Metallic Soaps—Their Uses, Preparation, and Properties - Industrial

Gas chromatographic study of interactions between aliphatic amines and metallic ions. R. C. Castells and J. A. Catoggio. Analytical Chemistry 1970 42 ...
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I-VDUSTRIAL A N D ENGINEERING CHEMISTRY

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Tol. 22. No. fi

Metallic Soaps-Their Uses, Preparation, and Properties'" Willett F. Whitmore and Michael Lauro TABPOLYTZCRNIC INSTITUTE, BROOKLYN. K. Y .

T

HE heavy-metal salts or soaps of the fatty acids occur-

suggested that nickel soaps and also mixtures of nickel and ring in vegetable and animal oils and fats are used for iron or copper soaps of fatty acids be used as cataIytic agents a great variety of purposes. We are accustomed to re- to promote intimate contact between catalyst and oil in this gard as soaps, in the common acceptance of the term, only the process. alkaline salts-those of soda, potash, and ammonia-of the I n the pharmaceutical field, the mercury, lead, and zinc fatty acids contained in fats and oils; rarely stopping to con- compounds, chiefly in the form of ointments and plasters, sider how restricted this view is and how many soaps may be have long been known and used. Lead oleate has great made from other metals. Oleic, palmitic, and stearic acids antiquity, having been described by Dioscorides, the Greek are common ingredients of fats, with lauric and myristic, physician and authority on materia medica. Copper oleate linolic and linoleic, erucic and cluponodonic occurring in is useful in the treatment of granulation and ulcers. The special cases, as in coconut, linseed, rape, and fish oils. With strontium soap of dibromobelienic acid is substituted for soda, potash, and ammonia their compounds are soluble in potassium bromide in special rases. Zinc stearate is a fawater, produce lather in abundance, and are cleansing or de- miliar facial or dusting powder, and incorporated with petrot e r g e n t i n a c t i o n . The latum serves as an antialkaline-earth and h e a v y s e p t i c e m o l l i e n t . The metal salts of t h e s e same of copper, zinc, and oleates A brief &sum6 of the utility of the heavy metal acids are insoluble in water mercury are formed by mixsoaps is presented. The normal silver, lead, copper, and manifest properties and ing the metallic oxides with mercury, nickel, and zinc salts or soaps of oleic, stearic, characteristics which difoleic a c i d . In fact, the palmitic, erucic, and lauric acids have been prepared ferentiate t h e m s h a r p l y s o a p s of p h a r m a c y are and studied. The data presented, although in some from the water-soluble alkali u s u a l l y prepared in this respects conflicting with what the literature records, soaps. way, by trituration of lard, are believed to be quite accurate inasmuch as great suet, or oleic acid with the Uses care was exercised to obtain the pure normal neutral oxides, and are therefore soap. A few soaps concerning which the literature Some of t h e s e heavychiefly mixtures rather than furnishes no information at all have been made and metal soaps have been used compounds in the form of studied. for ages with scarcely any pastes or ointments. knowledge of their coinHowever, the pure soaps position. Accident rather themselves are beginning to than research has been the chief cause of the discovery find a field of their own. The saturated fatty acid s a k of i e r of their valuable properties. They are the siccatives, or cury may find use as antiseptic and germicidal dusting powders driers, of the paint and varnish industry. At first, oxides, for wounds, since they do not possess the toxic effects of the borates, carbonates, and inorganic salts of such metals as usual mercury salts. Iron soaps, in pill or emulsion form, may lead, cobalt, and manganese were added to linseed and tung replace to advantage the various preparations of inorganic oils to aid in oxidation or drying. iron salts and peptized iron. Likewise, the silver soaps may Soaps of zinc, iron, nickel, cobalt, and chromium are used be substituted for preparations like argyrol, which stain, or for waterproofing leather and canvas and in coloring var- for the cauterizing of raw surfaces in place of the caustic, nishes. To render textile goods waterproof, soaps of alumi- nitrate or in lieu of the silver caseins, proteins, and vitellin?. num and magnesium find extensive application. Aluminum Research along these lines should reveal an astonishing numsoaps are especially useful in sizing papers which are to be ber of places where metallic soaps may be employed, dry or used as leather substitutes, also in the manufacture of cellu- emulsified and incorporated with other ingredients. loid, rubber, and insulating materials. They also serve in The dry-cleaner's art has found magnesium oleate of help mordanting, in the preparation of oil emulsions, solidified in preventing fire from the static generated by the withdrawal oils or solid lubricants, floor waxes, polishing compounds, of silk goods from the solvent bath, and is beginning to make etc. The oleate of aluminum is particularly useful as an use of oleates other than those of soda and potash for cleansoil thickener. ing purposes. The writers have discovered that a, few of the Copper and mercury soaps serve in fungicide sprays for lnetallic soaps-as, for example, nickel-in benzene or certain trees and plants, in disinfectants, and in antifouling paints. of the lacquer solvents will produce good lathering and deterThe latter applied to ship bottoms prevent the growth of gent effects. They offer, therefore, useful possibilities in barnacles by their toxic action and protect the metal of the the cleaning of such fabrics as are injured by water-soluble ship from corrosion. Small amounts added to the ordinary alkali soaps, and may possess considerable utility in the dryalkali soaps contribute antiseptic and germicidal properties. cleaning of silk and chamois. In the hydrogena.tion of vegetable oils small amounts of One of the most important properties of ordinary soap is nickel soaps are formed at high temperatures. It has been its ability t o form emulsions, and the alkali soaps have been used extensively for such. Of the insoluble soaps, lime and 1 Received January 8, 1930 Presented before the Division of Organic Chemistry a t the 77th Meeting of the American Chemical Society, Columaluminum are thus chiefly employed. The textile industry bus, Ohio, April 29 to May 3, 1929 draws on the former type for its softening agents, the petro2 This paper is based upon a thesis presented by Michael Lauro in partial leum industry on the latter for lubricants. The usual emulfulfilment of the requirements for the degree of master of science in chemGens are of oil dispersed in water with a small amount of istry a t the Polytechnic Institute of Brooklyn

