Open access peer-reviewed chapter

Intersecting Straight Lines: Titrimetric Applications

Written By

Julia Martin, Gabriel Delgado Martin and Agustin G. Asuero

Submitted: December 9th, 2016 Reviewed: March 27th, 2017 Published: September 27th, 2017

DOI: 10.5772/intechopen.68827

Chapter metrics overview

1,662 Chapter Downloads

View Full Metrics


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 lead to segmented linear titration curves, the endpoint being defined by the intersection of the two straight line segments. The estimation of the statistical 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 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 aim of 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.


  • titrimetric
  • straight lines
  • breakpoint

1. Introduction

Titrimetry is one of the older analytical methods [1], and it is still found [24] in a developing way. It plays an important role in various fields as in routine studies [59], being used widely in the analytical laboratory given their simplicity, speed, accuracy, good reproducibility, 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). Independent values of chemical quantities expressed in SI units are obtained through gravimetry and titrimetry (classical analysis).

In titrimetry, the quantity of tested components of a sample is assessed through 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); that is, spectrophotometry, conductimetry or potentiometry, for instance. The importance of titrimetric analysis has increased with the advance of the instrumental method of end point detection.

TechniqueMeasured property
Conductimetric titrationsElectrical conductivity
Potentiometric titrationsPotential of an indicating electrode
Spectrophotometric titrationsAbsorbance
Amperometric titrationsDiffusion current at a polarizable indicating (dropping mercury or rotating platinum) electrode

Table 1.

Instrumental end point detection technique more widely applied.

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 intersection 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 amperometric 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 [1214] and has received considerable attention in the statistical literature. The problem in question is known by a variety of names (Table 2) [1227]. Computer analysis [2831], elimination of outliers [32, 33], and confidence limits for the abscissa [22, 34, 35] have been subject to study.

BreakpointJones and Molitoris [12]; Shanubhogue et al. [15]
Changepoint problemCsörgo and Horváth [16]; Krishanaiah and Miao [17]
Common intersection pointRukhin [18]
Hockey stick regressionYanagimoto and Yamamoto [19]
Intersystem crossingKita et al. [20]
Piecewise linear regressionVieth [21]
Segmented regressionPiegorsch [22]
TransitionBacon and Watts [23]
Two phase linear regressionChristensen [24]; Lee et al. [25],; Seber [26, 27]; Shaban [14]; Sprent [13]

Table 2.

Name received in the literature for the intersecting point of two straight lines.

At the point of intersection (xI), the two lines have the same ordinate. The estimation of statistical uncertainty of end points obtained from linear segmented titration is the subject of this chapter. The topic is scarcely treated in [36, 37] analytical monographs. The method of least squares is the most common choice and the most appropriate 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 [3843], and methods bases 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 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 N1 observations on the first line


and N2 the observations on the second


where a1, b1, a2, b2 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 alternative most common. 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., wi = 1), is evident.

  • Equation


  • Weights


  • Explained sum of squares


  • Residual sum of squares


  • Mean


  • Sum of squares about the mean


  • Slope


  • Intercept


  • Weighted residuals


  • Correlation coefficient


  • Standard errors


  • sa2=sy/x2(wixi2)/(SXXwi)

  • sb2=sy/x2/SXX

  • Cov(a,b)=x¯sy/x2/SXX

Table 3.

Formulae for calculating statistics for weighted linear regression.

Note that in summation (1) and (2) by dividing by N1 and N2, respectively, we get


At the point of intersection, the lines (1) and (2) have the same ordinate y^1=y^2and the abscissa of intersection (denotes by x^I) is given by


Random error in the points produce 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%).


3. First-order propagation of variance for V[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^Ion this way. First-order propagation of variance retains only first derivatives in the Taylor expansions and this procedure leads to


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


Then, the standard error estimate of x^Iis as follows


The end point x^Idepends on four least squares parameters a1, a2, b1, b2 that are random variables. Segment one parameters depend only on measurements made along segment one and these are statistically independent of the measurements along 2. However, Δa, and Δb are correlated random variables because each involves b1 and b2. Note that, a1 and a2 are related to b1 and b2 by means of Eqs. (1) and (2).

