For a general discussion of the calculations of pH in solutions of acids, bases,
salts, and mixtures, the students is referred to Chapter 3. Application of these derivations
are made in this chapter.
ACID-BASE INDICATORS
The indicators used in acidimetry and alkalimetry are either weak organic acids
(indicator acids) or weak organic bases (indicator bases), the acid from of which has a
color different from that of the conjugate base form. Using the Bronsted concept of acids
and bases we denote the acid form of an indicator by HI, which can have a charge of
zero, or one or more positive or negative charges. The conjugate base, denoted by I, has a
charge one less positive than that of HI. At a given pH the equilibrium between HI and I
is given by
HI H+ + I
(1)
[H+][I] = KHI
[HI]
(2)
in which KHI is the indicator constant.
[ I ] [ form with alkaline color ] H HI  I


HI
[ form with acid color ]
[H  ]
(3)
When [H+] has the same numerical value as KHI the ratio of [I] to [HI] becomes equal to
unity, meaning that 50 per cent of the indicator is present in the acid form and 50 per cent
in the alkaline form. From Eq. (3) it is seen that at any drogen-ion concentration both the
forms I and HI are present. The human eye has a limited ability to detect either of the two
colors when one of the two predominates. If the acid form of the indicator is red, for
example, and the alkaline form is yellow and we compare a solution consisting entirely of
the red form with one consisting of a mixture of 99 per cent red and 1 per cent yellow, the
eye will not be able to detect any difference in color between the two solutions.
Generally, this will still be true when the ratio of [HI] to [I] is approximately equal to 10.
In any solution in which the ratio [I]: [HI] is smaller than 0.1, we obtain by visual.
The equilibrium should be expressed in terms of activities instead of
concentrations. In general, we can neglect activity coefficients in the application of
indicators in acid-base titrations. In the colorimetric pH determination activity
coefficients must be considered.
Observation the impression that the indicator is present completely in the acid
form. Similarly, we get the impression that the indicator is present completel acid form.
Similarly, we get the impression that the indicator is present completely in the alkaline
form when the ratio [I] to [HI] is greater than 10. (These limits of
1
10
and 10 are somewhat
arbitrary, depending on the color sensitivity of the eye and the particular indicator, but are
quite close to the truth.)
From the above, then, it is clear that we see only the acid color of the indicator
when
[I ]
 0.1
[ HI ]
and only the basic color when
[I ]
 10
[ HI ]
Between the ratios 0.1 and 10 we observe an intermediate color. An indicator,
therefore, does not change its color abruptly from that of the acid form to that of the
alkaline form at a definite hydrogen-ion concentration but changes gradually over a
certain range of [H+]. If we assume that the two limits are determined by the ratio [I]/[HI]
> 0.1 < 10, it is found [Eq. (3) that]
[I ]
K
 101  HI
[ HI ]
[H ]
or [ H  ]  10 K HI
(4)
[I ]
K
 10  HI
[ HI ]
[H ]
or [ H  ]  101 K HI
(5)
At [H+] = 10KHI the indicator is present in the form, and at [H+] = 101 K HI it is
present in the alkaline form. The indicator changes color between
[ H  ]  10 K HI and
( acid color)
1
HI
10
( alkalinecolor)
K
or, expressed in pH, between
pH=pKHI+I
in which pKHI is-log KHI.
This range of pH over which the color change of the indicator takes place is called
the color-change interval of the indicator. The color-change interval is determined
experimentally with the aid of buffer solutions; its exact limits will depend more or less
upon the subjective judgment of the observer. Table 34.1 indicates that the color-change
interval of methyl orange is located between pH 3.1 (red) and 4.4 (yellow-orange). This
means then that the pronounced color change takes place between these two pH 2.9 and
4.3.
An enormous number of compounds with indicator properties are found in nature
and among the products of the laboratory. A selected list of indicators for volumetric
purposes (and also for the colorimetric determination of pH) is given in Table 34.1.
Compare I. M. Kolthoff, Acid-Base Indicators (trans, by C. Rosenblum),
Macmillan, New York, 1937;pp. 103 ff.
Table 34-1.
Color-Change Interval of Selected Indicators and Values of KHI at Room
Temperature
Scientific Name
Diphenylamino p-benzene sodium
sulfonate
Tropeoline OO
Thymolsulfonphthalein
Thymol blue
Dimethylaminoazobenzene
Methyl yellow
Dimethylaminoazobenzene sodium
sulfonate
Methyl organ
Water
Water
90% alcohol
a
a
b
Red
Red
Red
Yellow
Yellow
Yellow
Water
b
Red
Yelloworange
Bromphenol
blue
Tetrabrom-m-cresolsulfonphthalein Bromcresol
green
Dimethylaminoazobenzene sodium Methyl red
carboxylate
(sodium)
Dichlorosulfonphthalein
Chlorophenol
red
Dibromthymolsulfonphthalein
Bromthymol
blue
Phenosulfonphthalein
Phenol red
Dimethyldiaminophenazin chloride
Neutral red
m-Cresolsulfonphthalein
Cresol purple
Thymoslsulfonphtalein
Thymol blue
Phenolphthalein
Water
a
Yellow
Purple
Water
a
Yellow
Blue
Water
Water
b
a
Red
Yellow
Yellow
Water
a
Yellow
Water
70% alcohol
Water
Water
70% alcohol
a
b
a
a
a
Thymolphthalein
p-Nitroanilline azosodium salicylate
90% alcohol
Water
a
a
Yellow
Red
Yellow
Yellow
Colorless Redvioletr
Coolness Blue
Yellow
Violet
Terabromphenolsulfonphthalein
Alizarine yellow
In Fig. 34-1 the color-change intervals of various indicators are represented
graphically. It is seen that the above set of indicators covers the pH range between 1 and
12. The color-change intervals in Table 34-1 and the K0HI values refer to a temperature of
250C. When HI is uncharged or has a negative charge K0HI hardly changes with the
temperature. Thus the color-change interval of the sulfophtaleins and the phthatleins
hardly changes between 0 and 1000C. On the other hand, when HI has a positive charge,
as with methyl orange, methyl yellow, and methyl red, the color-change interval changes
to lower pH values; for example, for methyl organ from 3.1 to 4.4 at 250C to 2.5-3.7 at
100C; for methyl yellow from 2.9 to 4.0 at 250C to 2.4-3.5 at 1000C, and for methyl red
from 4.2 to 6.3 at 250C to 4.0-6.0 at 1000C.
Preparation of Indicator Solutions
The stock solutions of the indicators contain 0.5-1g of indicator per liter of
solvent. Tropeoline ), methyl orange, methyl red sodium salt, and alizarine yellow can be
dissolved in water. Methyl yellow, neutral red, phenolphthalein, and thymolphalein are
dissolved in 70-90 per cent alcohol.
The sulfonphthaleins are soluble in water if enough sodium hydroxide is added to
neutralize the sulfonic group; 100mg of these indicators are ground in an agate mortar
with the specified volume of 0.1 N sodium hydroxide solution and then diluted with
water to 100ml. The following volumes of 0.1N sodium hydroxide are required for 100
mg of the sulfonphthaleins: thymol blue, 2.15 ml; bromphenol blue, 1.5 ml; bromeresol
green, 1.45 ml; chlorophenol red, 2.35 ml; bromthymol blue, 1.6 ml; phone red, 2.85 ml;
cresol purple, 2.65 ml.
Color change and structural change of indicators
The color change of an indicator is accompanied by a structural change. Below
the structural changes of some of the more representative indicators are given (for further
details see the literature).
THEORY OF ACID-BASE TITRATIONS
Phenol red:
Phenolphthalein:
Methyl orange:
Mixed Indicators
When the color change at the end point is not sharp, a mixture of two
indicators or of an indicator and a dye can often be used to advantage. If a proper
selection is made and the indicator and the dye are mixed in the proper ratio, the mixed
indicator will give a sharp and pronounced color change at a definite pH and may be used
in cases in which the color change is not easily seen with an ordinary indicator. A few of
these mixed indicators are given below.
Methyl orange-indigo carmine: A solution containing 1g of methyl orange and
2:5g of indigo carmine in 1 liter of water. The alkaline color is green. The mixture
exhibits a grayish neutral shade at pH 4.0 and changes toward violet in more acid
solutions. The stock solution should be kept in brown bottles, best in the dark.