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water-soluble soap added to keep the globules of oil from coalescing. With metallic soaps we. ma.y reverse the procedure and disperse water in oil. Thus are built up solidified oils, solid lubricants, and cutting compounds for lubrica,tion, textile softening, leather dressing, and metal working. It was universally believed a t one time that saponifia,ble matter in petroleum oils used for lubricatioli was a serious objection, because of its gumming under friction, and the corrosive effect on metal parts of contact, due either to the free fatt'y acids present in the vegetable or animal fat>sused, or produced by hydrolysis. In present-day practice we incorporate fatty compounds or, better, smaller amounts of their fatty acids in suitable proportions in various types of lubricants such as steam turbine oil, certain cylinder oils, cup grease, etc. These fatty ingredients seem to diminish the surface tension of the oil and enable it to spread more readily, and to wet surfaces that were not so efficiently reached before. Metallic soaps may now be substituted in part or entirely for the fats. As emulsions with mineral oils, they maintain continuous and permanent, films, wetting the surfaces of contact and spreading better, owing to the decreased interfacial tension between oil and metal because of their presence. The authors believe that there is considerable likelihood of the heavy-metal salts of the fatty acids serving a very useful purpose in quantitative separations and estimations of the fatty acids in animal and vegetable fate, waxes, and oils. This important application of these compounds will be discussed in a subsequent paper. 4 s Lewkowitsch so ably states ( I O ) : "A systematic study of the metallic soaps and their solubilities iii the usual solvents is greatly desired; and an investigation should well repay the time required for it, as new methods of separating fatty acids could be elaborated." Preparation of Metallic Soaps

In spite of their utility in the arts, both singly and in niixtures, there is an unusual lack of information on the pure compounds. The literature on some contains many discrepancies, while in the case of others no information ahatever is available (1 to 9, 11 to 15). The work herein reported was inaugurated to prepare the pure normal salts or soaps of six of the most importa.nt heavy metals, and to determine their properties, Oleic, stearic, palmitic, erucic, and lauric acids were selected as being typical for the majority of the commercial oils and fats, yet characteristic enough for a few groups, such as the kernel and the lish oils. The first three are common components of fat,s and oils in genera.1. Erucic acid is characteristic for rape and fish oils, and lauric of coconut and palmkernel oils. In the preparation of these metal poaps, silver, mercury, zinc, lead, nickel, and copper were employed. l h e oiily suitable method for the preparation of these soaps or salts was double decomposition from aqueous-alcoholic solution, using the acetate of the metal and the alkali salt of the fatty acid. The pure acids were obtained from Kahlbaum and the Eastman Kodak Company, and before using their physical and chemical constants were carefully determined as assurance of their purity. In the caw of oleic acid vacuum distillation was necessary to obtain a product sufficiently pure for oiir purpose. The salts of ihe heavy metals employed mere of c. P. grade and mere recrystallized twice before using. Silver nitrate was employed instead of the acetate. In general, molecular proportions of the fatty acids were weighed out, dissolved in ethyl alcohol, and carefully neutralized with alkali, some water was added, and then this solution was treated with a little more than the theo-