The variances of Δaand Δbare given by


and for the covariance between Δaand Δb, we get [63]


It should be noted that in the calculations, the variance regression estimates from both line segments are pooled into a single sp2, by using the following formula which weights each contribution according to the corresponding [6467] degrees of freedom


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 (sa2 = V[a]), we may derive


in which s2 is s(y/x) in Table 3; ∑wiis 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


Once the values of Va], Vb] and Covab] are known from Eqs. (16), (17) and (18), respectively, the estimate of the variance of the intersection abscissa of the two straight lines, V[x^I], is calculated by applying Eq. (8).


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 yiare assumed to have Gaussian (normal) distribution, the least squares parameters as well as Δa, and Δbare also normally distributed [68]. However, x^I, 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, and under these circumstances, confidence intervals may be calculated from the standard deviation of x^Iwhich is also accurate only when variances are small.

However, the construction of the confidence interval (limits) for the equivalence point by using the Student’s t-test


where tα/2 is the Student’s tstatistics at the 1 − αconfidence level (i.e., leaving an area of α/2 to the right) and for the number of degrees of freedom (N1+N2 − 4) inherent in the standard deviation of x^I, could be misleading. Note that because x^Iinvolves the ratio of random variables first-order propagation of variance is not exact [69]. Evidently, x^Iis a random variable not normally distributed unless s(x^I)is small enough. When the variances of the responses are not necessarily small, one a solution to this problem is to apply the called Fieller’s theorem [3843]. Another point of view focuses 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 [3843] is supported by two capital premises:

  1. Any linear combination zof normally distributed random variables is itself normally distributed.

  2. If the standardized variable zV[z]is distributed as N(0, 1), then zis distributed as t.

Consider now any pair of individual line segments written as a difference zas follows


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 zof all these zvalues is zero and zare normally distributed because it is formed as a linear combination of normally distributed variables. Taking into account that a1, a2, b1 and b2 are normally distributed, then zwill be normally distributed. Then, in the vicinity of the intersection point, zhas zero mean and its variance is


and therefore, zV^[z]is distributed as N(0, 1) and according to (ii)


This is called Fieller’s Theorem [34, 38]. The development of Eq. (11) leads to the equation


which on rearrangement leads to


which may be factored as


The solution of Eq. (25) gives the confidence limits for xIestimated where tα/2 is the appropriate value of the Student distribution at a αsignificance level (confidence level 1 − α) for N1 + N2 − 4 degrees of freedom. Note that in Eqs. (21), (23), (24) and (25), the corresponding values of Va], Vb] and Cova, Δb] are given by Eqs. (16), (17) and (18), respectively, as in the first-order propagation of variance for V[x^I].

The first and last group 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 xI2 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 in some other contexts in analytical and chemical literature (Table 4).

Arrhenius plotCook and Charnock [44]; Han [45]; Puterman et al. [46]
Calibration curvesBaxter [47]; Bonate [48]; Mandel y Linning [49]; Schwartz [5052]
Estimation of safe dosesYanagimoto and Yanamoto [19]
Estimation of uncertainty in binding constantsAlmansa López et al. [53]
Models for biologic half-life dataLee et al. [25]
Position and confidence limits of an extremumAsuero and Recamales [54]; Heilbronner [55]
Standard addition methodFranke et al. [56]

Table 4.

Some applications of Fieller theorem in analytical chemistry.