(a) Bromcresol green (0.1 per cent)- (b) methyl red (0.1 per cent): 3 parts of (a)
are mixed with 2 parts of (b). Pronounced color change at pH 5.1; acid color, red;
alkaline color, green.
(a) Phenolphthalein (0.1 per cent)-(b) Imethylene green (0.1 per cent): 1 part of
(a) mixed with 2 parts of (b). Color in acid medium, green; at pH 8.8, pale blue; at pH >
9.0, violet.
(a) Cresol red (0.1 per cent)- (b) thymol blue (0.1 per cent): 1 part of (a) mixed
with 3 parts of (b). Acid color, yellow; alkaline color, violet; pink at pH 8.2; violet at pH
8.4. (a) Cresol red (0.1 per cent)- (b) Methyl red (0.1 per cent) – (c) Methylene blue (0.2
per cent): 10ml of (a) + 20 ml of (b) + 4 ml of (e). At pH 8.8, violet; pH 8.6, gray-violet;
pH 8.35, gray-green; pH 8.25, green.
Neutralization of strong acids with strong bases. Neutralization curves
Since strong acids and bases are completely ionized, it is simple to calculate the
change in the hydrogen-ion concentration and the pH during the course of the
neutralization of a strong acid with a strong base, or vice versa.
If, for example, we start with 100 ml of 1 N hydrochloric acid and add 90 ml of 1
N sodium hydroxide solution, there will be 10 ml of 1 N hydrochloric acid left in a
volume of 190 ml. Therefore.
[H  ] 
10
x 1  5.3 x 10 2
190
and pH=1.3. When 100ml of 1 N sodium hydroxide has been added, the equivalence
point is reached; and the pH should be 7 if no impurities are present. With 101 ml of 1 N
sodium hydroxide there will be 1ml of 1N sodium hydroxide in excess in a volume of
201 ml; therefore, [OH-] = 5 x 10-3, [H+] = 2 x 10-12, and pH 11.7 etc.
In Table 34-2 the change in the pH during the neutralization of 100ml of 1N, 0.1
N, and 0.01 N solutions of a strong acid, respectively, after successive additions of
sodium hydroxide of corresponding strength is given. The data are plotted in
TABLE: 34-2: Titration of 100 ml of HCI with NaOH at Room Temperature
NaOH Added, ml
(=%
Acid
Neutralized)
0
50
90
98
99
99.8
99.9
E.P.a 100.0
100.1
100.2
101
102
110
N Solution
[ H  ] pH
0.1 N Solution
pH
0.01 N Solution
pH
1
3.3 x 10-1
5 x 10-2
1 x 10-2
5 x 10-3
1 x 10-3
5 x 10-4
10-7
2 x 10-11
1 x 10-11
2 x 10-12
1 x 10-12
2 x 10-13
1.0
1.5
2.3
3.0
3.3
4.0
4.3
7.0
9.7
10.0
10.7
11.0
11.7
2.0
2.5
3.3
4.0
4.3
5.0
5.3
7.0
8.7
9.0
9.7
10.0
10.7
0
0.5
1.3
2.0
2.3
3.0
4.3
7.0
10.7
11.0
11.7
12.0
12.7
E. P, equivalence point
S.A. Jimeno and A.A. Cheoux, Chim. Anal. (Paris) 46, 196 (1964).
Fig. 34-2; the curve obtained is called the neutralization curve. From the
quantitative viewpoint we are especially interested in the changes in the pH near the
equivalence point. For this reason, the part of the curve of Fig. 34-2, per cent before and
2 per cent after the equivalence point, is reproduced on an enlarged scale in Fig. 34-3.
The color-change intervals of some of the ordinary indicators are also indicated
on this graph. Inspection of Fig 34-3 or an analysis of the data in Table 34-2 reveals: (1)
the best titration conditions and (2) the error that is made if an indicator is used which
does not change color in the neighborhood of the equivalence point. Let us consider
methyl orange (or bromphenol blue) and phenolphthalein (or thymol blue) as indicators
covering the most extreme changes of pH.
Methyl organ changes color between pH 3.0 (red) and 4.2 (yellow-organge), and
phenolphthalein between pH 8.0 (colorless) and 9.8 (violet-red).
Titration of 1 N acid with 1 N sodium hydroxide. After the addition of 99.8 ml of
1 N sodium hydroxide, the pH is 3.0. Methyl orange is then still present in the red form.
with 99.9 ml of sodium hydroxide the pH is 3.3, and therefore most of the indicator is
still in the acid form. upon addition of the last 0.1 ml of sodium hydroxide, the pH
changes abruptly from 3.3 to 7. Therefore a very sharp color change is obtained with
methyl orange. If sodium hydroxide is added until the alkaline color just appears, there is
no indicator blank. At the equivalence point phenolphthalein is still present in the acid
form; addition of a very small amount of 1 N sodium hydroxide will change the pH
abruptly to the alkaline side. Thus the addition of 0.1 ml of 1 N base changes the pH from
7 to 10.7; at the latter pH phenolphthalein or thymol blue is present completely in the
alkaline form. Summarizing then, it may be said that in the titration of 1 N hydrochloric
acid with 1 N sodium hydroxide any indicator having a color-change interval lying
between that of methyl orange and phenolphthalein (and even thymolphthalein) can be
used. The indicator blanks are negligibly small, and the color changes are very
pronounced.
Fig. 34-2: Neutralization curves of 1 N, 0.1 N, and 0.01 N hydrochloric acid with
sodium hydroxide.
Fig. 34-2: Neutralization curves of 1 N, 0.1 N, and 0.01 N hydrochloric acid with 1 N,
0.1 N, and 0.01 N sodium hydroxide. From 2% before to 2% after the equivalence
points.
Titration of 0.1 N hydrochloric acid with 0.1 N sodium hydroxide. With more dilute
acids and base the pH changes near the equivalence point become less abrupt. After the
addition of 98 ml of 0.1 N sodium hydroxide to 100 ml of 0.1 N hydrochloric acid, the
pH has become equal to 3. On further addition of sodium hydrochloric acid, orange
beings to change color from red to the intermediate color. After 99.8 ml of sodium
hydroxide has been added, the pH is 4; and most of the methyl orange is present in the
alkaline form. if the titration is stopped here, we are still 0.2 ml from the equivalence
point; the error then amounts to 0.2 per cent, and the indicator blank is 0.2 ml. (In this
case the indicator blank must be added to the titration figure). In the titration of 0.1 N
hydrochloric acid with 0.1 N sodium hydroxide, it is advisable to add sodium hydroxide
until the indicator is present completely in the alkaline form. For most practical purposes
the error is then negligibly small. In the reverse titration of 0.1 N sodium hydroxide with
0.1 N. hydrochloric acid with methyl orange as indicator, an excess of acid must be added
to produce a change from yellow to pink (the first change in color should be considered
as the end point). If this color change takes place at pH. 4.0, the error again is 0.2 per
cent, and the indicator blank is 0.2 ml (which in this case is to be subtracted from the
amount of acid used). This indicator blank is experimentally determined in the following
way.
The solution is titrated in the ordinary manner until the color change of methyl
orange to the first pink is noticed. Then in another vessel the same amount of indicator as
used in the titration is added to 200 ml of 0.05 N sodium chloride solution, and standard
hydrochloric acid is added from a buret until the color of the solution matches that of the
titrated solution at the end point.
It is evident that methyl red will give a much shaper color change than methyl
orange in the titration of 0.1 N solutions. After the addition of 99.9 ml of 0.1 N sodium
hydroxide the pH is 4.3; and methyl red is still present in the acid form. upon the
addition of the last 0.1 ml of sodium hydroxide, the pH changes abruptly from 4.3 to 7,
the color of the indicator changes from red to yellow. The endpoint error is also
negligibly small with phenolphthalein as indicator. With 0.1 ml of 0.1 N sodium
hydroxide in excess, the pH is 9.7, and most of the phenolphthalein is present in the
alkaline form.
