Intersecting Straight Lines: Titrimetric Applications

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Chapter 3
Intersecting Straight Lines: Titrimetric Applications
Julia Martin, Gabriel Delgado Martin and
Agustin G. Asuero
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/intechopen.68827
Provisional chapter
Intersecting Straight Lines: Titrimetric Applications
Julia Martin, Gabriel Delgado Martin and
Agustin G. Asuero
Additional information is available at the end of the chapter
Abstract
Plotting two straight line graphs from the experimental data and determining the point of
their intersection solve a number of problems in analytical chemistry (i.e., potentiometric
and conductometric titrations, the composition of metal-chelate complexes and binding
interactions as ligand-protein). The relation between conductometric titration and the
volume of titrant added leads to segmented linear titration curves, the endpoint being
defined by the intersection of the two straight line segments. The estimation of the statis-
tical uncertainty of the end point of intersecting straight lines is a topic scarcely treated in
detail in a textbook or specialized analytical monographs. For this reason, a detailed
treatment with that purpose in mind is addressed in this book chapter. The theoretical
basis of a variety of methods such as first-order propagation of variance (random error
propagation law), Fieller’s theorem and two approaches based on intersecting confidence
bands are explained in detail. Several experimental systems described in the literature are
the subject of study, with the aimof gaining knowledge and experience in the application
of the possible methods of uncertainty estimation. Finally, the developed theory has been
applied to the conductivity measurements in triplicate in the titration of a mixture of
hydrochloric acid and acetic acid with potassium hydroxide.
Keywords:
titrimetric, straight lines, breakpoint
1. Introduction
Titrimetry is one of the oldest analytical methods [1], and it is still found [2–4] in a develop-
ing way. It plays an important role in various fields as well as routine studies [5–9], being
used widely in the analytical laboratory given their simplicity, speed, accuracy, good repro-
ducibility, and low cost. It is, together with gravimetry, one of the most used methods to
determine chemical composition on the basis of chemical reactions (primary method).
© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and eproduction in any medium, provided the original work is properly cited.
DOI: 10.5772/intechopen.68827
© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.

Independent values of chemical quantities expressed in SI units are obtained through gra-
vimetry and titrimetry (classical analysis).
In titrimetry, the quantity of tested components of a sample is assessed by the use of a solution
of known concentration added to the sample, which reacts in a definite proportion. To identify
the stoichiometric point, where equal amounts of titrant react with equal amounts of analyte,
indicators are used in many cases to point out the end of the chemical reaction by a color
change.
Information on reaction parameters is usually obtained from an analysis of the shape of the
titration curve, whose shapes depend on some factors such as the reaction of titration, the
monitored specie (indicator, titrant, analyte or formed product) as well as the chosen [10, 11]
instrumental technique (Table 1) i.e., spectrophotometry, conductimetry or potentiometry, for
instance. The importance of titrimetric analysis has increased with the advance of the instru-
mental method of end point detection.
Linear response functions are generally preferred, and when the response function is nonlinear, a
linearization procedure has been commonly used with a suitable change of variables. Plotting
two straight lines graphs from the experimental data and determining the point of their intersec-
tion solve a number of problems in analytical chemistry [10, 11] (Table 1). In segmented linear
titration curves, the end point is defined by the intersection of the two straight segments. In some
common examples in analytical chemistry (conductometric, spectrophotometric and ampero-
metric titrations), this intersection lies beyond the linear ranges, and deviations from linearity
are often observed directly at the end point. All curvature points should be excluded from the
computation. The accuracy and precision of the results of a titrimetric determination are
influenced not only by the nature of the titration reaction but also by the technique [10, 11] of
the end-point location.
The problem of finding the breakpoint of two straight lines joined at some unknown point has a
long statistical history [12–14] and has received considerable attention in the statistical literature.
The problem in question is known by a variety of names (Table 2 ) [12–27]. Computer analy-
sis [28–31], elimination of outliers [32, 33], and confidence limits for the abscissa [22, 34, 35] have
been subject to study.
At the point of intersection (x
I
), the two lines have the same ordinate. The estimation of
statistical uncertainty of end points obtained from linear segmented titration is the subject
Technique Measured property
Conductimetric titrations Electrical conductivity
Potentiometric titrations Potential of an indicator electrode
Spectrophotometric titrations Absorbance
Amperometric titrations Diffusion current at a polarizable indicator
(dropping mercury or rotating platinum) electrode
Table 1. Instrumental end point detection techniques more widely applied.
Advances in Titration Techniques60

of this chapter. The topic is scarcely treated in [36, 37] analytical monographs. The method of
least squares is the most common and appropriate choice and when the relative statistical
uncertainties of the xdata are negligible compared to the ydata. Single linear regression or
weighted linear regression may be applied depending on whether the variance of yis constant
or varies from point to point with the magnitude of the response y, respectively.
The theoretical basis of a variety of method such as first-order propagation of variance for
the abscissa or intersection, the application of Fieller’s method [38–43], and other methods
based on intersecting hyperbolic confidence bands as weighted averages [57, 58] of the
abscissas of the confidence hyperbolas at the ordinate of intersection will be dealt in detail
in this book chapter. In addition, several experimental systems will be the subject of study,
with the aims of gaining knowledge and experience in the application of these methods to
uncertainty estimation.
2. Theory
V-shaped linear titration curves (Table 1) are well known in current analytical techniques
such as conductimetry, radiometry, refractometry, spectrophotometric and amperometric
titrations as well as in Gran’s plot. In this kind of titrations, the end point is usually located
at the intersection of two lines when a certain property (conductance, absorbance, diffusion
current) is plotted against the volume xof titrant added to the unknown sample containing
the analyte.
Let N
1
observations on the first line
^
y
1
¼a
1
þb
1
xð1Þ
and N
2
observations on the second
Name Authors
Breakpoint Jones and Molitoris [12]; Shanubhogue et al. [15]
Changepoint Csörgo and Horváth [16]; Krishanaiah and Miao [17]
Common intersection point Rukhin [18]
Hockey stick regression Yanagimoto and Yamamoto [19]
Intersystem crossing Kita et al. [20]
Piecewise linear regression Vieth [21]
Segmented regression Piegorsch [22]
Transition Bacon and Watts [23]
Two phase linear regression Christensen [24]; Lee et al. [25],; Seber [26, 27]; Shaban [14]; Sprent [13]
Table 2. Names received in the literature for the intersecting point of two straight lines.
Intersecting Straight Lines: Titrimetric Applications
http://dx.doi.org/10.5772/intechopen.68827
61

^
y
2
¼a
2
þb
2
xð2Þ
where a
1
,b
1
,a
2
,b
2
are the usual least squares estimates of the kth line (k= 1, 2), respectively. As
it is stated in the introduction section when the relative statistical uncertainties of the xdata are
negligible compared to the ydata, the use of the least squares method is the most common
alternative. The ordinate variance can be considered on a priori grounds to vary systematically
as a function of the position along the curve, so that weighted least squares analysis is
appropriate. Formulae for calculating the intercept a, the slope band their standard errors by
weighted linear regression [59] are given in Table 3, where the analogy with simple linear
regression (i.e., w
i
= 1), is evident.
Note that in summation (1) and (2) by dividing by N
1
and N
2
, respectively, we get
y
1
¼a
1
þb
1
x
1
ð3Þ
y
2
¼a
2
þb
2
x
2
ð4Þ
At the point of intersection, the lines (1) and (2) have the same ordinate ^
y
1
¼^
y
2
and the
abscissa of intersection (denotes by ^
x
I
) is given by
a
1
þb
1
^
x
I
¼a
2
þb
2
^
x
I
ð5Þ
^
x
I
¼a
2
�a
1
b
1
�b
2
¼�Δa
Δbð6Þ
Random error in the points produces uncertainty in the slopes and intercepts of the lines, and
therefore in the point of intersection. The probability that a confidence interval contains the
true value is equal to the confidence level (e.g., 95%).
•
Equation
^
y
i
¼aþbx
i
•
Weights
w
i
¼1=s
2
i
•
Explained sum of squares
SS
Reg
¼Xw
i
ð^
y
i
�yÞ
2
•
Residual sum of squares
SSE ¼Xw
i
ðy
i
�^
y
i
Þ
2
•
Mean
x¼Xw
i
x
i
=Xw
i
y¼Xw
i
y
i
=Xw
i
•
Sum of squares about the mean
S
XX
¼Xw
i
ðx
i
¼xÞ
2
S
YY
¼Xw
i
ðy
i
¼yÞ
2
S
XY
¼Xw
i
ðx
i
¼xÞðy
i
�yÞ
•
Slope
b¼S
XY
=S
xx
•
Intercept
a¼y�bx
•
Weighted residuals
w
1=2
i
ðy
i
�^
y
i
Þ
•
Correlation coefficient
r¼S
XY
=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
XX
S
YY
p
•
Standard errors
s
2
y=x
¼
SSE
n�2
¼
S
YY
�b
2
S
XX
n�2
•
s
2
a
¼s
2
y=x
ðXw
i
x
2
i
Þ=ðS
XX
Xw
i
Þ
•
s
2
b
¼s
2
y=x
=S
XX
•
Covða, bÞ¼xs
2
y=x
=S
XX
Table 3. Formulae for calculating statistics for weighted linear regression.
Advances in Titration Techniques62

3. First-order propagation of variance for V½^
x
I

The precision of the point of intersection and the corresponding statistical confidence interval
can be found in the simplest way by considering the random error propagation law [60]. Some
authors [61, 62] evaluate the uncertainty in ^
x
I
on this way. First-order propagation of variance
retains only first derivatives in the Taylor expansions and this procedure leads to
V½^
x
I
¼
∂
^
x
I
∂Δa

2
V½Δaþ
∂
^
x
I
∂Δb

2
V½Δbþ2∂^
x
I
∂Δa

∂^
x
I
∂Δb

CovðΔa, ΔbÞ
¼
∂
Δa
Δb
ðÞ
∂Δa

2
V½Δaþ
∂
Δa
Δb
ðÞ
∂Δb

2
V½Δbþ2
∂Δa
Δb

∂Δa
0
B
B
@1
C
C
A
∂Δa
Δb

∂Δb
0
B
B
@1
C
C
ACovðΔa, ΔbÞ
¼V½Δa
Δb
2
þΔa
2
V½Δb
Δb
4
2ΔaCovðΔa, ΔbÞ
Δb
3
ð7Þ
valid in those cases, in which the standard deviations of the ordinate data are a small fraction
of their magnitude. Taking into account Eq. (6), Eq. (7) may be rewritten as follows
V½^
x
I
¼ 1
Δb
2
V½ΔaþΔa
2
Δb
2
V½Δb2Δa
ΔbCovðΔa, ΔbÞ