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retical quantity of the heavy metal salt. Usually an excess of from 2 to 5 per cent of the heavy metal salt was employed. The soaps produced by this method are as a rule voluminous, and dry down to fluffy powders of relatively large bulk. The following outline of the general procedure will illustrate the method employed: Two grams of the fatty acid were weighed out and dissolved in about 25 cc. of 95 per cent ethyl alcohol, then exactly neutralized with 0.2 N caustic soda using phenolphthalein as indicator. The bulk of the alcohol was evaporated off, and water was then added to bring the volume of the solution up to 150 cc. If any cloudiness appeared, due to hydrolysis, sufficient alcohol wm added to clarify the solution and inhibit the formation of acid soaps. The metal salts were dissolved separately in 30 cc. of water and added slowly, with stirring, to the solution of the alkali soaps, this procedure being preferable to adding the alkali soap solution to the solution of the metal salt. Precipitation of the metallic salt or soap occurred immediately. Most of these double decompositions were carried out at 60" e..-i. e., the solutions before mixing were heated to this temperature. The precipitate of the metallic soap was allowed to stand about an hour and then filtered through paper, washed with water several times, then with alcohol to remove the water and any free fatty acid present, and finally with a small amount of cold ether to facilitate subsequent drying. At the outset the filtration of these soaps presented quite a problem, due to fineness and slight greasiness of the soaps in general. With a Buchner funnel employing paper and suctioii the paper became plugged in a short time. Centrifuging threw the soaps out as hard, compact, difficultly workable masses. Filtration without suction through a fluted or even plain folded filter paper proved best under all conditions. After washing with plenty of water, the paper was flattened out, placed on a Buchner, slight suction applied, the contents washed with a little alcohol to free the material from fatty acids, and, if perniissible, washing finally with a little cold ether to expedite final drying. After they had been previously brought to a practically air-dry condition by the aid of suction, the drying of the pure soaps was accomplished by spreading out the papers containing the solid soaps upon the shelf of an oven kept at about 45" C. For those salts melting or softening below 45" C. and for the oleates particularly, because oxidizable, drying in vacuum or in desiccators at room temperature was employed. I n the preparation of the palmitates and stearates it was necessary to adhere strictly to the above procedure, since below 60" C. the sodium salts of these acids tend to jell and a t greater dilutions dissociate appreciably. The forination of neutral metallic soaps was desired, with as little occlusion of free fatty acid or metal salt as possible; and by maintaining an exactly neutral solution of the alkaline soaps, with no turbidity at optimuni temperature and concentration, it was hoped that the formation of acid or basic soaps could be prevented. In certain cases it wab found necessary to modify somewhat the procedure given above, owing to the varying physical characteristics of the products produced. All the silver salts or soaps mere prepared by this general niethod, and the soaps of this metal separate in fine curds, quickly settling to the lwttorn of the beaker. They are quite stable and are easily washed. purified, and dried. After drying they niay be ground in a mortar to fine voluminous pomdrrs. The oleate is slightly tacky and tends to lump together. The zinc and lead soaps. with the exception of the oIeates, come down like the silver salts, and when dried and ground are light powders. With the oleates of the metals other than silver. a temperature considerably lower than 60" C. should be used. I n fact, cold solution. containing ice are preferable.