The above interpretation is correct only when we work with solution which do not
contain carbon dioxide. Under practical conditions this is never the case, since standard
sodium hydroxide as a rule contains a small amount of carbonate. Carbon dioxide is such
a weak acid (see p. 787) that it has no effect upon the color of methyl orange; however, it
is titrated as a univalent acid with phenolphthalein as indicator. In working with carbon
dioxide-free solutions, the difference in the amounts of sodium hydroxide used with
methyl orange and with phenolphthalein as indicators is not larger than 0.15-0.2 ml of 0.1
N sodium hydroxide, when 100ml of. 0.1 N hydrochloric acid are titrated. A larger
difference will be due to carbon dioxide present in the solution after the end point to
methyl orange has been reached.
Titration of 0.01 N hydrochloric acid with 0.01 N sodium hydroxide. It is now
a simple matter to specify what indicators should be used in the neutralization of 0.01 N
hydrochloric acid with 0.01 N sodium hydroxide. Methyl orange does not give a sharp
color change because the indicator is present in the alkaline form before the equivalence
point is reached. After the addition of 90 ml of 0.01 N sodium hydroxide the pH is 3.3,
upon further addition of base the color of the indicator changes gradually to yelloworange. After the addition of 98-99 ml of 0.01N sodium hydroxide, the indicator is
practically completely present in the alkaline form. The end-point error then amounts to 1
or 2 per cent. In this case it is much better to use methyl red and to titrate to the yellow
color. Other indicators which change color between a pH of 5 and 9, such as bromthymol
blue, phenol red, or phenolphthalein, can be used as well. Quite generally, when very
dilute solutions of a strong acid must be titrated with a strong base (or vice versa), it is
preferable to carry out the titration at boiling temperature, using phenolphthalein, phenol
red, bromthymol blue, or methyl red as indicator. Any carbon dioxide present is expelled
by boiling the slightly acid solution. The latter should be cooled before the final color
adjustment.
Neutralization of weak acids with strong bases or weak bases with strong acids
In the following discussion the titration of 0.1 n solutions of weak acids with
strong bases or of 0.1 N solutions of weak bases with strong acids only will be
considered. It is a simple matter to make similar derivations for other concentrations.
Whereas in the titration of 0.1 N solutions of a strong acid with a strong base we can
select any indicator changing color between a pH of 4 and 10, we are much more nnuous
as regaus conoce of indicator in the titration of weak acids (with strong bases) or of weak
bases (with strong acids).
This is evident form a study of the neutralization curves. In Chapter 3 equations
for the calculation of pH of a solution of a weak acid (p.44), of mixtures of the weak acid
and its salt (p.51), and of solutions of the salt of he acid with a strong base (p.48) were
derived. These equations are used in the calculation of the neutralization curves.
In Table 34-3 the neutralization curves of 0.1N acetic acid (Ka=1.8 x 10-5), and of
a weaker acid with an ionization constant of 10-7 with 0.1N sodium hydroxide, are given.
Table 34-3: Neutralization Curves of 100ml 0.1N Acetic Acid (Ka=1.8 x 10-5) and
of 100ml 0.1N HA (Ka=10-7) with 0.1 N sodium Hydroxide at Room temperature
(Kw=10-14)
0.1 N NaOH Added, ml 0.1 N Acetic Acid, pH
(=% Acid Neutralized)
0
2.87
10
3.80
50
4.75
90
5.70
99.0
6.75
99.8
7.45
99.9
7.75
E.P. 100.0
8.72
100.1
9.7
100.2
10.0
101
10.7
0.1 N Acid, Ka=10-7, pH
4.0
6.0
7.0
8.0
9.0
9.57
9.70
9.85
10.00
10.19
10.7
These data are plotted in Fig. 34.4; for comparison the neutralization curve of 0.1
N hydrochloric acid is added. In addition, the color-change intervals of methyl orange
(M.O), methyl red (M.R.), and phenolphthalein (Phpht.) are indicated.
An inspection of the neutralization curve of acetic acid shows that methyl orange
cannot be used as indicator. It changes from red to yellow-orange between a pH of 3.0
and 4.2. At the latter pH only 30 per cent of the acid is neutralized, and the titration error
would amount to 70 per cent. Methyl red is not a good indicator either because it begins
to change color at pH of about 4.6, i.e., before 50 per cent of the acid has been
neutralized, and is present in the yellow form at a pH os 6.0 when about 93 per cent of
the acid has been neutralized. In order to get a good result it is necessary to use an
indicator which changes color in the neighborhood of the pH prevailing at the
equivalence point. Since the solution has a slightly alkaline reaction at the equivalence
point, an indicator changing color in alakaine medium must be used. It is evident that
phenolphthalein and thymol blue are the best indicators. Quite generally weak acids
should be titrated with phenolphthalein, thynol blue, or thymolpthalein as indicators.
The changes in pH near the equivalence point become less pronounced with a
decrease in the ionization constant of the acid, because hydrolysis then begins to play a
more important role. In the titration of 0.1N solutions of acids having an ionization
constant of the order of 10-7, it is seen from the table that even thymol
Fig. 34-4. Neutralization curves of 0.1 N acid and of 0.1 N ammonia (I) 0.1 N
HCL; (II) 0.1 N acetic acid, Ka=1.8 x 10-5, (III) 0.1 N HA, Ka=10-7; (IV) 0.1 N NH3, Ka=
1.8 x 10-5.
blue and phenolphthalein do not yield good end points. After 90 per cent of the
acid has been neutralized, the pH = 8.0; and upon further addition of sodium hydroxide
the two indicators will only gradually change their colors to the alkaline side, and no
sharp end point is obtained. Thymolphthalein is still useful in this case, since it beings to
change color at a pH of about 9.6, 0.2 per cent before the end point. Acids having an
ionization constant smaller than 10-7 cannot be satisfactorily titrated in 0.1 N solutions.
In Table 34-4 the neutralization curve of 0.1 N ammonium hydroxide with 0.1 N
hydrochloric acid given (see also Fig. 34.4).
Table 34-4. Neutalization Curve of 100 ml of 0.1 N
Ammonium Hydroxide (Kb= 1.8 x 10-5) with 0.1N
Hydrochloric Acid Room Temperature (Kw=10-14)
0.1 N HCL Added, ml
pH
(=% Base Neutralized)
0
10
50
90
99.0
99.8
99.9
E. P. 100.0
100.1
100.2
101.0
11.13
10.20
9.25
8.30
7.25
6.55
6.25
5.25
4.3
4.0
3.3
From the data it is evident that none indicators changing color in alkaline medium
will yield good end point. Phenolphthalein gives a very gradual color change and
becomes colorless when about 94 per cent of the ammonia has been neutralized. Methyl
red, methyl orange, and other indicators changing color at pH values between
approximately 6 and 3 should be used. Quite generally weak bases are titrated with
methyl red, chlorphenol red, bromcresol green, bronmphenol blue, or methyl orange as
indicators, using a comparison solution for the end-point determination.
In the titration of weak acids with bases (e.g., acetic acid with ammonia), the pH
changes near the quivalence point are always very gradual, and no sharp end point can be
found with any indicator. Useful results can be obtained by using an indicator which
assumes an intermediate color at the end point and employing buffer solutions with the
same pH the solution at the equivalence point, with the same amount of indicator, for
comparison. If possible, however, one should always titrate a weak acid with a strong
base and a weak base with a strong acid.
In the above examples the weak acid or weak based to be titrated was uncharged.
The calculations remain the same when the acid has a positive charge or the base has a
negative charge. For example, the ammonium ion is an acid with KHA= 5.5 x 10-10. no
sharp end point can be obtained in the titration of 0.1 N ammonium chloride with sodium
hydroxide. On the other hand, the cations of aromatic amines, such as anilinium and
pyridinium, are much stronger acids with KHA of the order of 10-5-10-4. Therefore, salts of
such weak bases can be titrated with sodium hydroxide, using thymol blue or
phenolphthalein as indicator.
Aquocations of the heavy metals are acids in the Bronsted sense. In addition, the
corresponding hydroxides or hydrous oxides are slightly soluble. Under suitable
conditions such cations can be titrated with sodium hydroxide. Examples include
A13++3 OH- A1(OH)3
Cu2+ + 2 OH- Cu(OH)2
Quite generally, the direct titration of these cations with a strong base yields an
end point considerably before the equivalence point because of the precipitation of basic
salts. Satisfactory results are usually obtained by the addition of an excess of base and
back-titration with acid. The method can even be applied to the determination of
magnesium in its salts. Although magnesium hydroxide is a strong base, it is slightly
soluble salt and the excess of base in the clear supernatant liquid or in the filtrate is backtitrated with standard acid.