¼V½Δaþ^
x
2I
V½Δbþ2^
x
I
CovðΔa, ΔbÞ
Δb
2
ð8Þ
Then, the standard error estimate of ^
x
I
is as follows
sð^
x
I
Þ¼ ffiffiffiffiffiffiffiffi
V½^
x
pð9Þ
The end point ^
x
I
depends on four least squares parameters a
1
,a
2
,b
1
,b
2
that are random
variables. Segment one parameters depend only on measurements made along segment one
and these are statistically independent of the measurements along segment two. However, Δa,
and Δb are correlated random variables because each involves b
1
and b
2
. Note that, a
1
and a
2
are related to b
1
and b
2
by means of Eqs. (1) and (2).
The variances of Δaand Δbare given by
V½Δa¼V½a
1
a
2
¼V½a
1
þV½a
2
ð10Þ
V½Δb¼V½b
1
b
2
¼V½b
1
þV½b
2
ð11Þ
and for the covariance between Δaand Δb, we get [63]
CovðΔa, ΔbÞ¼Covða
1
a
2
,b
1
b
2
Þ
¼Covy
1
b
1
x
1
ðy
2
b
2
x
2
Þ,b
1
b
2
¼Covðy
1
y
2
b
1
x
1
þb
2
x
2
,b
1
b
2
Þ
¼Covðy
1
y
2
,b
1
b
2
ÞCovðb
1
x
1
b
2
x
2
,b
1
b
2
Þ
¼Covðb
1
x
1
b
2
x
2
,b
1
b
2
Þ
¼
x
1
V½b
1
x
2
V½b
2

ð12Þ
Intersecting Straight Lines: Titrimetric Applications
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63

It should be noted that in the calculations, the variance regression estimates from both line
segments are pooled into a single s
p2
, by using the following formula which weights each
contribution according to the corresponding [64–67] degrees of freedom
s
2
p
¼X
N
1
i¼1
ðy
1i
y
1
Þ
2
þX
N
2
i¼1
ðy
2i
y
2
Þ
2
ðN
1
2ÞþðN
2
2Þ¼ðN
1
2Þs
2
1
þðN
2
2Þs
2
2
N
1
þN
2
4ð13Þ
The standard deviation in Eq. (13) is calculated on the assumption that the s
(y/x)
values for the
two lines are sufficiently similar to be pooled.
From expression in Table 3 for the variance of the intercept (s
a2
=V[a]), we may derive
V½a¼s
2
Xw
i
x
2
i
S
XX
Xw
i

2
43
5¼s
2
S
XX
þXw
i
x
i

2
Xw
i
S
XX
Xw
i

2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
¼s
2
S
XX
þXw
i

Xw
i
x
i
Xw
i
!
2
S
XX
Xw
i

2
6
6
6
6
6
4
3
7
7
7
7
7
5
¼s
2
S
XX
þXw
i

x
2
S
XX
Xw
i

2
43
5¼s
2
1
Xw
i
þx
2
S
XX
"#
ð14Þ
in which s
2
is s
(y/x)
in Tab l e 3 ;∑w
i
is the sum of weights, which simply reduces to N, the number
of points if the non-weighted least squares analysis is used. Taking into account Eq. (14), Eqs. (10)
and (11) lead to
V½Δa¼V½a
1
þV½a
2
¼ 1
Xw

1
þx
2
1
ðS
XX
Þ
1
2
6
43
7
5s
2
1
þ1
Xw

2
þx
2
1
ðS
XX
Þ
2
2
6
43
7
5s
2
2
ð15Þ
V½Δa¼ 1
Xw

1
þ1
Xw

2
þx
2
1
ðS
XX
Þ
1
þx
2
1
ðS
XX
Þ
2
2
6
43
7
5s
2
p
ð16Þ
V½Δb¼V½b
1
þV½b
2
¼ s
2
1
ðS
XX
Þ
1
þs
2
1
ðS
XX
Þ
2
¼1
ðS
XX
Þ
1
þ1
ðS
XX
Þ
2

s
2
p
ð17Þ
CovðΔa, ΔbÞ¼
x
1
V½b
1
x
2
V½b
2
¼ x
1
ðS
XX
Þ
1
þx
2
ðS
XX
Þ
2

s
2
p
ð18Þ
Once the values of V[Δa], V[Δb] and Cov[Δa,Δb] are known from Eqs. (16), (17) and (18),
respectively, the estimate of the variance of the intersection abscissa of the two straight lines,
V½^
x
I
, is calculated by applying Eq. (8).
Advances in Titration Techniques64

4. Confidence interval on the abscissa of the point of intersection of two
fitted linear regressions
The use of confidence intervals is another alternative to express the statistical uncertainty of ^
x
I
.
This method depends on the distribution function of the random variable ^
x
I
. If the ordinates y
i
are assumed to have Gaussian (normal) distribution, the least squares parameters as well as
Δa, and Δbare also normally distributed [68]. However, ^
x
I
, which even is regarded as the ratio
of two normally distributed variables, is not normally distributed and, indeed, becomes more
and more skewed [69] as the variance levels increase. For sufficiently small variance though,
^
x
I
, is approximately normally distributed. Under these circumstances, confidence intervals
may be calculated from the standard deviation of ^
x
I
, which is also accurate only when
variances are small.
However, the construction of the confidence interval (limits) for the equivalence point by using
the Student’st-test
^
x
I
t
α=2
sð^
x
I
Þð19Þ
where t
α/2
is the Student’ststatistics at the 1 αconfidence level (i.e., leaving an area of α/2 to
the right) and for the number of degrees of freedom (N
1
+N
2
4) inherent in the standard
deviation of ^
x
I
, could be misleading. Note that because ^
x
I
involves the ratio of random vari-
ables, first-order propagation of variance is not exact [69]. Evidently, ^
x
I
is a random variable
not normally distributed unless sð^
x
I
Þis small enough. When the variances of the responses are
not necessarily small, a solution to this problem is to apply the called Fieller’s theorem [38–43].
Another point of view is focused on the problem in the calculation of the limits of the
confidence intervals by using the confidence bands for the two segmented branches.
5. The Fieller’s theorem
This theorem [38–43] is supported by two capital premises:
i. Any linear combination zof normally distributed random variables is itself normally
distributed.
ii. If the standardized variable
z
ffiffiffiffiffiffi
V½z
pis distributed as N(0, 1), then zis distributed as t.
Consider now any pair of individual line segments written as a difference zas follows
z¼½a
1
þb
1
x
I
½a
2
þb
2
x
I
¼Δaþx
I
Δbð20Þ
Note that for any such pair of lines, the difference zis not, in general, zero, because the “best”
end point cannot be the one for each pair of lines of the collection. However, the mean 〈z〉of all
these zvalues is zero and zare normally distributed because it is formed as a linear combina-
tion of normally distributed variables. Taking into account that a
1
,a
2
,b
1
and b
2
are normally
Intersecting Straight Lines: Titrimetric Applications
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65

distributed, then zwill be normally distributed. Then, in the vicinity of the intersection point, z
has zero mean and its variance is
V½z¼V½ΔaþxIΔb¼V½Δaþx2
IV½Δbþ2xICovðΔa, ΔbÞð21Þ
and therefore, z
ffiffiffiffi
^
V
p½z
is distributed as N(0, 1) and according to (ii)
z
ffiffiffiffi
^
V
p½z¼tð22Þ
This is called Fieller’s Theorem [34, 38]. The development of Eq. (11) leads to the equation
ðΔaþxIΔbÞ2
V½Δaþx2
IV½Δbþ2xICovðΔa, ΔbÞ¼t2ð23Þ
which on rearrangement leads to
ðΔaÞ2þ2xIΔaΔbþx2
IðΔbÞ2¼t2V½Δaþt2x2
IV½Δbþ2t2xICovðΔa, ΔbÞð24Þ
which may be factored as
ðΔaÞ2t2V½Δaþ2xIΔaΔbt2CovðΔa, ΔbÞþx2
IðΔbÞ2t2V½Δb¼0ð25Þ
The solution of Eq. (25) gives the confidence limits for x
I
estimated, where t
α/2
is the appropri-
ate value of the Student distribution at a αsignificance level (confidence level 1 α) for N
1
+N
2
4 degrees of freedom. Note that in Eqs. (21), (23), (24) and (25), the corresponding values of V
[Δa], V[Δb] and Cov[Δa,Δb] are given by Eqs. (16), (17) and (18), respectively, as in the first-
order propagation of variance for V½^
xI.
The first and last groups of symbols enclosed in braces in Eq. (25) has the form of hypothesis
tests, that is, two-tailed tests, for significant difference of intercepts and significant difference of
slopes, respectively. When the hypothesis test for different slope fails, the coefficient of x
I2
Topic Reference
Arrhenius plot Cook and Charnock [44]; Han [45]; Puterman et al. [46]
Calibration curves Baxter [47]; Bonate [48]; Mandel y Linning [49]; Schwartz [50–52]
Estimation of safe doses Yanagimoto and Yanamoto [19]
Estimation of uncertainty in binding constants Almansa López et al. [53]
Models for biologic half-life data Lee et al. [25]
Position and confidence limits of an extremum Asuero and Recamales [54]; Heilbronner [55]
Standard addition method Franke et al. [56]
Table 4. Some applications of Fieller theorem in analytical chemistry.
Advances in Titration Techniques66

becomes negative finding two complex roots [22], so Fieller confidence interval embraces the
entire x-axis (the lower and upper limits should strictly be set to ∞and ∞, respectively) at the
chosen level of confidence.
This method has been extensively described in some other contexts in analytical and chemical
literature (Table 4).
6. Use of hyperbolic confidence bands for the two linear branches
Several procedures dealing with hyperbolic confidence bands approximate them by straight
lines and give symmetric confidence intervals for estimated x
I
[58, 61, 70–72]. Evidently, the
best confidence interval would be obtained by the projection on the abscissa of the surface
between the four hyperbolic arcs [73].
Because a confidence band, bounded by two hyperbolic arcs, is associated with each regression
line, it is obvious that the point of intersection, x
I
, is only a mean value, with which a certain
confidence interval is associated. If the signal values both before and after the point of inter-
section are normally distributed around the line with a constant standard deviation, the point
of intersection and its statistical confidence interval will be estimated by the projection of the
intersection onto the abscissa. The confidence interval (x
l,
x
u
) for the true value of the equiva-
lence point is given by the projection on the abscissa of the common surface delimited by the
four hyperbolic arcs.
For the first line, we get:
y
01
t
1
s
y
01
ð26Þ
and for the second line:
y
02
t
2
s
y
02
ð27Þ
t
1
and t
2
are the corresponding tStudent values for α/2 = 0,05 and N
1
2 and N
2
2 degrees of
freedom, respectively. Hence, the lower value x
l
of the confidence interval is obtained by
solving the following equation:
y
01
t
1
s
y
01
¼y
02
þt
2
s
y
02
ð28Þ
The higher value x
u
is obtained from the equation:
y
01
þt
1
s
y
01
¼y
02
t
2
s
y
02
ð29Þ
From Eqs. (1) and (3) we get
^
y
1
¼y
1
þb
1
ðxx
1
Þð30Þ
and then the variance of the fitted y
1
value will be given by
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67

