**Intersecting Straight Lines: Titrimetric Applications**

DOI: 10.5772/intechopen.68827

Intersecting Straight Lines: Titrimetric Applications

Julia Martin, Gabriel Delgado Martin and Agustin G. Asuero Julia Martin, Gabriel Delgado Martin and

Additional information is available at the end of the chapter Agustin G. Asuero

http://dx.doi.org/10.5772/intechopen.68827 Additional information is available at the end of the chapter

#### Abstract

[75] Gran G. Equivalence volumes in potentiometric titrations. Analytica Chimica Acta 1988;

[76] Schwartz LM. Advances in acid-base Gran plot methodology. Journal of Chemical Edu-

[77] de Levie R. Principles of Quantitative Chemical Analysis. New York: McGraw-Hill; 1997 [78] de Levie R. Aqueous Acid-Base Equilibria and Titrations. Oxford: Oxford University

[79] Asuero AG, Gonzalez G. Fitting straight lines with replicated observations by linear regression. III weighting data. Critical Reviews in Analytical Chemistry. 2007;37(3):

[80] Asuero AG, Bueno JM. Fitting straight lines with replicated observations by linear regression. IV. Transforming data. Critical Reviews in Analytical Chemistry. 2011;41(1):36–69

[81] Ivaska A and Wanninen E. Potentiometric titration of weak acids. Analytical Letters.

[82] Michalowski T, Asuero AG, Wybraniec S. The titration in the kjeldahl method of nitrogen determination: Base or acid as titrant. Journal of Chemical Education. 2013;90(2):191–197

[83] 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

[84] Asuero AG. Inflexion points of the titration curve for two step overlapping equilibria.

[85] Harris DC. Quantitative Chemical Analysis. 8th ed. New York: W.H. Freeman and Com-

[86] Schwartz LM. Uncertainty of a Titration equivalence point. A graphical method using spreadsheets to predict values and detect systematic errors. Journal of Chemical Education.

[87] Baes CF, Mesmer RE. The Hydrolysis of Cations. Malabar, FL: Krieger; 1976. pp. 104–111

206:111–123

58 Advances in Titration Techniques

Press, 1999

143–172

1973;6:961–967

pany; 2010

1992;69(11):879–883

Chemistry. 2007;37(4):269–301

Microchemical Journal. 1993;47:386–398

cation. 1987;64(11):947–950

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 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 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 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.

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 developing 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 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).

© 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.

© 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.

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 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 instrumental 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 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 [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 analysis [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 (xI), the two lines have the same ordinate. The estimation of statistical uncertainty of end points obtained from linear segmented titration is the subject


Table 1. Instrumental end point detection techniques more widely applied.


Table 2. Names received in the literature for the intersecting point of two straight lines.

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 x data are negligible compared to the y data. Single linear regression or weighted linear regression may be applied depending on whether the variance of y is 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

Independent values of chemical quantities expressed in SI units are obtained through gra-

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

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-

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 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 analysis [28–31], elimination of outliers [32, 33], and confidence limits for the abscissa [22, 34, 35] have

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

Technique Measured property Conductimetric titrations Electrical conductivity

Spectrophotometric titrations Absorbance

Table 1. Instrumental end point detection techniques more widely applied.

Potentiometric titrations Potential of an indicator electrode

Amperometric titrations Diffusion current at a polarizable indicator

(dropping mercury or rotating platinum) electrode

vimetry and titrimetry (classical analysis).

60 Advances in Titration Techniques

mental method of end point detection.

the end-point location.

been subject to study.

change.

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 x of titrant added to the unknown sample containing the analyte.

Let N<sup>1</sup> observations on the first line

$$
\hat{y}\_1 = a\_1 + b\_1 \mathbf{x} \tag{1}
$$

and N<sup>2</sup> observations on the second

$$
\hat{y}\_2 = a\_2 + b\_2 \text{x} \tag{2}
$$

where a1, b1, a2, b<sup>2</sup> 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 x data are negligible compared to the y data, 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 b and their standard errors by weighted linear regression [59] are given in Table 3, where the analogy with simple linear regression (i.e., w<sup>i</sup> = 1), is evident.

Note that in summation (1) and (2) by dividing by N<sup>1</sup> and N2, respectively, we get

$$
\overline{y}\_1 = a\_1 + b\_1 \overline{x}\_1 \tag{3}
$$

$$
\overline{y}\_2 = a\_2 + b\_2 \overline{x}\_2 \tag{4}
$$

At the point of intersection, the lines (1) and (2) have the same ordinate y^<sup>1</sup> ¼ y^<sup>2</sup> and the abscissa of intersection (denotes by x^I) is given by

$$a\_1 + b\_1 \hat{\mathbf{x}}\_I = a\_2 + b\_2 \hat{\mathbf{x}}\_I \tag{5}$$

$$
\hat{\mathbf{x}}\_I = \frac{a\_2 - a\_1}{b\_1 - b\_2} = -\frac{\Delta a}{\Delta b} \tag{6}
$$

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%).

Table 3. Formulae for calculating statistics for weighted linear regression.