Anions in salts of weak acids are bases and can be titrated with standard
hydrochloric acid, provided the liberated acid is weak enough to give a distinct end point.
The basic strength of anions such as cyanide and metaborate is of the same order of
magnitude as that of ammonia (see Appendix) and they can be titrated with a strong acid.
Borax is a salt, which in solution can be considered as half-neutralized boric acid:
Na2B4O7 + 5 H2O
2 H2BO3 + 2 H+
2 H3BO3 + H2BO3 + 2 Na+
2 H3BO3
The end point is so sharp that borax can be used as a standard substance for
hydrochloric acid.
Polyprotic acids and their salts. Mixtures of two acids
Consider a diprotic acid with dissociation constants K1 and K2. For all practical
purposes we can use for the calculation of [H+] at the first equivalence point of the
equation (see p. 54).
[H+]= (K1K2)1/2
At the second equivalence point the same equation can be used as fro a
monoprotic acid. In an equivalent mixture of two weak acids with dissociation constants
K1 and K2, Eq. (6) holds at the first equivalence point. In Table 34-5 is presented
Table 34-5. Titration with Standard Sodium Hydroxide of a Mixture
0.1 N in Acetic Acid and 0.1 N in Ammonium Chloride
(K1 = 1.8 x 10-5, K2 = 5.5 x 10-10)
NaOH Added Expressed in
Equivalent % of Acetic Acid
[H+]
pH
50
90
98
1.8 x 10-5
2.0 x 10-6
3.7 x 10-7
4.75
5.70
6.43
pH
CNaOH
2.2 x 10-7
1.4 x 10-7
1.0 x 10-7
7.1 x 10-8
4.5 x 10-8
99
99.6
100
100.4
101
6.65
6.85
7.00
7.15
7.35
0.22
0.33
0.38
0.33
the change of pH in the titration of a mixture of 0.1 N in acetic acid and 0.1N in
ammonium (chloride) or 0.1 N in boric acid with standard sodium hydroxide.
It is seen that no large change in pH occurs at the first equivalence point; this
means that no sharp color change of any indicator can be observed at the end point.
Between 0.4 per cent before and 0.4 per cent after the equivalence point the change in pH
is only 0.3. The naked eye can detect such a difference with the aid of a suitable indicator
when a comparison solution is used which has the same pH and which contains the same
amount of indicator as the titrated solution at the equivalence point. In order to eliminate
ionic strength effects, which have been neglected in our derivation, it is best to use a
comparison solution which is closely similar to the titrated solution at the first
equivalence point. In the example of Table 34-5 0.1 n solution of pure ammonium acetate
can be used. Mixed indicators (p. 695) are particularly useful in titrations in which the
change in pH at the equivalence point is small.
The larger the ratio K1/K2, the more sharply the end point can be detected. When
this ratio is smaller than 1 per cent.
Phosphoric acid can be titrated as a mono-or diprotic acid. The pH at the first
equivalence point is
1
2
(pK1 + pK2) = 4.66. Bromocresol green or methyl yellow can be
used as indicators. It is advisable to use a solution of pure NaH2PO4 for comparison in the
detection of the end point. The pH at the second equivalence point is 12 ( pK2  pK3 )  9.7
At this pH phenolphthalein and thymol blue are present in the alkaline form and a color
change of these indicators is observed before the equivalence point. Thymolphthalein is a
better indicator for this titration, as it begins to change color at pH 9.6.
The third ionization constant of phosphoric acid is of the order of 5 x 10 -13.
Therefore, a solution of Na3PO4 reacts strongly alkaline. A titration to the third
equivalence point is not possible, unless the trivalent phosphate ion is removed. This is
done by addition of a large excess of calcium chloride after reaching the second
equivalence point.
2 Na2HPO4 + 3 CaCI2
Ca3(PO4)2 + 4 NaCI + 2 HCI
Carbonic acid (hydrated carbon dioxide) is a dibasic acid. A solution of alkali
bicarbonate has a pH of 12 ( pK1  pK2 )  8.40.
Carbon dioxide can be titrated as a monobasic acid, using phenolphthalein or
thymol blue (or, better, a mixed indicator, see pp. 695 and 787) as indicator. The color
change is not sharp, and it is necessary to use a solution of freshly prepared pure sodium
bicarbonate with the same amount of indicator for comparison.
The second ionization constant of carbonic acid is so small that it cannot be
determined as a dibasic acid by a direct titration. However, the carbonate ion can be
removed by precipitation. A measured excess of barium hydroxide is added to the
solution of carbon dioxide or bicarbonate:
H2CO3 + Ba(OH)2
NaHCO3 + Ba(OH)2
BaCO3 + 2 H2O
BaCO3 + NaOH + H2O
and the excess is titrated back with standard acid, using phenolopthalein or thymol blue
as indicator, without filtering off the barium carbonate.
In Table 34-6 data for the neutralization curve of carbon dioxide to NaHCO3 and
Na2CO3 are given (see also Fig. 34-5). It is clear that a solution of sodium carbonate can
be titrated as a mono-or diprotic base.
Table 34-6. Data for Neutralization Curve of Carbon Dioxide (100ml of 0.05 M carbon
dioxide titrated with 0.1 N NaOH; K1 = 3 x 10-7, K2 = 6 x 10-11)
0.1 N NaOH Added, ml
0
0.5
5
25
45
49
49.5
50 NaHCO3
50.5
51
55
75
95
100 Na2CO3
105
pH
3.9
4.5
5.5
6.5
7.5
8.08
8.21
8.35 [Phpht.; thymol blue (mixed indicator)]
8.54
8.67
9.27
10.2
11.2
11.4 (no good indicator here)
11.75
Fig. 34-5: Titration of 100 ml. 0.05 M H2CO3 with 0.1 N sodium hydroxide.
Hydrolytic precipitation and complex formation reactions
A solution of potassium cyanide is alkaline.
CN- + H2O OH-  +HCN)
If cyanide ions are removed by the formation of a slightly soluble or slightly
dissociated compound, the solution will not become alkaline until cyanide is in excess.
Mercuric cyanide is a substance which is much less dissociated than mercuric chloride.
Therefore, the following reaction runs quantitatively to the right:
HgCI2 + 2 CN- Hg(CN)2 + 2 CIUpon titration of an alkali cyanide solution with mercuric chloride, the reaction
remains alkaline until the end point. A solution of mercuric chloride reacts acid toward
methyl red. A sharp color change from yellow to red is observed at the end point using
methyl red as indicator.
A solution of an alkali cyanide can be titrated with standard sliver nitrate, using
an acid-base indicator such as phenolphthalein or phenol red for the detection of the end
point. The cyanide is transformed into the complex Ag(CN) 2 ion:
2 CN- + Ag+ Ag (CN) 2
which is the anion of a strong acid. The solution of its alkali salt accordingly has a neutral
reaction. Therefore, after the removal of the last trace of cyanide with silver ions, the
reaction changes from alkaline to neutral.
An application of a hydrolytic precipitation react is made in the determination of
the hardness of water (sum of calcium and magnesium). Potassium palmitate solutions
(soap) are strongly hydrolyzed and have an alkaline reaction. They form insoluble
palmitrates with calcium, magnesium, barium, strontium, and some other cations:
2 KPalm + Ca++
Ca(Palm)2 + 2 K+
For the determination of the sum of calcium and magnesium in water, the latter is
neutralized with acid to methyl orange (titration of the bicarbonate). After boiling off the
carbon dioxide, the solution will have a neutral reaction after cooling. It is then titrated
with the soap solution, suing phenolphthalein as indicator.
One might expect that sodium carbonate would be a useful reagent in the
determination of the alkaline earhs. The carbonates of calcium, strontium, and barium are
very slightly soluble, and their saturated solutions have a pH of about 9. If an excess of
the alkaline earth ion is present, the pH is less than 9; and with an excess of sodium
carbonate it is greater. Actually the precipitation of the alkaline earth carbonates during
titration with sodium carbonate is rather slow; direct titration, therefore, is not very
useful. Addition of excess of the standard carbonate solution and back titration of the
filtration, however, yields good results.