V½^
y
1
¼V½y
1
þðxx
1
Þ
2
V½b
1
¼ V½y
1

Xw

1
þðxx
1
Þ
2
V½y
1

ðS
XX
Þ
1
¼1
ðXwÞ
1
þðxx
1
Þ
2
ðS
XX
Þ
1
!
V½y
1
ð31Þ
Note that the variance of the (weighted) mean of the values
V½y
1
¼VXw

1
y
1
Xw

1
2
6
43
7
5¼VXðffiffiffiffi
w
pÞ
1
ðffiffiffiffi
w
pÞ
1
y
1

Xw

1
2
6
43
7
5
¼Xw

1
V½ð ffiffiffiffi
w
pÞ
1
y
1

Xw

1

2
¼V½y
1

Xw

1
ð32Þ
and that the mean y
1
value and the slope b
1
are uncorrelated random variables (property,
which was also applied in Eq. (12) without further demonstration) as shown as follows. Taking
into account that
y
1
¼Xb
1
ðffiffiffiffi
w
pÞ
1
y
1
and b
1
¼Xc
1
ðffiffiffiffi
w
pÞ
1
y
1
where
b
1
¼ðffiffiffiffi
w
pÞ
1
ðXwÞ
1
c
1
¼ðffiffiffiffi
w
pÞ
1
ðxx
1
Þ
ðS
XX
Þ
1
ð33Þ
and then
Covðb
1
,c
1
Þ¼ Xa
1
c
1

V½ffiffiffiffi
w
p

1
y
1
¼ Xðffiffiffiffi
w
pÞ
1
ðxx
1
Þ
ðS
XX
Þ
1
Xw

1
0
B
@1
C
AV½y¼0ð34Þ
From Eq. (31), we get for the standard error of the fitted value
s
y01
¼V½^
y
1i
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

1
þðxx
1
Þ
2
ðS
XX
Þ
1
v
u
u
t
0
B
@1
C
As
1
y
01
¼^
y
1i
;s
1
¼ffiffiffiffiffiffiffiffiffiffiffi
V½y
1

q

ð35Þ
Thus, the lower value, x
l
, for the confidence interval is obtained by solving the equation
a
1
þb
1
x
l
þt
1
s
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

1
þðx
l
x
1
Þ
2
ðS
XX
Þ
1
v
u
u
t¼a
2
þb
2
x
l
þt
2
s
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

2
þðx
l
x
2
Þ
2
ðS
XX
Þ
2
v
u
u
tð36Þ
by, for example, successive approximations with an Excel spreadsheet. The higher value x
u
is
obtained in the same way from the equation that follows also by successive approximations
Advances in Titration Techniques68

a
1
þb
1
x
u
þt
1
s
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

1
þðx
u
x
1
Þ
2
ðS
XX
Þ
1
v
u
u
t¼a
2
þb
2
x
u
þt
2
s
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

2
þðx
u
x
2
Þ
2
ðS
XX
Þ
2
v
u
u
tð37Þ
The point of view of Liteanu et al. [57, 58] is very interesting: the authors consider the point of
intersection x
I
as belonging to the linear regression before the equivalence point. Then, a
certain interval is associated with it. If it is regarded as belonging to the linear regression after
the equivalence point, however, another interval is associated with it. As the equivalence point
belongs concurrently to both linear regressions, the confidence interval of the two segments
can be got by taking the weighted averages of the branches of the two separate sets of
confidence intervals. So, we obtain the ultimate confidence interval (x
Il
,x
Iu
) where
ðN
1
2Þðx
I
Þ
l
1
þðN
2
2Þðx
I
Þ
l
2
N
1
þN
2
4¼x
l
I
ð38Þ
ðN
1
2Þðx
I
Þ
u
1
þðN
2
2Þðx
I
Þ
u
2
N
1
þN
2
4¼x
u
I
ð39Þ
The two values of the limits of the confidence interval will be given by the two solutions of the
equations
y
I
¼a
1
þb
1
ðx
I
Þ
1
t
1
s
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

1
þðx
I
Þ
1
x
1

2
ðS
XX
Þ
1
v
u
u
u
u
tð40Þ
y
I
¼a
2
þb
2
ðx
I
Þ
2
t
2
s
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Xw

2
þðx
I
Þ
2
x
2

2
ðS
XX
Þ
2
v
u
u
u
u
tð41Þ
As the estimation method used assumes the worst case in combining random error of the two
lines, the derived confidence limits are on the pessimistic (i.e., realistic) side.
On rearrangement Eq. (40) and squaring, we have
y
I
a
1
b
1
ðx
I
Þ
1

2
¼t
2
1
s
2
1
1
Xw

1
þðx
I
Þ
1
x
1

2
ðS
XX
Þ
1
0
B
@1
C
Að42Þ
which by simple algebra it may be ordered in powers of x
l
as
b
2
1
t
2
1
s
2
1
ðS
XX
Þ
1

ðx
I
Þ
2
1
2b
1
ðy
I
a
1
Þt
2
x
1
s
2
1
ðS
XX
Þ
1

ðx
I
Þ
I
þðy
I
a
2
1
Þt
2
1
s
2
1
1
Xw

1
þx
2
ðS
XX
Þ
1
0
B
@1
C
A¼0ð43Þ
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69

and taking into account the values of V[b
1
], Cov[a
1
,b
1
] and V[a
1
] (see Table 3), we get finally
ðb
2
1
t
2
1
V½b
1
Þðx
I
Þ
2
1
2b
1
ðy
I
a
1
Þþt
2
1
Covða
1
,b
1
Þðx
I
Þ
1
þðy
I
a
1
Þ
2
t
2
1
V½a
1
¼0ð44Þ
whose roots give the two values of ðx
I
Þ
1
. Since the point of intersection x
I
belongs to one of the
response functions, then a certain confidence interval is associated with it.
Similarly, if it is regarded as belonging to the other response function, there is another confi-
dence interval associated with it
ðb
2
2
t
2
2
V½b
2
Þðx
I
Þ
2
2
2b
2
ðy
I
a
2
Þþt
2
2
Covða
2
,b
2
Þðx
I
Þ
2
þðy
I
a
2
Þ
2
t
2
2
V½a
2
¼0ð45Þ
Because the intersection point belongs concomitantly to the two response functions, the two
segments which together compose the confidence interval, will be obtained by averaging the
segments of the two separate confidence intervals, Eqs. (40) and (41). The two values of the limits
of the confidence interval will be the two solutions of the second degree Eqs. (44) and (45).
The bands mentioned in this section are [63] for the ordinate of the true line at only a single point.
If we desire the confidence bands for the entire line, the critical constant ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2F
α
2
,
n2
qshould be
substituted for t
α/2
, originating wider bands.
7. Statistical uncertainty of endpoint differences
When we are dealing with the titration of a mixture of a strong and a weak acid that is,
hydrochloric and acetic acids, then if x
I
is the volume at which the straight lines one and two
intersect and x
II
the volume at which the two and three lines intersect, the difference x
II
–x
I
denoted as Δx, is given by
Δx¼^
x
II
^
x
I
¼Δa
2
Δb
2
þΔa
1
Δb
1
ð46Þ
By multiplying Δxby the molarity of titrant, we have the amount in millimoles of the second
acid, that is, acetic acid, in the reaction mixture.
First-order propagation of variance applied to Δxleads to [65] the following expression
V½Δx¼V½^
x
I
þV½^
x
II
þCovðΔa
1
,Δa
2
ÞþCovðΔa
1
,Δb
2
ÞþCovðΔa
2
,Δb
1
ÞþCovðΔb
1
,Δb
2
Þð47Þ
where
CovðΔa
1
,Δa
2
Þ¼2V½y
2
þx
2
2
V½b
2

Δb
1
Δb
2
 ð48Þ
Advances in Titration Techniques70

CovðΔa
1
,Δb
2
Þ¼2x
2
^
x
II
V½b
2

Δb
1
Δb
2
 ð49Þ
CovðΔa
2
,Δb
1
Þ¼2x
2
^
x
I
V½b
2

Δb
1
Δb
2
 ð50Þ
CovðΔb
1
,Δb
2
Þ¼2^
x
I
^
x
II
V½b
2

Δb
1
Δb
2
 ð51Þ
The standard error estimate is given by
s
Δx
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V½^
x
I
þV½^
x
II
þX
4
Cov
rð52Þ
where ∑
4
is the sum of Eqs. (48)–(51).
Attempts to derive confidence limits for Δxas we get in the previous Fieller’s theorem
section fails because the quantity analogous to zof Eq. (20) involves products of random
variables. Therefore, this quantity is not normally distributed and so exact confidence
limits cannot be found in terms of Student’stdistribution. Because in this case the exact
confidence limits cannot be calculated, we use the small variance confidence interval
C:I:¼2t
α=2
s
Δx
ð53Þ
8. Application to experimental system
A bibliographic search allows us to demonstrate the importance of conductivity measure-
ments despite their antiquity. The general fundamentals of this technique are collected in
Gelhaus and Lacourse (2005) [74] and Gzybkoski (2002) [75]. Its importance in the educa-
tional literature has been highlighted [76, 77] and many examples have been recently
published in the Journal of Chemical Education i.e., studies on sulfate determination [78];
the identification and quantification of an unknown acid [79], electrolyte polymers [80, 81],
acid and basic constants determinations [82], its use in general chemistry [83], microcom-
puter interface [84] and conductometric-potentiometric titrations [85]. An accurate method
of determining conductivity in acid-base reactions [86], the acid-base properties of weak
electrolytes [87], and those of polybasic organic acids [88] have also been recently subject of
study.
The relation between conductometric and the volume of titrant added leads to segmented
linear titration curves, the endpoint being defined by the intersection of the two straight lines
segments. What follows is the application of the possible methods of uncertainty estimation of
the endpoint of data described in the literature as well as experimental measurements carried
out in the laboratory.
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8.1. Conductometric titration of 100 mL of a mixture of acids with potassium hydroxide 0.100 M
Tab l e 5 shows the data [conductance (1/R), volume (x)] published by Carter et al. [69]; Schwartz
and Gelb [65]) and corresponding to the conductometric titration of a mixture of acids, perchloric
acid and acetic acid with potassium hydroxide 0.100 M as titrant agent. The points recorded
belong to the three branches of the titration curve; the first (branch A) corresponds to the
neutralization of perchloric acid, the second (branch B) to the neutralization of acetic acid, and
the third (branch C) to the excess of potassium hydroxide.
Let us focus first on the perchloric acid titration. The plot of conductance data (1/R) versus
volume (x), in general, is not linear due to the dilution effect of the titrant. So that, as it is
carried out in the usual way, it is plotted the product (1/R)(100 + x) versus x(see Figure 1).
Firstly, Schwartz and Gelb [65] select 13 points, six (volume 4–14 mL) for branch A and seven
(volume 20–32 mL) for branch B. The points near to the endpoint of perchloric acid are deviating
from linearity and discarded in the first instance. It is also considered that the data have a
different variance V(100 + x
i
)
2
, being the weighting factor (100 + x
i
)
2
(see Table 5 ).
In the case of acetic titration, six points (volume 35–44 mL) are selected for branch C, at first.
The points of branch B near to the acetic acid endpoint are discarded. Figures 2 and 3show the
straight line segments with the corresponding selected points.
Figure 1. Conductometrictitration of a mixt ureof pe rchloricand acetic acids with potassium hydroxide (data shown in Ta ble 5 ).
Advances in Titration Techniques72

