The details of saponification with sulfuric acid are of no interest
from an analytical point of view.[307]
=341. Saponification in an Open Dish.=—The simplest method of
saponifying fats is to treat them with the alkaline reagent in an
open dish. In all cases the process is accelerated by the application
of heat. Vigorous stirring also aids the process by securing a more
intimate mixture of the ingredients. This method of decomposing
glycerids, however, is not applicable in cases where volatile ethers
may be developed. These ethers may escape saponification and thus
prevent the formation of the maximum quantity of soap. While not suited
to exact quantitive work, the method is convenient in the preparation
of fat acids which are to be the basis of subsequent analytical
operations, as, for instance, in the preparation of fat acids for
testing with silver nitrate. Large porcelain dishes are conveniently
used and the heat is applied in any usual way, with care to avoid
scorching the fat.
=342. Saponification under Pressure.=—The method of saponification
which has given the best satisfaction in my work and which has been
adopted by the Association of Official Agricultural Chemists is
described below.[308]
_Reagents._—The reagents employed are a solution of pure potash
containing 100 grams of the hydroxid dissolved in fifty-eight grams of
recently boiled distilled water, alcohol of approximately ninety-five
per cent strength redistilled over caustic soda, and sodium hydroxid
solution prepared as follows:
One hundred grams of sodium hydroxid are dissolved in 100 cubic
centimeters of distilled water. The caustic soda should be as free as
possible from carbonates, and be preserved from contact with the air.
_Apparatus._—A saponification flask; it has a round bottom and a ring
near the top, by means of which the stopper can be tied down. The flask
is arranged for heating as shown in Fig. 101. A pipette graduated to
deliver forty cubic centimeters is recommended as being more convenient
than a burette for measuring the solutions: A pipette with a long stem
graduated to deliver 5.75 cubic centimeters at 50°.
_Manipulation._—The fat to be examined should be melted and kept in a
dry warm place at about 60° for two or three hours, until the water
has entirely separated. The clear supernatant fat is poured off and
filtered through a dry filter paper in a jacket funnel containing
boiling water. Should the filtered fat, in a fused state, not be
perfectly clear, it must be filtered a second time. The final drying is
accomplished at 100° in a thin layer in a flat bottom dish, in partial
vacuum or an atmosphere of inert gas.
The saponification flasks are prepared by thoroughly washing with
water, alcohol, and ether, wiping perfectly dry on the outside,
and heating for one hour at the temperature of boiling water. The
hard flasks used in moist combustions with sulfuric acid for the
determination of nitrogen are well suited for this work. The flasks
should be placed in a tray by the side of the balance and covered with
a silk handkerchief until they are perfectly cool. They must not be
wiped with a silk handkerchief within fifteen or twenty minutes of the
time they are weighed or else the electricity developed will interfere
with weighing. The weight of the flasks having been accurately
determined, they are charged with the melted fat in the following way:
[Illustration: FIG. 101.—APPARATUS FOR SAPONIFYING UNDER PRESSURE.]
The pipette with a long stem, marked to deliver 5.75 cubic centimeters,
is warmed to a temperature of about 50°. The fat, having been poured
back and forth once or twice into a dry beaker in order to thoroughly
mix it, is taken up in the pipette, the nozzle of the pipette having
been previously wiped to remove any externally adhering fat, is carried
to near the bottom of the flask and 5.75 cubic centimeters of fat
allowed to flow into the flask. After the flasks have been charged
in this way they should be re-covered with the silk handkerchief and
allowed to stand for fifteen or twenty minutes, when they are again
weighed.
=343. Methods of Saponification.=—_In the Presence of Alcohol._—Ten
cubic centimeters of ninety-five per cent alcohol are added to the fat
in the flask, and then two cubic centimeters of the sodium hydroxid
solution. A soft cork stopper is inserted and tied down with a piece
of twine. The saponification is completed by placing the flask upon
the water or steam-bath. The flask during the saponification, which
should last one hour, should be gently rotated from time to time, being
careful not to project the soap for any distance up its sides. At the
end of an hour the flask, after having been cooled to near the room
temperature, is opened.
