Gravimetric Determination of lithium by Precipitation as Trilithium Phosphate EARLE R. C4LEY AND GEORGE .4. SIMMONS, J R . ~ McPherson Chemical Laboratory, T h e Ohio State Unicersity, Columbus, Ohio l'reiioirs attempts to separate and determine lithium quantitatively as phosphate have not been successful. 4 method is described by H-hich lithium is precipitated as phosphate in 50% 2-propanol with choline phosphate as a reagent. By this method lithium may be accurately determined in the presence of potassium, sodium, or hoth, even when these ions are in widely different ratios.
T
HE separation and determination of lithium as phosphate, first advocated by Berzelius (I), was a method in common use a t one time, but adverse criticism as to its accuracy and reliability, ar well as the development of other methods, finally caused it to be abandoned. Becording to the procedure of Berzelius, phosphoric acid and an excess of sodium carbonate were added to a solution of the mixed alkali chlorides, and the \Thole was evaporated to dryness. After leaching the residue with cold water, the insoluble matter, considered b y Berzelius to he a douhle phosphate of lithium and sodium, was weighed and the amount of lithium calculated. Rammelsberg (6) examined the procedure critically and found that the wash water contained lithium and that the insoluble matter contained variable proportions of lithium and sodium in accordance with the proportions of thp two elements originally present. He therefore concluded that t'he method was entirely unreliable. On the other hand, Mayrr ( 4 ) ol)tained only trilithium phosphate on adding sodium phosphat,e and sodium hydroxide to the solution of a lithium salt, evaporating to dryness, and washing out the soluble salts. He found that if sodium carbonate was used instead of sodium hydroside the insoluble matter then cont'ained lithium carbonate. ;is a wash liquid, hiayer employed dilut,e ammonium hydroxide, since the lithium phosphate was more insoluble in this than in ivater. With large precipitates he found that extensive washing was required to remove the sodium from the insoluble residue and that it was then necessary to evaporate the washings in order to recover the lithium phosphate that had keen dissolved. Furthermore. when the amounts of potassium and sodium in the sample w r e very high as compared to the lithium it was necessary to of these other two alkalies hefore proceeding to ~ ~ ' i n o vmost e sepai,ate the lithium as phosphate. Though the results of nine t e s t tipterminations by Mayer were regarded by him as evidence procedure was satisfactory, recalculation of his data on s of present atomic weights shows a ranee of error in terms of lithium from -1.4 mg. to +2.5 mg., with a mean of $0.1 mg. Rammelsberg ( 7 ) examined the lithium phosphate obtained by JIayer's procedure and insisted that it always contained sodium. Fresmius (3) repeated some of the work of Mayer and introduced sonip minor refinements in his procedure. He recommended that. the Iiulk of the potassium and sodium as chlorides be separated from the lithium chloride by treatment with alcohol before procwding. Furthermore, his experiments showed that as many as four successive evaporations of the washings with intervening filtration are necessary in order to recover all the lithium. He confirnied the finding of RIayer that the insoluble matter was always trilithium phosphate, and found that, it could either be dried a t 100" C. or ignited a t red heat for weighing. Two test determinations were made by Fresenius, one of which, on the basis of present atomic weights, showed an error of f 0 . 8 mg. and 1 Present address, Department of Chemistry, Purdue University, West Lafayette, Ind.