1/R xy W
i
1/R xy W
i
6.975 4 0.7254 9.246E05 3.633 24 0.4505 6.504E05
6.305 6 0.6683 8.900E05 3.742 26 0.4715 6.299E05
5.638 8 0.6089 8.573E05 3.840 28 0.4915 6.104E05
5.020 10 0.5522 8.264E05 3.946 30 0.5130 5.917E05
4.432 12 0.4964 7.972E05 4.052 32 0.5349 5.739E05
3.865 14 0.4406 7.695E05 4.097 33 0.5449 5.653E05
3.610 15 0.4152 7.561E05 4.145 34 0.5554 5.569E05
3.415 16 0.3961 7.432E05 4.280 35 0.5778 5.487E05
3.328 17 0.3894 7.305E05 4.445 36 0.6045 5.407E05
3.330 18 0.3929 7.182E05 4.772 38 0.6585 5.251E05
3.370 19 0.4010 7.062E05 5.080 40 0.7112 5.102E05
3.420 20 0.4104 6.944E05 5.380 42 0.7640 4.959E05
3.522 22 0.4297 6.719E05 5.680 44 0.8179 4.823E05
Table 5. Data conductance (1/R) and volume (x) corresponding to the titration of a mixture of perchloric acid and acetic
acid with potassium hydroxide.
Figure 2. Conductometric titration of perchloric acid in the mixture (branches A and B).
Intersecting Straight Lines: Titrimetric Applications
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Table 6 includes the intermediate results obtained in the calculation of the first endpoint,
corresponding to the neutralization of perchloric acid (Figure 2), in order to follow the pro-
cedures previously detailed. The first endpoint is located at 16.367 mL and therefore
1.637 mmol of HClO
4
. The estimated standard error at the endpoint, using the first-order
propagation of variance, is 0.039 mL. The confidence limits are calculated using t= 2.262
(9 degrees of freedom) and correspond to 16.455 and 16.279 mL, respectively, for the upper
and lower limits, being the confidence interval equal to 0.176 mL. The application of Fieller's
theorem leads to the values of 16.455 and 16.278 mL, respectively. Carter et al. [67] give values
of 16.455 and 16.279 mL, identical to the first ones indicated.
The second endpoint, corresponding to the complete neutralization of both perchloric and
acetic acids, is located at 34.197 mL. If x
(I)
is the volume in which lines A and B intersect, and
x
(II)
the volume in which lines B and C intersect, the difference x
(II)
–x
(I)
(34.1971–16.3665 mL)
corresponds to acetic acid in the sample, 17.831 mL. If the above methodology is used forlines,
B and C (Figure 3) give x
(II)
s
d[x(II)]
equal to 34.197 0.0478, and 34.305 and 34.089 mL for the
confidence limits.
Figure 3. Conductometric titration of acetic acid in the mixture (branches B and C).
Advances in Titration Techniques74

1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
6.975 4 0.7254 9.246E05 3.420 20 0.4104 6.944E05
6.305 6 0.6683 8.900E05 3.522 22 0.4297 6.719E05
5.638 8 0.6089 8.573E05 3.633 24 0.4505 6.504E05
5.020 10 0.5522 8.264E05 3.742 26 0.4715 6.299E05
4.432 12 0.4964 7.972E05 3.840 28 0.4915 6.104E05
3.865 14 0.4406 7.695E05 3.946 30 0.5130 5.917E05
4.052 32 0.5349 5.739E05
N
1
=6 [ΣW
i
]1= 5.065E04 N
2
=7 [ΣW
i
]2= 4.423E04
x
1
(mean)= 8.786 y
1
(media)= 0.5881 x
2
(mean)= 25.746 y
2
(mean)= 0.4690
[S(xx)]1= 5.8987E03 [S(xx)]2= 7.0636E03
[S(xy)]1= 1.683E04 [S(xy)]2= 7.323E05
[S(yy)]1= 4.804E06 [S(yy)]2= 7.594E07
b
1
=0.028535824 Δb= 0.038903 b
2
= 0.010367102
a
1
= 0.838792976 Δa=0.636705 a
2
= 0.202087987
[R2]1= 0.99988 x(I)= 16.3665 [R2]2= 0.99974
V[y/x]1= 1.45580E10 V[y/x]2= 3.96396E11
V(b
1
)= 2.468E08 V[pooled]= 8.67244E11 V(b
2
)= 5.612E09
V(a
1
)= 2.193E06 V(a
2
)= 3.809E06
Cov(a
1
,b1)= 2.168E07 Cov(a
2
,b
2
)= 1.445E07
V[Δa]= 6.002E06 V[Δa]= 9.640E06 V[x(I)]= 0.0015 FIELLER ax2+bx +c=0
V[Δb]= 3.029E08 V[Δb]= 2.698E08 s[x(I)]= 0.0389 a= 1.513E03 V(u)= 16.455
cov(Δa,Δb)= 3.613E07 cov(Δa,Δb)= 4.453E07 t(0, 05, 9)= 2.262 b=4.953E02 V(l)= 16.278
Pooled variances ts[x(I)]= 0.0880 c= 4.053E01
Table 6. Intermediate results obtained in the calculation of the first endpoint (titration of perchloric acid with potassium hydroxide (Figure 2)).
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However, as it is indicated in the section on “statistical uncertainty of endpoint differences,”
the statistical uncertainty of Δxis not a simple combination of uncertainties for x
(I)
and x
(II)
. The
attempt to deduce equations analogous to Eqs. (22) and (25) in order to calculate the confi-
dence limits for Δx, is not applicable since the magnitude analogous to zin Eq. (20) implies, in
this case, the product of random variables.
This quantity is not normally distributed, and therefore, no exact confidence limits can be
calculated in terms of the Student tdistribution. The application of (first-order) propagation
of the variance is nonetheless feasible, leading this procedure to an expression for the standard
error of Δxof the same type as Eq. (9) for a single endpoint.
The latter methodology is applied to the optimal case detailed by Schwartz and Gelb [65]. The
corresponding data areshown in Figure 4, and the calculations necessary to locate the equivalence
points, first and second, are shown in Tab l e 7. The results obtained are: first equivalence point
(perchloric acid): x
(I)
= 16.358 mL, s[x
(I)
] = 0.035 mL, ts[x
(I)
] = 0.078 mL, [I.C.]
I
= 0.156 mL. Second
equivalence point (mixture of perchloric and acetic acids): x
(II)
= 34.244 mL, s[x
(II)
] = 0.027 mL, ts
[x
(II)
] = 0.061 mL, [IC]
II
= 0.122 mL. This latter is notcorrect because it does not take into account the
covariances described in Section 7. If covariances are incorporated into the calculations, we get for
the second point (acetic acid): x= 17.887 mL, s[Δx]=0.040mL;ts[Δx] = 0.086 mL, [IC]
Δx
= 0.172 mL.
The confidence interval, as expected, is higher than that found for x
(II)
, despite decreasing the
value of Student's tby increasing the number of degrees of freedom: N
1
+N
2
N
3
23=13).
Some points near to the endpoint appear to deviate slightly from linearity. However, it is not
always clear whether or not to omit these problem points in the analysis, which can be done by
Figure 4. Illustrative example described by Schwartz and Gelb [65] as optimal. Numerical data are shown in Table 5. First
branch (A), volumes of 4–12 mL, 5 points. Second branch (B), volumes of 22–34 mL, 8 points. Third branch (C), volumes of
35–44 mL, 6 points.
Advances in Titration Techniques76