## 3. First-order propagation of variance for V½x^I�

y^<sup>2</sup> ¼ a<sup>2</sup> þ b2x ð2Þ

y<sup>1</sup> ¼ a<sup>1</sup> þ b1x<sup>1</sup> ð3Þ

y<sup>2</sup> ¼ a<sup>2</sup> þ b2x<sup>2</sup> ð4Þ

a<sup>1</sup> þ b1x^<sup>I</sup> ¼ a<sup>2</sup> þ b2x^<sup>I</sup> ð5Þ

<sup>Δ</sup><sup>b</sup> <sup>ð</sup>6<sup>Þ</sup>

• Slope

• s<sup>2</sup> <sup>a</sup> <sup>¼</sup> <sup>s</sup><sup>2</sup> <sup>y</sup>=<sup>x</sup>ð <sup>X</sup>wix<sup>2</sup>

• s<sup>2</sup> <sup>b</sup> <sup>¼</sup> <sup>s</sup><sup>2</sup> <sup>y</sup>=<sup>x</sup>=SXX

• Covða, bÞ ¼ xs<sup>2</sup>

b ¼ SXY=Sxx • Intercept a ¼ y � bx • Weighted residuals w<sup>1</sup>=<sup>2</sup> <sup>i</sup> ðyi � y^<sup>i</sup> Þ • Correlation coefficient <sup>r</sup> <sup>¼</sup> SXY<sup>=</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SXXSYY p • Standard errors s2 <sup>y</sup>=<sup>x</sup> <sup>¼</sup> SSE

<sup>n</sup>�<sup>2</sup> <sup>¼</sup> SYY�b<sup>2</sup>SXX n�2

<sup>i</sup> Þ=ðSXX

<sup>y</sup>=<sup>x</sup>=SXX

<sup>X</sup>wi<sup>Þ</sup>

where a1, b1, a2, b<sup>2</sup> 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 x data are negligible compared to the y data, 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 b and their standard errors by weighted linear regression [59] are given in Table 3, where the analogy with simple linear

At the point of intersection, the lines (1) and (2) have the same ordinate y^<sup>1</sup> ¼ y^<sup>2</sup> and the

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

¼ � <sup>Δ</sup><sup>a</sup>

<sup>x</sup>^<sup>I</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> � <sup>a</sup><sup>1</sup> b<sup>1</sup> � b<sup>2</sup>

Note that in summation (1) and (2) by dividing by N<sup>1</sup> and N2, respectively, we get

regression (i.e., w<sup>i</sup> = 1), is evident.

62 Advances in Titration Techniques

abscissa of intersection (denotes by x^I) is given by

true value is equal to the confidence level (e.g., 95%).

2

Þ 2

2

2

Table 3. Formulae for calculating statistics for weighted linear regression.

<sup>X</sup>wi

= <sup>X</sup>wi • Sum of squares about the mean SXX <sup>¼</sup> <sup>X</sup>wiðxi <sup>¼</sup> <sup>x</sup><sup>Þ</sup>

SXY <sup>¼</sup> <sup>X</sup>wiðxi <sup>¼</sup> <sup>x</sup>Þðyi � <sup>y</sup><sup>Þ</sup>

SYY <sup>¼</sup> <sup>X</sup>wiðyi <sup>¼</sup> <sup>y</sup><sup>Þ</sup>

• Equation y^<sup>i</sup> ¼ a þ bxi • Weights wi <sup>¼</sup> <sup>1</sup>=s<sup>2</sup> i • Explained sum of squares SSRe<sup>g</sup> <sup>¼</sup> <sup>X</sup>wiðy^<sup>i</sup> � <sup>y</sup><sup>Þ</sup>

• Mean

• Residual sum of squares SSE <sup>¼</sup> <sup>X</sup>wiðyi � <sup>y</sup>^<sup>i</sup>

<sup>x</sup> <sup>¼</sup> <sup>X</sup>wixi<sup>=</sup>

<sup>y</sup> <sup>¼</sup> <sup>X</sup>wiyi

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^<sup>I</sup> on this way. First-order propagation of variance retains only first derivatives in the Taylor expansions and this procedure leads to

$$\begin{split} V[\dot{X}\_{I}] &= \left(\frac{\partial \dot{\hat{x}}\_{I}}{\partial \Delta a}\right)^{2} V[\Delta a] + \left(\frac{\partial \dot{\hat{x}}\_{I}}{\partial \Delta b}\right)^{2} V[\Delta b] + 2 \left(\frac{\partial \dot{\hat{x}}\_{I}}{\partial \Delta a}\right) \left(\frac{\partial \dot{\hat{x}}\_{I}}{\partial \Delta b}\right) \text{Cov}(\Delta a, \Delta b) \\ &= \left(\frac{\partial \left(\frac{-\Delta b}{\Delta b}\right)}{\partial \Delta a}\right)^{2} V[\Delta a] + \left(\frac{\partial \left(\frac{-\Delta b}{\Delta b}\right)}{\partial \Delta b}\right)^{2} V[\Delta b] + 2 \left(\frac{\partial \left(\frac{-\Delta a}{\Delta b}\right)}{\partial \Delta a}\right) \left(\frac{\partial \left(\frac{-\Delta a}{\Delta b}\right)}{\partial \Delta b}\right) \text{Cov}(\Delta a, \Delta b) \\ &= \frac{V[\Delta a]}{\Delta b^{2}} + \frac{\Delta a^{2} V[\Delta b]}{\Delta b^{4}} - 2 \frac{\Delta a \text{Cov}(\Delta a, \Delta b)}{\Delta b^{3}} \end{split} \tag{7}$$