_Without the Use of Alcohol._—To avoid the danger of loss from the
formation of ethers, and the trouble of removing the alcohol after
saponification, the fat may be saponified with a solution of caustic
potash in a closed flask without using alcohol. The operation is
carried on exactly as indicated above for saponification in the
presence of alcohol, using potassium instead of sodium hydroxid
solution. For the saponification, use two cubic centimeters of the
potassium hydroxid solution which are poured on the fat after it has
solidified in the flask. Great care must be taken that none of the
fat be allowed to rise on the sides of the saponifying flask to a
point where it cannot be reached by the alkali. During the process of
saponification the flask can only be very gently rotated in order to
avoid the difficulty mentioned. This process is not recommended with
any apparatus except a closed flask with round bottom. Potash is used
instead of soda so as to form a softer soap and thus allow a more
perfect saponification.
The saponification may also be conducted as follows: The alkali and fat
in the melted state are shaken vigorously in the saponification flask
until a complete emulsion is secured. The rest of the operation is then
conducted as above.
=344. Saponification in the Cold.=—By reason of the danger of loss
from volatile ethers in the hot alcoholic saponification, a method
for successfully conducting the operation in the cold is desirable.
Such a process has been worked out by Henriques.[309] It is based
upon the previous solution of the fat in petroleum ether, in which
condition it is so easily attacked by the alcoholic alkali as to make
the use of heat during the saponification unnecessary. The process is
conveniently conducted in a porcelain dish covered with a watch glass.
Five grams of the fat are dissolved in twenty-five cubic centimeters
of petroleum ether and treated with an equal quantity of four per cent
alcoholic soda lye. The process of saponification begins at once and
is often indicated by the separation of sodium salts. It is best to
allow the action to continue over night and, with certain difficultly
saponifiable bodies, such as wool fat and waxes, for twenty-four hours.
In the case of butter fat an odor of butyric ether may be perceived at
first but it soon disappears. After the saponification is complete,
the excess of alkali is determined by titration in the usual way with
set hydrochloric acid, using phenolphthalien as indicator. For the
determination of volatile acids, the mixture, after saponification is
complete, is evaporated rapidly to dryness, the solid matter being
reduced to powder with a glass rod, after which it is transferred
to a distilling flask and the volatile acids secured by the usual
processes. In comparison with the saponification and reichert-meissl
numbers obtained with hot alcoholic potash, the numbers given by the
cold process are found to be slightly higher with those fats which give
easily volatile ethers. On account of the simplicity of the process
and the absence of danger of loss from ethers, it is to be recommended
instead of the older methods in case a more extended trial of it should
establish the points of excellence claimed above.
=345. Saponification Value.=—The number of milligrams of potassium
hydroxid required to completely saturate one gram of a fat is known as
the saponification value of the glycerid. The process of determining
this value, as worked out by Koettstorfer and modified in the
laboratory of the Department of Agriculture, is as follows:[310]
The saponification is accomplished with the aid of potassium hydroxid
and in the flask and manner described in the preceding paragraph. About
two grams of the fat will be found a convenient quantity. Great care
must be exercised in measuring the alkaline solution, the same pipette
being used in each case and the same time for draining being allowed in
every instance. Blanks are always to be conducted with each series of
examinations. As soon as the saponification is complete, the flask is
removed from the bath, allowed to cool and its contents are titrated
with seminormal hydrochloric acid and phenolphthalien as indicator.
The number expressing the saponification value is obtained by
subtracting the number of cubic centimeters of seminormal hydrochloric
acid required to neutralize the alkali after saponification from
that required to neutralize the alkali of the blank determinations,
multiplying the result by 28.06 and dividing the product by the number
of grams of fat employed.
_Example._—Weight of sample of fat used 1.532 grams: Number of cubic
centimeters half normal hydrochloric acid required to saturate blank,
22.5: Number of cubic centimeters of half normal hydrochloric acid
required to saturate the alkali after saponification 12.0: Difference,
10.5 cubic centimeters:
Then 10.50 × 28.06 ÷ 1.532 = 192.3.
This latter number represents the saponification value of the sample.
=346. Saponification Equivalent.=—Allen defines the saponification
equivalent as the number of grams of fat saponified by one equivalent,
_viz._, 56.1 grams of potassium hydroxid.[311] The saponification
equivalent is readily calculated from the saponification value using
it as a divisor and 56100 as a dividend. Conversely the saponification
value may be obtained by dividing 56100 by the saponification
equivalent. No advantage is gained by the introduction of a new term so
nearly related to saponification value.