Various other reagents are used in hydrolytic precipitation and complex formation
reactions:
Potassium chromate for barium: CrO4-- + Ba++
CrO4- + H2O
Aluminum chloride for fluoride:
Sodium sulfide for zinc:
BaCrO4
HCrO 4 + OH -
Al3 + 6F-
ALF 36 
Al3 + H2O
AlOH++ + H+
Zn++ + S--
ZnS
S--+ H2O
HS- + OH-
Acid salts of chelating agents such as disodium ethylenediaminetetracetate (Na2H2Y)
react with many cation with libration of hydrogen ions:
H2Y- - + Zn + +
ZnY- - + 2H +
End-point error. This corresponds to the difference between the amount of standard
reagent used between the equivalence point and the point. the error often called titration
error is expressed in per cent or ppt of the number of equivalents of substance titrated.
The end point error in the titration of a strong acid with a strong bas has discussed. When
0.01N strong acid is titrated with a strong base, the end point error (neglecting volum
change) is only  0.1 per cent when pH=5 or 9 at the end point, and  0.01 per cent
when pH is 6 or 8 at the end point. The end point error is also small in the titration of
carboxylic acid (KHA  10 -5) with a strong base. The analytical concentration of the acid
is dented by Ca and that corresponding to the amount of standard base by Ca. for
convenience it is assumed that no volume change occurs during the titration. At the
equivalence point, Ca=Cb. At other points, Ca-Cb which can be positive or negative, gives
the end point error. At any point during the titration
Ca = [HA] + [A-]
Cb= [A-] + [OH-] – [H+]
And
In the titration under consideration the solution is weakly alkaline at near the end point
and [H+] can then be neglected. At the equivalence point
Ca = Cb = [A-] + [HA] = [A-] + [OH-]
[HA] + [OH-]
Or
This result, of course, is simply arrived at when considering that at the equivalence point
the solution is composed of pure Na A:
A- + H2O
HA + OH –
Consider the titration of 0.1M acetic acid (KHA = 1.8 x 10-5) and, assuming no volume
change, that pH at the end point deviates  1 from that the equivalence point. at the
equivalence point [HA] = [OH] = 7.5 x 10-6 and pH = 8.88 (KH2O= 10-14). When pH at
the end point is 9.88 (9.9)[HA] = 7x 10-7 and [OH-] = 7.6x 10-5. the end point error in per
cent is then
Cb – Ca = {[OH] –[HA]} x 100= 0.07 percent
Ca
Evidently the titration error is negligibly small when the change in pH at the equivalence
point is pronounced and the proper indicator is used. When the break in the pH at the
equivalence pint is gradual, no sharp end sharp end point can be observed with any
indicator. Under such conditions a comparison solution of the same pH, and preferably of
the same composition as the titrated solution at the equivalence point, must be used with
an indicator which is partially in the acid partially in the purpose. Assume that a color
difference between the solution titrated and the comparison solution can be detected
corresponding to a difference of  0.5 percent.
In the titration with standard base of polypro tic acids, such as phosphoric or
carbonic, or their salts to an acid salt with acid, no sharp color change at the equivalence
point is observed. To get the best and most precise results, a comparison solution must be
used. Consider the titration of sodium carbonate of molar concentration Cs to bicarbonate
with standard acid. In the following A--denotes carbonate and HA- bicarbonate ion. Kl =
3.5 x 10 -7, K2 = 4.5 x x10-11 KH2O = 10-14, and CS = 0.1 at the equivalence point have a
solution of NaHA and
[H+] = (K1K2) 1 2 = 4 x 10-9, pH 8.40
It is easily seen that at the equivalence point, [H+] and [OH-] are negligibly small as
compared to [H2A] and [A--]:
[H2A] = [H+ ] [HA-] = 1.1 x 10-3
K1
[A--] = [HA-] K2= 1.1 x 10-3
[H+]
At any point titration
CS = [A--] + [HA-] + [H2A]
And
Ca = 2[H2A] + [HA-] + [H+] – [OH-]
 2[H2A] + [HA-]
Neglecting [H+] and [OH-], it follows from the two equation that the equivalence point
when Ca = Cs, [H2A] = {A--].
Assume that the end point can be detected within  0.1 pH of the pHat the equivalence
point, e g., at pH 8.30. at this pH,
[A--] = 9 x 10 [H2A] – [A--] = 0.5 x 10-3
Ca – Cs = [H2A] – [A--] = 0.5 x 10-3
And the end point error is [(0.5 x 10-3)/10-1] x 100 =0.5 per cent. For the titration of
0.01M sodium carbonate the same end point error is calculated as for 0.1M solution.
When the dilution becomes so that approximation that [H+] and [OH-] are negligibly
small as compared to [H2CO3] and [CO3--] no longer holds, and the equations becomes
very involved.
Acid-base titrations in organic analysis
Below are presented some reactions that are applied in the determination of some
important organic functional groups and which involve acid-base titrations. Details and
procedure may be found in monographs on organic analysis and other references.
Acids and bases: The titration curves are similar to those of inorganic acids and bases.
Many acids and bases which are too weak to be titrated in aqueous medium can be
titrated in a no aqueous solvent. This is discussed in the next section.
Esters: These compounds are transformed into the corresponding alkali salts by heating
with alkali hydroxide, e.g.,
CH3COOC2H5 + Na+ + OH- CH3COO- + Na+ + C2H5OH
The reaction is called saponification. Most esters are slightly soluble in water and, for
quantitative purposes, they are usually saponified by heating with an excess of standard
alcoholic alkali. The excess alakali is back-titrated, using phenolphthalein or thymol blue
as an indicator. A blank is run with the reagent.
In the analysis of a mixture of two esters use can often be made of the difference
in rates of specification of the two esters. For example, the rate constant of saponification
of ethyl acetate is considerably greater than that of isopropyl acetate. The sum of the two
esters is found from the amount of alkali consumed in the saponification of both esters.
From a kinetic analysis of the rate of saponification the amounts of each ester can be
found.
Acid anhydrides: In general, these compounds react slowly with water to yield the
corresponding acids:
(RCO)2O + H2O
2 RCOOH
The reaction is catalyzed by a base. To a weighed amount of the substance in a flask is
added an excess of standard barium hydroxide or carbonate-free sodium hydroxide. The
flask is attached to a reflux condenser which has an efficient carbon dioxide trap, and the
liquid is refluxed until all the anhydride is hydrolyzed. The excess of base is titrated with
standard hydrochloric acid, using phenolphthalein or thymol blue as indicator.
Carbonyl compounds
Oximation reaction: Reaction with hydroxylamine hydrochloride (oximation) is the basis
of the most widely used method for the determination of carbonyl compounds:
R  C  O  NH2OH .HCI
RCH  NOH  H2O  HCI
H
R
C  O  NH2OH .HCI
RR' C  NOH  H2O  HCI
R
J. F. Fritz and G. F. Hammond, Quantitative Organic Analysis, Wiley, New York, 1957.
S. Siggia, Quantitative Organic Analysis, Wiley, New York, 3rd ed., 1963. J. Mitchell,
Jr., I. M. Kolthoff, E. S. Proskauer, and A. Weissberger (eds), Organic Analysis, vols. 1,
2, 3, and 4, Wiley-Interscience, New York, 1953-1960. I M. Kolthoff and P.J. Elving
(eds,), Treatise on Analytical Chemistry, part II, vols. 12 and 13, Wiley-Interscience,
New York, 1965-1965-1966.
T.S. Lee and I. M. Kolthoff, Ann. N. Y. Acad. Sci. 53, 1093 (1951).
See T. S. Lee in Organic Analysis, vol. 2, P. 237.
The released hydrochloric acid is titrated with standard alkali to the proper pH. (The
hydroxylamine salt has an acid reaction, pH4.)
Reaction with sodium sulfite: The carbonyl compound reacts with an excess of
sodium sulfite with the liberation of free alkali:
RC  O  SO 3  H 2O 
OH
R
 OH 
C
H
SO3
H
After completion of the reaction the alkali formed is titrated with standard acid. Since
sodium sulfite has an alkaline reaction toward phenolphthalein, it is recommended to use
thymolphthalein as indicator.