the other an emor of -0.2 me. in terms of lithium when the phosphate was dried at 100" C. When the phosphate was ignited, the respective errors were +0.2 nig. and -0.8 mg. The mean of his four measurements showed no error. However, later tests of Mayer's procedure by Ranzoli (8) and by llurman ( 5 ) yielded results that were much poorer. There are a t least three serious defects in Mayer's procedure and its modifications. Evaporation to dryness almost ensures contamination of the lithium phosphate; the lithium phosphate is appreciably soluble in the aqueous solvent; and the procedure is very slow and tedious. ;ill three are avoided by the precipitation procedure described in this papw. CONDITIOKS AND REAGEKTS FOR QUANTITATIVE PRECIPITATION
In order to obtain any precipitate at, all in a dilute lithium solution the concentrations of both phosphate and hydroxide must be high. Because i t provides an insufficient concentration of hydroxide, even when ammonium hydroxide is added. diammonium phosphate is unsuitable as a reagent. Trisodium phosphate, especially with added sodium hydroside, prPcipitat,es lithium as trilithium phosphate from dilutc lithium solutions, and precipitation occurs sooner and is more complete in hot solutions; but. regardless of variat,ionsin conditions, precipitation in aqueous solution is never quant,itative because the solubility of the trilithium phosphate is too great. However, it, is sufficiently less soluble in alcoholic solutions of the proper concentration t.o be precipitated completely with sodium triphosphate as is indicated by the results in Table I. Similar results are ohtained in methanol or ethanol solutions, but %propanol is preferred because the rate of loss from hot solutions is less. In these experiments 15 meq. of trisodium phosphate and 5 meq. of sodium hydroxide in the form of 0.3 M solutions were added to each aqueous lithium chloride solut,ion containing the stated amount of lithium. One hour after precipit,ation began in the warmed aqueous solution, the 2-propanol was added, and the mixture was allowed to digest on a steam plate for 2 hours. This heating procedure was necessary to obtain an easily filterable precipitate. The volumes of reagent solutions, lithium solution, and 2-propanol were adjusted to a final volume of 100 ml. Each precipitate was collected in a porcelain filter crucible, washed with a 2-propanol-water solution of the same concentration as that used in the precipitation, ignited in a muffle a t low red heat for half an hour, and weighed as anhydrous trilithium phosphate. The result obtained by precipitation in 50% 2-propanol is much too high. This is caused by coprecipitation of sodium phosphate. The apparently quantitative result in 40% 2-propanol is also caused by coprecipitation of sodium phosphate, which compensates for the slightly incomplete precipitation of the lithium. Obviously, trisodium phosphate is not a suitable reagent for precipitat,ion in alcoholic solutions.
1386
V O L U M E 2 5 , NO. 9, S E P T E M B E R 1 9 5 3 Table I. Precipitation of Lithium as Phosphate in Alcoholic Solutions with Trisodium Phosphate Lithium 2-Propano1,
Sd. g., -30
30 40 50
60
Table 11.
Sample
Taken,
Found,
mg. 19 6 19 6 19 6 19.6
mg.
18.1 19 0 19 8 32 5 Two-phase separation
Difference, Mg.
-1 J -0 .6
+t12o 29
Impurities in Solutions of Various Quaternary Ammonium Hydroxides (Expressed a8 weight per 100 ml.) Eodium, Silver, Iodide, -wg. Mg. Mg.
Silica, hIg.
What is needed is a phosphate of a strong base that is more soluble in alcoholic solutions, and preferably also one that will be decomposed and volatilized on ignition. These conditions are fulfilled by phosphates of quaternary ammonium bases. Unfortunately. such phosphates are not commerciallpavailable, and even quaternary ammonium bases from which they may be made are not generally available in sufficient purity for use as quantitative analytical reagents. Significant impurities in a sample of a 10% solution of tetramethylammonium hydroxide obtained from a prominent supplier of organic chemicals are shown in Table I1 (Yo. 1). The figure for the silica content does not actually represent the total silica in the product, as considerable suspended matter, most of it probably silica, was filtered off before the solution was analyzed. Since such products are packaged in glass bottles. and probably are prepared in glass vessels, the source of the sodium and silica present as impurities is obvious, Even tetramethylammonium hydroxide carefully prepared in the labratory in borosilicate glass vessels by the classical method of reacting silver oside with tetramet.hylammonium iodide contains too many impurities for a quantitative analytical reagent, as is indicated in Table I1 (No. 2). This sample was taken from a freshly prepared 0.85 M solution. However. a method was developed for preparing tetramethylammonium hydroxide solutions of satisfactory purity by the use of an eschange resin. The details of this method of preparation will be published elsewhere. The only commercially available quaternary ammonium hytlroside of sufficient purity for preparing a reagent, appeared to 1x1 the choline supplied as a 50% solution hg the Rohm & Haas Co. of Philadelphia. A sample of this product Fas rereivetl in a gallon glass container, diluted with an equal volume of water to render it less viscous, filtered to remove suspended matter, and stored in polydiylene 1)ottles. The results of tests O H the diluted solution arc shown in Table I1 (No. 3). In spite of hring oricinallp packagcd in glass, there mas present only a small amount of sodium, and no dissolved silica. Probably these favorable red t , s are due to it short time of storage in the original glass container. All of the suhsequent experiments and procedures are lmserj on it,suse. However, R s shown by other espcriments, tetramethylaninionium hydroside of sufficient purit,y gives equallj- good results, and it is prohal>le that still other pure quaternary ammonium hydroxides n-oulcl be equally satisfactory. EXPERIMENTS UITH CHOLINE PHOSPHATE A S A REAGEST
The reagent that gave the best results was prepared bv adding 4.5 ml. of 85% phosphoric acid to 100 ml. of diluted choline solution (about 25% concentration which corresponds to about 2 If). I n Table I11 are shown the results of precipitations in 2-propanol-water solutions with choline phosphate as a rragent.