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 xI[58, 61, 7072]. Evidently, the best confidence interval would result 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, xI, is only a mean value, with which a certain confidence interval is associated. If the signal values both before and after the point of intersection 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 (xl, xu) for the true value of the equivalence 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:


and for the second line:


t1 and t2 are the corresponding tStudent values for α/2 = 0, 05 and N1 − 2 and N2 − 2 degrees of freedom, respectively. Hence, the lower value xl of the confidence interval is obtained by solving the following equation:


The higher value xuis obtained from the equation:


From Eqs. (1) and (3) we get


and then the variance of the fitted y1 value will be given by


Note that the variance of the (weighted) mean of the one values


and that the mean y1 value and the slope b1 are uncorrelated random variables (property which was also applied in Eq. (12) without further demonstration) as shown in that follows. Taking into account that

y¯1=b1(w)1y1and b1=c1(w)1y1where


and then


From Eq. (31), we get for the standard error of the fitted value


Thus, the lower value, xl, for the confidence interval is obtained by solving the equation


by, for example, successive approximations with an Excel spreadsheet. The higher value xuis obtained in the same way from the equation that follows also by successive approximations


The point of view of Liteanu et al. [57, 58]. is very interesting: the authors consider the point of intersection xIas 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 (xIl, xIu) where


The two values of the limits of the confidence interval will be given by the two solutions of the equations


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


which by simple algebra it may be ordered in powers of xl as


and taking into account the values of V[b1], Cov[a1, b1] and V[a1] (see Table 3), we get finally


whose roots give the two values of (xI)1. Since the point of intersection xIbelongs 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 are another confidence interval associated with it


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 2F2,n2αshould be substituted for tα/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 xIis the volume at which the straight lines one and two intersect and xIIthe volume at which the two and three lines intersect, the difference xIIxIdenoted as Δx, is given by


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




The standard error estimate is given by


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 of 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’s tdistribution. Because in this case the exact confidence limits cannot be calculated, we use the small variance confidence interval


8. Application to experimental system

A bibliographic search allows us to demonstrate the importance of conductivity measurements 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 educational 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], microcomputer 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 object of recent 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 end point of data described in the literature as well as to experimental measurements carried out in the laboratory.

8.1. Conductometric titration of 100 mL of a mixture of acids with potassium hydroxide 0.100 F

Table 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 F 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.


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.

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).

Figure 1.

Conductometric titration of a mixture of perchloric and acetic acids with potassium hydroxide (data shown inTable 5).

Firstly, Schwartz and Gelb [65] select 13 points, six (volume from four to 14 mL) for branch A and seven (volume from 20 to 32 mL) for branch B. The points near to the end point of perchloric acid are deviating from linearity and are discarded in the first instance. It is also considered that the data have a different variance V⋅(100 + xi)2, being the weighting factor (100 + xi)−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 end point are discarded. Figures 2 and 3 show the straight lines segments with the corresponding selected points.

Figure 2.

Conductometric titration of perchloric acid in the mixture (branches A and B).

Figure 3.

Conductometric titration of acetic acid in the mixture (branches B and C).

Table 6 includes the intermediate results obtained in the calculation of the first end point, corresponding with the neutralization of perchloric acid (Figure 2), in order to follow the procedures previously detailed. The first end point is located at 16.367 mL and therefore 1.637 mmol of HClO4. The estimated standard error at the end point, 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 limit, 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.

1/Rxy= (1/R)(100 + x)(100 + xi)−21/Rxy= (1/R)(100 + x)(100 + xi)−2
Cov(a1, b1)=−2.168E−07Cov(a2, b2)=−1.445E−07
Va]=6.002E−06Va]=9.640E−06V[x(I)]=0.0015FIELLERax2 + bx+ c= 0
cova, Δb)=−3.613E−07cova, Δb)=−4.453E−07t(0, 05, 9)=2.262b=−4.953E−02V(l)=16.278
Pooled variancest s[x(I)]=0.0880c=4.053E−01

Table 6.

Intermediate results obtained in the calculation of the first end point (titration of perchloric acid with potassium hydroxide (Figure 2)).

The second end point, 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)(34.1971–16.3665mL) corresponds to acetic acid in the sample, 17.831 mL. If the above methodology is used for lines, B and C (Figure 3) give x(II) ± sd[x(II)] equal to 34.197 ± 0.0478, and 34.305 and 34.089 mL for the confidence limits.

