of iR drop on stationarv electrode polarography of little interest. However, recent experiments (6) have indicated that, even with three-electrode certain conditions the uncompcnsated iR drop can seriously distort stationary electrode polarograms. Furthermore, the effects of iR drop on stationary electrode polarography in some cases are similar to kinetic effects, and may give rise to misleading diagnostic information about the electrode mechanism. Qualitatively, the effect of iR drop on st'ationary electrode polarography is twofold. First, the polarograms are displaced along the voltage axis (cathodically for a cathodic scan) in the usual way. Second, the uncompensated iR drop causes the potential scan to be nonlinear. This results, for example, in deviations from espected dependence of VOL. 37, NO. 6, MAY 1965
a
667
ing dimensionless nonlinear equation is obtained:
I
integral
I
n
1
1
+ yBSx(at)exp[Hx(at)1
(4)
The solution of Equation 4 is related to the current by
i(t) = nFrZDo1'2(nFv/RT)1'2CC* d n ~ ( Hut) ,
(5) Parameter H is defined as
H
=
(nF/RT)nFA X
( T D ~ u ) " ~ C ~ * R(6), 120 POTENTIAL, mv.
Figure 1. Stationary electrode polarograms in the presence of uncompensated iR drop Potential scale is
(E -
.€l/*)n
-
(RT/F)HX(at)
current on the square root of scan rate (for the reversible case), and also changes the shape of the polarograms slightly. These effects have been discussed briefly by Delahay (4),who calculated qualitatively the decrease of peak current with iR drop. The only case considered here is reversible charge transfer a t a plane electrode in the absence of kinetic effects. The boundary value problem for this case is similar to one given previously (reversible case, 7 ) , except that equations defining the potential variation are modified to account for ohmic potential losses. The Nernst equation boundary condition (Equation 5A, 7 ) therefore becomes: t>o,z = 0
+ i(t)R,nF/RT] for eexp[at - 2aX + i(t)R,nF/RT
eexp[ -ut
CO/CR
=
The following definitions apply :
e
=
exp[(nF,'RT)(E, - E " ) ]
a
=
nFv/RT
(2)
(3)
Here E , is the initial potential, E" is the formal potential, v is the scan rate, A is the time a t which the direction of potential scan is reversed, i(t) is the instantaneous current, R, is the uncompensated resistance, and the remaining terms have their usual significance (IO). Dprivation of an integral equation from Fick's law and Equation 1 was described previously for the particular case R, = 0 ( 7 ) . Using the same changes of variable ( 7 ) ,the follow668
ANALYTICAL CHEMISTRY
As expected, from Equations 4 and 6 the influence of iR drop will vary linearly with bulk concentration, electrode area, and uncompensated resistance, and with the square root of scan rate. Numerical solution of Equation 4 is essentially as described previously ( 7 ) . Briefly, the range of integration is divided into 9 aubintervals of width 6 , the singular point in the kernel is removed through integration by parts, and the Riemann-Stieltjes integral is replaced by its finite sum. This results in a system of -1'nonlinear equations which are to be solved for x(n)[ = ~ ( 6 % ) = ut)]. The result of these operations on Equation 4 is :
i=n-1
+ 2 4 6 c x ( i ) [dn- i + 1 i=l
d n - i] = 0 ( 7 ) wheren = 1 , 2 , . . .llV. The only unknown in Equation 7 is ~ ( n ) Although . Equation 7 cannot be < X =
for t
eSx(at)exp[i(t)R,nF/RT] (1)
>X
solved analytically for ~ ( n )it, is in convenient form for numerical evaluation by the Newton-Raphson method. I n the first iteration an initial guess of 0.001 is made arbitrarily for ~ ( 1 ) .The initial guess for a subsequent calculation of ~ ( nis) always x(n - 1). The value of 6 used in these calculations was 0.04, and the results are estimated accurate to better than 1% of the peak value. Results of Calculations. Cyclic polarograms calculated for two values of H are shown in Figure 1. The horizontal avis is the applied voltage ( E - El& - (RT/F)Hx(at), so that the polarograms appear as they would experimentally. The curve for H =
60 (E
0
- E,,*)n,
-60 my.