1387 These experiments parallel those with trisodium phosphate as a precipitant, the results of which were shown in Table I. In these experiments, 15 meq. of phosphate and 5 meq. of extra hydroxide were present in each precipitation, but here the hydroxide was added in the form of choline. The results are obviously much better than those obtained with trisodium phosphate, there being no high results due to coprecipitation, and no separation of the solution into two phases a t a concentration of 60% %propanol. However, the results are still not quantitative. Additional experiments were therefore performed to find the cause of these low results. In Table IV are shown the results of experiments on the effect of various concentrations of phosphate and choline on completeness of precipitation in 50% %propanol. These were performed in the same general way as those just described. Evidently a considerable excess of either reagent, or both, does not promote completeness of precipitation, If anything, an excess of choline causes the results to be lower. In Table V are shown the results of experiments on the effect of total volume of solution on completeness of precipitation in 50% 2-propanol. In these precipitations no choline was added. With smaller amounts of lithium, a t least, the total volume of the solution is a factor of some importance. In fact, for the first four results in the table the deficiency is proportional to the total volume of solution. This seems to indicate that a small amount of lithium remains unprecipitated due to solubility, but it might also be due to the dissolution of a small amount of precipitate on washing with 50% 2-propano1, or to some combination of the two. A solubility correction of +0.3 mg. per 50 ml. of total solution applied to the first four results would yield very good quantitative resalts. The effect of concentration of 2-propanol on obtaining quantitative results with small amounts of lithium is shown in Table VI. In these experiments also, no choline was added, and the final volume of solution was maintained a t 50 ml. Precipitation is obviously incomplete in 40% 2-propanol, and precipitation in 60% 2-propanol does not give any better results than in 50%
Table 111. Precipitation of Lithium as Phosphate in Alcoholic Solutions with Choline Phosphate Lithium ~
2-Proiianol, Val. 70
Taken,
Found, mg.
mg.
Difference, llg.
Table IV. Precipitation of Lithium as Phosphate in 50% 2-Propanol Solution With Choline Phosphate and Choline Phosphate Present, Meq.
Choline Present, JIeq.
Lithium Taken, Found, mg.
mg.
Difference, Mg.
Table V. Effect of Total Volume of Solution on Precipitation of Lithium as Phosphate in 50% 2-Propanol d t h Choline Phosphate Total Yolurne, 111.
50 60 100
100
100
Lithium Taken, mg. 19.6
49.1 49.1 99.5 249.5
Found, mg. 19.3 48.8 48.4 99.0 249.8
Difference. Mg.
-0.3 -0.3 -0.7 -0.6 $0.3
ANALYTICAL CHEMISTRY
1388 Table VI. Effect of 2-Propanol Concentration on Precipitation of Small Amounts of Lithium Lithium %Propanol.
Taken, mg.
Found, mg.
40 50 50 60 40 50
2.0 2.0 4.9 4.9 10.0 10.0 10.0
1.2 1.7 4.6 4.5 9.5 9.7 9.7
VOl. %
00
~~
Difference, Mg. -0.8 -0.3 -0.3
-0.4
-0.5 -0
3
-0.3
Table VII. Lithium Determinations in Presence of Potassium in Chloride Solution by Single Precipitation Potsasium Present, Mg.
10 10 50
50 100
100 10 ..
10 50 50 100 100
10 10 ~~
50 50
100 100
Lithium Taken, mg.
Found, mg.
Difference, Mg.