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 confidence limits for Δxis not applicable since the magnitude analogous to zin Eq. (20) implies, in this case, the product of random variables.

Therefore, 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 the Eq. (9) for a single end point.

The latter methodology is applied to the optimal case detailed by Schwartz and Gelb [65]. The corresponding data are shown in Figure 4, and the calculations necessary to locate the equivalence points, first and second, are shown in Table 7. The results obtained are: first equivalence point (perchloric acid): x(I) = 16.358 mL, s[x(I)] = 0.035 mL, t s[x(I)] = 0.078 mL, [I.C.]I= 0.156 mL. Second equivalence point (mixture perchloric and acetic acids): x(II) = 34.244 mL, s[x(II)] = 0.027 mL, t s[x(II)] = 0.061 mL, [IC]II= 0.122 mL. This latter is not correct 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, sx] = 0.040 mL; t sx] = 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: N1 + N2N3 − 2 × 3 = 13).

1/Rxy= (1/R)(100 + x)(100 + xi)−21/Rxy= (1/R)(100 + x)(100 + xi)−2
cov(a1, b1)=-2.373E−07cov(a2, b2)=-4.321E−08
Va]=3.356E−06Va]=5.160E−06V[x(I)]=0.0012FIELLERax2 + bx + c= 0
cova, Δb)=−2.805E−07cova, Δb)=−2.463E−07t(0, 05, 9)=2.262b=−5.029E−02V(l)=16.280
Pooled variancest s[x(I)]=0.0784c=4.113E−01
V[x(I)]=0.0007t(0, 05, 9)=2.262V(u)=34.305
b3=0.026637531s[x(I)]=0.0269t s[x(I)]=0.0610V(l)=34.183
V(a3)=4.605E−06Cova1, Δa2)=3.871E−03
cov(a3, b3)=−1.172E−07Cova1, Δb2)=−4.674E−03
Cova2, Δb1)=−2.233E−03
Va]=5.761E−12Va]=6.659E−06Covb1, Δb2)=2.696E−03
cova, Δb)=−1.294E−07cova, Δb)=−1.797E−07Vx]=1.587E−03t(0, 05, 13)=2.160V(u)=17.973
Pooled variancessx]=3.984E−02t sx]=0.086V(l)=17.801

Table 7.

Evaluation of end points in the titration of a mixture of HClO4 and HCH3COO with KOH 0.100 F, optimum case (Figure 4).

Figure 4.

Illustrative example described by Schwartz and Gelb [65] as optimal. Numerical data are shown inTable 5. First branch (A), volumes of 4–12 mL, 5 points. Second branch (B), volumes of 22–34 mL, points. Third branch (C), volumes of 35–44 mL, 6 points.

Some points near to the end point appear to deviate slightly from linearity, but it is not always clear whether or not to omit these problem points in the analysis, which can be done by 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 regression, 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].

t(0.05;6)=1.943t1 s(y/x)1=0.0201t(0.05;3)=2.353t2 s(y/x)2=0.0048

Table 8.

Hyperbolic confidence intervals for the two lines: successive approximations.

Figure 5.

Conductometric titration of hydrochloric acid 0.1 M with sodium hydroxide 0.1 M as a titrant (data are shown inTable 8).

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 xlof the confidence interval is obtained by solving Eq. (36), which in this case (Table 8) is


And leading to xl= 16.264 mL. The highest value is obtained by solving (Eq. (37))


which leads to xu= 16.564 mL. Both equations θl= 0 and θu= 0 are resolved by successive approximations. Different values are tested for the lower and upper limits until to get a change of sign in θland θu.