Figure 2. Stationary electrode polarograms corrected for iR drop
0.05 (ipR, = 0.3,'n mv.) is identical with the reversible case discussed previously ( 7 ) . As expected, when H increases the curves become dram n out and displaced along the voltage axis. This behavior is similar qualitatively to the quasi-reversible case on varying the standard rate constant ( 5 ) . The data of Figure 2 illustrate that the effect of iR drop is more than simple displacement of the polarograms along the voltage axis. The solid curve in Figure 2 for H = 10 has been corrected for iR drop in the usual way [thus the horizontal axis has become ( E - E l 2)n]. The dashed curvc included for comparison is the reversible case ( H = 0 ) . Therefore, with iR drop the polarograms not only are lowered, but also are distorted near the peak. This broadening is caused by the fact that the effective scan rate continuously decreases before the peak, but progressively increases after the peak. The extent to which peak current is decreased by uncompensated iR drop can be seen from Table I, vchere peak currents are compared for several values of iR drop a t the peak. These data are in reasonably good agreement with the estimates previously made by Delahay
(4). Table I. Variation of Peak Current with Uncompensated Ohmic Potential Losses
ipR,n, mv. 0.3 0.6 3.2
6.4
30.1 57.0
104.1
ip/('&)Ru-a
1.0 0.998 0.993 0,984 0.933 0.879 0.805
t>o,z = 0 Do(aco/az)
I
0.5
+ DR(acR/ax)
=
0 (12)
CO/CR = exp[(nF/RT)(E, - vt - E ” ) ] = Oexp (- at)
(13)
The same definitions given for the previous case apply here. The integral equation representation of Equations 8, 10, and 11 is ( I O ) :
coz= o 120
60 0 -60 POTENTIAL, rnv.
=
co* -
-120
The solution of Equation 9 is easily obtained, and when combined with Equations 12 and 13 can be expressed as
Figure 3. Stationary electrode polarograms fcr succeeding chemical reaction
coz= o
-- Dimerization, potential scale defined by Equation 20 - - - - - - - - - First order, potential scale de-
=
[ i ( t ) , w1213~ (15)
(RT/3F)ln[(3,o,a)/(anRkC,*)].
A t this point Equations 14 and 15 are
fined by Equation 21
combined to give the following nonlinear integral equation:
defined by Equation 2 2
[i(t)/nFA]2/3[3j2DRk]1/3 X Oexp(-at)
STATIONARY ELECTRODE POLAROGRAPHY
FOR CASE OF DIMERIZATION FOLLOWING CHARGE TRANSFER
A second example of a boundary value problem described by a nonlinear integral equation is stationary electrode polarography for the case of dimerization following charge transfer. This mechanibm is a particularly important one, since for many organic electrode processes the charge transfer reactions involve free radical formation. The mechanism for this case is represented by
+ ne
PoPotentia1,s tentia1,a mv. G + ( a t ) mv. &+(at) 120 0.002 -5 0.513 10 n ,524 100 0.005 90 0.009 13.4 0.526 80 0.016 15 0,526 70 0.028 20 0.520 60 0.050 -25 0.508 50 0.085 30 0.493 40 0.140 35 0.476 30 0.220 40 0.456 25 0.268 50 0.420 20 0.318 -60 0.387 15 0.370 80 0.333 10 0.417 100 0.294 5 0.459 120 0.266 0 0 491 140 0.245 Potential scale is ( E - E”)n + 0
(3/2D~k)”~Oexp( -Ut)
- - Reversible case, potential scale
0
Table II. Current Functions d F @ ( a t ) for the Case of Rapid Dimerization Following Charge Transfer
-
=
1
Co* i(7)dT
nFAdzSo
7r7 (16)
With the change of variable z = a7
and the following definitions
i(t) = n F A 2 / x C o * @ ( ( a t )
(17)