10.0 10.0 10.0 10.0 10.0 10.0 49 1 49 1 49 1 49 1 49 1 49 1 99 6 99.6 99.6 99.6 99.6 99 6
9.8 9.7 9.9 9.9 10.0
-0.2 -0.3 -0.1 -0.1 0.0
10.0
49.1 49.0 49.1 49.1 49.2 49.1 99.8 99 5 99 6 99 6 99 8 99 6
n_ n_
0.0 -0.1 0.0 0.0 +o. 1 0.0 $0.2 -0.1 0.0 0.0 +0.2 0.0
%propanol. Here also the application of a solubility correction
of $0.3 mg. to the results obtained in 50% solution would yield very good quantitative results. EFFECT OF VARIATIONS I N iMANIPULATION
The only way by which easily filterable precipitates could be obtained in a reasonable time was by precipitation in hot solution followed by digestion in hot solution. When precipitation and digestion w-ere both performed a t room temperature, as long as 20 hours was required before precipitates were in a suitable condition for easy filtration. Aklso,too long a time was required when precipitation was done in hot solution followed by digestion a t room temperature. and vice versa. The ratio of amount of reagent to amount of lithium to be precipitated also has a significant effect on the filterability of the trilithium phosphate. When the amount of lithium is small, a considerable excess of reagent favors the formation of a precipitate that is easily filtered, whereas when the amount of lithium is large, a better precipitate is obtained when the reagent is not in large excess. Though contamination of the precipitate with silica might be expected in glass vessels because of the well-knopm corrosive action of alkaline phosphate solutions on glass, no such contamination was in fact observed when the time of digrstion was no longer than necessary and when only the necessary amount of reagent was used. Precipitations made in ordinary borosilicate glass beakers and in beakers of alkali resistant glass (Corning No. 728) gave equally satisfactory results. .kpparentlv the amount of attack on such glass by weakly alkaline 50% 2-propanol solutions containing a relatively low concentration of phosphate is not significant in the short time of contact needed for precipitation and digestion. Washing the precipitate with a saturated solution of trilithium phosphate in 50% 2-propanol rather than with 50% %propanol alone was found to be advantageous With small precipitates containing 20 mg. or less of lithium, the amount of precipitate dissolved by washing with 50% 2-propanol was not significant when a minimum of wash liquid was used, but with larger precipitates the loss was signifirant, probably because of the larger surface of the precipitate in contact with the wash liquid. Ese of a
wash solution saturated with the precipitate permitted the use of more than the minimum of wash liquid in all cases, and no high results due to the use of this wash solution were obtained. However, the use of 50% 2-propanol saturated with the precipitate did not eliminate low results of the order of those shown in Table VI, and, therefore, it seems probable that such low results are primarily due to the inherent solubility of the precipitate in the sohtion in which it is precipitated rather than to loss on washing. Hence, the use of the small solubility correction previously mentioned is desirable even though the precipitate is washed with a saturated solution of the precipitate. Attempts to filter off the trilithium phosphate on paper and ignite in the usual way were not successful. Dark grey precipitates that were too high in weight were obtained, and neither their appearance nor their weight could be corrected by prolonged ignition. Sometimes grayish precipitates were obtained when filter crucibles were used, There seemed to be no correlation between variations in manipulation and the occurrence of such discolored precipitates, Possibly the cause of the discoloration is organic dust collected during filtration. However, when the discolored precipitates were only a light gray, the final results were not significantly different from those obtained with pure white precipitates. RECOMMENDED PROCEDURE
Reduce the chloride or sulfate solution of the alkali metals to a volume of about 17 ml. when the expected amount of lithium is 50 mg. or less, to about 34 ml. when the expected amount of lithium is over 50 mg. but not over 150 mg., and to about 25 ml. when the expected amount is over 150 mg. If the volume 1s about 17 ml. add 8 ml. of choline phosphate reagent rapidly, and if about 34 ml., add 16 ml. rapidly. If the volume is 25 ml., add 25 ml. of reagent slowly and with constant stirring. After the reagent has been added, cover the beakers with well fitting watch glasses and place on a steam plate or on an electric hot plate of similar temperature. One hour after the first appearance of a precipitate, add an equal volume of Bpropanol to the solution and stir. Two hours after the addition of the Zpropanol, filter while still hot through a porcelain