=347. Saponification Value of Pure Glycerids.=—The theoretical
saponification values of pure glycerids are given in the following
table.[312]
Molecular Saponification
Name. Symbol. weight. value.
Butyrin C₃H₅(O.C₄H₇O)₃ 302 557.3
Valerin C₃H₅(O.C₅H₉O)₃ 344 489.2
Caproin C₃H₅(O.C₆H₁₁O)₃ 386 438.3
Caprin C₃H₅(O.C₁₀H₁₉O)₃ 554 305.0
Laurin C₃H₅(O.C₁₂H₂₃O)₃ 638 263.8
Myristin C₃H₅(O.C₁₄H₂₇O)₃ 722 233.1
Palmitin C₃H₃(O.C₁₆H₃₁O)₃ 806 208.8
Stearin C₃H₅(O.C₁₈H₃₅O)₃ 890 189.1
Olein C₃H₅(O.C₁₈H₃₃O)₃ 884 190.4
Linolein C₃H₅(O.C₁₈H₃₁O)₃ 878 191.7
Ricinolein C₃H₅(O.C₁₈H₃₃O₂)₃ 932 180.6
Euricin C₃H₅(O.C₂₂H₁₄O)₃ 1052 160.0
From the above table it is seen that in each series of glycerids the
saponification equivalent falls as the molecular weight rises.
=348. Acetyl Value.=—Hydroxy acids and alcohols, when heated with
glacial acetic acid, undergo a change which consists in substituting
the radicle of acetic acid for the hydrogen atom of the alcoholic
hydroxyl group. This change is illustrated by the equations below:[313]
_For a Fat Acid_:
Ricinoleic acid. Acetic anhydrid.
C₁₇H₃₂(OH).COOH + (C₂H₃O)₂O =
Acetyl-ricinoleic acid. Acetic acid.
C₁₇H₃₂(O.C₂H₃O)COOH + HC₂H₃O₂.
_For an Alcohol_:
Cetyl alcohol. Acetic anhydrid. Cetyl acetate. Acetic acid.
C₁₆H₃₃.OH + (C₂H₃O)₂O = C₁₆H₃₃.C₂H₃O + HC₂H₃O₂.
_Determination._—The method of determining the acetyl value of a fat or
alcohol has been described by Benedikt and Ulzer.[314] The operation is
conducted on the fat acids and not on the glycerids containing them.
The insoluble fat acids are prepared as directed in paragraph =340=.
From twenty to fifty grams of the fat acids are boiled with an equal
volume of acetic anhydrid, in a flask with a reflux condenser, for two
hours. The contents of the flask are transferred to a larger vessel
of about one liter capacity, mixed with half a liter of water and
boiled for half an hour. To prevent bumping, some bubbles of carbon
dioxid are drawn through the liquid by means of a tube drawn out to
a fine point and extending nearly to the bottom of the flask. The
liquids are allowed to separate into two layers and the water is
removed with a syphon. The oily matters are treated several times with
boiling water until the acetic acid is all washed out. The acetylated
fat acids are filtered through a dry hot jacket filter and an aliquot
part, from three to five grams, is dissolved in absolute alcohol.
After the addition of phenolphthalien the mixture is titrated as in
the determination of the saponification value. The acid value thus
obtained is designated as the acetyl acid value. A measured quantity
of alcoholic potash, standardized by seminormal hydrochloric acid,
is added, the mixture boiled and the excess of alkali determined by
titration. The quantity of alkali consumed in this process measures the
acetyl value. The sum of the acetyl acid and the acetyl values is the
acetyl saponification value. The acetyl value is therefore equal to
the difference of the saponification and acid values of the acetylated
fat acids. In other words, the acetyl value indicates the number of
milligrams of potassium hydroxid required to neutralize the acetic acid
obtained by the saponification of one gram of the acetylated fat acids.
_Example._—A portion of the fat acids acetylated as described, weighing
3.379 grams, is exactly neutralized by 17.2 cubic centimeters of
seminormal potassium hydroxid solution, corresponding to 17.2 × 0.02805
= 0.4825 gram of the hydroxid, hence 0.4825 × 1000 ÷ 3.379 = 142.8, the
acetyl acid value of the sample.