accurate construction of theoretical polarograms are listed in Table 11. For comparison with the present case of succeeding dimerization, Figure 3 also includes theoretical polarograma for the corresponding first order case (7, Case VI, k j a = lo), and the case with no succeeding reaction (7, Case I). To illustrate all three of these cases n-ith the same figure, the potential axis is defined differently for each caqe. For dimerization the definition is Equation 20; for the first order reaction, the definition is (see 7, Equation 82) :
exp(U) = ( ~ T D o U / ~ D R ~ C , * ) ”(18) ~O Equation 16 is converted to dimensionless form :
Finally, for the case with no succeeding reaction, the potential axis is
[ @ ( ~ t ) ] ~ / ~ e x-p (at) u =
i-’ R
k
2R
+
2
Although a general mathematical treatment of this problem would involve solving a nonlinear partial differential equation, this difficulty can be avoided if the half life for the dimerization is small with respect to the time of electrolysis. -4s a first approximation [based on results for the first order case ( r ) ] , this assumption should be valid whenever
v < 3kC0*/n, mv./sec. With this restriction formulation of the boundary value prohlem follows the work of Saveant and Vianello (11):
aCo ’at
=
Do(a2Co’ax2)
DR(a2cR ax2) = kCR2
(8) (9)
+
t = o,x>o
CO
=
cO*;CR
t>O,z-
=
CR*(=o)
(10)
( E - E”)n (R Tj3F)In [ (3 xDoa)/ (2DRkCo*)] (20)
(11)
A polarogram calculated on this basis is shown in Figure 3, and data for the
m
co
+
eo*;
Given a value of u, theoretical polarograms can be calculated by solving Equation 19 for @((at). The numerical procedure is the same one used t’o solve Equation 4. Calculations were performed using 6 = 0.01, and are estimated accurate to better than 1%. Results of Calculations. By solving Equation 19 for several values of u , it was found t h a t provided In(u) was greater than 6.5, the calculated values of @((at) were independent of the quantity ln(u) - at. Experimentally this condition corresponds to the usual procedure of selecting an initial 110tential a t t’he foot of the polarographic wave. Thus, calculated values of @(at) are independent of initial potential in terms of a potential axis defined by
CR
+
0
Qualitatively the effect of the dimerization is twofold. First, there is a displacement of the polarograms along the potential axis relative to the reversible case. These shifts-which in effect have been removed from Figure 3-are defined quantitatively by Equations 20 and 21, which have been discussed previously in some detail (7, I I), The second effect evident from Figure 3 is a change in the shape of the polarograms. First, in going from the reversible case to the dimerization, the polarograms become narrower-i.e.. they occur over a smaller potential range. This effect ir defined quantitatively by coni1)aring values of ( E , - E,l2)n for the three cases: reversible, 56 mv.; first-order! 48 mv.; second-order, 39 mv. Combined nith this effect is an increase in 1)eak height, consistent with the idea that the total amount of charge transferred for the three cases should be nearly constant. Thus, for the case of dimerization the peak is about 207, higher than for the reversible case. VOL. 37, N O . 6. MAY 1965
669
Stationary electrode polarograms are most conveniently characterized experimentally by peak current and peak potential. Theoretical expressions for these two parameters can be obtained from Equations 17 and 20, and Table
z s r m , mv.
AE, *
I1 :
i,
=
0.526nFA Do1/2Co* (nFv/R T) 1 1 2
0.7-
(23)
E , = E" -
im,rnv.
0.6 -
c
I
I
I
I
I
I
1
0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
(RT/3nF)ln [ ( 4 . 7 8 ~ 3 0 ~ ) / ( 2 ]0 ~) (RT/3nF)ln (a/kCo*) (24)
0.5
Use of the equations and theory presented here to evaluate kinetic parameters will be discussed elsewhere in connection with experimental studies.
t/T0
Figure 4.