filter crucible of medium porosity. Wash the precipitate with successive small portions of a saturated solution of trilithium phosphate in 50% 2-propanol until a total of about 40 ml. has been used. Draw air through the crucible for a few seconds to remove most of the wash liquid. If more than about 10 mg. of sodium is known or suspected to be present in the original solution, redissolve this first precipitate and reprecipitate the lithium. To dissolve the precipitate, add 2 ml. of 4 ilf hydrochloric acid to the precipitate in the crucible, allow to stand about half a minute, and draw the solution into the beaker used for the first precipitation. Repeat with two more 2-ml. portions of acid, and then wash out the crucible with four or five successive small portions of water. Concentrate the solution to a volunie of about 5 ml., add a drop of phenolphthalein indicator, and neutralize with 2 Jf choline to a distinct pink color. Adjust the volume of the solution and precipitate as before. Ignitr in a muffle a t a low led heat for a half to one hour, cool, and weigh. ;1Iultiply the weight of the precipitate by 0.1798 to obtain the weight of the lithium in it, and add 0.3 mg. of lithium as a solubility correction if the volume of solution was 50 ml., and 0.6 mg. if the volume was 100 ml. NOTES ON PROCEDURE
Even with no prior knowledge of the approximate amount of lithium present, it is easy to determine how much reagent to use. If,on first adjusting the volume of solution to 17 ml. and adding a drop of reagent, a precipitate imnlediately appears, the amount of lithium is more than about 150 mg., and the volume should be increased to 25 ml. for precipitation with 25 ml. of reagent. If no precipitate appears with the first drop of reagent, the whole 8 ml. of reagent should be added. If no precipitate then appears in about half a minute, less than about 50 mg. of lithium is present, and no further adjustment of the volumes of solution and reagent is necessary. If a precipitate appears in much less than half a minute the amount of lithium is in the range 50 to 150 mg., and the volumes should be adjusted accordingly.
V O L U M E 25, NO. 9, S E P T E M B E R 1 9 5 3
1389
If no precipitate appears after the solution has been heated for about 10 minutes, less than about 5 mg. of lithium is present, and it is then advisable, in order to save time, to add 5 or 10 ml. of 2propanol to start the precipitation. If it is known that less than 5 mg. of lithium is present, such volumes of 2-propanol may be added to the water solution before the reagent is added. When 2-propanol is added in either of these two ways, the volume added a t the end should be reduced accordingly so that the final conrentration is 50%. All the volumes mentioned in the procedure and in these notes need be measured no more accurately than is possible with a graduate or by means of volume marks on small beakers. Care should be taken not to boil the solutions, as this may result in too much reduction in volume. If the level of the solution has fallen appreciably by half an hour before filtration, the original volume should he restored by adding 2-propanol. If more than about 100 mg. of lithium has been precipitated, it i5 desirable as a precaution to dry the precipitate in an oven a t 110" C. for a few minutes before placing it in the muffle, as this may avoid loss of precipitate by the sudden evolution of water and alcohol vapor. The temperature of the muffle should be a t least 550" C. but should not exceed 800" C. About 2 mg. of lithium is the smallest amount that can be determined satisfactorily by this procedure. The possible upper limit is about 250 mg., though better results are obtained if determinations are restricted to amounts considerably below this upper limit, A11 metals other than the alkali metals interfere in this procedure and must be completely removed beforehand. TEST DETERMINATIONS
Results of test determinations with this procedure are shown in Tables VII, VIII, IX, and X. For these determinations lithium solutions of accurately known concentration were prepared by
Table VIII. Lithium Determination in Presence of Sodium in Chloride Solution by Single Precipitation Lithium ._ .__
Sodium Present, Mg.
Tnken,
Found,
mg.
10 10 50
mg.
10.0
9 9
10 0
9.9 10.1 10.1 10.4 10.4 49.1 49.1 49.6 j9.5 00.1 49.9 100.0 99.9 100.6 100.5 101 2 100.9
10.0
5..0
10.0 10.0 10.0 49.1 49.1 49.1 49.1 49.1 49.1 99.6 99.6 99.6 99.6 99.6 99.6
100
100 10 10 50
50 100
100 10 !O Jo
50 100 100
IMierence, AIg.
- n-
1.
-0.1 +o 1 +o. 1 +0.4 +0.4 0.0
o n
+o 5 +o -I
+I 0
+0 8
+o +o
1 3
to
9
+I 0
aig. 10 10
50 50 100 100
10 10 50 50 100
100
Sodium Present, mg. , .
,..
... ... 10 10 50 50 100 100 10
i n. .
50 50 100 100
Lithium Determinations in Presence of Large Amounts of Sodium by Reprecipitation
Sodium Present, Mg.