In the weighted mean method (Table 8), the following equations are solved


Resulting from squaring and reordering the Eqs. (44) and (45), respectively, (expressed as a function of the variances of a1, b1 and of the covariance between a1 and b1). Once calculated the solutions of the Eqs. (56): 16.630 and 16.487 mL, and (57): 16.206 and 16.339 mL, we have


8.3. Experimental measurements: conductometric titration of 100 mL of a mixture of hydrochloric acid and acetic acids with potassium hydroxide 0.100 F

8.3.1. Reagents

Acetic acid (C2H4O2) 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 (C8H5KO4) M= 204.23 g/mol (MERCK > 99.5%).

8.3.2. Instruments

Analytical balance (Metler AE200) (4 decimals), conductimeter Crison (EC-Metro GLP 31), calibrated by standards of 147 μS/cm, 1413 μS/cm, 12.88 mS/cm. Burette of 50 mL (Brand) (±0.1 at 20°C).

8.3.3. Solutions

  1. - Hydrochloric and acetic acids mixture 0.015 M.

  2. - Potassium hydroxide 0.1 M.

8.3.4. Experimental

About 100 mL of hydrochloric and acetic acids mixture ≈ 0.015 M is transferred to a 250 mL volumetric flask with 100 mL of distilled water. Then, the mixture is titrated conductometrically with KOH 0.0992 ± 0.0001 M (n= 3), (previously standardized with potassium hydrogen phthalate). Table 9 shows the data [conductance, volume] as well as the product of the conductance 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)

Table 9.

Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (first assay).

* Conductivity⋅((100 + V)/100).

Figure 6.

Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are shown inTable 9, first assay). Branch A: V [0–15]. Branch B: V [1628]. Branch C: V [29.1–45].

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 neutralization of acetic acid, and the third (branch C) to the excess of potassium hydroxide.

Figures 7 (hydrochloric acid) and 8 (hydrochloric acid + acetic acid), respectively, are the graphs corresponding to the estimation of the end points. The points represented in the graph and then 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.

Figure 7.

Conductometric titration of hydrochloric acid in the mixture (branches A and B).

Figure 8.

Conductometric titration of acetic acid in the mixture (branches B and C).

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 end point), 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 has been previously explained in Section 7). Three decimals were considered to compare and check calculations.

cov(a1, b1)=−6.572E−06cov(a2, b2)=−1.201E−05
Va]=3.279E−04Va]=3.564E−04V[x(I)]=0.0007FIELLERax2 + bx+ c= 0
cova, Δb)=−1.859E−05cova, Δb)=−1.932E−05t(0, 05, 27)=2.052b=−3.221E + 00V(l)=15.278
Pooled variancest s[x(I)]=0.0562c=2.469E + 01

Table 10.

Evaluation of end points in the titration of a mixture of HCl and HCH3COO with KOH 0.0992 F (data Table 9).

V KOH (mL)Conductance (mS/cm)Conductance*(mS/cm)V KOH (mL)Conductance (mS/cm)Conductance* (mS/cm)

Table 11.

Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (second assay).

* Conductivity⋅((100 + V)/100).

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 difference 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 used in the detailed calculations are colored in the tables. The results obtained (and intermediate calculations) for the second experience 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)

Table 12.

Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (third assay).

* Conductivity⋅((100 + V)/100).

cov(a1, b1)=−1.600E−06cov(a2, b2)=−7.774E−06
Va]=1.927E−04Va]=1.590E−04V[x(I)]=0.0004FIELLERax2 + bx+ c= 0
cova, Δb)=−9.374E−06cova, Δb)=−8.459E−06t(0, 05, 24)=2.064b=−3.420E+00
Pooled variancest s[x(I)]=0.0414c=2.550E+01
V[y/x]3=1.792E−04V[x(II)]=0.0034t(0, 05, 26)=2.056
V(b3)=6.400E−07s[x(II)]=0.0587t s[x(II)]=0.1207
cov(a3, b3)=−2.368E−05Vol(l)=29.251
Cova1, Δa2)=7.014E−03
Cova1, Δb2)=−9.265E−03
Cova2, Δb1)=−4.704E−03
Covb1, Δb2)=6.224E−03
Va]=1.583E−07Va]=9.320E−04Vx]=3.116E−03t(0, 05, 37)=2.026
Vb]=9.902E−07Vb]=1.014E−06sx]=5.583E−02t sx]=0.113
cova, Δb)=−2.893E−05cova, Δb)=−2.952E−05Vol(u)=14.571
Pooled variancesVol(l)=14.345

Table 13.

