l H l H” l I I _ ' H II I I I I ll I m I I I I A CRITICAL EVALUATION OF FOUR METHODS 0F SODIUM ANALYSIS IN FOODS WITH SPECIAL EMPHASIS ON THE USE OF THE SODIUM ION ELECTRODE Thesis for the Degree of M. (S. MICHIGAN STATE UNIVERSITY Yolanda Nifi'o-Orozco 1964 THESIS LIBRARY Michigan State University nBSTRACT A CRITICAL EVALUATION OF FOUR.METHODS OF SODIUM ANALYSIS IN FOODS WITH SPECIAL EMPHASIS ON THE USE OE THE SODIUM ION ELECTRODE by Yolanda NiHo-Orozco Sodium concentration in foods varies from traces in certain fresh vegetables to more than ten thousand parts per million in some processed foods. The sodium analysis in foods is important since sodium restricted diets are used in the treatment of certain diseases. Four methods were evaluated in the determination of sodium in foods of widely different composition: The standard A.O.A.C. magnesium-uranyl-acetate and flame photometry methods were used as reference. Alpha- methoxy-phenyl acetic acid (MOPA) as well as a sodium ion electrode were compared to the above. Very good precision was obtained when using the reference methods. When working with milk and buttermilk, the magnesium- uranyl acetate method gave much higher results than those found with other methods. The determination of sodium in foods by the MOPA method was not successful except where the concentrations were above 20x102 ppm. 2 The sodium ion electrode method proved to be fast and convenient. In the proportions usually found in foods, fats, carbohydrates and proteins did not affect the reSponse of the electrode, but some changes in the recovery of added sodium occurred during canning and freezing. Some modifica- tions will be necessary for the analysis of meats, since in this product the results obtained differed widely from the standard methods. Overall, the sodium ion electrode presents a good possibility as a standard method of analysis in foods. A mis padres. A Critical Evaluation of Four Methods of Sodium Analysis in Foods with Special Emphasis on the Use of the Sodium Ion Electrode bv . r‘ , ' fi“l" I .‘f..- Yolanda Nino-Orozco Submitted to Michigan State University in partial fulfillment of the requirements for the degree of IVIhSTER OF SCIENCE Department of Food Science 1964 27‘ 3 {/[g/ég/ AWIJOL‘ILEDGEZ‘IELJT The writer wishes to eXpress her sincere gratitude to Dr. Sigmund H. Schanderl for his tireless efforts in guiding the work of this study. Appreciation is extended to Dr. Pericles Markakis for his valuable suggestions during the course of this research, and to Dr. Dorothy Arata for reviewing the manu- script. The author wishes also to thank Dr. Clifford L. Bedford for his worthy assistance. TABLE OF CONTENTS Page __i_ INTRODUCTION . o o . . o . o o o . . . . . . o . . . . l LITEm—sTURERE's/IEW..-................4 a) Sodium in foods . . . . . . . . . . . . . . 4 b) Effects of heating in sodium content . . . 5 c) Methods for sodium analysis . . . . . . . . 6 l. Gravimetric methods. . . . . . . . . 6 2. Physical-chemical methods . . . . . . 9 3. Volumetric methods . . . . . . . . . 14 METHODS AND MATERIALS . . . . o o . . o . o o o o . . 15 a) Magnesium-uranyl-acetate . . . . . . . . . 16 b) Flame photometry . . . . . . . . . . . . . .16 c) Sodium ion electrode . . . . . . . . . . . .17 d) Alpha-methoxy phenyl acetic acid . . . . . .24 RESULTS ANDIDISCUSSION . . . . . o . . o o o o . . . . 27 SUMMARY AND CONCLUSIONS . . . . . . 5 . o . . . . . . 40 APPENDIX . o . o . . o o . . o . o o . . o o . . . . . 42 BBLIOGWHY O O O O O O O O O O“ O O O O O O O O O O O 49 ii LIST OF TABLES Variation in the Reading with Increasing the Time allowed for Sodium Ion Electrode to Reach Equilibrium . . . . . . . . . Effect of the Protein on the ReSponse of the Electrode . . . . . . . . . . . . . . . Effect of Carbohydrate Content on the Electrode ReSponse . . . . . . . . . . . . . . . Influence of Mineral Mixtures on the Electrode ReSponse . . . . . . . . . . . . . . . Influence of the Sample Dilution on the Sodium Electrode Response . . . . . . . . . . Effect of Processing Operations on the ReSponse for Sodium . . . . . . . . . .. . . . . 