Taken, mg.
Found, mg.
ME.
250
10.0
10.3 10.2 10.1
f0.3 f0.2
500
10.0
10 2 10.2 10.1
+0.2 +0.2 +o. 1
250
19.6
20.1 19.9 19.8
+0.3 +0.2
500
Lithium
19.6
Difference,
+O. 1
+0.5
20.0
+0.4 +0.3 f0.3
19.9 19.9
dissolving weighed portions of lithium carbonate, purified by the method of Caley and Elving (Z), in a minimum of hydrochloric acid and diluting to volume in calibrated flasks. The concentrations of these solutions were checked by evaporating with sulfuric acid in platinum and weighing the residues as lithium sulfate. Solutions of the chlorides and sulfates of potassium and sodium were prepared by dissolving weighed quantities of the reagentgrade salts and making up to volume in calibrated flasks. For the test determinations, accurately measured volumes of these standard solutions were evaporated to the volumes specified in the procedure. As shown in Table VII, the results of determinations of lithium in the presence of potassium in chloride solution by single precipitation were all satisfactory. However, there is a slight tendency for the results to be higher with increasing amounts of potassium, and it may be that, in determining lithium in the presence of much larger amounts of potassium, reprecipitation would be necessary. As shown in Table VIII. results of determinations of lithium in the presence of sodium in chloride solution by single precipitation were not satisfactory when the amount of sodium exceeded 10 mg., unless the amount of lithium was also small. The need for reprecipitation to avoid high of so- results from coprecipitation dium is obvious, In Table IX are shown results of determinations of lithium in solutions containing sulfate, introduced in the form of solutions of potassium sulfate, sodium sulfate, or both. This point was tested because of the possible difficulty that might arise from the fact that the alkali sulfates are more insoluble in alcoholic solutions than the chlorides, and because sulfate solutions are en. countered in practice. As may be seen by comparing the results in this table with those in Tables VI1 and YIII, the presence of sulfate does not affect the results. Moreover, when both potassium and sodium are present, any high results that may be obtained are caused by sodium alone. By reprecipitation, determinations of lithium in the presence of sodium become as satisfactory as those for lithium in the presence of potassium by single precipitation. Even when the ratio of sodium to lithium is high, acceptable results are still obtained, as indicated in Table X. I
_
f l 6 AI 3
~ _ _ _ ________ -~ Table IX. Lithium Determinations in Presence of Potassium, Sodium, or Both, in Sulfate Solution by Single Precipitation Potassium Present,
Table X.
ACKNOWLEDGMENT
Part of the work described in this paper was supported by a grant in aid to The Ohio State University by the Du Pont Co.
Lithium Taken, mg. 49 1 49.1 49 1 49.1 49.1 49 1 49.1 49.1 49.1 49.1 49 1 49.1 49 1 49 1 49 1 49.1 49.1 49 1
Found, Mg. 49.0 48.9 49.1 49.1 49.2 49.1 49.3 49 2 49.7 49.6 50.0 50.0 49.4 49.4 49.7 49.5 50.1 49.8
Difference, Mg. -0.1 -0.2 0.0 0.0
+O. 1 0.0 +0.2
+O. 1 +0.6 +0.5
+O. 9 +O.S
+0.3 +0.3
+0.6 +0.4 +1.0
+0.7
LITERATURE CITED
11) Berzelius, Ann. Physik Chem., 4, 245-9 (1825). (2) Booth, H. S., ed., "Inorganic Syntheses," 1'01. I, pp. 1-2, New York, 14cGraw-Hill Book Co., 1939. ( 3 ) Fresenius, 2. anal. Chem., 1, 42-6 (1862). 14) hlayer, Ann., 98, 193-212 (1856) (5) Murman, E., 2. anal. Chem., 50, 171-4 (1911). (6) Rammelsberg, Ann. Physik Chem., 66, 79-91 (1845). (7) Rammelsberg, Ibid., 102, 441-4 (1857); I b i d . , [ 2 ] 7, 167-8 (1879). (8) Ranzoli, Gazz. chim. ital., 311, 40-8 (1901). RECEIVED for review March 13, 1953. .4ccepted June 18, 1953. Constructed from a dissertation presented to the Graduate School of The Ohio State University in partial fulfillment of the requirement for the Ph.D. degree, August 1952.