Evaluation of end points in the titration of a mixture of HCl and HCH3COO with KOH 0.0992 F (data Table 11).

Figure 9.

Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are shown inTable 10, second assay).

Figure 10.

Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are shown inTable 9, third assay).

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 to tabulated for P= 0.05, that is, Qtab = 1.155 and Gtab = 1.15 (although the Gexp and Gtab values are practically the same). The mean ± confidence limits of the values are 15.093 ± 0.217 mL for hydrochloric acid (first end point) and 14.417 ± 0.038 mL for acetic acid (difference), which leads to molarity values of the solutions of hydrochloric and acetic acid of 0.01497 ± 0.00022 M and 0.01430 ± 0.00004 M. If the most distant value were 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 the slope of the straight lines obtained by the least squares method is not taken into account, the propagation 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 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 end point detection increases the importance and the worth of titrimetric analysis. Physicochemical methods are intensively developed nowadays. However, in spite of this, titration continues to maintain its importance for chemical analysis. Plotting two straight lines graphs from the conductivity versus volume added experimental data and determining the corresponding intersection point of the two branches allow locating the end point in a conductometric titration. The estimation of uncertainty of end point 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 applied to the two branches of the conductometric titration curves leads to similar results 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 of 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 of the small number of times 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 fail in this case, being necessary to resort to first-order propagation of variance (random error propagation law). However, the algebra associated in this case though simple is 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 propagation law) and Fieller’s theorem methods preferable to methods based on intersecting confidence bands.