42 44 .45 46 47 48 Introduction Since Sodium plays a very important role in human metabolism and Sodium restricted diets have been necessary in the treatment of some cardiac and renal diseases, the quantitative analysis of this element in foods has become a very important topic. Several methods have been used for sodium analysis, but because of the high cost of the equipment and reagents of some, and to the time consumed by others, none of these methods has been universally accepted. Sodium is important in foods, 1. For the maintenance of the normal water balance and distribution, 2. for the maintenance of the normal acid-base equilibrium, 3. for the maintenance of the osmotic equilibrium, 4. for the maintenance of the normal muscle irritability. Balance eXperiments showed that approximately a liter of water is retained for every 6 or 7 g. of sodium chloride accumulated in the body (64). When an excess of sodium chloride is ingested by a normal person, swelling of the body occurs. When plasma proteins are low, the excessive use of salt also produces edema (64). In some cardiac patients it was observed that the sodium ingested was not excreted in the urine, nor was an increase-of sodium in the blood noticed. It was concluded that the sodium accumulated with water formed edema. (66) 2 Sodium exerts its buffer action usually reacting with either bicarbonate or Chloride ions (58), thus maintaining the acid base equilibrium of the body. Sodium ions have an effect on irritable tissues, such as muscle, which is counteracted by the presence of calcium ions (64). This is, at least in part, one of the bases for the recommendations of dietary salt restrictions in cases of hypertension and cardiac diseases. Sodium ions are found mainly in the extracellular, and potassium in the intracellular, fluids, both working together to maintain the normal osmotic pressure of the tissues and cells. When potassium is restricted in the diet, sodium may take its place without presenting any trouble until a critical point is reached. Most sodium is absorbed in the small intestine but some is absorbed in the stomach (64). Sodium is mainly ex- creted in the urine and perSpiration. Sodium deficiency in rats will adversely affect appetite, growth, and synthesis of both fat and protein, and finally produce death. (64). In humans, undesirable effects may be caused as a result of deficiency of sodium. However, no case has been reported in Which a diet caused death be- Cause of its deficiency in sodium. The analysis of sodium in foods was made by using in- direct methods, until around 1930 when application was made of the Streng's reagent for the precipitation of sodium as the triple salt: sodium-magnesium-uranyl-acetate. In the 3 past ten years, extensive study using flame SpectrOphotometry for sodium analysis in foods has been develOped. The method is based upon the measure of the characteristic light emitted when a solution of a salt of sodium is atomized as a mist into a gas flame. The work described here gives the results of a study of four methods applied to the analysis of sodium in foods - the above mentioned already studied, and two whose applic- ability has not yet been proved, namely, the alpha-methoxy- phenyl acetic acid, and the sodium ion electrode. The purpose of this study is to evaluate the two new methods as possible standards for sodium analysis in foods. LITERATURE REVIEW A. Sodium in foods In general, one can say that food of animal origin contains much sodium, while food of vegetable origin usually contains little sodium before processing. In the processing Operations to which foods are sub- jected, some sodium is generally added. Processed meats and fish usually contain much more sodium than the fresh products, because of the addition of salt either in the canning or freezing. Sometimes this addition is made in the washing of the foods before processing. The most common sodium compound generally added is sodium chloride, which is used as preservative and also to improve the taste of the food. Many manufactured foods which are not intentionally salted contain somewhat more sodium than the natural foods from which they are prepared, because the preservatives and additives contain compounds of sodium. Marked differences in the sodium content of lettuce, onion and tomatoes between different producing areas and between different seasons were found by Homer (35). If this is true for all vegetables, no generalization should be made about the sodium content of different items of the same family, and the data found for sodium content in foods would have only a very limited application. The following dataere the result of a series of analyses made by Bills, NcDonald, 5 and Hiedermeier (7). They found sodium to be present only as a trace element in unprocessed fruits, grain, nuts and legume seeds. Excluding coconut, the average content is about 2 mg per 100 9. Coconut is somewhat high in sodium, around 35 mg per 100 g. In leaf, stalk and root vegetables the sodium content varied. White potatoes contained