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Under these conditions the oleates will come down flocculent instead of as oils. At room temperature nickel oleate is an oil and the other oleates are very soft and waxy. As a class the oleates of the six metals are thrown out as thick curds, massing together and difficult to wash and dry because of their soft and sticky nature. Mercuric oleate was particularly troublesome, inasmuch as it holds water very much like lanolin, and had to be squeezed against the walls of the beaker each time after washing to remove the water. Nickel oleate, which is an oil, is easily treated in a separatory funnel. The metal content of all the soaps was determined, and in no case was the differential between found and theory greater than 0.4 per cent. The oleates showed the greatest discrepancies. The nickel soaps are somewhat soluble in water and alcohol. Copper oleate is also quite soluble. These soaps should accordingly be made in cold solutions with minimum volume and hurried washings, using as little alcohol as possible and no ether a t all. The mercury laurate and erucate are likewise better prepared in the cold, with the least alcohol possible left in the alkaline soap solution before precipitation. The salts of mercury should not be washed with ether, since they tend to decompose with this solvent. Properties of Metallic Soaps

All the soaps prepared in this investigation, with the exception of nickel oleate, are solids a t room temperature. Nickel oleate is an oil. The mercuric and zinc oleates are somewhat plastic. Copper formed soaps with the characteristic blue color. I n the nickel soaps, except for the oleate which was deeply colored, there was just the slightest suggestion of nickel green. The remaining metallic soaps were white, varying from the consistency and transparency of wax to that of chalk. The folloKing gives in detail the physical characteristics of the thirty soaps prepared: SILVER Oleate-Gray-white t o yellowish powder, slightly tacky, dries down satisfactorily, and can be ground. Odor of oleic acid or of rancid fat. Stearate-Fine, fluffy talc-like powder, easily ground, and has no odor. Palmitate-White powder, slightly greasy to the touch. Erucate-Gray-white, fluffy powder, somewhat "tacky" and waxy, with odor of erucic acid. Laurate-White powder, slightly greasy, odorlesos. These salts melt a t temperatures above 200 C . with decomposition, charring, and emitting vapors, and cooling to a metallic looking solid of bluish black luster. Under the conditions of preparation the salts are amorphous. They can, howeyer, be prepared crystalline, with the exception of the oleate, by precipitation in alcoholic solution instead of in water, and using alcoholic nitrate of silver. MERCURY Oleate-Semi-liquid a t room temperature. A white plastic solid that softens on standing, changing to a creamy yellow color, and hydrolyzing under the influence of heat and light to free metallic mercury and oleic acid, unstable, and melts a t no definite temperature. Stearate-Fine white powder, melting a t 112.2" C. to a clear water-white liquid and solidifying to a hard, opaque white mass. Palmitate-Fine white powder melting above 105' C. to a brilliant, colorless liquid, cooling to a hard, opaque, white mass that is easily crumbled. Erucate-White, translucent paraffin-like solid, melting below 50' C. to a clear oil, with the odor of erucic acid. Laurate-Compact white powder melting around 100 C., with odor of grease, to a water-white liquid, and solidifying to a semi-opaque paraffin-like mass. O

LEAD Oleate-A white powder to a semi-plastic cream-white, sticky mass, melting around 50" C. to a yellow oil. Stearate-Chalky white, fluffy powder, smooth and talc-like t o the touch, melting a t 115' C. to a clear iiquid, and cooling t o a hard, opaque, white brittle solid, easily crumbled and powdered.