  1. 1. Beck CM. Classical analysis. A look at the past, present and future. Analytical Chemistry. 1994;66(4):224A–239A
  2. 2. Granholm K, Sokalski T, Lewenstam A, Ivaska A. Ion-selective electrodes in potentiometric titrations: A new method for processing and evaluating titration data. Analytica Chimica Acta. 2015;888:36–43
  3. 3. Thordason P. Determining association constants from titration experiments in supramolecular chemistry. Chemical Society Reviews. 2011;40(3):1305–1323
  4. 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. 5. Asuero AG and Michalowski T. Comprehensive formulation of titration curves for complex acid-base systems and its analytical Implications. Critical Reviews in Analytical Chemistry. 2011;41(2):151–187
  6. 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. 7. Beck CM. Toward a revival of classical analysis. Metrología. 1997;34(1):19–30
  8. 8. Felber H, Rezzonico S, Máriássy M. Titrimetry at a metrological level. Metrología. 2003;40(5):249–254
  9. 9. King B. Review of the potential of titrimetry as a primary method. Metrología. 1997;34(1):77–82
  10. 10. Ortiz-Fernandez MC, Herrero-Gutierrez A. Regression by least median squares, a methodological contribution to titration. Chemometrics and Intelligent Laboratory Systems. 1995;27:241–243
  11. 11. Kupka K, Meloun M. Data analysis in the chemical laboratory II. The end-point estimation in instrumental titrations by nonlinear regression. Analytica Chimica Acta. 2001;429:171–183
  12. 12. Jones RH, Molitoris BA. A statistical method for determining the breakpoint of two lines. Analytical Biochemistry. 1984;41(1):287–290
  13. 13. Sprent P. Some hypothesis concerning two phase regression analysis. Biometrics. 1961;17(4):634–645
  14. 14. Shaban SA. Change point problem and two-phase regression: An annotated bibliography. International Statistical Review. 1980;48(1):83–93
  15. 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. 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. 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. 18. Rukhin AL. Estimation and testing for the common intersection point. Chemometrics and Intelligent Laboratory Systems. 2008;90(2):116–122
  19. 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. 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 American Chemical Society. 1999;121(40):9265–9275
  21. 21. Vieth E. Fitting piecewise linear regression functions to biological responses. Journal of Applied Physiology. 1989;67(1):390–396
  22. 22. Piegorsch WW. Confidence intervals on the joint point in segmented regression. Biometrical Unit Technical Reports BU-785-M. 1982
  23. 23. Bacon DW, Watts DG, Estimating the transition between two intersecting straight lines. Biometrika. 1972;58(3):525–534
  24. 24. Christensen R. Plane answers to complex questions. The Theory of Linear Models. 4th ed., New York: Springer, 2011
  25. 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
  26. 26. Seber GAF. Linear regression analysis. 7.6 Two-phase linear regression. New York: Wiley, 1977. pp. 205–209
  27. 27. Seber GAF, Lee AJ. Linear regression analysis. 6.5 Two-phase linear regression. 2nd ed., New York: Wiley, 2003. pp. 159–161
  28. 28. Cook DA, Charnock JS. Computer-assisted analysis of functions which may be represented by two intersecting straight lines. Journal of Pharmacological Methods. 1979;2(1):13–19
  29. 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. 30. Hubálovsky S, Sedivú J. Algorithm for estimates determination of parameters of regression straight polyline with objective elimination of outliers. 2nd International Conference on Information Technology (ICIT) 2010 Gdansk, 28–30 June 2010. pp. 263–266
  31. 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. 32. Jehlicka V, Mach V. Determination of intersection of regression straight lines with elimination of outliers. Chemical Papers. 1999;53(5):279–282
  33. 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. 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. 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. 36. Meier PC, Zünd RE. Statistical methods in analytical chemistry. 2nd ed., New York: Wiley, 2000. pp. 127–128, 374
  37. 37. Miller JN, Miller JC. Statistics and Chemometrics in Analytical Chemistry. Harlow, England: Pearson, 2010. pp. 140–141
  38. 38. Fieller EC. The biological standardization of insulin. Journal of the Royal Statistical Society. 1940–1941;7(1):1–64
  39. 39. Fieller EC. A fundamental formula in the statistics of biological assay, and some applications. Quarterly Journal of Pharmacy and Pharmacology. 1944;17:117–123
  40. 40. Fieller EC. Some problems in interval estimation. Journal of the Royal Statistical Society: Series B. 1954;16(2):175–185
  41. 41. Koschat MA. A characterization of the Fieller solution. Annals of Statistics. 1987;15:462–468
  42. 42. Wallace DL. The Behrens-Fischer and Fieller-Creasy problems. In Berger J, Fienberg S, Gani J, Krickeberg K, Olkin I. R.A. Fischer, an appreciation, editors, Berlin: Springer-Verlag, 1980
  43. 43. Wilkinson GN. On resolving the controversy in statistical inference. Journal of the Royal Statistical Society: Series B. 1977;39(2):119–171