Coulostatic potential-time curves for E, =
El/
COULOSTATIC METHOD W I T H DIFFUSIONAL RELAXATION
As a final example of the numerical solution of nonlinear integral equations we consider the coulostatic relaxation technique ( 3 , 9). To simplify the treatment the theoretical model chosen assumes electrochemical equilibrium is maintained throughout the experiment. Thus, diffusion is the only relaxation process involved. Although this case is admittedly uninteresting from the kinetic point of view, it' nevertheless serves to illustrate t'he method used. In addition it provides a quantitative estimate of t,he effect of linearization of the integral equation. Mathematical formulation of the problem follows Reinmuth's treatment (9). The boundary value problem to be solved is the following:
acolat= Do(azCo/axz)
(25)
aCR/at = DR(a2cR/aX2)
(26)
across the electrical double layer; and 6 ( B , t ) is the Dirac delta function. For details on the suitability of Equation 31 as a boundary condition, the reader is referred to Reinmuth's derivation (9). The integral equation representations of Equations 25 and 26 are ( 7 ,IO) :
The result is
coz=o=
Although Equation 40 cannot be solved analytically, a solution can be obtained if it is reduced to a linear form. The usual procedure involves expanding the exponential? as a Taylor series in which only first order terms are retained:
Co* -
s'
aFAdRDO C&=O = CR*
+
0
l/t
- r
(32)
exIJ [ $ ( t ) 1 By combining Equations 30, 31, 32, and 33, the following integro-differential equation is derived
s'
%$6(6,7)dT ~
0 d
C
T
sot
=
t>O,z+
= cR*
(27)
dt-7
ro1/2
TR'"
+ DR(aCR/aX) = 0
CO/CR = exp[(nF/RT) ( E
=
(28)
t >O,r = 0
Do(dC0jaz)
(29)
=
RTcl/nZFzLl\/aTRCR* (36)
$ ( t ) = (nF/RT)AE(t)
(37)
$, = (nF/RT)AE,
(38)
ANALYTICAL CHEMISTRY
dTO/TR
'T(d70+ d
erfc[fidr(dK
$(.WE(41)
2
2
+ 4;)1
1
(42)
Equation 42 is identical with Reinmuth's result (9) when the different definitions used for the various relaxation constants are considered. Theoretical potential-time curves for large values of $ also can be obtained by linearization of Equation 40. Thus, when the following approximations are satisfied exp($) - 1 = exp($)
+
(31)
The following parameters have not been defined previously: cl is the double layer capac,ity (assumed independent of potential); A E ( t ) is the amount the electrode potential has departed from its initial equilibrium value a t time t; AE%is the maximum potential developed
+
S,
L/r0
__
R T ~ , / ~ ~ F ~ A ~ * D(35) , C ~ * $ ( t ) = $,expit
(Co*,'CR*)exp[(nF/RT)AE(t) 1 (30)
e
1
The solution of Equation 41 can be shown to be :
Equation 34 can be reduced to a nonlinear integral equation by replacing t with X and integrating from zero to t:
- E")] =
Do(aC0jas) = ( c i / n F d ) d A E ( t ) / d t(cl/nFA4)AEo
C O
=
(rR/d1/2exp(4) For the purpose of subsequent numerical calculations it is useful to put Equation 39 in dimenhionless form. This is accomplished by the change of variable
5
1
= (TR/TO)1/2eXI)($)
Equation 40 becomes
$,2dt0 (43)
X/To
The solution of Equation 43 is easily shown to be [c(t
7O)l
1
#+%
-
=
[22/tiro]/i$%?T2/7R/Tg
(44)
This last case corresponds to the coulostatic method when the potential is a t a value iii the limiting current region foi the electrode piocess This case and it-. anal> tical applications have been di.cu\+ed b? Delahay (6) Vaiiation- of $ betneen these t n o limitiiig ca.ei can only be obtained by direct solution of Equation 40 We have u.ed the yam? numerical procedure d e w ibed for the t n o previous cases. .\I1 calculations n ere performed T$ ith 6 = 0 01 and iatios ’$z are estimated acruiate to + 0 001 Results of Calculations. From Equation 40 value. of as a function of t 7” can be calculated for any given and r n T O It nil1 be recalled that 4% is the maximum value can attain, and that fiom the definitions of T~ and T R :
+
+
+%
+
Here E , i b the initial equilibrium potential and El is the polarographic half-nave potential Thus, the parameter ( T R T ~ )in ~ Equation 40 simply defines the initial potential for the experiment in terms of the conventional polarographic M ave Typical relaxation curves calculated from Equation 40 are shonn in Figure 4 for several values of 4%and E , = El 2 . For convenience these data were nor-
malized by plotting $,‘+%. As expected, the results are for small values of identical to those calculated with Equation 42. Thus, for $%= 1 0 . 0 4 ( A E = +ll.O, n mv. at 25” C.) the values of $/ICt agree with Equation 42 to 10.001. However, even for values of $, as large as 1.0 ( A E = i 2 6 / n mv. at 25” C.), the values of $/$&in Equation 40 are only about 0.01 larger than those from Equation 42 at a given TO. Thus, in this case half-relaxation times calculated from Equation 42 are valid a t the 2% error level for potential excursions as large as 25’n mv. Actually this apparent good agreement is partly fortuitous, since the nonlinearized results (say for AE = +25 mv.) depend on both the sign of A E and on initial potential-Le., on the ratio of 7 0 and sR-whereas the linearized results depend only on the sum of relaxation constants ( T = ro TR). For example, for initial potentials anodic of El,z, the potential for large anodic excursions decays slower than Equation 41 predicts, whereas for cathodic excursions the decay is faster. The difference amounts to 15 to 2097, under some conditions. Kevertheless for many applications (especially oscillographic measurements) Equation 41 should be useful for somewhat larger potential excursions than assumed previously (9). For values of $L greater than about 5.0 the system is driven well into the limiting current region, and then the potential decay is linear with PI2, in good agreement with results predicted by Equation 44. The result that Equation 41 is useful for potential excursions greater than 2 to $$
+
3 mv. is especially important if the same situation is found for the case involving simultaneous diffusional and charge transfer relaxation. From an experimental point of view the use of potential variations larger than 1 or 2 mv. often is desirable. Thus, the possibility of using fairly large potential variations still described adequately by the relatively simple closed-form solutions, would make the coulostatic method even more attractive. LITERATURE CITED
(1) Berzins, T., Delahay, P., J . A m . Chem. Soc. 77, 6448 (1955). (2) Delahay, P., ANAL.CHEU.34, 1267
(1962). (3) Delahav. P.. J . Phvs. Chem. 66. 2204 (1962). (4) Delahay, P., “New Instrumental Methods in Electrochemistry,” p. 132, Interscience, Sew York, 1954. ( 5 ) Matsitda, H., Ayabe, Y., 2. Elektrochem. 59, 494 (1955). (6) Sicholson, R. S., unpublished work. (7) Nicholson, R. S., Shain, I., ASAL. CHEM.36. 706 11964). (8) Oldham; K. B., J : Electrochem. Soc. 107, 766 (1960). (9) Reinmuth, W. H., ANAL.CHEM.34, 1272 (1962). (10) Zbid., p. 1446. 111) Saveant. J. M.. Yianello. E.. C o m ~ t . Rend. 256,’2597 (1963). ‘ (12) Scarborough, J. B., “Numerical Mathematical Analysis,” p. 192, Johns Hopkins Press, Baltimore, 1950. (13) Smith, D. E., ANAL. CHEM. 35, 602 (1963). (14) Vielstich, W.$Delahay, P., J . A m . Chem. Soc. 79, 1874 (1957). ~
RECEIVED for review December 24, 1964. Accepted February 25, 1965. Presented in part at the Division of Analytical Chemistry, 149th Sleeting ACS, Detroit, hlich., April 1965.
Direct Potentiometric Titration of Polyethylene Glycols and Their Derivatives with Sodium Tetraphenylboron ROBERT J. LEVINS and ROBERT
M. IKEDA
Philip Morris Research Center, Richmond, Va.
b A simple, direct potentiometric titration has been developed for polyethylene glycols (PEG’s) and their derivatives, using sodium tetraphenylboron (NaTPB) as titrant in the presence of barium ions. A combined titrimetric and gravimetric procedure demonstrates that PEG’s 600 to 4000 react stoichiometrically to form complex precipitates containing 2 moles of TPB and 10.4 f 0.2 moles of ethylene oxide for each mole of barium. The approximate molecular weight of an unknown PEG may b e obtained from the infrared spectrum of its precipitate. The titration is applicable to other polyethylene oxide derivatives.
P
(PEG’s) and their derivatives have been among the more difficult classes of organic compounds to analyze. Gravimetric procedures for their determination are based on precipitation with reagents containing large anions, such as potassium ferrocyanide ( 9 ) , potassium bismuth iodide (171, and heteropoly acids (12) or sodium tetraphenylboron (KaT P E ) in the presence of barium ions ( 7 , 8, 1 1 ) . Several colorimetric methods have been investigated, most of which involve precipitation of the P E G as a colored complex which is subsequently extracted into a n organic solvent for photometric readout ( 1 , 3, I S ) . Alternatively, the precipitate is decomposed OLYETHYLEXE GLYCOLS
and one of its constituents is colorimetrically determined (4, 6). A gas chromatographic method has been reported in which the lower molecular weight PEG’S (to mol. a t . 400) have been run as their methyl ethers (2). Most of these methods require considerable manipulation and are time consuming. We desired a simple, rapid method suitable for routine use. The gravimetric procedures of Neu ( 7 ) and Seher (IO)suggested the possibility of a rapid titrimetric precipitation method for PEG’S and their derivatives. PEG’S form oxonium ions in the presence of barium, which are instantaneously precipitated by NaTPB. The complex precipitates are extremely insoluble in VOL. 37, NO. 6, M A Y 1965
e
671