1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
6.975 4 0.7254 9.246E05 3.522 22 0.4297 6.719E05
6.305 6 0.6683 8.900E05 3.633 24 0.4505 6.504E05
5.638 8 0.6089 8.573E05 3.742 26 0.4715 6.299E05
5.020 10 0.5522 8.264E05 3.840 28 0.4915 6.104E05
4.432 12 0.4964 7.972E05 3.946 30 0.5130 5.917E05
4.052 32 0.5349 5.739E05
4.097 33 0.5449 5.653E05
4.145 34 0.5554 5.569E05
N
1
=5 [ΣW
i
]1= 4.296E04 N
2
=8 [ΣW
i
]2= 4.850E04
x
1
(mean)= 7.852 y
1
(media)= 0.6145 x
2
(mean)= 28.362 y
2
(mean)= 0.4962
[S(xx)]1= 3.4319E03 [S(xx)]2= 8.221E03
[S(xy)]1= 9.856E05 [S(xy)]2= 8.624E05
[S(yy)]1= 2.831E06 [S(yy)]2= 9.048E07
b
1
=0.028717584 Δb= 0.039208 b2= 0.010490769
a1= 0.839983924 Δa=0.641362 a2= 0.198621917
[R2]1= 0.99989 x(I)= 16.3578 [R2]2= 0.99992
V[y/x]1= 1.03732E10 V[y/x]2= 1.25234E11
V(b
1
)= 3.023E08 V[pooled]= 4.29264E11 V(b
2
)= 1.523E09
V(a
1
)= 2.105E06 V(a
2
)= 1.251E06
cov(a
1
,b
1
)= -2.373E07 cov(a
2
,b
2
)= -4.321E08
V[Δa]= 3.356E06 V[Δa]= 5.160E06 V[x(I)]= 0.0012 FIELLER ax2 + bx + c=0
V[Δb]= 3.175E08 V[Δb]= 1.773E08 s[x(I)]= 0.0347 a= 1.537E03 V(u)= 16.436
cov(Δa,Δb)= 2.805E07 cov(Δa,Δb)= 2.463E07 t(0, 05, 9)= 2.262 b=5.029E02 V(l)= 16.280
Pooled variances ts[x(I)]= 0.0784 c= 4.113E01
V(u)= 16.436 V(l)= 16.279
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1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
1/Rx y= (1/R)(100 + x) (100 + x
i
)
2
4.280 35 0.5778 5.487E05 Δb= 0.016147
4.445 36 0.6045 5.407E05 Δa=0.552936
4.772 38 0.6585 5.251E05 x(I)= 34.2444
5.080 40 0.7112 5.102E05 Δx= 17.8866
5.380 42 0.7640 4.959E05
5.680 44 0.8179 4.823E05 V[pooled]= 1.12632E11
N
3
=6 [ΣW
i
]1= 3.103E04
x
3
(mean)= 39.022 y
3
(media)= 0.6851
[S(xx)]1= 3.1199E03
[S(xy)]1= 8.311E05
[S(yy)]1= 2.214E06
V[x(I)]= 0.0007 t(0, 05, 9)= 2.262 V(u)= 34.305
b3= 0.026637531 s[x(I)]= 0.0269 ts[x(I)]= 0.0610 V(l)= 34.183
a3= 0.354313932
[R2]3= 0.99998
V[y/x]3= 9.37285E12 V[x(I)]= 1.201E03
V(b3)= 3.004E09 V[x(II)]= 7.262E04
V(a3)= 4.605E06 Cov(Δa
1
,Δa
2
)= 3.871E03
cov(a
3
,b
3
)= 1.172E07 Cov(Δa
1
,Δb
2
)= 4.674E03
Cov(Δa
2
,Δb
1
)= 2.233E03
V[Δa]= 5.761E12 V[Δa]= 6.659E06 Cov(Δb
1
,Δb
2
)= 2.696E03
V[Δb]= 4.528E09 V[Δb]= 4.980E09
cov(Δa,Δb)= 1.294E07 cov(Δa,Δb)= 1.797E07 V[Δx]= 1.587E03 t(0, 05, 13)= 2.160 V(u)= 17.973
Pooled variances s[Δx]= 3.984E02 ts[Δx]= 0.086 V(l)= 17.801
Table 7. Evaluation of endpoints in the titration of a mixture of HClO
4
and CH
3
COOH with KOH 0.100 M, optimum case (Figure 4).
Advances in Titration Techniques78

trial and error. The optimal point set (Figure 4) is one that minimizes, for example, the
confidence interval [63].
The weighting factors are very similar so that the values obtained by weighted linear regres-
sion and the simple one become equivalent.
8.2. Conductometric titration of hydrochloric acid 0.1 M with sodium hydroxide 0.1 M
The data corresponding to the two branches of the conductometric titration of 0.1 M HCl with
0.1 M NaOH is shown in the upper part of Table 8 and plot in Figure 5. The cut-off point of
both lines is (6.414, 0.358) [57, 58, 89].
Table 8 also shows all the operations required to calculate the minimum and maximum values
of the confidence interval by the use of hyperbolic confidence bands for the two linear
branches. The limit x
l
of the confidence interval is obtained by solving Eq. (36), which in this
case (Table 8) is
Θ
l
¼1:403 0:0637x
l
þ1:943 0:01034 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
8þðx
l
9Þ
2
168
s
þ0:4908 0:0517x
l
þ2:353 0:0024 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
5þðx
l
20:2Þ
2
32:8
s¼0
ð54Þ
leading to xl = 16.264 mL. The highest value is obtained by solving (Eq. (37))
Θ
u
¼1:403 0:0637x
u
þ1:943 0:01034 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
8þðx
u
9Þ
2
168
s
þ0:4908 0:0517x
u
þ2:353 0:0024 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
5þðx
u
20:2Þ
2
32:8
s¼0
ð55Þ
which leads to x
u
= 16.564 mL. Both equations θ
l
= 0 and θ
u
= 0 are resolved by successive
approximations. Different values are tested for the lower and upper limits to get a change of
sign in θ
l
and θ
u
.
In the weighted mean method (Table 8), the following equations are solved
0:00405ðx
I
Þ
2
1
0:1332ðx
I
Þ
1
þ1:09179 ¼0ð56Þ
0:00267ðx
I
Þ
2
2
0:08776ðx
I
Þ
2
þ0:720 ¼0ð57Þ
being resulted from squaring and reordering the Eqs. (44) and (45), respectively (expressed as a
function of the variances of a
1
,b
1
and of the covariance between a
1
and b
1
). Once calculated the
solutions of the Eqs. (56): 16.630 and 16.487 mL, and (57): 16.206 and 16.339 mL, we have
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79

xy x y
2 1.265 17 0.388
4 1.141 18 0.441
6 1.028 20 0.544
8 0.906 22 0.644
10 0.777 24 0.752
12 0.641
14 0.51
16 0.372
N1= 8 N2= 5
MEAN1= 9 0.83 MEAN2= 20.2 0.5538
[SXX]1= 168 [SXX]2= 32.8
a
1
= -0.06367 1.40300 =a
0
a
1
= 0.05171 -0.49081 =a
0
s(a
1
)= 0.00080 0.00806 =s(a
0
)s(a
1
)= 0.00036 0.00726 =s(a
0
)
R2= 0.99906 0.01034 =s(y/x)R2= 0.99986 0.00204 =s(y/x)
x(I)= 16.414 y(I)= 0.358
t(0.05;6)= 1.943 t
1
s(y/x)1= 0.0201 t(0.05;3)= 2.353 t
2
s(y/x)2= 0.0048
θDIFF-1 DIFF-2 θDIFF-1 DIFF-2
16.25 0.00164 0.036126 16.50 0.02736 0.007436
16.26 0.00048 0.034979 16.51 0.02852 0.006289
16.261 0.00037 0.034864 16.52 V0.02968 0.005141
16.262 0.00025 0.034749 16.53 0.03084 0.003994
16.263 0.00014 0.034634 16.54 0.03200 0.002846
16.264 0.0000191 0.034520 16.55 0.03316 0.001699
16.2641 0.0000075 0.034508 16.56 0.03432 0.000552
16.2642 0.0000041 0.034497 16.561 0.03443 0.000437
16.265 0.00010 0.034405 16.562 0.03455 0.000322
16.266 0.00021 0.034290 16.563 0.03467 0.000207
16.267 0.00033 0.034175 16.564 0.03478 0.000093
16.268 0.00044 0.034060 16.5648 0.03487 0.000001
16.269 0.00056 0.033946 16.5649 0.03489 0.000011
16.27 0.00068 0.033831 16.565 0.03490 0.000022
16.265 16.414 16.565
Table 8. Hyperbolic confidence intervals for the two lines: successive approximations.
Advances in Titration Techniques80

x
u
¼ð82Þ16:630 þð52Þ16:487
8þ54¼16:583 mL ð58Þ
x
l
¼ð82Þ16:206 þð52Þ16:339
8þ54¼16:250 mL ð59Þ
8.3. Experimental measurements: conductometric titration of 100 mL of a mixture of
hydrochloric acid and acetic acids with potassium hydroxide 0.100 M
8.3.1. Reagents
Acetic acid (C
2
H
4
O
2
)M= 60 g/mol (MERCK > 99.5%; 1.049 g/mL); hydrochloric acid (HCl) 1 M
(MERCK, analytical grade); potassium hydroxide (KOH) 1 M (MERCK, analytical grade);
potassium hydrogen phthalate (C
8
H
5
KO
4
)M= 204.23 g/mol (MERCK > 99.5%).
8.3.2. Instruments
4-decimal point analytical balance (Metler AE200), conductivity meter Crimson (EC-Metro GLP
31), calibrated by standards of 147 μS/cm, 1413 μS/cm, 12.88 mS/cm. Digital burette of 50 mL
(Brand) (accuracy: 0.2%, precision: <0.1%, resolution: 0.01 mL, with standard vent valve at 20

C).
8.3.3. Solutions
- Mixture of hydrochloric and acetic acids 0.015 M.
- Potassium hydroxide 0.1 M.
Figure 5. Conductometric titration of hydrochloric acid 0.1 M with sodium hydroxide 0.1 M as a titrant (data are shown
in Table 8).
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81

8.3.4. Experimental
About 100 mL of mixture of hydrochloric and acetic acids 0.015 M is transferred to a 250 mL
volumetric flask containing 100 mL of distilled water. Then, the mixture is titrated conductome-
trically with KOH 0.0992 0.0001 M (n= 3), (previously standardized with potassium hydrogen
phthalate). Tabl e 9 shows the data [conductance, volume] as well as the product of the conduc-
tance by (100 + x)/100 to correct the dilution effect of the titrant. The data are plotted in Figure 6.
V KOH (mL) Conductance
(mS/cm)
Conductance*
(mS/cm)
V KOH
(mL)
Conductance
(mS/cm)
Conductance*
(mS/cm)
0.0 5.64 5.6400 21.1 2.22 2.6884
1.3 5.30 5.3689 21.5 2.25 2.7338
2.1 5.10 5.2071 22.0 2.27 2.7694
4.6 4.44 4.6442 23.0 2.34 2.8782
5.5 4.21 4.4416 24.0 2.39 2.9636
6.1 4.05 4.2971 25.0 2.45 3.0625
7.0 3.82 4.0874 26.1 2.51 3.1651
8.0 3.58 3.8664 27.0 2.56 3.2512
9.0 3.32 3.6188 28.0 2.61 3.3408
10.0 3.07 3.3770 29.1 2.67 3.4470
11.0 2.84 3.1524 30.0 2.72 3.5360
12.1 2.58 2.8922 31.0 2.88 3.7728
13.1 2.36 2.6692 32.0 3.03 3.9996
14.0 2.16 2.4624 33.0 3.19 4.2427
15.0 2.01 2.3115 34.0 3.35 4.4890
15.5 1.97 2.2765 35.0 3.50 4.7250
16.0 1.96 2.2736 36.0 3.63 4.9368
16.5 1.97 2.2951 37.0 3.79 5.1923
17.1 1.99 2.3303 38.0 3.91 5.3958
17.5 2.01 2.3618 39.0 4.04 5.6156
18.0 2.04 2.4072 40.0 4.19 5.8660
18.5 2.06 2.4411 41.0 4.32 6.0912
19.0 2.09 2.4871 42.1 4.46 6.3377
19.5 2.12 2.5334 43.0 4.57 6.5351
20.0 2.15 2.5800 44.0 4.70 6.7680
20.5 2.18 2.6269 45.0 4.82 6.9890
* Conductivity((100 + V)/100).
Table 9. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids
with potassium hydroxide (first assay).
Advances in Titration Techniques82