  44. 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. 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. 46. Puterman ML, Hrboticky N, Innist SM. Nonlinear estimation of parameters in biphasic Arrhenius plots. Analytical Biochemistry. 1988;170(2):409–420
  47. 47. Baxter DC. Evaluation of the simplified generalised standard additions method for calibration in the direct analysis of solid samples by graphite furnace atomic spectrometric techniques. Journal of Analytical Atomic Spectrometry. 1989;4(5):415–421
  48. 48. Bonate PL. Approximate confidence intervals in calibration using the bootstrap. Analytical Chemistry. 1993;65(10):1367–1372
  49. 49. Mandel J, Linning FJ. Study of accuracy in chemical analysis using linear calibration curves. Analytical Chemistry. 1957;29(5):743–749
  50. 50. Schwartz LM. Nonlinear calibration curves. Analytical Chemistry. 1977;49(13):2062–2066
  51. 51. Schwartz LM. Statistical uncertainties of analyses by calibration of counting measurements. Analytical Chemistry. 1978;50(7):980–984
  52. 52. Schwartz LM. Calibration curves with nonuniform variance. Analytical Chemistry. 1979;51(6):723–727
  53. 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. 54. Asuero AG, Recamales MA. A bilogarithmic method for the spectrophotometric evaluation of acidity constants of two-step overlapping equilibria. Analytical Letters 1993;26(1):163–181
  55. 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. 56. Franke JP, de Zeeuw RA, Hakkert R. Evaluation and optimization of the standard addition method for absorption spectrometry and anodic stripping voltammetry. Analytical Chemistry. 1978;50(9):1374–1380
  57. 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. 58. Liteanu C, Rica I. Statistical Theory and Methodology of Trace Analysis. Chichester: Ellis Horwood Ltd, 1980. pp. 166–172
  59. 59. Asuero AG, Sayago A, González AG. The correlation coefficient: an overview. Critical Reviews in Analytical Chemistry. 2006;36(1):41–59
  60. 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. 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. 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. 63. Draper NR, Smith H. Applied Regression Analysis. New York: Wiley, 1998
  64. 64. McCormick D, Roach A. Measurement, Statistics and Computation, ACOL. Chichester: Wiley, 1987. pp. 312–315
  65. 65. Schwartz LM, Gelb RI. Statistical uncertainties of end points at intersecting straight lines. Analytical Chemistry. 1984;56(8):1487–1492
  66. 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. 67. Sharaf MA, Illman DL, Kowalski BR. Chemometrics. New York: Wiley, 1986
  68. 68. Brownlee KA. Statistical Theory and Methodology in Science and Engineering. 2nd ed., Malabar, FL: R.E. Krieger, 1984
  69. 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. Analytical Chemistry. 1991;63(13):1270–1278
  70. 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. 71. Jandera P, Kolda S, Kotrly S. End-point evaluation in instrumental titrimetry-II Confidence intevals in extrapolation of linear titration curves. Talanta. 1970;17(6):443–454
  72. 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 kleinsten Quadrate. Revue Roumaine de Chimie. 1965;10:361–376
  73. 73. Sharaf MA, Illman DL, Kowalski BR. Chemometrics. New York: Wiley, 1986
  74. 74. Gelhaus SL, Lacourse WR. Measurement of Electrolytic Conductance. In Analytical Instrumentation Handbook. New York: Marcel Dekker, 2005. pp. 561–580
  75. 75. Gzybkovski W. Conductometric and potentiometric titrations. Politechnika Gdańska, Gdansk, 2002. pdf[accessed on August 5, 2017]
  76. 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. 77. Donkersloot MCA. Teaching conductometry. Another perspective. Journal of Chemical Education. 1991;68(2):136–137
  78. 78. Garcia J, Schultz LD. Determination of sulphate by conductometric titration: an undergraduate laboratory experiment. Journal of Chemical Education. 2016;93(5):910–914
  79. 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. 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. 81. Farris S, Mora L, Capretti G, Piergiovanni L. Charge density quantification of polyelectrolyte polysaccharides by conductimetric titration: an analytical chemistry experiment. Journal of Chemical Education. 2012;89(1):121–124
  82. 82. Nyasulu F, Moehring M, Arthasery P, Barlag R. Ka and Kb from pH and conductivity measurements: a general chemistry laboratory exercise. Journal of Chemical Education. 2011;88(5):640–642
  83. 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. 84. Holdsworth DK. Conductivity titrations - a microcomputer approach. Journal of Chemical Education. 1986;63(1):73–74
  85. 85. Rosenthal LC, Nathan LC. A conductimetric-potentiometric titration for an advanced laboratory. Journal of Chemical Education. 1981;58(8):656–658
  86. 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. 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. 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. 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

Written By

Julia Martin, Gabriel Delgado Martin and Agustin G. Asuero

Submitted: December 9th, 2016 Reviewed: March 27th, 2017 Published: September 27th, 2017