C
C
O
These epoxides react with hydrochloric acid, forming a chlorohydrain:
The difference between the amount of acid added and the amount unconsumed, as
deterined by titration with standard base, is a measure of the epoxide.
Hydroxyl groups: Compounds containing hydroxyl groups react with an excess of acetyl
chloride to form an acetate ester and hydrochloric acid:
ROH + CH3COOI
CH3COOR + HCI
The excess of acetyl chloride is hydrolyzed to yield two equivalents of acid:
CH3COOI + H2O
CH3COOH + HCI
In the actual determination an exactly measured amount of acetyl chloride dissolved in
dry toluene is placed in a glass-stoppered flask; an excess of pyridine, which catalyzes
the reaction, is added, followed by the (water-free) sample. After completion of the
reaction at 600C, water is added to decompose the excess of reagent and the acid is
titrated with standard alkali using phenolphthalein as indicator. A blank which is run in a
similar way yields two equivalents of acid, while I mole of alcohol liberates only one
equivalent of acid. The hydroxyl content is calculated from the difference in the amounts
of alkali used in the blank and the procedure. For details see smith and Bryant. Another
procedure makes use of the amount of acid formed in the acetylation with acetic
anhydride:
(CH3CO)2O + ROH
CH3COOR + CH3COOH
Acid-base titrations in nonaqueous media
We distinguish between amphiprotic solvents, which have distinct acid and basic
properties, and approtic solvents, which are much weaker acids and usually much weaker
bases than water (see Chapter 5). Among the amphiprotic solvents we distinguish
D.M. Smith and W. M. D. Dryant, J. Am. Chem. Soc. 57, 61 (1935).
among (1) hydroxylic solvents (water, alcohols), (2) protogenic solvents (acetic, sulfuric
acid), and (3) protophylic solvents (ethylenediamine, dimethylformamide, ammonia).
Hydroxylic solvents have acid and basic character comparable to that of water. However,
they have a much smaller dielectric constant, DC, than water (DC of water is 78.5, of
methanol 32.6, of ethanol 24.3, and of butanol 11.5). The dielectric constant of the
solvent HS has a large effect on the dissociation of uncharged acids and bases:
HA + HS
H2S+ + ABH+ + S-
B + HS
in which H2S+ is the solvated proton (lynium ion) and S- the solvent minus a proton (lyate
ion). The electrostatic attraction between ions increases very much with decreasing
dielectric constant. As a consequence KHA and KB decrease with decreasing dissociation
constant (see Table 5-2).
Similarly, the autoprotolysis constant KHS=[H2S+][S-] decreases with decreasing
dissociation constant if acid and basic properties of the solvent remain about unchanged.
For example, the dissociation constant of uncharged acids and bases in ethanol is about
10 times smaller than that in water, and so is the autoprotolysis constant of ethanol (KHS=
8 x 10-20) (see Table 5.2). This means that the changes in pH at the end point in the
titration of weak uncharged acids with a strong base an of weak uncharged bases with a
strong acid are the same as in water. On the other hand, if we consider the protolysis of a
cation acid or an anion base:
BH+ + HS
A- + HS
B + H2S+
HA + S-
There is no dissociation into ions and KBH+ and KA- remain about the same as in water.
This is actually found to be the situation in ethanol. We have seen that in the titration of a
weak cation acid, such as ammonium ion in water, no sharp break in pH occurs at the end
point. Similarly, in the titration of an anion base, such as acetate ion, no sharp break at
the end point in the titration with a strong acid occurs. By calculations similar to those
made in aqueous medium it is easily shown that a sharp break in pH occurs in the
titration in ethanol of ammonium with sodium ethylate and of acetate with ethanolium
chloride (HCI) in ethanol.
Extensive investigations on acid-base titrations in tertiary butanol have been made
by Marple and Fritz. This solvent has a diclectric constant of only 11.5. At this low
dissociation constant all electrolytes, even those which are completely dissociated in
water, are incompletely dissociated. In such solvents positive and negative ions combine
to form ion pairs, which makes calculation of acid-base equilibrium more involved than
in water.
Among the protogenic solvents glacial acetic acids is a very popular solvent for
the titration of weak bases. All bases that have a Kb in water greater than 10-9 become
strong bases in acetic acid (leveling effect by acid character of solvent) and yield about
the same break in pH at the end point in titrations with perchloric acid.
The titrated base at the end point is present in the form of BH+CIO4, which is
partially dissociated into its ions. The more the eaction.
BH+ + HS
H2S+ + B
proceeds to the right the less sharply the end point can be detected. Acetic acid is a much
weaker base than water, and the above reaction proceeds much less to the right in acetic
acid than in water. This accounts for the fact that bases, which are too weak to be titrated
in water, can be titrated successfully in acetic acid. Similarly, it is easily
L. Marple and J. S. Fritz, Anal, Chem. 35, 1223, 1431 (1963).
Seen that acids which are too weak to be titrated in water can be titrated in solvents
which are weaker acids than water. In such a solvent the reaction.
HA + S- A- + HS
goes much less to the right than in water.
Acetic acid has a dielectric constant of only 6.1 and ion-pair formation is very
pronounced. For example, the dissociation constant of perehloric acid is about 10-5, of
sulfuric acid K1 is 6 x 10-8, of alkali and amine acetates of the order of 10-7, and of
sodium perchlorate of 3 x 10-6. The autoprotolysis constant of acetic acid is 3 x 10-15.
From the equations presented in Chapter 5 it is simple to calculate the changes of pH on
the entire neutralization curves of bases with perchloric acid.
Protophilic solvents such as dimethylformamide (DC 27) and ethylenediamine
(DC 12.9) have gained popularity of the titration of weak acids such as phenols with
tetraalkylammonium hydroxide. Pyridine (DC 12.5) (an aprotic protophylic solvent – it
has no acid character) is particularly suitable for such titrations.
Aprotic protophobic solvents are much weaker bases than water and are useful for
the titration of very weak bases with perchloric acid. However, many strong electrolytes
are insoluble in inert solvents and often soluble in acetic acid or protophylic solvents. For
example, sodium salts of carboxylic acids are soluble in glacial acetic acid and insoluble
in protophobic aprotic solvents. For this reason mixtures of such a solvent with acetic
acid are often used to advantage when the protophobic solvents alone are less suitable
because of solubility relations.
Aprotic protophobic solvents that have been used in titrations of weak bases are
acetone (dielectric constant 20.7), acetonitrile (36), methyl isobutyl ketone (13.1), acetic
anhydride (20.7), and nitromethane (35.9). acetonitrile has no leveling effect on bases. It
is highly recommendd as a solvent for titration of amines and carboxylates with standard
perchloric acid in acetic acid. A very sharp end point is observed with p-naphtholbenzein
from yellow to green. Salts that are insoluble in acetonitrile are dissolved in acetic acid.
The color change and the break in pH at the equivalence point in acetonitrile are much
sharper than in glacial acetic acid. Hydrocarbons, including bezene, toluene, and
chloroform are less suitable; they have a very low dielectric constant, and most
electrolytes are insoluble in such solvents.
The shape of neutralization curves of most acids is quite different in approtic
protophobic than in amphiprotic solvents and in aprotic protophylic solvents.
Amphiprotic solvents are acids and stabilize anions of an acid by hydrogen bonding:
(a)
HA + HS
(b)
A- + HS
overall dissociation: HA + 2 HS
H2S+ + AA- - HS
H2S+ + A- -HS. (For convenience only one HS is
written in the partial equations; many molecules of HS may participate). Reaction (b)
cannot occur in approtic protophobic solvents because they have no hydrogen-bonding
donating properties. When no hydrogen-bond donors other than HA are present in the
solution, the following reaction, called homoconjugation, occurs:
(c)
A- + HA
HA 2
with a homoconjugation constant
KHA 2 
[ HA 2 ]
[ A][ HA]
R. H. Cundiff and P. C. Markunas, Anal. Chem. 28, 792 (1956).
For details see I. M. Kolthoff, M. K. Chantooni, Jr., and S. Bhowmik, Anal. Chem. 39,
1627 (1967).