The points recorded belong to the three branches of the titration curve; the first (branch A)
corresponds to the neutralization of hydrochloric acid, the second (branch B) to the neutraliza-
tion of acetic acid, and the third (branch C) to the excess of potassium hydroxide.
Figures 7 (hydrochloric acid) and 8(hydrochloric acid + acetic acid) are the graphs corre-
sponding to the estimation of the endpoints. The points represented in the graph and then
Figure 6. Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are
shown in Table 9, first assay). Branch A: V [0–15]. Branch B: V [16–28]. Branch C: V [29.1–45].
Figure 7. Conductometric titration of hydrochloric acid in the mixture (branches A and B).
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used in the calculations are colored yellow (branch A), green (branch B) and blue (branch C) in
Table 9, thus avoiding proximity to the breakpoints. The values obtained for the intersections
of the abscissa are 15.334 mL for hydrochloric acid and 29.743 mL for the sum of hydrochloric
and acetic acids. So, acetic acid corresponds to the difference, 14.410 mL. From the data of
Figure 6, without discarding of points, somewhat different values are obtained: 15.383, 29.582
and 14.189 mL.
Table 10 shows in detail all the calculations necessary to estimate the confidence limits of the
abscissa of the breakpoint. The first-order variance propagation method [60] leads to the
following volumes confidence limits: 15.334 0.0619 (first endpoint), 29.743 0.151 (second
endpoint), and 14.410 0.142 (difference). In the second case, the confidence limits cannot
refer to the difference (acetic acid), since the covariates involved are not taken into account (as
previously explained in Section 7). Three decimal numbers were considered to compare and
check calculations.
The application of Fieller’s theorem leads to the same results as those obtained by the law of
propagation of errors, not being applicable to the estimation of confidence limits of the differ-
ence of volumes. The fundamentals of the first-order variance propagation method and
Fieller’s theorem are much stronger than those based on the use of hyperbolic confidence
bands, which lead to higher confidence intervals and limits (not applied in this case).
The conductometric titration was carried out in triplicate, on different days, obtaining the
results included in Tables 11 and 12, and also represented in Figures 9 and 10. Again, the data
Figure 8. Conductometric titration of acetic acid in the mixture (branches B and C).
Advances in Titration Techniques84

N
1
= 13 [ΣW
i
]1= 13 N2= 18 [ΣW
i
]2= 18
x
1
(mean)= 7.985 y
1
(media)= 3.8527 x
2
(mean)= 21.350 y
2
(media)= 2.7175
[S(xx)]1= 194.9369 [S(xx)]2= 209.5250
[S(xy)]1= 44.9940 [S(xy)]2= 19.5426
[S(yy)]1= 10.3870 [S(yy)]2= 1.8246
b
1
=0.230813226 Δb= 0.324084 b
2
= 0.093270863
a
1
= 5.695614832 Δa=4.969405 a2= 0.726209860
[R2]1= 0.99983 x(I)= 15.3337 [R2]2= 0.99897
V[y/x]1= 1.60452E04 V[y/x]2= 0.000117906
V(b1)= 8.231E07 V[pooled]= 0.000135239 V(b2)= 5.627E07
V(a1)= 6.482E05 V(a2)= 2.631E04
cov(a
1
,b
1
)= 6.572E06 cov(a
2
,b
2
)= 1.201E05
V[Δa]= 3.279E04 V[Δa]= 3.564E04 V[x(I)]= 0.0007 FIELLER ax2+bx +c=0
V[Δb]= 1.386E06 V[Δb]= 1.339E06 s[x(I)]= 0.0274 a= 1.050E01 V(u)= 15.390
cov(Δa,Δb)= 1.859E05 cov(Δa,Δb)= 1.932E05 t(0, 05, 27)= 2.052 b=3.221E + 00 V(l)= 15.278
Pooled variances ts[x(I)]= 0.0562 c= 2.469E + 01
V(u)= 15.390 V(l)= 15.278
Table 10. Evaluation of endpoints in the titration of a mixture of HCl and CH
3
COOH with KOH 0.0992 M (data Table 9).
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used in the detailed calculations are colored in the tables. The results obtained (and intermedi-
ate calculations) for the second assessment are shown in Table 13: 14.913 0.041 (propagation
of errors and Fieller), 29.372 0.120 (approximate method of propagation of errors) and 14.458
0.113 (propagation of errors). In the third assessment: 15.032 0.043, 29.414 0.146, and
14.383 0.140 mL.
V KOH
(mL)
Conductance
(mS/cm)
Conductance*
(mS/cm)
V KOH
(mL)
Conductance
(mS/cm)
Conductance*
(mS/cm)
0.0 5.71 5.7100 22.0 2.27 2.7694
1.0 5.47 5.5247 23.0 2.33 2.8659
2.0 5.17 5.2734 24.0 2.39 2.9636
3.0 4.90 5.0470 25.0 2.45 3.0625
4.0 4.61 4.7944 26.0 2.51 3.1626
5.0 4.34 4.5570 27.0 2.57 3.2639
6.0 4.06 4.3036 28.0 2.62 3.3536
7.0 3.79 4.0553 29.0 2.67 3.4443
8.0 3.52 3.8016 30.0 2.78 3.6140
9.0 3.28 3.5752 31.0 2.94 3.8514
10.0 3.02 3.3220 32.0 3.10 4.0920
11.1 2.75 3.0553 33.0 3.27 4.3491
12.0 2.53 2.8336 34.0 3.42 4.5828
13.0 2.29 2.5877 35.0 3.57 4.8195
14.0 2.10 2.3940 36.0 3.73 5.0728
15.0 1.96 2.2540 37.0 3.88 5.3156
15.5 1.94 2.2453 38.0 4.03 5.5614
16.0 1.95 2.2585 39.0 4.17 5.7963
16.5 1.96 2.2869 40.0 4.30 6.0200
17.0 1.99 2.3248 41.0 4.44 6.2604
17.5 2.01 2.3618 42.0 4.57 6.4894
18.0 2.04 2.4072 43.0 4.70 6.7210
19.0 2.10 2.4990 44.0 4.83 6.9552
20.0 2.16 2.5920 45.0 4.95 7.1775
21.0 2.22 2.6862
* Conductivity((100 + V)/100).
Table 11. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic
acids with potassium hydroxide (second assay).
Advances in Titration Techniques86

If the series corresponding to the first equivalence point are analyzed: 15.334, 14.913 and 15.032,
one of the data seems to be very distant from the other two, but the values of Qof Dixon 0.717
and of Gof Grubbs 1.110 are lower than tabulated values for P= 0.05, that is, Q
tab
= 1.155 and G
tab
= 1.15 (although the G
exp
and G
tab
values are practically the same). The mean confidence limits
of the values are 15.093 0.217 mL for hydrochloric acid (first endpoint) and 14.417 0.038 mL
for acetic acid (difference), which leads to molarity values of the solutions of hydrochloric and
acetic acids of 0.01497 0.00022 M and 0.01430 0.00004 M. If the most distant values were
V KOH
(mL)
Conductance
(mS/cm)
Conductance*
(mS/cm)
V KOH
(mL)
Conductance
(mS/cm)
Conductance*
(mS/cm)
0.0 5.81 5.8100 22.0 2.29 2.7938
1.0 5.51 5.5651 23.0 2.34 2.8782
2.0 5.22 5.3244 24.0 2.40 2.9760
3.0 4.94 5.0882 25.0 2.45 3.0625
4.0 4.66 4.8464 26.0 2.52 3.1752
5.0 4.38 4.5990 27.0 2.57 3.2639
6.0 4.11 4.3566 28.0 2.63 3.3664
7.0 3.85 4.1195 29.0 2.68 3.4572
8.0 3.59 3.8772 30.0 2.75 3.5750
9.0 3.32 3.6188 31.0 2.92 3.8252
10.0 3.07 3.3770 32.0 3.09 4.0788
11.0 2.82 3.1302 33.0 3.26 4.3358
12.0 2.58 2.8896 34.0 3.42 4.5828
13.0 2.33 2.6329 35.0 3.56 4.8060
14.0 2.13 2.4282 36.0 3.71 5.0456
14.5 2.05 2.3473 37.0 3.86 5.2882
15.0 1.99 2.2862 38.0 4.00 5.5200
16.0 1.96 2.2748 39.0 4.14 5.7546
16.5 1.98 2.3020 40.0 4.28 5.9920
17.0 2.00 2.3377 41.0 4.40 6.2040
17.5 2.03 2.3853 42.0 4.54 6.4468
18.0 2.05 2.4190 43.0 4.67 6.6781
19.0 2.11 2.5109 44.0 4.80 6.9120
20.0 2.17 2.6040 45.0 4.92 7.1340
21.0 2.23 2.6983
* Conductivity((100 + V)/100).
Table 12. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic
acids with potassium hydroxide (third assay).
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87

discarded, the results obtained would be very close to 14.973 0.084 M and 14.421 0.053 M,
although the accuracy would improve considerably in the first case.
It is worth noting the fact that when the covariance between the intercept and slope of the
straight lines obtained by the least squares method is not taken into account, the propagation
Figure 9. Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are
shown in Table 10, second assay).
Figure 10. Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are
shown in Table 9, third assay).
Advances in Titration Techniques88