but traces, while celery, beets, and chard contained 100-200 mg per 100 g. The fresh meats, fish, and fowl mostly contained from 50 to 100 mg of Na per 100 g . of edible portion. In each of the five fowls studied by Bills — chicken, duck, goose, quail and turkey - the breast meat always contained substan- tially less Na than the leg meat. Na may be related to the higher metabolism of these tissues. Four Species of widely used fish - catfish, cod, halibut and salmon - were analyzed in the fresh state. These Species were similar in Na con- tent ranging from 48 to 60 mg of Na per 100 g. Most of the Na in eggs is in the white. Cow's milk contained around 50 mg of sodium per 100 g. Butter and cheese contained more sodium because of the salt added. B. _§§fects of heating in sodium content 1. No matter what method of cooking is employed, meats always shrink and eXpress a portion of their juices as fats: as ordinarily prepared, meats may lose a fourth to a third of their raw weight. The readily soluble ions (includ- ing Na) are lost in prOportion to the expressed juices. 6 2. hhen cooking is by steam, weight loss or shrinkage is very close to the percentage loss of water and of soluble salts. 3. In boiling water, a leaching effect causes addition— al loss of soluble salts, but this loss is not nearly as large as might be eXpected. Neat just cooked through by boiling may lose only a negligible additional amount of soluble salts from leach. 4. In roasting and frying, fat is expressed and evaporation of water is intense: however, salt losses are low. (44). C. Methods for Na Analysis The determination of Na may be made by gravimetric methods, volumetric methods, and by using physical-chemical methods. 1. Gravinetric methods. Among gravimetric methods for Na analysis we find indirect and direct methods. a. Indirect methods were used first, based on the precipitation of Na and K as salts (usually as chlorides or sulfates): potassium was then separated, and by difference, sodium was calculated. I The first sodium determination was made in the 19th century when Fresenius (27) separated Ea and K as chlorides. Iotassium was precipitated from the mixture as chlorOplati- nate, and by difference sodium was estimated. This method was later modified by Talapatra (61) for sodium analysis in foods, by precipitating potassium as K0104. 7 At the beginning of the 20th centuryaineW'method for sodium analysis appeared. Sodium and potassium were precipitated as sulfates, and by difference after estimation of potassium as cobalto cyanide, sodium was calculated (26). It was not until 1913 that the Na analysis in foods began to take on interest to the scientists. The first estimations were made by Forbes (26). he moistened the sample with concentrated sulfuric or nitric acid, and ashed to destroy organic matter. PhOSphorus, magnesium, calcium, and barium interferred and were eliminated. Phosphorus was precipitated with magnesium hydroxide, calcium, and barium as carbonates and oxalates. Ammonium sulfate was used to pre- cipitate sodium and potassium as sulfates. The latter were filtered, washed, and weighed. Potassium sulfate was found from the precipitation of potassium as chlorOplatinate, and subtracted from the weight of the combined sulfates to obtain the amount of sodium in the sample. The method was later modified by Rhue (53), who suggested altering the process of ashing and the precipitation of calcium and iron: and by Husband (37), who introduced in 1927, the precipitation of potassium as cobalto cyanide instead of using chlorOplatinate. b. Qirgct Methods. Indirect methods were used until 1923 when Blanchetiere made application of the Streng's reagent for the detection and precipitation of sodium as the triple salt sodium-magnesium-uranyl-acetate (8). In 1942, 8 Fruit (48) presented a modification of this method, prolong- ing the precipitation period from 40 minutes to 24 hrs. at 20°C, and preparing the magnesium uranyl acetate solution so that it was saturated at this temperature. The new'method became official for fruits and fruit products, and later for foods in general. Many different methods have been deve10ped based on the magnesium uranyl acetate method, such as the colorimetric method described by Foulk (12), and the volumetric method described by Vuk and Sandor (65). Lack of correlation of the results obtained when using magnesium uranyl acetate was pointed out by Crepaz (l9), Perietzeana (47), and Barber and Kolthoff (2). The latter suggested the use of zinc uranyl acetate method instead of the magnesium uranyl acetate, since in the first case the analysis is faster, and provision is made for the precipita- tion of potassium. The zinc uranyl acetate, or Barber and Kolthoff's method, consists in the precipitation of sodium as sodium zinc uranyl acetate previous to the removal of potassium with ammonium perchlorate in 72% alcohol solution. According to Kolthoff (3), lithium, phosphate, and thiosulfate ions inter- fere and should be removed. The method has been widely studied by Collins (18) and Clifford, and Winkler (15, 16). 