The overall dissociation of the acid is then the sum of reactions (a) and (c):
(d)
2 HA H+ + HA 2
with an overall dissociation constant
K 2 HA 
[ H  ][ HA 2 ]
[ HA]2
whereas the hydrogen-ion concentration of a weak acid in hydroxylic solvent changes in
proportion to [HA] 1 2 . We find that in an aprotic protophobic solvent it changes in
proportion to [HA]. Thus, on tenfold dilution the hydrogen-ion concentration decreases
10 times, just as that of a strong acid in a hydroxylic solvent. The neutralization curve is
also greatly affected by the homoconjugation. When KHA2 and HA are large enough, [H+]
in the beginning of the neutralization curve is given by
[H  ] 
[ HA]2
K 2 HA
HA 2
or, when Ca represents the initial acid con contraction and Cb the concentration of strong
base added.
(f)
[H  ] 
[ HA]2
K 2 HA
[ HA 2 ]
and
(g)
K 2 HA  K HA K HA
2
Again we notice a difference between the behavior in a hydroxylic and an approtic
protophobic solvent. In a hydroxylic solvent.
[H  ] 
Ca  Cb
K HA
Cb
and dilution of the mixture of a weak acid and its salt slight affects [H+] because of a
slight change of the activity coefficients. In the aprotic protophobic solvent, [H+]
decreases with dilution, in proportion to the change of C0-Cb. When the acid is 50 per
cent neutralized (often called in the literature the half-neutralization potential, HNP, with
reference to the glass electrode),
[ H  ] 1 2  K HA
(h)
It is immaterial what the value of K HA
is. As a result of the homoconjugation,
2
neutralization curves of weak acids with a strong base in inert solvents have quite a
different shape from those in which no homoconjugation occurs. This can easily be
demonstrated by a simple example.
Take K HA =10, and Ca-Cb = 0.01 and Cb=0.001. Then [H+] = 10-1x 10-4 x KHA=
2
103KHA. Without homoconjugation [H+] = 10KHA. With or without homoconjugation
[H+]1/2= KHA. When Cb is greater than
1
2
Ca (after 50 per cent neutralization), there is a
marked concentration of unconjugated A-. This will react with HA to give HA 2 and
[ HA] 
[ HA 2 ]
1
x

[ A ] K HA
2
when Ca-Cb=0.001, practically all HA is present in the form of its homoconjugate:
[ HA] 
0.001
1
x 4  1.1x10 5
0.009 10
and
[ A ]  9 x 103
Hence
[H  ] 
1.1 x 105
K HA 1.2 x 103 K HA
3
9 x 10
as compared to
[H  ] 
0.001
K HA  101 K HA
0.01
in water. Without homoconjugation pH changed between 9 and 91 per cent neutralization
from pKHA- 1 to pKHA + 1. In our example (Ca = 0.011, KHA2= 104) pH changes from
pKHA- - 3 to pKHA + 3.
Figure 34-6 gives the shape of some neutralization curves of weak acids with a
strong base in acetonitrile. Curve V refers to an acid the anion of which does not
homoconjugate, e.g., the picrate ion. The other curves refer to the neutralization of weak
acids, keeping the volume unchanged; i.e., Ca + Cs remains constant, Cs denoting the salt
concentration. The salt has been considered to be completely dissociated. If curve 1
represents the shape of the neutralization curve of a 0.1 M acid with K HA  10 4 , = then
2
curve 2 is that of 0.01 M, curve 3 of 0.001 M, and curve M of 0.0001 M acid. A more
extensive and more exact discussion is found in the papers by Kolthoff and Chantooni.
In aprotic solvents the ratio of K1 to K2 of diprotic acids is much greater than that
in water. In water most dibasic acids do not give a break in pH after neutralization of the
first proton, only one break being observed after complete neutralization of the acid. In
aprotic solvents distinct breaks in pH are observed after neutralization of the first
Fig. 34-6:
The neutralization curves of some weak acids with a strong base in
acetonitrile.
I.M. Kolthoff and M. Chantooni, Jr. J. Am. Chem. Soc. 87, 4428 (1965). In this
paper are found references to other papers dealing with homoconjugation.
and second protons. A typical example is the titration in acetone of succinic acid with
tetralkylammonium hydroxide. The neutralization curve is presented in Fig. 34-7.
Sulfuric acid also gives a pronounced break in pH after neutralization of the first proton.
pK2 of sulfuric acid in water is 2.0, but in acetonitrile 25.9 and in dimethyl-sulfoxide
14.5. Several examples can be given of differentiating titrations in aprotic solvents, i.e.,
titration of a mixture of two acids, which only exhibit one break in pH in aqueous
solution after neutralization of both acids, while in an inert solvent a break in pH is also
observed after neutralization of the stronger acid. In an aprotic protophylic solvent such
as dimethyl sulfoxide or pyridine, homoconjugation is very much less pronounced than in
aprotic protophobic solvents. The reason is that the protophyylic solvents hydrogen bond
the acid HA:
HA + S AH---S
Hence the anion A- competes with the solvent in hydrogen-bonding HA:
AH---S + A- AH--- A- + S
Because of small homoconjugation, aprotic protophobic solvents are recommended as
media for the titration of acids or mixtures of acids. On the other hand aprotic
protophobic solvents (nitroalkanes, ketones, acetonitrile) are recommended for titrations
of weak bases.
REFERENCES
There is an abundant literature on acid-base titrations and equilibria in nonaqueous
solvents. below is given a select list of references to theoretical and experimental
treatment of the subject.
Theoretical
I.M. KOLTHOFF and S. BRUCKENSTEIN, in Treatise of Analytical Chemistry (I. M.
KOLTHOFF and P.J. ELVING, eds.) part I, vol. 1, p. 475, Wiley-Interscience, New
York, 1959.
J. S. Fritz and S. S. Yamamura, Anal. Chem. 29, 1079 (1957).
R. P. BELi, The Proton in Chemistry, Cornell Univ. Press, Ithaca, N.Y., 1959. Chemistry
in Noaqueous Solvents Vol. 4 (G. ZANDER, H. SPANDAU, and C.C. ADDISON, eds),
a multivolume treatise, Wiley-Interscience, New York, 1963.
I. GYENES, Titration in Nonaqueous Media, Van. I, Principles and Techniques, J. J.
LAGOWSKI, ed., Austin, Academic Press, 1966.
W. HUBER, Titrations in Nonaqueous Solvents, Academic Press, New York,
1967.
Experimental
Biannual reviews of Analytical Chemistry.
J.S. FRITZ, Acid Base Titrations in Nonaqueous Solvents, G.F. Smith Chemical Co.,
Columbus, Ohio, 1952.
S. R. PALET, M. N. DAS, and G.R. SOMAYULU, Nonaqueous Titrations,
Indian Association for Cultivation of Science, Calecutta, India, 1959.
T. KUCHARSKY and L. SAFARIK, Titrations, in Non-aqueous Solvents,
Elsevier, Amsterdam, 1965.
PROBLEMS
1.
Derive the neutralization curve of 0.2 N hydrochloric acid with 0.2 N sodium
hydroxide at room temperature. What is the titration error when methyl orange
is used as indicator (pH at end point=4.0), methyl red (pH at end point =
10.0)?
2.
Derive the neutralization curve of 0.1 N formic acid with 0.1 N sodium
hydroxide at room temperature. What is the titration error with methyl red (pH
at end point = 6.0), phenol red (pH at end point = 7.0), and phenolphthalein
(pH at end point = 9.0)? Ka= 10-4, Kw= 10-14.
3.
Derive the neutralization curve of 1 N ammonium hydroxide with 1 N
hydrochloric acid at room temperature. What is the titration error with methyl
orange (pH at end point = 4.0), methyl red (pH at end point = 6.0), phenol red
(pH at end point = 7.0), and phenolphthalein (pH at end point = 9)? KNH3 =
1.8 x 10-5, Kw=10-14.
4.
Sulfurous acid is a dibasic acid. K1= 1 x 10-2, K2= 1.0 x 10-7. Can it be titrated
as a monobaic acid? What is the pH of a solution of NaHSO3? What indicators
could be used? Can it be titrated as a dibasic acid? What is the pH of a 0.1 M
solution of Na2SO3? (Kw=10-14). What indicator should be used?
5.
0.1 N borax solution is to be titrated with 0.1 N hydrochloric acid. What will
be the pH at the end point? What indicator should be used? Ka=10-9.
6.