N
1
= 13 [ΣW
i
]1= 13 N
2
= 15 [ΣW
i
]2= 15
x
1
(mean)= 7.008 y
1
(mean)= 4.0562 x
2
(mean)= 22.200 y
2
(mean)= 2.8029
[S(xx)]1= 182.8092 [S(xx)]2= 244.9000
[S(xy)]1= -44.7709 [S(xy)]2= 22.9477
[S(yy)]1= 10.9651 [S(yy)]2= 2.1514
b
1
=0.24491 Δb= 0.33861 b
2
= 0.09370
a
1
= 5.77243 Δa=5.04972 a
2
= 0.72272
[R2]1= 0.99996 x(I)= 14.9132 [R2]2= 0.99948
V[y/x]1= 4.175E05 V[y/x]2= 8.57549E05
V(b
1
)= 2.284E07 V[pooled]= 6.5584E05 V(b
2
)= 3.502E07
V(a
1
)= 1.443E05 V(a
2
)= 1.783E04
cov(a
1
,b
1
)= 1.600E06 cov(a
2
,b
2
)= 7.774E06
V[Δa]= 1.927E04 V[Δa]= 1.590E04 V[x(I)]= 0.0004 FIELLER ax2+bx +c=0
V[Δb]= 5.785E07 V[Δb]= 6.266E07 s[x(I)]= 0.0200 a= 1.147E01
cov(Δa,Δb)= 9.374E06 cov(Δa,Δb)= 8.459E06 t(0, 05, 24)= 2.064 b=3.420E+00
Pooled variances ts[x(I)]= 0.0414 c= 2.550E+01
Vol (u)= 14.955 Vo l (u)= 14.955
Vol (l)= 14.872 Vo l (l)= 14.872
N
3
= 15 [ΣW
i
]1= 15
x
3
(media)= 37.000 y
3
(media)= 5.3001 Δb= 0.145555
[S(xx)]1= 280.0000 Δa=4.275169
[S(xy)]1= 66.9920 x(II)= 29.3716
[S(yy)]1= 16.0306 Δx= 14.4584
b
3
= 0.2393 V[pooled]= 0.000132481
a
3
=3.5525
[R2]3= 0.9999
V[y/x]3= 1.792E04 V[x(II)]= 0.0034 t(0, 05, 26)= 2.056
V(b
3
)= 6.400E07 s[x(II)]= 0.0587 ts[x(II)]= 0.1207
V(a
3
)= 8.881E04 Vol (u)= 29.492
cov(a
3
,b
3
)= 2.368E05 Vol (l)= 29.251
V[x(I)]= 4.017E04
V[x(II)]= 3.447E03
Cov(Δa
1
,Δa
2
)= 7.014E03
Cov(Δa
1
,Δb
2
)= 9.265E03
Cov(Δa
2
,Δb
1
)= 4.704E03
Cov(Δb
1
,Δb
2
)= 6.224E03
V[Δa]= 1.583E07 V[Δa]= 9.320E04 V[Δx]= 3.116E03 t(0, 05, 37)= 2.026
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89

of the error leads to values of much larger confidence limits, 0.429 in the example of Massart
(1997) versus 0.104, or 0.648 by Liteanu and Rica [58] versus only 0.113, in this book chapter,
for the same data. As in many monographs, the covariance in the propagation of errors is not
taken into account, and this is perhaps the reason why the estimates of the uncertainties of the
intersection abscissa in the analytical literature do not abound.
9. Final comments
The advance of instrumental methods of endpoint detection increases the importance and
the worth of titrimetric analysis. Physicochemical methods are intensively developed
nowadays. However, titration continues to maintain its importance for chemical analysis.
Plotting two straight line graphs from experimental data i.e., the conductivity versus
volume added and determining the corresponding intersection point of the two branches
allow locating the endpoint in a conductometric titration. The estimation of uncertainty
of endpoint from linear segmented titration curves may be easily carried out by first-
order propagation of variance, that is, by applying random error propagation law.
The weighted linear regression procedure as being applied to the two branches of the
conductometric titration curves leads to results similar to those obtained by the
unweighted (single) linear regression procedure. The weighting factors are very similar
to each other.
The covariance of measurements can be as important as the variance and both contribute
significantly to the total analytical error. In particular, the strong correlation existing
between the estimated slope and intercept of a straight line obtained by the least squares
method must not be ignored. The inclusion of the covariance term on this respect is of vital
importance, being usually a subtractive character lowering, in this case, the confidence limits
of the abscissa of the intersection point. Perhaps this omission, which leads to too greater
uncertainties, may be the cause for a small number of times that uncertainty is reported in
this context.
The algebra associated with the Fieller’s theorem is simple, and no problem is observed
with its derivation in this particular case of intersecting straight lines. However, the
statistical uncertainty of endpoint differences is a complex problem. Attempt to derive
the confidence limits by applying Fieller’s theorem fails in this case, being necessary to
resort to the first-order propagation of variance (random error propagation law).
N
1
= 13 [ΣW
i
]1= 13 N
2
= 15 [ΣW
i
]2= 15
V[Δb]= 9.902E07 V[Δb]= 1.014E06 s[Δx]= 5.583E02 ts[Δx]= 0.113
cov(Δa,Δb)= 2.893E05 cov(Δa,Δb)= 2.952E05 Vol (u)= 14.571
Pooled variances Vol (l)= 14.345
Table 13. Evaluation of endpoints in the titration of a mixture of HCl and CH
3
COOH with KOH 0.0992 M (data
Table 11).
Advances in Titration Techniques90

Nevertheless, the algebra associated in this case is simple but cumbersome, as some
terms in covariance need to be derived. As a matter of fact, greater accuracy and firmer
statistical justification make first-order propagation of variance (random error propaga-
tion law) and Fieller’s theorem methods preferable to methods based on intersecting
confidence bands.
Author details
Julia Martin, Gabriel Delgado Martin and Agustin G. Asuero*
*Address all correspondence to: asuero@us.es
Department of Analytical Chemistry, Faculty of Pharmacy, The University of Seville, Seville,
Spain
References
[1] Beck CM. Classical analysis. A look at the past, present and future. Analytical Chemistry.
1994;66(4):224A–239A
[2] Granholm K, Sokalski T, Lewenstam A, Ivaska A. Ion-selective electrodes in potentiomet-
ric titrations: A new method for processing and evaluating titration data. Analytica
Chimica Acta. 2015;888:36–43
[3] Thordason P. Determining association constants from titration experiments in supramo-
lecular chemistry. Chemical Society Reviews. 2011;40(3):1305–1323
[4] Winkler-Oswatitsch R and Eigen M. The art of titration. From classical end points to
modern differential and dynamic analysis. Angewandte Chemie International Edition
English. 1979;18(1):20–49
[5] Asuero AG and Michalowski T. Comprehensive formulation of titration curves for com-
plex acid-base systems and its analytical Implications. Critical Reviews in Analytical
Chemistry. 2011;41(2):151–187
[6] Asuero AG. Buffer capacity of a polyprotic acid: First derivative of the buffer capacity and
pKa values of single and overlapping equilibria. Critical Reviews in Analytical Chemistry.
2007;37:269–301
[7] Beck CM. Toward a revival of classical analysis. Metrología. 1997;34(1):19–30
[8] Felber H, Rezzonico S, Máriássy M. Titrimetry at a metrological level. Metrología. 2003;40(5):
249–254
[9] King B. Review of the potential of titrimetry as a primary method. Metrología. 1997;34(1):77–82
Intersecting Straight Lines: Titrimetric Applications
http://dx.doi.org/10.5772/intechopen.68827
91

[10] Ortiz-Fernandez MC, Herrero-Gutierrez A. Regression by least median squares, a meth-
odological contribution to titration. Chemometrics and Intelligent Laboratory Systems.
1995;27:241–243
[11] Kupka K, Meloun M. Data analysis in thechemical laboratory II. The end-point estimation in
instrumental titrations by nonlinear regression. Analytica Chimica Acta. 2001;429:171–183
[12] Jones RH, Molitoris BA. A statistical method for determining the breakpoint of two lines.
Analytical Biochemistry. 1984;41(1):287–290
[13] Sprent P. Some hypothesis concerning two phase regression analysis. Biometrics. 1961;17
(4):634–645
[14] Shaban SA. Change point problem and two-phase regression: An annotated bibliogra-
phy. International Statistical Review. 1980;48(1):83–93
[15] Shanubhogue A, Rajarshi MB, Gore AP, Sitaramam V. Statistical testing of equality of two
break points in experimental data. Journal of Biochemical and Biophysical Methods.
1992;25(2-3):95–112; Erratum 1994;28(1): 83
[16] Csörgo M, Horváth L. Nonparametric methods for changepoint problems. In Krishanaiah
PR, Rao CR, editors, Handbook of Statistics, Vol. 7, Amsterdam:Elsevier, 1988. pp.403–425
[17] Krishanaiah PR, Miao BQ. Review about estimation of change points. In Krishanaiah PR,
Rao CR, editors. Handbook of Statistics. Vol. 7, Amsterdam: Elsevier, 1988. pp. 375–402
[18] Rukhin AL. Estimation and testing for the common intersection point. Chemometrics and
Intelligent Laboratory Systems. 2008;90(2):116–122
[19] Yanagimoto T, Yamamoto E. Estimation of safe doses: Critical review of the Hockey Stick
regression method. Environmental Health Perspectives. 1979;32:193–199
[20] Kita F, Adam W, Jordam P, Nau WM, Wirz J. 1,3-Cyclopentanedyl diradicals: Substituent
and temperature dependence of triplet-singlet intersystem crossing. Journal of the Amer-
ican Chemical Society. 1999;121(40):9265–9275
[21] Vieth E. Fitting piecewise linear regression functions to biological responses. Journal of
Applied Physiology. 1989;67(1):390–396
[22] Piegorsch WW. Confidence intervals on the joint point in segmented regression. Biomet-
rical Unit Technical Reports BU-785-M. 1982
[23] Bacon DW, Watts DG, Estimating the transition between two intersecting straight lines.
Biometrika. 1972;58(3):525–534
[24] Christensen R. Plane answers to complex questions. The Theory of Linear Models. 4th
ed., New York: Springer, 2011
[25] Lee ML, Poon WY, Kingdon HS. A two-phase linear regression model for biologic half-
life data. Journal of Laboratory and Clinical Medicine. 1990;115(6):745–748
Advances in Titration Techniques92

[26] Seber GAF. Linear regression analysis. 7.6 Two-phase linear regression. New York: Wiley,
1977. pp. 205–209
[27] Seber GAF, Lee AJ. Linear regression analysis. 6.5 Two-phase linear regression. 2nd ed.,
New York: Wiley, 2003. pp. 159–161
[28] Cook DA, Charnock JS. Computer-assisted analysis of functions which may berepresented
by two intersecting straight lines. Journal of Pharmacological Methods. 1979;2(1):13–19
[29] Hubálovsky S. Processing of experimental data as educational method of development of
algorithmic thinking. Procedia - Social and Behavioral Sciences. 2015;191:1876–1880
[30] Hubálovsky S, Sedivú J. Algorithm for estimates determination of parameters of regres-
sion straight polyline with objective elimination of outliers. 2nd International Conference
on Information Technology (ICIT) 2010 Gdansk, 28–30 June 2010. pp. 263–266
[31] Ouvrard C, Berthelot M, Lamer T, Exner O. A program for linear regression with a common
point of intersection. Journal of Chemical Information and Modeling. 2001;4(5):1141–1144
[32] Jehlicka V, Mach V. Determination of intersection of regression straight lines with elimi-
nation of outliers. Chemical Papers. 1999;53(5):279–282
[33] Jehlicka V, Mach V. Determination of estimates of parameters of calibration straight line
with objective elimination of remote measurements. Collect. Collection of Czechoslovak
Chemical Communications. 1995;60(12):2064–2073
[34] Kastenbaum MA. A confidence interval on the abscissa of the point of intersection of two
fitted straight lines. Biometrics. 1959;15(3):323–324
[35] Filliben JJ, McKinney JE. Confidence limits for the abscissa of intersection of two linear
regression. Journal of Research of NIST. 1972;76B(3-4):179–192
[36] Meier PC, Zünd RE. Statistical methods in analytical chemistry. 2nd ed., New York:
Wiley, 2000. pp. 127–128, 374
[37] Miller JN, Miller JC. Statistics and Chemometrics in Analytical Chemistry. Harlow,
England: Pearson, 2010. pp. 140–141
[38] Fieller EC. The biological standardization of insulin. Journal of the Royal Statistical
Society. 1940–1941;7(1):1–64
[39] Fieller EC. A fundamental formula in the statistics of biological assay, and some applica-
tions. Quarterly Journal of Pharmacy and Pharmacology. 1944;17:117–123
[40] Fieller EC. Some problems in interval estimation. Journal of the Royal Statistical Society:
Series B. 1954;16(2):175–185
[41] Koschat MA. A characterization of the Fieller solution. Annals of Statistics. 1987;15:462–468
[42] Wallace DL. The Behrens-Fischer and Fieller-Creasy problems. In Berger J, FienbergS, Gani
J, Krickeberg K, Olkin I. R.A. Fischer, anappreciation, editors, Berlin: Springer-Verlag, 1980
Intersecting Straight Lines: Titrimetric Applications
http://dx.doi.org/10.5772/intechopen.68827
93