After the addition of 32.8 cubic centimeters more of the seminormal
potash solution, the mixture is boiled to saponify the acetylated
fat acids. The residual potash requires 14.2 cubic centimeters of
seminormal hydrochloric acid. The quantity of potash required for the
acetic acid is therefore 32.8 - 14.3 = 18.5 cubic centimeters or 18.5 ×
0.02805 = 0.5189 gram of potassium hydroxid. Then 0.5189 × 1000 ÷ 3.379
= 153.6 = acetyl value of sample. The sum of these two values, _viz._,
142.8 and 153.6 is 296.4, which is the acetyl saponification value of
the sample. As with the iodin numbers, however, it is also found that
acids of the oleic series give an acetyl value when treated as above,
and it has been proposed by Lewkowitsch to determine, in lieu of the
data obtained, the actual quantity of acetic acid absorbed by fats.[315]
This is accomplished by saponifying the acetylated product with
alcoholic potash and determining the free acetic acid by distillation,
in a manner entirely analogous to that used for estimating volatile fat
acids described further on.
The rôle which the acetyl value plays in analytical determinations is
interesting, but the data it gives are not to be valued too highly.
=349. Determination of Volatile Fat Acids.=—The fat acids which are
volatile at the temperature of boiling water, consist chiefly of
butyric and its associated acids occurring in the secretions of the
mammary glands. Among vegetable glycerids cocoanut oil is the only
common one which has any notable content of volatile acids. The boiling
points of the above acids, in a pure state, are much higher than the
temperature of boiling water; for instance, butyric acid boils at about
162°. By the expression volatile acids, in analytical practice, is
meant those which are carried over at 100°, or a little above, with
the water vapor, whatever be their boiling point. The great difficulty
of removing the volatile from the non-volatile fat acids has prevented
the formulation of any method whereby a sharp and complete separation
can be accomplished. The analyst, at the present time, must be content
with some approximate process which, under like conditions, will give
comparable results. Instead, therefore, of attempting a definite
determination, he confines his work to securing a partial separation
and in expressing the degree of volatile acidity in terms of a standard
alkali. To this end, a definite weight of the fat is saponified, the
resulting soap decomposed with an excess of fixed acid, and a definite
volume of distillate collected and its acidity determined by titration
with decinormal alkali. The weight of fat operated on is either two and
a half[316] or five grams.[317]
Numerous minor variations have been proposed in the process, the most
important of which is in the use of phosphoric instead of sulfuric
acid in the distillation. An extended experience with both acids has
shown that no danger is to be apprehended in the use of sulfuric acid
and that on the whole it is to be preferred to phosphoric.[318]
The process as used in this laboratory and as adopted by the official
agricultural chemists is conducted as follows:[319]
=350. Removal of the Alcohol.=—The saponification is accomplished in
the manner already described, (=341-344=) and when alcoholic potash is
used proceed as follows:
The stopper having been laid loosely in the mouth of the flask, the
alcohol is removed by dipping the flask into a steam-bath. The steam
should cover the whole of the flask except the neck. After the alcohol
is nearly removed, frothing may be noticed in the soap, and to avoid
any loss from this cause or any creeping of the soap up the sides of
the flask, it should be removed from the bath and shaken to and fro
until the frothing disappears. The last traces of alcohol vapor may be
removed from the flask by waving it briskly, mouth down, to and fro.
_Dissolving the Soap._—After the removal of the alcohol the soap
should be dissolved by adding 100 cubic centimeters of recently
boiled distilled water, or eighty cubic centimeters when aqueous
potassium hydroxid has been used for saponification, and warming on the
steam-bath, with occasional shaking, until the solution of the soap is
complete.
_Setting free the Fat Acids._—When the soap solution has cooled to
about 60° or 70°, the fat acids are separated by adding forty cubic
centimeters of dilute sulfuric acid solution containing twenty-five
grams of acid in one liter, or sixty cubic centimeters when aqueous
potassium hydroxid has been used for saponification.
_Melting the Fat Acid Emulsion._—The flask is restoppered as in the
first instance and the fat acid emulsion melted by replacing the flask
on the steam-bath. According to the nature of the fat examined, the
time required for the fusion of the fatty acid emulsions may vary from
a few minutes to several hours.