9 As in the case of the magnesium uranyl acetate method, variations in the method have been made such as the colorimetric methods described by McCance and Schipp (43) for high sodium foods, and the modified zinc uranyl acetate method, converting the precipitate to uranyl potassium ferrocyanide, as given by Bradbury (10). In 1931, Basset (4) used, for the first time, the Barber and Kolthoff method for the analysis of sodium in diets. In 1954, Clifford (16) found very close agreement when he compared the results obtained with the zinc uranyl acetate method to those obtained by SpectrOphotometry, while the magnesium uranyl acetate seemed to fail when low sodium quantities accompanied by much potassium (as in the case of legumes) were encountered. It should be noted that, in Spite of the fact that the magnesium uranyl acetate method is still the standard method for sodium analysis in foods given by the S.O.A.C., most of the research work on sodium analysis in foods, done in the last 30 years, has been carried out by following either zinc uranyl or flame photometric methods. Physical chemical methods. among the physical- chemical methods of sodium analysis are: a) The different colorimetric techniques using either magnesium-uranyl-acetate or zinc-uranyl—acetate. Caley (12) lO pointed out a method wherein sodium is precipitated with magnesium uranyl acetate, as in the gravimetric method, the precipitate dissolved in warm water and the depth of the color compared with a standard. McCance (43) made the precipitation with zinc uranyl acetate, and evaluated the uranium colorimetrically with potassium ferrocyanide. barnell (20) used the sane reagent, and washed the precipitate with ethyl acetate and acetic acid, then photometrically measured the color develOped by sulfosalicylic acid and sodium acetate. b) Determination Cf sodium by using flame photometry is based on the measurement of the energy emitted by a sarple at the sodium wave length as revised by Farrow (23). Flame spectrosc0py measures the light absorbed, at a selected wave length, when a light ray from a cathode is passed through the flame (52). -J C) By using radio-activation, Smales, using a Geiger counter, measured the intensity of the radiation from sodium24 as compared with a standard irradiated simultaneously (EO). Amiel and Feisch (1), employing the same principle, counted the photoneutrons in an assembly consisting of D20 and neutron detector. Castro and Smith (14) measured sodium in citrus fruit juices by activating neutrons, using the General atomic TRIGA reactor. d) The determination of sodium by ion exchange resins: Gabrielson (23) separated alkali metals from complex ll cyanides, using cation exchange resins of sulfonic type. Sodium and potassium were eluted as chlorides and measured either volumetrically or gravimetrically. e) Using column chromatography: Erlenmeyer (22) analyzed sodium and potassium in blood samples, by means of a thin column chromatography made of a Special peeking, which produced a sharp red zone with sodium. The length of the zone was directly preportional to the sodium concentration in the sample. P. Hsieh (46) used Amberlite IR-120., and measured the amount of sodium and potassium indirectly by titrating the excess of hydrochloric acid used in the elution. f) By means of the sodium ion electrode: the sodium ion activity is determined by measuring the potential whidh is developed between the sample and the solution which is inside the body of the electrode, as published by Leonard (39). a. Flame Photometry The principle of the instrument is that the sample is introduced into a flame, and the characteristic energy emitted at certain wave length, is measured by means of a photometer. This method is by far the most accepted, and has found wideSpread application, because it is simple and a minimum of preparation of the sample is required. 12 Lundergarth, in 1930, was the first to apply flame photometry to sodium analysis in plants and biological materials. Numerous investigations have been carried out on this subject, eSpecially in order to overcome interferences by altering the sample preparation (23) or by introducing modi- fications in the instruments as reviewed by Robinson (52). Bills (7) analyzed sodium in more than 100 foods and waters, applying, for the first time, flame photometry to the analysis of sodium in foods. Clifford (16) found very good correlation between results obtained with flame photometric and zinc uranyl acetate methods. Shoga Ogawa (57) obtained lower results than eXpected from the references, when he analyzed sodium in 31 foods by means of flame photometry. In 1956, Have (32) presented the results of the com- parison of flame photometry and zinc uranyl acetate method for sodium analysis in milk. His results gave good agree- ment, and suggested the adaptation of flame photometry as a simpler method for sodium analysis in foods. It should be pointed out that Have added calcium, in a concentration similar to that found in milk, to the Na81 solutions used for the drawing of the standard curve in the flame photo- metric method. Santini (55) found that flame photometry works better for food and feces after wet ashing with nitric acid - l3 perchloric acid mixture instead of dry ashing. sagakova (54) determined sodium in canned foods and raw materials by using a selenium photoelement with OS 12 orange filter, and 7.5% solutbn of COpper sulfate in sulfuric acid as additional filter. The determination took from 20- 30 mins (excluding ashing time). B. Sodium Ion_§;ectrode As early as 1923, Horowitz (36) studied the res- ponse of glass electrodes to monovalent ions other than hydro- gen - e.g., sodium and potassium. Sohiller extended his wofl: (55). However, the first comprehensive study of this subject was made in 1934 by Lengyel and Blum (38). Their aim was to establish the relationship between composition and the deveIOpment of hydrogen or sodium functions in the response of electrodes made from soda-lime glasses. They discovered the important fact that the trivalent oxides (especially those of aluminum and Boron) cause glasses to develop a sodium function when used to replace part of silica in the glass structure. Goremykin (30) studied some Lengyel and Blum glasses, and reported some correlation between composi- tion, pre-conditioning and reSponse time. In 1957, Eiseman and his co-workers (21) extended, Lengyel and Blum's work to sodium alumina silicate glasses containing as much as 18% (mol.) alumina. They found this glass to be highly Specific for sodium, but too refractory for practical manufacture. Leonard (39) found that a glass 14 with much lower alumina content (4% mol.) has about the same sensitivity and is much more adaptable to quantity manufacture of electrodes. Mattock (42) designed in 1962, two new electrodes which were highly Specific for sodium, and proved workable in 5 molar. Sulfate, the range from one molar down to nearly 10- calcium, and barium ions should be tolerated at high excess over sodium concentration. PhOSphate was tolerable at 50 fold excesses, over sodium at pH 7. He pointed out the pos- sibility that phOSphate ions would not be tolerable beyond this limit. Bower (9) proved that the sodium ion electrode method gives accurate results for sodium analysis in saline solutions. 3. Volumetric Hethods Aside from the volumetric method using magnesium uranyl acetate used by vuk (65) for sodium analysis in milk, before 1957 the literature does not mention any other volu- metric method which has had considerable importance. Alpha Lethoxy_pheny1 acetic acid. Wilkin Reeve presented, in 1957, a new reagent for sodium analysis: Alpha methoxy phenyl acetic acid (50). Sodium forms a precipitate (an acid salt) with an aqueous solution of this reagent, half-neutralized with tetra-methyl ammonium, or potassium hydroxide. The salt is titrated with sodium hydroxide. Oesper (45) used this method to analyze sodium in blood serum. 15 According to Reeve (50), lithium, potassium, and magnesium do not interfere. Calcium, strontium, and barium interfere and should be removed. methods and Katerials The methods for sodium analysis used here are the official magnesium-uranyl acetate method, the flame photo- metric method, the electrode method, and the alpha-methoxy phenyl acetic acid method. Some regulations, however, may be applied to all of them in general, such as the use of de- mineralizing water whenever water is necessary, and pyrex glassware, or any other material which does not free sodium which may contaminate the sample. The chemicals used were always C.P. or reagent quality. The storage flasks used were all made of plastic or of no—solvit glass. 1. Magnesium Uranyl Acetate Method This method is based on using magnesium uranyl acetate to precipitate sodium as the triple salt sodium- magnesium-uranyl acetate (Ea Mg (U02)3 (CI-I3COO)9 6HZQQ7. The procedure followed for the determination was the one given for foods by the A.O.A.C. official methods of Analysis (pp. 77). The sample was weighed (1 to 10 g.) in a quartz crucible, then moistened with (1+10) sulfuric acid (5 ml), dried in an oven, and ignited at BOO-600°C in order to destroy organic matter. The ash was dissolved in (2 to 5 ml”) l6 concentrated hydrochloric acid, and the phosphates precipi- tated and discarded as calcium phOSphate by adding calcium ions (as calcium chloride). Ammonium hydroxide was used to assure precipitation of the calcium phOSphate. The sodium salt was then precipitated with the reagent, allowed to stand for 24 hrs., filtered (over No. 5 Whatman filter paper) using Buchner funnel, washed with alcohol containing reagent, dried (at 105°C for half an hour), and weighed. The sodium content in ppm in the sample was calcu- lated according to the following equation: Wt of the ppte X;;0153 X 106 ppm Of SOdlum = Wt of the sample The value .0153 is the ratio at.Wt. of sodium/mol.wt. of sodium-magnesium-uranyl acetate, and it converts the grams precipitated salt to grams of sodium. Following the above procedure, samples of different foods were analyzed for sodium. 2. The Flame:photometric Method A Beckman D.B. SpectrOphotometer with flame attachment was used in the analysis. The standard method for analysis of plant materials was followed.1 ”he sample was weighed (0.5 to 5.0 g. according to the sodium content in the sample) into a 25 m1 volumetric ls.o.a.c. Standard methods of Analysis, pp. 76 (1961). 17 flask with glass stOpper, and ammonium oxalate2 stock solu- tion was added up to the mark. The solution was allowed to react for 15 minutes, Shaking at frequent intervals to assure the reaction between calcium and oxalate. The pre— cipitate was filtered off (through No. 2 Whatman filter paper) and the filtrate used for the determination. A standard curve was made with sodium chloride, at concentrations ranging from 0 to 0.02 Normal. The instrument was calibrated to read zero (0) with the extracting solution (ammonium oxalate), and 100 with 0.02 N sodium chloride solution, at 577 mp which proved to be the wave length of maximum absorption in this instrument. (The wave length suggested in the references was 589.0 mp). The standard curve is shown in graph No. 1. 3. Sodium Ion Electrode. a Beckman sodium ion electrode #39278, attached to a Beckman model 176 eXpanded scale pH meter, was the instrument used in measuring sodium in this method. This electrode is sensitive to sodium, ammonium, potassium, silver and hydrogen ions, giving the same theoretical relationship between electromotive forces and cationic activities as the already known pH meter. The main parts of the Beckman glass electrode are a sodium-alumino-silicate glass bulb filled with 0.1 N NaCl, 2Ammonium oxalate.24N (made dissolving 17g. of C 0 H 0 in water and dilute to one liter.) (EH45 6 4 2 -185 D 9 u . ‘ v -H..—_.V..._.- * _-. 3 . GRAPH No.1 FLAME SPECTROPHOTOMETRY LUMINGSITY .Vs. CONCENTRATION 577m) ( J 4 . H . . _ L _ . r r r L s . .LI . a J); .. q . . 1 . e . P1. . . a _ r - - Z :1 (IO- . 5 o O . V . O . . _, m . O » O . .. T...” +11! :0 . I) 0" ll, '1 O i . ,. . . A . l j)r| l1 0.)" I £41'lll . — . . . O . . . .I a ”in v1 I I1 aP'vllnrlllllnhPJ 4 p a a . . . o . i... .. . . . . . ~ -.»|+v) I .Iol. T3. ‘. l i'nolhl. . . . _ . . . . . . e L > . . . _ . ML 1 l I- 90 I O. 3. A ‘ i . oil . .IA | -Il v o .1. )‘Ill’xll’l!’ IOIII . . )0 ......|o. I 1. H.141} 4. . . . M. .. .14.-..J .34 i . . _ . u I ‘40 30 607080904190" - crv . ,.. . -._._..... - my...” W..- —._{. A .l 1.. . .. , . I i . '- —¢-..__._ _. _._—__..-_-ao—- -Hb—n _ I o I w . l | . __.____.____._— _- __ . _ . . _ . .... . . . . ... H . . A . . .. A 00‘ v _ . . . . . _ o _ b. . . q . .u. -.o‘».bll. . . _ ... a 30 10 Luminosity nritlunlc, 2 x 2 Cycles l9 ir1t:<:) which is immersed a silver-Silver chloride electrode. £3 sstzandard saturated potassium chloride calomel half cell, wvj;t:r1 fiber-type bridge, serves as the reference electrode. The measurement of the sodium ion activity is accom- plished with this electrode by determining the potential vikugixjh is develOped between the sample in the beaker and the ergpeecial solution inside the body of the electrode (0.1 N sodium chloride). The pdential observed for any given activity of sac>