[43] Wilkinson GN. On resolving the controversy in statistical inference. Journal of the Royal
Statistical Society: Series B. 1977;39(2):119–171
[44] Cook DA, Charnot JS. Computer assisted analysis of functions which may be represented
by two intersecting straight lines. Journal of Pharmacological and Toxicological Methods.
1979;2(1):13–19
[45] Han MH. Non-linear Arrhenius plots in temperature-dependent kinetic studies of enzyme
reactions. I. Single transition processes. Journal of Theoretical Biology. 1972;35(3):543–568
[46] Puterman ML, Hrboticky N, Innist SM. Nonlinear estimation of parameters in biphasic
Arrhenius plots. Analytical Biochemistry. 1988;170(2):409–420
[47] Baxter DC. Evaluation of the simplified generalised standard additions method for cali-
bration in the direct analysis of solid samples by graphite furnace atomic spectrometric
techniques. Journal of Analytical Atomic Spectrometry. 1989;4(5):415–421
[48] Bonate PL. Approximate confidence intervals in calibration using the bootstrap. Analyt-
ical Chemistry. 1993;65(10):1367–1372
[49] Mandel J, Linning FJ. Study of accuracy in chemical analysis using linear calibration
curves. Analytical Chemistry. 1957;29(5):743–749
[50] Schwartz LM. Nonlinear calibration curves. Analytical Chemistry. 1977;49(13):2062–2066
[51] Schwartz LM. Statistical uncertainties of analyses by calibration of counting measure-
ments. Analytical Chemistry. 1978;50(7):980–984
[52] Schwartz LM. Calibration curves with nonuniform variance. Analytical Chemistry.
1979;51(6):723–727
[53] Almanda Lopez E, Bosque-Sendra JM, Cuadros Rodriguez L, García Campaña AM, Aaron
JJ. Applying non-parametric statistical methods to the classical measurement of inclusion
complex binding constants. Analytical and Bioanalytical Chemistry. 2003;375(3):414–423
[54] Asuero AG, Recamales MA. A bilogarithmic method for the spectrophotometric evalua-
tion of acidity constants of two-step overlapping equilibria. Analytical Letters 1993;26
(1):163–181
[55] Heilbronner E. Position and confidence limits of an extremum. The determination of the
absorption maximum in wide bands. Journal of Chemical Education. 1979;56(4):240–243
[56] Franke JP, de Zeeuw RA, Hakkert R. Evaluation and optimization of the standard addi-
tion method for absorption spectrometry and anodic stripping voltammetry. Analytical
Chemistry. 1978;50(9):1374–1380
[57] Liteanu C, Rica I, Liteanu V. On the confidence interval of the equivalence point in linear
titrations. Talanta. 1978;25(10):593–596
[58] Liteanu C, Rica I. Statistical Theory and Methodology of Trace Analysis. Chichester: Ellis
Horwood Ltd, 1980. pp. 166–172
Advances in Titration Techniques94

[59] Asuero AG, Sayago A, González AG. The correlation coefficient: an overview. Critical
Reviews in Analytical Chemistry. 2006;36(1):41–59
[60] Asuero AG, Gonzalez G, de Pablos F, Ariza JL. Determination of the optimum working
range in spectrophotometric procedures. Talanta. 1988;35(7):531–537
[61] Ceasescu D, Eliu-Ceasescu VZ. Die Sicherheit der Ergebnisse im Falle von linearen
physikalisch-chemischen Titrationskurven. Das Vertrauensintervall des Äquivalenzpunktes.
Zeitschrift für Analytische Chemie. 1976;282(1):45–46
[62] Ceasescu D, Eliu-Ceasescu V. Equivalence point and confidence interval of linear titration
curves in outlook of normal bidimensional distribution of branches. Revue Roumaine de
Chimie. 1977;22(4):563–567
[63] Draper NR, Smith H. Applied Regression Analysis. New York: Wiley, 1998
[64] McCormick D, Roach A. Measurement, Statistics and Computation, ACOL. Chichester:
Wiley, 1987. pp. 312–315
[65] Schwartz LM, Gelb RI. Statistical uncertainties of end points at intersecting straight lines.
Analytical Chemistry. 1984;56(8):1487–1492
[66] Sayago A, Asuero, A.G. Fitting straight lines with replicated observations by linear
regression: Part II. Testing for homogeneity of variances. Critical Reviews in Analytical
Chemistry. 2004;34(3–4):133–146
[67] Sharaf MA, Illman DL, Kowalski BR. Chemometrics. New York: Wiley, 1986
[68] Brownlee KA. Statistical Theory and Methodology in Science and Engineering. 2nd ed.,
Malabar, FL: R.E. Krieger, 1984
[69] Carter KN, Scott DM, Salmon JK, Zarcone GS. Confidence limits for the abscissa of
intersection of two least squares lines such as linear segmented titration curves. Analyti-
cal Chemistry. 1991;63(13):1270–1278
[70] Eliu-Ceasescu V, Ceasescu D. Statistische untersuchung des Äquivalenzpunktes vonlinearen
Titrationskurven mit Hilfe numerischer Simulierung. Fres. Zeitschrift für Analytische
Chemie. 1978;291(1):42–46
[71] Jandera P, Kolda S, Kotrly S. End-point evaluation in instrumental titrimetry-II Confi-
dence intevals in extrapolation of linear titration curves. Talanta. 1970;17(6):443–454
[72] Liteanu C, Cormos D. Beiträge zum Problem der äquivalenzpunktbestimmung. VII Über
die Präzision der äquivalenzpunktbestimmung bei der physikalisch-chemischen linearen
titration mit Hilfe der analytischen Methode und der kleinstenQuadrate. Revue Roumaine
de Chimie. 1965;10:361–376
[73] Sharaf MA, Illman DL, Kowalski BR. Chemometrics. New York: Wiley, 1986
[74] Gelhaus SL, Lacourse WR. Measurement of Electrolytic Conductance. In Analytical
Instrumentation Handbook. New York: Marcel Dekker, 2005. pp. 561–580
Intersecting Straight Lines: Titrimetric Applications
http://dx.doi.org/10.5772/intechopen.68827
95

[75] Gzybkovski W. Conductometric and potentiometric titrations. Politechnika Gdańska,
Gdansk, 2002. http://fizyczna.chem.pg.edu.pl/documents/175260/14212622/chf_epm_lab_1.
pdf [accessed on August 5, 2017]
[76] Cáñez-Carrrasco MG, García-Alegría AM, Bernal-Mercado AT, Federico-Pérez RA,
Wicochea-Rodríguez JD. Conductimetría y titulaciones, ¿cuando, por qué y para qué?.
Educación química. 2011;22(2):166–169
[77] Donkersloot MCA. Teaching conductometry. Another perspective. Journal of Chemical
Education. 1991;68(2):136–137
[78] Garcia J, Schultz LD. Determination of sulphate by conductometric titration: an under-
graduate laboratory experiment. Journal of Chemical Education. 2016;93(5):910–914
[79] Smith KC, Garza A. Using conductivity measurements to determine the identities and
concentrations of unknown acids: an inquiry laboratory experiment. Journal of Chemical
Education. 2015;92(8):1373–1377
[80] Compton OC, Egan M, Kanakaraj R, Higgins TB, Nguyen ST. Conductivity through
polymer electrolytes and its implications in lithium-ion batteries: real-world application
of periodic trends. Journal of Chemical Education. 2012;89(11):1442–1446
[81] Farris S, Mora L, Capretti G, Piergiovanni L. Charge density quantification of polyelec-
trolyte polysaccharides by conductimetric titration: an analytical chemistry experiment.
Journal of Chemical Education. 2012;89(1):121–124
[82] Nyasulu F, Moehring M, Arthasery P, Barlag R. K
a
and K
b
from pH and conductivity
measurements: a general chemistry laboratory exercise. Journal of Chemical Education.
2011;88(5):640–642
[83] Smith KC, Edionwe E, Michel B. Conductimetric titrations: a predict - observe -explain
activity for general chemistry. Journal of Chemical Education. 2010;87(11):1217–1221
[84] Holdsworth DK. Conductivity titrations - a microcomputer approach. Journal of Chemi-
cal Education. 1986;63(1):73–74
[85] Rosenthal LC, Nathan LC. A conductimetric-potentiometric titration for an advanced
laboratory. Journal of Chemical Education. 1981;58(8):656–658
[86] Rodríguez-Laguna N, Rojas-Hernández A, Ramírez-Silva MT, Hernández-García L,
Romero-Romo M. An exact method to determine the conductivity of aqueous solutions
in acid-base titrations. Journal of Chemistry. 2015, Article ID 540368, pp. 13
[87] Selitrenikov AV, Zevatskii Yu E. Study of acid-base properties of weak electrolytes by
conductometric titration. Russian Journal of General Chemistry. 2015;85(1):7–13
[88] Apelblat A, Bester-Rogac M, Barthel J, Neueder R. An analysis of electrical conductances
of aqueous solutions of polybasic organic acids. Benzenehexacarboxylic (mellitic) acid
and its neutral and acidic salts. Journal of Physical Chemistry B. 2006;110(17):8893–8906
[89] Massart D, Vandeginste B, Buydens L, De Jong S, Lewi PJ, Smeyers-Verbeke J. Handbook
of Chemometrics and Qualimetrics: Part A, no. 20A in Data Handling in Science and
Technology. Amsterdam: Elsevier Science, 1997. p. 305
Advances in Titration Techniques96