_The Distillation._—After the fat acids are completely melted, which
can be determined by their forming a transparent, oily layer on the
surface of the water, the flask is cooled to room temperature, and
a few pieces of pumice stone added. The pumice stone is prepared by
throwing it, at a white heat, into distilled water, and keeping it
under water until used. The flask is connected with a glass condenser,
Fig. 102, slowly heated with a naked flame until ebullition begins, and
then the distillation continued by regulating the flame in such a way
as to collect 110 cubic centimeters of the distillate in, as nearly as
possible, thirty minutes. The distillate should be received in a flask
accurately marked at 110 cubic centimeters.
[Illustration: FIG. 102.—APPARATUS FOR THE DISTILLATION OF VOLATILE
ACIDS.]
_Titration of the Volatile Acids._—The 110 cubic centimeters of
distillate, after thorough mixing, are filtered through perfectly
dry filter paper, 100 cubic centimeters of the filtered distillate
poured into a beaker holding about a quarter of a liter, half a cubic
centimeter of phenolphthalien solution added and decinormal barium
hydroxid solution run in until a red color is produced. The contents
of the beaker are then returned to the measuring flask to remove any
acid remaining therein, poured again into the beaker, and the titration
continued until the red color produced remains apparently unchanged for
two or three minutes, The number of cubic centimeters of decinormal
barium hydroxid solution required should be increased by one-tenth to
represent the entire distillate.
The number thus obtained expresses, in cubic centimeters of decinormal
alkali solution, the volatile acidity of the sample. In each case
blank distillations of the reagents used should be conducted under
identical conditions, especially when alcoholic alkali is used for
saponification. It is difficult to secure alcohol which will not yield
a trace of volatile acid in the conditions named. The quantity of
decinormal alkali required to neutralize the blank distillate is to be
deducted from that obtained with the sample of fat.
=351. Determination of Soluble and Insoluble Fat Acids.=—The volatile
fat acids are more or less soluble in water, while those which are
not distillable in a current of steam are quite insoluble. It is
advisable, therefore, to separate these two classes of fat acids, and
the results thus obtained are perhaps more decidedly quantitive than
are given by the distillation process just described. The methods used
for determining the percentage of insoluble acids are essentially those
of Hehner.[320] Many variations of the process have been proposed,
especially in respect of the soluble acids.[321]
The process, as conducted in this laboratory and approved by the
Association of Official Agricultural Chemists, is as follows:
_Preparation of Reagents.—Sodium Hydroxid Solution._—A decinormal
solution of sodium hydroxid is used. Each cubic centimeter contains
0.0040 gram of sodium hydroxid and neutralizes 0.0088 gram of butyric
acid (C₄H₈O₂).
_Alcoholic Potash Solution._—Dissolve forty grams of good caustic
potash in one liter of ninety-five per cent alcohol redistilled over
caustic potash or soda. The solution must be clear and the potassium
hydroxid free from carbonates.
_Standard Acid Solution._—Prepare accurately a half normal solution of
hydrochloric acid.
_Indicator._—Dissolve one gram of phenolphthalien in 100 cubic
centimeters of ninety-five per cent alcohol.
_Determination.—Soluble Acids._—About five grams of the sample are
placed in the saponification flask already described, fifty cubic
centimeters of the alcoholic potash solution added, the flask stoppered
and placed in the steam-bath until the fat is entirely saponified. The
operation may be facilitated by occasional agitation. The alcoholic
potash is always measured with the same pipette and uniformity further
secured by allowing it to drain the same length of time (thirty
seconds). Two or three blank experiments are conducted at the same time.
In from five to thirty minutes, according to the nature of the fat, the
liquid will appear perfectly homogeneous and, when this is the case,
the saponification is complete and the flask is removed and cooled.
When sufficiently cool, the stopper is removed and the contents of
the flask rinsed with a little ninety-five per cent alcohol into an
erlenmeyer, of about 200 cubic centimeters capacity, which is placed on
the steam-bath together with the blanks until the alcohol is evaporated.
The blanks are titrated with half normal hydrochloric acid, using
phenolphthalien as indicator, and one cubic centimeter more of the half
normal hydrochloric acid than is required to neutralize the potash in
the blanks is run into each of the flasks containing the fat acids. The
flask is connected with a reflux condenser and placed on the steam-bath
until the separated fat acids form a clear stratum on the upper surface
of the liquid. The flask and contents are cooled in ice-water.