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Palmitate-Chalky white powder, like talc when rubbed between the fingers, melting a t 113' C. t o an opalescent liquid and cooling t o a brittle solid of vitreous luster. Erucate-White granular powder, melting a t about 100" C. to a pasty liquid and cooling to a tacky and finally wax-like solid. Laurate-Chalky white, granular powder, smooth to the touch, melting a t 106' C. to a clear liquid, and cooling to a hard, opaque, brittle solid, easily powdered. ZINC Oleate-Cream-colored, lumpy, wax-like, amorphous mass, which cannot be ground. It is somewhat sticky and softens with rising temperature, but is harder than the corresponding lead soap. It melts a t an indefinite temperature around 70" C. Stearate-Light, fluffy, white, voluminous powder, talc-like to the touch, melting about 130O C. to a clear water-white liquid, and cooling to a translucent glassy mass of crystalline appearance which is brittle and easily powdered. Palmitate-Like the stearate in appearance, melting a t 129 C. and cooling to a semi-opaque brittle solid. Erucate-White powder, slightly waxy. It will not grind to a loose powder, but tends to lump, melting a t 94' C. with the odor of erucic acid, to a clear water-white liquid, and cooling to a hard white wax. Laurate-Fluffy white powder, silk-like to the touch, melting a t 128" C. to a water-white clear liquid, and cooling to an opaque white mass. COPPER Oleate-Hard crayon-like masses, amorphous, deeply colored a copper-blue, with the odor of oleic acid and a fracture like pitch. It has no definite melting point and fuses below 100 C . It will dissolve completely in ether to form "liquid crystals." Stearate-Fine, fluffy, voluminous powder, of pale blue-green color and no odor. It melts indefinitely a t about 125" C. to a clear deep blue liquid, hardening to a brittle, opaque mass. Palmitate-Fine, fluffy, voluminous powder, of light blue color, melting a t 120" C. to a clear deep blue liquid, and cooling to a hard, brittle, opaque mass. It is quite soluble in water. Erucate-Tacky, deep copper blue powder, with odor of erucic acid, melting indefinitely around 100" C. to a somewhat sticky, blue-green oil, which will decompose with heat to a brown liquid. It solidifies to a sticky clear glass. It is sparingly soluble in water. Laurate-Very fluffy, fine voluminous powder, of light blue color and no odor, melting with the production of grease odor to a semi-opaque and somewhat waxy mass, easily crumbled when cooled. It slightly soluble in water. It melts over a range of l l l ' t o 113 . NICKEL Oleate-A deeply colored, bluish green oil a t ordinary temperature, sticky, and very easily oxidized. It is appreciably soluble in water. It melts indefinitely a t from 18" to 20". Stearate-A light greenish fluffy powder, smooth to the touch, melting a t about 100' C. to a pasty green liquid, hardening to a clear brittle glass. Heat causes a partial decomposition with liberation of free stearic acid, rendering the melting point uncertain. Palmitate-A very light greenish, almost white, powder, not so fluffy as the stearate but somewhat granular, and talc-like to the touch. It melts indefinitely near 80' C. t o a light green liquid which cools to a paraffin-like crystalline solid, with surface coating of white and colored interior. It appears to decompose somewhat with heat, with the liberation of free acid. Erucate-A white crystalline solid, with the odor ok erucic acid, softening a t low temperature, and melting a t about 35" C. similar to coconut fat between the fingers. Laurate-White crystalline flakes, paraffin-like, greasy, melting sharply to a clear liquid a t 44' C. Except for the oleate, the nickel salts are, or can be, prepared crystalline from solutions with many of the solvents. F u r t h e r Work in Progress

The authors are at present engaged in studying the solubilities of these soaps in the organic solvents, old and new, in the hope that methods for the separation of the fatty acids, based on differentials in solubility of these heavy metal salts, may be suggested. This work will be reported later. Literature Cited (1) Albuquerque, J . Chem. Soc , [i], 118, 216 (1920); Rezs ckrm. Pura a p p l t cada (11) I , reprint (1916).

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(2) Andre, Comfit. r e n d . , 175, 107 (1922). (3) Elsdon, “Chemistry and Examination of Edible Oils and Fats,” 1926. (4) Gardner, Coleman, Paint Mfrs. Assocn., U. S., Tech, Circ. 120 (1921). ( 5 ) Gates, J . Phys. Chem., 15, 101 (1911). (6) Jacobson and Holmes, J . Biol. Chem., 26, 55 (1916). (7) Klimont, J . grukt .Chem., 109, 266 (1925). (8) Koling, J . A m . Chem. SOL.,36, 951 (1916). (9) Lee, .4nn. R e p t . .4gr. Mort. Res. .?tu., Unri.. Bristol, 1917, 39 (1918).

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(10) Lewkowitsch, “Chemical Technology and Analysis of Oils, Fats, and Waxes,” Vol. I (1912). (11) Seave, Anolrst, 37, 399 (1921). (12) Oudemans, J . p r a k t . Chem., [ l ] , 81, 356 (1860); 99 (1866). (13) Rather and Reed, Arkansas Agr. Expt. S a . , Bull. 156, 32 (1918). (14) Veimarn, von. J . Russ. Phys. Chem. .Tot., Proc. 1916, 48, 703. (15) Wright and Mitchell, “Oils, Fats, Waxes, and Their Manufactured Products,” 1921

Motor-Fuel Volatility’ 11--S tarting Volatili ty*,3 George Granger Brown, Charles L. Nickols, and Paul Bigby UNIVERSITY OF MICHIGAN, ANN ARBOR,b f I C H .

HE m o s t i m p o r t a n t characteristic of a fuel is its ability to start the motor in which it is to be u s e d . M o t o r - c a r drivers readily recognize difft,’rences in the effect which various fuels have on the ease of starting and frequently judge the value of a fuel almost entirely on this basis. It is extremely important for both c o n s u m e r and producer to u n d e r s t a n d what qualities m a k e a fuel effective i n starting the motor easily and quickly.

In the first paper of this series, published previously

On the assumption that t h e decrease in pressure in the satisfactory method was described for its determinamanifold tends to compensate tion. In the papers comprising the present group, for the decreased temperature motor-fuel volatility has been interpreted in terms of due to the latent heat of engine performance. vaporization and also for the The second paper of the series, which is the first of short time interval allowed the present group, relates equilibrium volatility to for engine distillation, the ease of starting as determined by actual engine tests. r e s u l t a n t mixture attained The third paper defines effective volatility under driving in the equilibrium air-distilconditions and indicates how it may be determined lation may be regarded as from the equilibrium volatility or A. S. T. M. distillathe ideal attainable only in tion data. The fourth paper relates effective volainfinite time in the engine tility to engine performance and suggests the A. S. T. M . d i s t i l l a t i o n . Thus the distillation characteristics required for satisfactory efficiency of the engine disperformance under different conditions of atmospheric tillation may be defined as temperature. The fifth paper discusses the relation the ratio of the air-vapor of vapor pressure to vapor lock and suggests the fuel ratio attained in the equilibPrevious Investigations characteristics requisite to insure freedom from this r i u m d i s t i l l a t i o n to that trouble. During the past six years attained in the engine distilan extensive investigation of lation. This efficiencv deengine starting has been conducted at the Bureau of Stand- pends only slightly upon such factors as temperature; fuel, ards as part, of the cooperative fuel research program. Cragoe mixture-ratio supplied, and cranking speed, but does depend and Eisinger (5)identify the vaporization of fuel in bhe car- almost entirely upon the number of engine revolutions. buretor and manifold of a motor during the cranking period Cragoe and Eisinger concluded that about ten revolutions as “engine distillation.” This engine distillation is not a of the engine were required to obtain an explosive mixture in simple equilibrium distillation, such as was described in the cylinders when a mixture was supplied a t the lower end Part I of this series 07). It does not take place at at’mos- of the manifold, which would give a resultant air-vapor pheric pressure nor necessarily at atmospheric temperature. mixture of 12 to 1 under equilibrium conditions. They On account of the choke and throttle, the absolute pressure derived the following equation which fitted their experiin the manifold during t he cranking period may be considerably mental data with an accuracy of about *25 per cent: less than atmospheric. This would tend to cause more of 2 log R E ~ = D 1.301 - y7 the fuel to vaporize than would be the case in an equilibrium *V air distillation. The latent heat of vaporization of the fuel where -READ = resultant air-vapor mixture attained in equilibrium-air distillation must come eit’her from the surrounding metal part of the carN = number of engine revolutions buretor and manifold, from t’he air, or from the liquid fuel itself. The absorption of bhis latent heat lowers the tem- This equation corresponds to a 20-to-1 air-vapor mixture reperature of the mixture and tends t o decrease the amount of quired in the cylinders for starting. fuel vaporized. Furthermore, the short time interval for the Cold-Room Tests completion of the engine distillation clearly indicates that equilibrium conditions would not be obtained during the Since these conclusions were based entirely upon tests cranking period. made over a limited temperature range and with the special set-up described by Eisinger, it seemed desirable to 1 The investigations reported in this series of papers have been carried out during the past four years in the Chemical Engineering Laboratory make a thorough investigation of the ease of starting a cold of the University of Michigan. Owing to the interdependence of various motor with widely different fuels when the motor was driven phases of the work, publication of the results has been delayed. More by the electric starter supplied from the same battery as was complete information than can be given in this series of papers will he used for ignition and with the standard carburetor mounted found in Bulletin 14 of the Department of Engineering Research of the University of Michigan, available in the near future. on the manifold instead of the special fuel system used in * Received April 26, 1930. E * Part of a thesis submitted by Charles I,. Nickols in partial fulfilment the tests a t the Bureau of Standards. ilPPARATuS-The tests reported in this investigation were of the requirements for the degree of doctor of philosophy at the University made on a small, water-cooled, four-cylinder, valve-in-head of Michigan.

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(2, Part I), equilibrium volatility was defined and a