To 50.00 ml of 0.1012 N magnesium sulfate solution 25.00 ml of 0.2514 N
sodium hydroxide solution is added, and the mixture is diluted to 100 ml in a
volumetric flask. With 0.1046 N hydrochloric acid 50.00 ml of the filtrate is
back-titrated. How much acid will be required? What indicator should be
used? How much magnesium remains in solution after the addition of the
sodium hydroxide? [Mg++][OH-]2 = 10-11.
Ans. 5.85 ml of acid; [Mg++]=
6.9 x 10-8 M.
7.
A solution is 0.1 N in hydrochloric acid and 0.2 N (M) in boric acid. To 50 ml
of this solution is added 50 ml of 0.1 N sodium hydroxide. What is the pH of
the mixture after addition of the sodium hydroxide? (KH3BO3=10-9). What
indicator should be used in the titration of hydrochloric acid in the presence of
boric acid?
8.
Ans. pH=5.0.
A solution of 0.1 N hydrochloric acid is titrated with 0.1 N sodium hydroxide.
The pH at the end point is 4.0. Calculate the titration error.
9.
Ans. 0.2 per cent.
Pure water exposed to carbon dioxide was found to have a pH of 4.52. (a)
Calculate the carbon dioxide concentration of the water. (K1H2CO3= 3 x 10-7)
(b) How much 0.01 N barium hydroxide is necessary to precipitate all the
carbon dioxide in 100 ml of the water?
CHAPTER 37 OXIDATION REDUCTION TITRATIONS
Potentiometric Titration Curves
A potentiometric titration curve portrays the change of reduction potential in the
course of a titration of a reductant with an oxidant or vice versa. In Chapter 8 the
general equations have been presented which give the relation between reduction
potential and the concentrations of the constituents which determine it. Most redox
titrations are carried out at relatively large ionic strength. Therefore expressions for
formal potentials will be used. (See appendix table for values of Ef).
Ox + ne
Red
Ox + mH+ + ne
Red
E = Ef +
0.59
[Ox ]
log
(25oC)
n
[Re d ]
E = Eo’ +
0.059m
log [H+]
n
+
0.059
[Ox ]
log
n
[Re d ]
=Ef +
0.059
[Ox ]
log
n
[Re d ]
Titration of ferrous iron with ceric cerium in 1 N H2SO4
we are dealing with the partial systems
Fe(III) + e
Fe (II)
E = EfFe (III)/(II) + 0.059
= ).68 + 0.59 log
Ce (IV) + e
Ce(III)
[ Fe( III )]
[ Fe( II )]
[ Fe( III )]
[ Fe( II )]
E = 1.44 + 00059 log
[Ce( IV )]
[Ce( III )]
The Roman numerals indicate the valence state of the various ions, and the
bracketed valence state refer to the analytical concentration of the species and not to the
single ion concentration. The equilibrium constant K of reaction (1) is given by (see
Appendix)
Log K = log
EEp =
[ FE ( III )][Ce( III )] 1.44  0.68
=
= 12.9
0.059
[ Fe( II )][Ce( IV )]
K = 8x1012
Efce( IV )  EfFe( III ) / Fe( II ) 1.44  0.68

 1.06
2
2
In which EEP is the potential (versus NHE) at the equivalence point. At this point
[ Fe( III )] [Ce( III )]
=
=(8x1012) 1/2 2.8 x106
[ Fe( II )]
[Ce( IV )]
And the reaction is quantitative.
It is a simple matter now to calculate the entire potentiometric titration curve.
When 9 percent of the iron (II) has been oxidized [Fe(III)]/[Fe(II)] = 0.1 and E=0.680.059=0.62. after addition of 99.9 percent of the equivalent amount of cerium E=0.68 =
0.86. with 0.1 percent cerium in excess E=1.44-0.18=1.26. thus the potential changes
from 0.86 to 1.26 between 0.1 percent before and 0.1 percent after the end point. Any
oxidation-reduction indicator with Ef of the order of 1.05  0.1 will indicate a sharp end
point. The change in reduction potential in the titration of ferrous iron with ceric sulfate is
presented in Table 37-1 and graphically in Fig.37-1
Table 37-1.Titration of Iron(II)with cerium(IV) in 1 N sulfuric Acid
Cerium(IV) Added in % of
Ration
Iron(II)
E
Fe( III )
Fe( II )
9
0.1
0.62
50
1
0.68
91
10
0.74
99
100
0.80
99.9
1000
0.86
100
2.5x106
1.06(EEP)
Ratio
Ce( IV )
Ce( III )
0.001
100.1
1.26
0.01
101
1.32
0.1
110
1.38
With a given initial concentration of iron(II) and taking the volume charge into
account [Fe(II)], [Fe(III)], [Ce(III)], are [Ce(IV) are easily calculated at any point during
the titration.
The equilibrium constant of the reaction
2Ox1 + Red2
Red1 + Ox2
With the partial reactions
Ox1 + e
Ox2 + 2e
Red1
Red2
has been derived in Chapter 8 (p.165). It has been also been there that in a mixture
containing equivalent amounts of Ox1 and Red2 (corresponding to the equivalence point
in a titration) that
E  EEP 
E f ox1 / Re d1  2 Eoxf 1 / Re d1
3
As an example we consider the titration of tin (II) with iron (II) in 1 N hydrochloric acid.
EFef ( III ) / Fe( II )  0.68
E  0.68  0.059 log
ESnf ( IV ) / Sn( II )
[ Fe( III )]
0.059
[ Sn( IV )]
 0.14 
log
[ Fe( II )]
2
[ Sn( II )]
The values of E in the above titration are readily calculated. The data are presented in
Table 37-2. It is of interest to note that in the titration of iron (II) with cerium (IV)
Table 37.2. Titration of Tin(II) with Iron(III) in I N Hydrochloric Acid
Iron(III) Added in % of
Sn(II)
9
50
91
99
99.9
100
Ratio
0.1
1
10
100
1000
Ratio
100.1
101
110
Sn( IV )
Sn( II )
E
0.11
0.14
0.17
0.20
0.23
0.32 (EEP)
[ Fe( III )]
[ Fe( II )]
0.001
0.01
0.1
0.50
0.56
0.62
the change in potential between 0.1 per cent before the equivalence point and the
equivalence point is equal to that between the equivalence point and 0.1 per cent after the
equivalence point. The curve is symmetrical at both sides of the equivalence point. The
curve is symmetrical at both sides of the equivalence point. This is no longer true in the
titration of tin (II) with iron (III). Here the change in potential between 0.1 per cent
before and at the equivalence point is 0.09 V, but 0.18 V between that at the equivalence
point and 0.1 per cent after the equivalence point. In the present chapter we are concerned
with the visual detection of the end point. In Chapter 52 potentionmetric titrations are
discussed in which E is measured during the titration.
It is a simple matter to calculate titration curves of a mixture of two reducing
agents with an oxidant or of a mixture of two or more oxidants with a reductant.
Consider, for example, the titration of a mixture of tin(II) and iron(II) with cerium(IV).
Until the first equivalence point the values of E are the same as in Table 37-2. After the
first equivalence point iron(II) is oxidized and the potentials are the same as in Table 371 until the second equivalence point. At and after the second equivalence point the
concentration of Ce(III) formed during the titration of tin(II) must be taken into
consideration. This has only a small effect on EEP and potentials thereafter.
Oxidation-reduction (Redox) indicators
An oxidation-reduction indicator is a substance the oxidized form of which has a color
different from the reduced form. the oxidation and reduction of the indicator should be
reversible.
Iox = ne IRed
Iox is the oxidized form of the indicator, which has a color different from that of
the reduced form, IRed. A dilute iodine or iodide solution, for example, may be considered
as an indicator for oxidation-reduction reactions. The iodide solution is oxidized by
substances that have a more positive reduction potential than iodine, whereas substances
with a more negative reduction potential reduce iodine to iodide. If a strong reducing
agent is titrated with a strong oxidizing agent, iodine will be liberated at the end point. In
general, the action of an oxidation-reduction indicator does not depend upon the specific
nature of the oxidant or reducant titrated but upon the reduction potentials of the
indicator and of the system titrated.
The properties of some oxidation-reduction indicators which are very useful in
titrations with strong oxidizing agents such as ceric sulfate and potassium dichromate are
discussed below.