The fat acids having quite solidified, the liquid contents of the
flask are poured through a dry filter into a liter flask, taking care
not to break the cake. Between 200 and 300 cubic centimeters of water
are brought into the flask, the cork with the condenser reinserted
and the flask placed on the steam-bath until the cake of acid is
thoroughly melted. During the melting of the cake of fat acids, the
flask should occasionally be agitated with a rotary motion in such a
way that its contents are not made to touch the cork. When the fat
acids have again separated into an oily layer, the flask and its
contents are cooled in ice-water and the liquid filtered through the
same filter into the same liter flask as before. This treatment with
hot water, followed by cooling and filtration of the wash water, is
repeated three times, the washings being added to the first filtrate.
The mixed washings and filtrate are made up to one liter, and 100
cubic centimeters thereof in duplicate are titrated with decinormal
sodium hydroxid. The number of cubic centimeters of sodium hydroxid
required for each 100 cubic centimeters of the filtrate is multiplied
by ten. The number so obtained represents the volume of decinormal
sodium hydroxid neutralized by the soluble fat acids of the fat, plus
that corresponding to the excess of the standard acid used, _viz._, one
cubic centimeter. The number is therefore to be diminished by five,
corresponding to the excess of one cubic centimeter of half normal
acid. This corrected volume multiplied by 0.0088 gives the weight of
soluble acids as butyric acid in the amount of fat saponified.
_Insoluble Acids._—The flask containing the cake of insoluble fat acids
from the above determination and the paper through which the soluble
fat acids have been filtered are allowed to drain and dry for twelve
hours, when the cake, together with as much of the fat acids as can be
removed from the filter paper, is transferred to a weighed evaporating
dish. The funnel, with the filter, is then placed in an erlenmeyer and
the paper thoroughly washed with absolute alcohol. The flask is rinsed
with the washings from the filter paper, then with pure alcohol, and
the rinsings transferred to the evaporating dish. The dish is placed
on the steam-bath until the alcohol is evaporated, dried for two hours
at 100°, cooled in a desiccator and weighed. It is again placed in the
air-bath for two hours, cooled as before and weighed. If there be any
considerable decrease in weight, reheat two hours and weigh again. The
final weighing gives the weight of insoluble fat acids in the sample,
from which the percentage is easily calculated.
The quantity of non-volatile and insoluble acids in common glycerids
is from ninety-five to ninety-seven parts in 100. The glycerids yield
almost the same proportion of fat acids and glycerol when the acids are
insoluble and have high molecular weights. When the acids are soluble
and the molecular weight low the proportion of acids decreases and that
of glycerol increases.
In the following table will be found the data secured by quantitive
saponification and separation of soluble and insoluble acids found in
the more common glycerids:[322]
Yield per 100 parts
Molecular weight of of glycerid.
Glycerid. Fat acid. Glycerid. Fat acid. Fat acid. Glycerol.
Stearin Stearic 890 284 95.73 10.34
Olein Oleic 884 282 95.70 10.41
Palmitin Palmitic 806 256 95.28 11.42
Myristin Myristic 722 228 94.47 12.74
Laurin Lauric 638 200 94.95 14.42
Caprin Capric 594 172 93.14 15.48
Caproin Caproic 386 116 90.16 23.83
Butyrin Butyric 302 88 87.41 30.46
The general expression for the saponification of a neutral fat is
C₃H₅O₃.R₃ + 3H₂O = 3R.OH + C₃H₈O₃, in which R represents the acid
radicle. It is evident from this that the yield of more than 100 parts
of fat acids and glycerol given by glycerids is due to the absorption
of water during the reaction.
=352. Formulas for General Calculations.=—For calculating the
theoretical yields of fat acids and glycerol, the following general
formulas may be used:
Let _M_ = the molecular weight of the fat acid:
_K_ = saponification value:
_F_ = the quantity of free fat acids in the glycerid:
_N_ = the quantity of neutral fat in the glycerid:
_A_ = the number of milligrams of potassium hydroxid required
to saturate the free acid in one gram of the
sample.
The free acid is determined by the method given below.
_M_ grams of a fat acid require 56100 milligrams of potassium hydroxid
for complete neutralization while _F_ grams corresponding to 100 grams
of fat are saturated by 100 × _A_ milligrams of the alkali.
Then _M_ : 56100 = _F_ : 100_A_.
_AM_
Whence _F_ = ------ (1).
561
Likewise since _M_ grams of fat acid require the quantity of potassium
hydroxid mentioned above we have: