order for X to be constant, as is observed, it would be necessary for the concentration of the complexing agent-Le., monomethyl oxalic acid-to be virtually constant during the course of the precipitation. This does not appear likely. Thus, neither complex formation nor the activity term appears t o account for the difference between the experimental and the calculated values of X. Even a test for self-consistency of the data fails to support the equilibrium model. According to this model the quotient of the observed values of X for the neodymium-cerium and ytterbiumcerium pairs should be equal t o the value of for the ytterbium-neodymium system, if the data are self-consistent. The ratio of the observed values is 0.56, while the measured value for the ytterbium-neodymium system is 0.69, nhich is about 23% greater. This difference is outside the limits of experimental error. Because the evidence indicates that the equilibrium model does not hold, other models must be examined and, in particular, a kinetic model. Evidence by Gordon, Reimer, and Burtt (9), and by Hermann (14),indicates that the rate of precipitation has an effect on the value of A. Similar results were obtained in this investigation. It will be shown that a relationship can exist between X and the rate of precipitation for systems which obey the DoernerHoskins relationship. The term, rate of precipitation, is used h6re in a very general sense. The precipitation very likely involves several steps, The rate may be determined by a n y one step or combination of steps. be the respective Let R,, and total rates of precipitation of the rare earths A and B, in moles per unit time, and R8-o and Rs-b be the respective total rates of solution of 4 and B, in moles per unit time. The net rates of precipitation are:
where V is the volume of the solution. If the rate of precipitation is much greater than the rate of solution, and the rate of precipitation is proportional to the concentration of the rare earth ion in solution, then, dividing Equation 13 by 14 yields
rate constants. Dividing the differential form of the Doerner-Hoskins equation b y dt gives :
dt
which is identical with Equation 15 if
Thus, the Doerner-Hoskins equation can be derived from a nonequilibrium model and is then proportional to the ratio of the rate constants (which may include concentrations and other quantities which might not vary in a series of experiments or in a single precipitation). Herniann’s investigation (24) of the coprecipitation of americium with lanthanum precipitated with dimethyl oxalate, a n investigation similar in many respects t o the present one, clearly shows the dependence of X on the rate of precipitation. However, it will apparently be necessary to obtain data showing the effect of a rare earth on the hydrolysis of dimethyl oxalate before the precise dependence of X on the rate of precipitation can be established. Although the kinetics of the hydrolysis of dimethyl oxalate have been studied (gd), a calculation employing the published rate constants in conjunction with the present experimental conditions indicates that the rare earths are precipitated a t a faster rate than oxalate is produced according to the given rate law. Thus, the rare earths either accelerate the hydrolysis or engage in some direct reaction with the ester. I n any event, the available evidence in the case of the rare earth oxalates seems to be in favor of the vien. that the magnitude of X is rate controlled. The present system has shown clearly that the rare earth systems studied obey the Doerner-Hoskins distribution law, and not the homogeneous distribution law as v a s concluded by Keaver (dd), who also investigated several pairs of rare earths. The validity of the objections raised by Salutsky and Gordon (20) to Weaver’s conclusions are borne out by the rcsulti: of this investigation. LITERATURE CITED
(1) Cork, J. RI., Brice, RI. K., RIartin, D. JT., Schmid, L. C., Helmer, R. G., Phys. k e v . 101,1042 (1956). (2) Crouthamel, C. E., Rlartin, D. S., J . Am. Chem. Soc. 72, 1382 (1950). (3) Ibid., 73,569 (1951). ’
dt
where KP-,, and K,,
are the respective
(4) Den’aard, H., Phil. Mug. 46, 445 (1955). (5) Doerner, H., Hoskins, W., J . A m . Chem. SOC.47, 662 (1925). (6) Feihush, A. >I., Ph. D. thesis, Syracuse University, Syracuse, N . Y., 1956. ( 7 ) Feihush, A. AI., Rowley, K., Gordon, L., A N A L . CHEM. 30, 1610 (1958). (8) Gordon, L., Feibush, A. RI., A N A L . CHEM.27,1050 (1955). (9) Gordon, L., Reimer, C. C., Burtt, B. P., Zbid., 26, 842 (1934). (10) Gordon, L., Rowley, K., Ibzd., 29, 34 (1957). (11) Greenhaus, H. L., Feibush, A. A I , Gordon, L., Ibzd., 29, 1531 (1957). (12) H $ n , O., “Applied Radlochem-
istrv. Cornel1 University Press, Ithaca,
N. y., 1936. (13) Hans, H. S., Saraf, B., Mandiville, C. E., Phys. Rev. 97,1267 (1953). (14) Hermann, J. 4., Ph. D. thesis, Cniversity of Sew Mexico, 1955. (15) Jones, J. T., Jr., Jensen, E. S . , Phys. ReLi. 97, 1031 (1955). (16) Kahn, B., Lyon, W. S., Ibid., 97, 58 (1955). (17) Marmier, P., Boehm, F., Zbzd., 97, 103 (1955). (18) Moeller, T., “Inorganic Chemistry,” p. 442, Wiley, lien- York, 1952. 119) Ratner, A. P., J . Chem. Phgs. 1, 789
R e v . 97, 102 (1955). f24) \ - - , Weaver. B.. AKAL.CHEM.26, 474, 479 (1954)’. ’ (25) JJ7illard, H. H., Gordon, L., Ibid., 20, 165 (1948). ~
RECEIVEDfor reviem- August 14, 1957. Accepted May 28, 1958. Division of Analytical Chemistry, 132nd Meeting, ACS, Xew York, N. Y., September 1?5i. Research supported in part by the L.S. Atomic Energy Commission.
Apparatus for Spectrotitra tion of Submilligram SamplesCorrection I n the article on “Apparatus for Spectrotitration of Submilligram Samples” [H. E. Boaz and J. W. Forbes, ASAL. CHEX 30, 456 (195S)l the captions for Figures 1 and 2 should have been “Titration Assembly Used with the Cary Model 14 Spectrophotometer” and “Detail Showing Absorption Cell and llagnetic Stirrer,” respectively. I n adclition, the authors should have been listed a s H. E. Boaz, Eli Lilly and Co., Indianapolis 6, Ind., and J. IT. Forbes, Shell Development Co., Emeryville, Calif., since the apparatus s h o m in the illustrations was constructed a t Shell Development Co. Hen-ever, both authors mere a t Eli Lilly and Co. during development and construction of the original apparatus.
VOL. 30, NO. 10, OCTOBER 1958
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