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

$$\begin{split} V[\hat{\mathbf{x}}\_{I}] &= \frac{1}{\Delta b^{2}} \left( V[\Delta a] + \frac{\Delta a^{2}}{\Delta b^{2}} V[\Delta b] - 2 \frac{\Delta a}{\Delta b} \text{Cov}(\Delta a, \Delta b) \right) \\ &= \frac{V[\Delta a] + \hat{\mathbf{x}}^{2} V[\Delta b] + 2 \hat{\mathbf{x}}\_{I} \text{Cov}(\Delta a, \Delta b)}{\Delta b^{2}} \end{split} \tag{8}$$

Then, the standard error estimate of x^<sup>I</sup> is as follows

$$\mathbf{s}(\hat{\mathbf{x}}\_{l}) = \sqrt{V[\hat{\mathbf{x}}]} \tag{9}$$

The end point x^<sup>I</sup> depends on four least squares parameters a1, a2, b1, b<sup>2</sup> 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<sup>1</sup> and b2. Note that, a<sup>1</sup> and a<sup>2</sup> are related to b<sup>1</sup> and b<sup>2</sup> by means of Eqs. (1) and (2).

The variances of Δa and Δb are given by

$$V[\Delta a] = V[a\_1 - a\_2] = V[a\_1] + V[a\_2] \tag{10}$$

$$V[\Delta b] = V[b\_1 - b\_2] = V[b\_1] + V[b\_2] \tag{11}$$

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

$$\begin{aligned} \text{Cov}(\Delta a, \Delta b) &= \text{Cov}(a\_1 - a\_2, b\_1 - b\_2) \\ &= \text{Cov}\left(\overline{y}\_1 - b\_1 \overline{x}\_1 - (\overline{y}\_2 - b\_2 \overline{x}\_2), b\_1 - b\_2\right) = \text{Cov}(\overline{y}\_1 - \overline{y}\_2 - b\_1 \overline{x}\_1 + b\_2 \overline{x}\_2, b\_1 - b\_2) \\ &= \text{Cov}(\overline{y}\_1 - \overline{y}\_2, b\_1 - b\_2) - \text{Cov}(b\_1 \overline{x}\_1 - b\_2 \overline{x}\_2, b\_1 - b\_2) \\ &= -\text{Cov}(b\_1 \overline{x}\_1 - b\_2 \overline{x}\_2, b\_1 - b\_2) \\ &= -\overline{x}\_1 V[b\_1] - \overline{x}\_2 V[b\_2] \end{aligned} \tag{12}$$

It should be noted that in the calculations, the variance regression estimates from both line segments are pooled into a single sp 2 , by using the following formula which weights each contribution according to the corresponding [64–67] degrees of freedom

$$\mathbf{s}\_p^2 = \frac{\sum\_{i=1}^{N\_1} (y\_{1i} - \overline{y}\_1)^2 + \sum\_{i=1}^{N\_2} (y\_{2i} - \overline{y}\_2)^2}{(N\_1 - 2) + (N\_2 - 2)} = \frac{(N\_1 - 2)\mathbf{s}\_1^2 + (N\_2 - 2)\mathbf{s}\_2^2}{N\_1 + N\_2 - 4} \tag{13}$$

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 (sa <sup>2</sup> = V[a]), we may derive

$$V[a] = s^2 \left[ \frac{\sum w\_i x\_i^2}{S\_{\text{XX}} \left( \sum w\_i \right)} \right] = s^2 \left[ \frac{S\_{\text{XX}} + \frac{\left( \sum w\_i x\_i \right)^2}{\sum w\_i}}{S\_{\text{XX}} \left( \sum w\_i \right)} \right] = s^2 \left[ \frac{S\_{\text{XX}} + \left( \sum w\_i \right) \left( \sum \frac{w\_i x\_i}{\sum w\_i} \right)^2}{S\_{\text{XX}} \left( \sum w\_i \right)} \right] \tag{14}$$
 
$$= s^2 \left[ \frac{S\_{\text{XX}} + \left( \sum w\_i \right) \overline{\mathbf{x}}^2}{S\_{\text{XX}} \left( \sum w\_i \right)} \right] = s^2 \left[ \frac{1}{\sum w\_i} + \frac{\overline{\mathbf{x}}^2}{S\_{\text{XX}}} \right]$$

in which s <sup>2</sup> is s(y/x) in Table 3; ∑wi 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

$$V[\Delta a] = V[a\_1] + V[a\_2] = \left[ \frac{1}{\left(\sum w\right)\_1} + \frac{\overline{\mathbf{x}}\_1^2}{\left(\mathbf{S}\_{\text{XX}}\right)\_1} \right] s\_1^2 + \left[ \frac{1}{\left(\sum w\right)\_2} + \frac{\overline{\mathbf{x}}\_1^2}{\left(\mathbf{S}\_{\text{XX}}\right)\_2} \right] s\_2^2 \tag{15}$$

$$V[\Delta a] = \left[ \frac{1}{\left(\sum w\right)\_1} + \frac{1}{\left(\sum w\right)\_2} + \frac{\overline{\chi}\_1^2}{\left(\mathcal{S}\_{\text{XX}}\right)\_1} + \frac{\overline{\chi}\_1^2}{\left(\mathcal{S}\_{\text{XX}}\right)\_2} \right] s\_p^2 \tag{16}$$

$$V[\Delta b] = V[b\_1] + V[b\_2] = \frac{s\_1^2}{(S\_{XX})\_1} + \frac{s\_1^2}{(S\_{XX})\_2} = \left[\frac{1}{(S\_{XX})\_1} + \frac{1}{(S\_{XX})\_2}\right] s\_p^2 \tag{17}$$

$$\text{Cov}(\Delta a, \Delta b) = -\overline{\mathbf{x}}\_1 V[b\_1] - \overline{\mathbf{x}}\_2 V[b\_2] = -\left[ \frac{\overline{\mathbf{x}}\_1}{(\mathbf{S}\_{\text{XX}})\_1} + \frac{\overline{\mathbf{x}}\_2}{(\mathbf{S}\_{\text{XX}})\_2} \right] s\_p^2 \tag{18}$$

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, V½x^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 yi are assumed to have Gaussian (normal) distribution, the least squares parameters as well as Δa, and Δb are 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's t-test

$$
\hat{\mathfrak{x}}\_I \pm t\_{\alpha/2} \mathbf{s}(\hat{\mathfrak{x}}\_I) \tag{19}
$$

where tα/2 is the Student's t statistics at the 1 � α confidence level (i.e., leaving an area of α/2 to the right) and for the number of degrees of freedom (N1+N<sup>2</sup> � 4) inherent in the standard deviation of x^I, could be misleading. Note that because x^<sup>I</sup> involves the ratio of random variables, first-order propagation of variance is not exact [69]. Evidently, x^<sup>I</sup> is a random variable not normally distributed unless sðx^IÞ is 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

It should be noted that in the calculations, the variance regression estimates from both line

<sup>ð</sup>N<sup>1</sup> � <sup>2</sup>ÞþðN<sup>2</sup> � <sup>2</sup><sup>Þ</sup> <sup>¼</sup> <sup>ð</sup>N<sup>1</sup> � <sup>2</sup>Þs<sup>2</sup>

The standard deviation in Eq. (13) is calculated on the assumption that the s(y/x) values for the

<sup>X</sup>wixi � �<sup>2</sup>

<sup>2</sup> is s(y/x) in Table 3; ∑wi is the sum of weights, which simply reduces to N, the number

x2 1 ðSXXÞ<sup>1</sup>

> 2 þ

s2 1 ðSXXÞ<sup>2</sup>

x2 1 ðSXXÞ<sup>1</sup> þ

> <sup>¼</sup> <sup>1</sup> ðSXXÞ<sup>1</sup> þ

ðSXXÞ<sup>1</sup> þ

<sup>¼</sup> <sup>s</sup><sup>2</sup>

<sup>X</sup>wi

<sup>X</sup>wi � �

" #

of points if the non-weighted least squares analysis is used. Taking into account Eq. (14), Eqs. (10)

, by using the following formula which weights each

SXX <sup>þ</sup> <sup>X</sup>wi

SXX

1 Xw � �

> x2 1 ðSXXÞ<sup>2</sup>

� �

� �

x2 ðSXXÞ<sup>2</sup>

2 6 4

2 þ

1 ðSXXÞ<sup>2</sup>

s 2

s 2

x2 1 ðSXXÞ<sup>2</sup>

<sup>p</sup> ð16Þ

<sup>p</sup> ð17Þ

<sup>p</sup> ð18Þ

<sup>2</sup> ð15Þ

<sup>1</sup> þ ðN<sup>2</sup> � <sup>2</sup>Þs<sup>2</sup>

<sup>2</sup> = V[a]), we may derive

<sup>X</sup>wi � �

� � <sup>X</sup>

2 <sup>N</sup><sup>1</sup> <sup>þ</sup> <sup>N</sup><sup>2</sup> � <sup>4</sup> <sup>ð</sup>13<sup>Þ</sup>

> X wixi wi

!<sup>2</sup>

ð14Þ

2

<sup>2</sup> þX<sup>N</sup><sup>2</sup> i¼1 ðy2<sup>i</sup> �y2Þ 2

contribution according to the corresponding [64–67] degrees of freedom

segments are pooled into a single sp

X<sup>N</sup><sup>1</sup> i¼1 ðy1<sup>i</sup> �y1Þ

two lines are sufficiently similar to be pooled.

<sup>X</sup>wix<sup>2</sup> i

> <sup>X</sup>wi � �

SXX <sup>þ</sup> <sup>X</sup>wi � �

> <sup>X</sup>wi � �

<sup>V</sup>½Δa� ¼ <sup>V</sup>½a1� þ <sup>V</sup>½a2� ¼ <sup>1</sup>

<sup>V</sup>½Δa� ¼ <sup>1</sup>

2 6 4

<sup>V</sup>½Δb� ¼ <sup>V</sup>½b1� þ <sup>V</sup>½b2� ¼ <sup>s</sup><sup>2</sup>

V½x^I�, is calculated by applying Eq. (8).

SXX

3 <sup>5</sup> <sup>¼</sup> <sup>s</sup><sup>2</sup>

SXX

2 4

From expression in Table 3 for the variance of the intercept (sa

x2

3 <sup>5</sup> <sup>¼</sup> <sup>s</sup><sup>2</sup>

SXX þ

SXX

X 1 wi þ x2 SXX

Xw � �

> 1 þ

CovðΔa,Δb޼�x1V½b1� � <sup>x</sup>2V½b2�¼� <sup>x</sup><sup>1</sup>

2 6 4

Xw � � 1 þ

1 ðSXXÞ<sup>1</sup> þ

1 Xw � �

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,

s 2 <sup>p</sup> ¼

64 Advances in Titration Techniques

<sup>V</sup>½a� ¼ <sup>s</sup><sup>2</sup>

<sup>¼</sup> <sup>s</sup><sup>2</sup>

in which s

and (11) lead to

2 4

This theorem [38–43] is supported by two capital premises:


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

$$z = [a\_1 + b\_1 \mathbf{x}\_l] - [a\_2 + b\_2 \mathbf{x}\_l] = \Delta a + \mathbf{x}\_l \Delta b \tag{20}$$

Note that for any such pair of lines, the difference z is 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 z values is zero and z are normally distributed because it is formed as a linear combination of normally distributed variables. Taking into account that a1, a2, b<sup>1</sup> and b<sup>2</sup> are normally distributed, then z will be normally distributed. Then, in the vicinity of the intersection point, z has zero mean and its variance is

$$V[z] = V[\Delta a + \mathbf{x}\_l \Delta b] = V[\Delta a] + \mathbf{x}\_l^2 V[\Delta b] + 2\mathbf{x}\_l \text{Cov}(\Delta a, \Delta b) \tag{21}$$

and therefore, <sup>z</sup> ffiffiffiffi <sup>V</sup>^ <sup>p</sup> ½z� is distributed as N(0, 1) and according to (ii)

$$\frac{z}{\sqrt{\hat{V}}[z]} = \mathbf{t} \tag{22}$$

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

$$\frac{\left(\Delta a + \mathbf{x}\_{l}\Delta b\right)^{2}}{V[\Delta a] + \mathbf{x}\_{l}^{2}V[\Delta b] + 2\mathbf{x}\_{l}\mathrm{Cov}(\Delta a, \Delta b)} = \mathbf{t}^{2}\tag{23}$$

which on rearrangement leads to

$$\left(\Delta a\right)^{2} + 2\mathbf{x}\_{I}\Delta a\Delta b + \mathbf{x}\_{I}^{2}\big(\Delta b\big)^{2} = t^{2}V[\Delta a] + t^{2}\mathbf{x}\_{I}^{2}V[\Delta b] + 2t^{2}\mathbf{x}\_{I}\text{Cov}(\Delta a, \Delta b) \tag{24}$$

which may be factored as

$$\left(\left(\Delta a\right)^2 - t^2 V[\Delta a]\right) + 2\mathbf{x}\_l \left(\Delta a \Delta b - t^2 \text{Cov}(\Delta a, \Delta b)\right) + \mathbf{x}\_l^2 \left(\left(\Delta b\right)^2 - t^2 V[\Delta b]\right) = 0 \tag{25}$$

The solution of Eq. (25) gives the confidence limits for xI estimated, where tα/2 is the appropriate value of the Student distribution at a α significance level (confidence level 1 � α) for N<sup>1</sup> + N<sup>2</sup> � 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 firstorder propagation of variance for V½x^I�.

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 xI 2


Table 4. Some applications of Fieller theorem in analytical chemistry.

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 xI [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, 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:

distributed, then z will be normally distributed. Then, in the vicinity of the intersection point, z

is distributed as N(0, 1) and according to (ii)

z ffiffiffiffi <sup>V</sup>^ <sup>p</sup> ½z�

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

ðΔa þ xIΔbÞ

2

<sup>I</sup> V½Δb� þ 2xICovðΔa,ΔbÞ

V½Δa� þ t

CovðΔa,ΔbÞ

The solution of Eq. (25) gives the confidence limits for xI estimated, where tα/2 is the appropriate value of the Student distribution at a α significance level (confidence level 1 � α) for N<sup>1</sup> + N<sup>2</sup> � 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-

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 xI

Calibration curves Baxter [47]; Bonate [48]; Mandel y Linning [49]; Schwartz [50–52]

Arrhenius plot Cook and Charnock [44]; Han [45]; Puterman et al. [46]

2 x2

> � <sup>þ</sup> <sup>x</sup><sup>2</sup> I � ðΔbÞ <sup>2</sup> � <sup>t</sup> 2 V½Δb� �

<sup>I</sup> V½Δb� þ 2xICovðΔa,ΔbÞ ð21Þ

¼ t ð22Þ

<sup>2</sup> <sup>ð</sup>23<sup>Þ</sup>

xICovðΔa,ΔbÞ ð24Þ

¼ 0 ð25Þ

2

¼ t

2

<sup>I</sup> V½Δb� þ 2t

<sup>V</sup>½z� ¼ <sup>V</sup>½Δ<sup>a</sup> <sup>þ</sup> xIΔb� ¼ <sup>V</sup>½Δa� þ <sup>x</sup><sup>2</sup>

<sup>V</sup>½Δa� þ <sup>x</sup><sup>2</sup>

<sup>I</sup>ðΔbÞ

<sup>2</sup> <sup>¼</sup> <sup>t</sup> 2

ΔaΔb � t

2

<sup>2</sup> <sup>þ</sup> <sup>2</sup>xIΔaΔ<sup>b</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup>

has zero mean and its variance is

ffiffiffiffi <sup>V</sup>^ <sup>p</sup> ½z�

which on rearrangement leads to

order propagation of variance for V½x^I�.

Topic Reference

Models for biologic half-life data Lee et al. [25]

Standard addition method Franke et al. [56]

Table 4. Some applications of Fieller theorem in analytical chemistry.

Estimation of safe doses Yanagimoto and Yanamoto [19] Estimation of uncertainty in binding constants Almansa López et al. [53]

Position and confidence limits of an extremum Asuero and Recamales [54]; Heilbronner [55]

ðΔaÞ

which may be factored as

� ðΔaÞ <sup>2</sup> � <sup>t</sup> 2 V½Δa� � þ 2xI �

and therefore, <sup>z</sup>

66 Advances in Titration Techniques

$$y\_{01} \pm t\_1 s\_{y\_{01}} \tag{26}$$

and for the second line:

$$y\_{02} \pm t\_2 s\_{y\_{02}} \tag{27}$$

t<sup>1</sup> and t<sup>2</sup> are the corresponding t Student values for α/2 = 0,05 and N<sup>1</sup> � 2 and N<sup>2</sup> � 2 degrees of freedom, respectively. Hence, the lower value x<sup>l</sup> of the confidence interval is obtained by solving the following equation:

$$y\_{01} - t\_1 s\_{y\_{01}} = y\_{02} + t\_2 s\_{y\_{02}} \tag{28}$$

The higher value xu is obtained from the equation:

$$y\_{01} + t\_1 s\_{y\_{01}} = y\_{02} - t\_2 s\_{y\_{02}} \tag{29}$$

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

$$
\hat{y}\_1 = \overline{y}\_1 + b\_1(\mathbf{x} - \overline{\mathbf{x}}\_1) \tag{30}
$$

and then the variance of the fitted y<sup>1</sup> value will be given by

$$\begin{split} V[\hat{y}\_1] &= V[\overline{y}\_1] + (\mathbf{x} - \overline{\mathbf{x}}\_1)^2 V[\mathbf{b}\_1] = \frac{V[y\_1]}{\left(\sum w\right)\_1} + (\mathbf{x} - \overline{\mathbf{x}}\_1)^2 \frac{V[y\_1]}{\left(\mathbf{S}\_{\text{XX}}\right)\_1} \\ &= \left(\frac{1}{\left(\sum w\right)\_1} + \frac{(\mathbf{x} - \overline{\mathbf{x}}\_1)^2}{\left(\mathbf{S}\_{\text{XX}}\right)\_1}\right) V[y\_1] \end{split} \tag{31}$$

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

$$\begin{split} V[\overline{y}\_1] &= V \left[ \frac{\left(\sum w\right)\_1 y\_1}{\left(\sum w\right)\_1} \right] = V \left[ \frac{\left(\sum \left( (\sqrt{w})\_1 (\sqrt{w})\_1 \right) y\_1 \right)}{\left(\sum w\right)\_1} \right] \\ &= \frac{\left(\sum w\right)\_1 V[(\sqrt{w})\_1 y\_1]}{\left(\left(\sum w\right)\_1\right)^2} = \frac{V[y\_1]}{\left(\sum w\right)\_1} \end{split} \tag{32}$$

and that the mean y<sup>1</sup> value and the slope b<sup>1</sup> are uncorrelated random variables (property, which was also applied in Eq. (12) without further demonstration) as shown as follows. Taking into account that

 $\overline{y\_1} = \sum b\_1 (\sqrt{w})\_1 y\_1$  and  $b\_1 = \sum c\_1 (\sqrt{w})\_1 y\_1$  where 
$$b\_1 = \frac{(\sqrt{w})\_1}{(\sum w)\_1} \quad c\_1 = \frac{(\sqrt{w})\_1 (x - \overline{x}\_1)}{(S\_{X\overline{X}})\_1} \tag{33}$$

and then

$$Cov(b\_1, c\_1) = \left(\sum a\_1 c\_1\right) V[(\sqrt{w})\_1 y\_1] = \left(\sum \frac{(\sqrt{w})\_1 (x - \overline{x}\_1)}{(S \ge x)\_1 \left(\sum w\right)\_1}\right) V[y] = 0 \tag{34}$$

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

$$s\_{y01} = V[\hat{y}\_{1i}] = \left(\sqrt{\frac{1}{\left(\sum w\right)\_1} + \frac{\left(x - \overline{x}\_1\right)^2}{\left(S\_{\text{XX}}\right)\_1}}\right) s\_1 \left(y\_{01} = \hat{y}\_{1i}; \quad s\_1 = \sqrt{V[y\_1]}\right) \tag{35}$$

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

$$a\_1 + b\_1 \mathbf{x}\_l + t\_1 \mathbf{s}\_1 \sqrt{\frac{1}{\left(\sum \mathbf{w}\right)\_1} + \frac{\left(\mathbf{x}\_l - \overline{\mathbf{x}}\_1\right)^2}{\left(S\_{\text{XX}}\right)\_1}} = a\_2 + b\_2 \mathbf{x}\_l + t\_2 \mathbf{s}\_2 \sqrt{\frac{1}{\left(\sum \mathbf{w}\right)\_2} + \frac{\left(\mathbf{x}\_l - \overline{\mathbf{x}}\_2\right)^2}{\left(S\_{\text{XX}}\right)\_2}}\tag{36}$$

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

Intersecting Straight Lines: Titrimetric Applications http://dx.doi.org/10.5772/intechopen.68827 69

$$a\_1 + b\_1 \mathbf{x}\_u + t\_1 \mathbf{s}\_1 \sqrt{\frac{1}{\left(\sum w\right)\_1} + \frac{(\mathbf{x}\_u - \overline{\mathbf{x}}\_1)^2}{(\mathbf{S}\_{\text{XX}})\_1}} = a\_2 + b\_2 \mathbf{x}\_u + t\_2 \mathbf{s}\_2 \sqrt{\frac{1}{\left(\sum w\right)\_2} + \frac{(\mathbf{x}\_u - \overline{\mathbf{x}}\_2)^2}{(\mathbf{S}\_{\text{XX}})\_2}} \tag{37}$$

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

$$\frac{(N\_1 - 2)(\mathbf{x}\_I)\_1^l + (N\_2 - 2)(\mathbf{x}\_I)\_2^l}{N\_1 + N\_2 - 4} = \mathbf{x}\_I^l \tag{38}$$

$$\frac{\left(\text{N}\_1-\text{2}\right)\left(\text{x}\_l\right)\_1^u + \left(\text{N}\_2-\text{2}\right)\left(\text{x}\_l\right)\_2^u}{\text{N}\_1+\text{N}\_2-\text{4}} = \text{x}\_l^u\tag{39}$$

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

$$y\_I = a\_1 + b\_1(\mathbf{x}\_I)\_1 \pm t\_1 \mathbf{s}\_1 \sqrt{\frac{1}{\left(\sum w\right)\_1} + \frac{\left((\mathbf{x}\_I)\_1 - \overline{\mathbf{x}}\_1\right)^2}{(\mathbf{S}\_{\mathbf{XX}})\_1}}\tag{40}$$

$$y\_I = a\_2 + b\_2(\mathbf{x}\_I)\_2 \pm t\_2 \mathbf{s}\_2 \sqrt{\frac{1}{\left(\sum w\right)\_2} + \frac{\left((\mathbf{x}\_I)\_2 - \overline{\mathbf{x}}\_2\right)^2}{(S\_{\rm XX})\_2}}\tag{41}$$

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

V½y^1� ¼ V½y1�þðx � x1Þ

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

2 6 4

¼

<sup>w</sup><sup>p</sup> <sup>Þ</sup>1y<sup>1</sup> and <sup>b</sup><sup>1</sup> <sup>¼</sup> <sup>X</sup>c1<sup>ð</sup> ffiffiffiffi

Covðb1, c1Þ ¼ <sup>X</sup>a1c<sup>1</sup>

� ¼

1 Xw � �

sy<sup>01</sup> ¼ V½y^1<sup>i</sup>

a<sup>1</sup> þ b1xl þ t1s<sup>1</sup>

<sup>¼</sup> <sup>1</sup> ð <sup>X</sup>wÞ<sup>1</sup>

V½y1� ¼ V

into account that

68 Advances in Titration Techniques

<sup>y</sup><sup>1</sup> <sup>¼</sup> <sup>X</sup>b1<sup>ð</sup> ffiffiffiffi

and then

2

<sup>þ</sup> <sup>ð</sup><sup>x</sup> � <sup>x</sup>1<sup>Þ</sup>

!

Xw � �

Xw � �

Xw � �

1 y1

1 <sup>V</sup>½ð ffiffiffiffi <sup>w</sup><sup>p</sup> <sup>Þ</sup>1y1�

� <sup>X</sup><sup>w</sup> � �

<sup>b</sup><sup>1</sup> <sup>¼</sup> <sup>ð</sup> ffiffiffiffi <sup>w</sup><sup>p</sup> <sup>Þ</sup><sup>1</sup>

ð <sup>X</sup>wÞ<sup>1</sup>

> <sup>V</sup><sup>½</sup> ffiffiffiffi <sup>w</sup> � � <sup>p</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> <sup>ð</sup><sup>x</sup> � <sup>x</sup>1<sup>Þ</sup>

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

2

ðSXXÞ<sup>1</sup>

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

1

1 Xw � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> <sup>ð</sup>xl � <sup>x</sup>1<sup>Þ</sup>

ðSXXÞ<sup>1</sup> vuut <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup>2xl <sup>þ</sup> <sup>t</sup>2s<sup>2</sup>

1

� �

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

vuut 0 B@

1

ðSXXÞ<sup>1</sup>

3 7 <sup>5</sup> <sup>¼</sup> <sup>V</sup>

1

<sup>w</sup><sup>p</sup> <sup>Þ</sup>1y<sup>1</sup> where

and that the mean y<sup>1</sup> value and the slope b<sup>1</sup> are uncorrelated random variables (property, which was also applied in Eq. (12) without further demonstration) as shown as follows. Taking

<sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>ð</sup> ffiffiffiffi

<sup>1</sup>y1� ¼ <sup>X</sup> <sup>ð</sup> ffiffiffiffi

0 B@

2

1

<sup>V</sup>½b1� ¼ <sup>V</sup>½y1�

V½y1�

�X�

�<sup>2</sup> <sup>¼</sup> <sup>V</sup>½y1�

2 6 4

2

Xw � �

1

<sup>ð</sup> ffiffiffiffi <sup>w</sup><sup>p</sup> <sup>Þ</sup>1<sup>ð</sup> ffiffiffiffi <sup>w</sup><sup>p</sup> <sup>Þ</sup><sup>1</sup> � y1 �

Xw � �

<sup>w</sup><sup>p</sup> <sup>Þ</sup>1ð<sup>x</sup> � <sup>x</sup>1<sup>Þ</sup> ðSXXÞ<sup>1</sup>

<sup>w</sup><sup>p</sup> <sup>Þ</sup>1ð<sup>x</sup> � <sup>x</sup>1<sup>Þ</sup>

Xw � �

1

; s<sup>1</sup> ¼

1 Xw � �

� � q

1 CA

ðSXXÞ<sup>1</sup>

CA<sup>s</sup><sup>1</sup> <sup>y</sup><sup>01</sup> <sup>¼</sup> <sup>y</sup>^1<sup>i</sup>

Xw � �

1

1

þ ðx � x1Þ

<sup>2</sup> V½y1� ðSXXÞ<sup>1</sup>

> 3 7 5

ð31Þ

ð32Þ

ð33Þ

ð35Þ

V½y� ¼ 0 ð34Þ

ffiffiffiffiffiffiffiffiffiffiffi V½y1�

<sup>þ</sup> <sup>ð</sup>xl � <sup>x</sup>2<sup>Þ</sup>

ðSXXÞ<sup>2</sup> vuut <sup>ð</sup>36<sup>Þ</sup>

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

$$\left(y\_I - a\_1 - b\_1(\mathbf{x}\_I)\_1\right)^2 = t\_1^2 s\_1^2 \left(\frac{1}{\left(\sum w\right)\_1} + \frac{\left((\mathbf{x}\_I)\_1 - \overline{\mathbf{x}}\_1\right)^2}{(\mathbf{S}\_{\mathbf{XX}})\_1}\right) \tag{42}$$

which by simple algebra it may be ordered in powers of x<sup>l</sup> as

$$\begin{aligned} \left(b\_1^2 - \frac{l\_1^2 s\_1^2}{(\mathcal{S}\_{\text{XX}})\_1}\right) (\mathbf{x}\_I)\_1^2 - 2\left(b\_1(y\_I - a\_1) - l^2 \frac{\overline{\mathbf{x}\_1} s\_1^2}{(\mathcal{S}\_{\text{XX}})\_1}\right) (\mathbf{x}\_I)\_I \\ + (y\_I - a\_1^2) - l\_1^2 s\_1^2 \left(\frac{1}{\left(\sum w\right)\_1} + \frac{\overline{\mathbf{x}^2}}{(\mathcal{S}\_{\text{XX}})\_1}\right) = 0 \end{aligned} \tag{43}$$

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

$$\left( (b\_1^2 - t\_1^2 V[b\_1])(\mathbf{x}\_l) \right)\_1^2 - 2 \left( b\_1 (y\_l - a\_1) + t\_1^2 \text{Cov}(a\_1, b\_1) \right) (\mathbf{x}\_l)\_1 + (y\_l - a\_1)^2 - t\_1^2 V[a\_1] = 0 \tag{44}$$

whose roots give the two values of ðxIÞ1. Since the point of intersection xI 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 confidence interval associated with it

$$\left(b\_2^2 - t\_2^2 V[b\_2]\right) \left(\mathbf{x}\_l\right)\_2^2 - 2\left(b\_2(y\_1 - a\_2) + t\_2^2 \text{Cov}(\mathbf{a}\_2, b\_2)\right) \left(\mathbf{x}\_l\right)\_2 + \left(y\_1 - a\_2\right)^2 - t\_2^2 V[\mathbf{a}\_2] = 0 \tag{45}$$

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<sup>α</sup> 2,n�2 q should 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 xI is the volume at which the straight lines one and two intersect and xII the volume at which the two and three lines intersect, the difference xII–xI denoted as Δx, is given by

$$
\Delta \mathbf{x} = \hat{\mathbf{x}}\_{II} - \hat{\mathbf{x}}\_{I} = \frac{-\Delta a\_{2}}{\Delta b\_{2}} + \frac{\Delta a\_{1}}{\Delta b\_{1}} \tag{46}
$$

By multiplying Δx by 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 Δx leads to [65] the following expression

$$V[\Delta \mathbf{x}] = V[\hat{\mathbf{x}}\_{I}] + V[\hat{\mathbf{x}}\_{II}] + \mathbb{C}\text{ov}(\Delta a\_{1}, \Delta a\_{2}) + \mathbb{C}\text{ov}(\Delta a\_{1}, \Delta b\_{2}) + \mathbb{C}\text{ov}(\Delta a\_{2}, \Delta b\_{1}) + \mathbb{C}\text{ov}(\Delta b\_{1}, \Delta b\_{2}) \tag{47}$$

where

$$\text{Cov}(\Delta a\_1, \Delta a\_2) = 2\left(\frac{V[\overline{y}\_2] + \overline{x}\_2^2 V[b\_2]}{\Delta b\_1 \Delta b\_2}\right) \tag{48}$$

#### Intersecting Straight Lines: Titrimetric Applications http://dx.doi.org/10.5772/intechopen.68827 71

$$\text{Cov}(\Delta a\_1, \Delta b\_2) = -2 \left( \frac{\overline{\mathbf{x}}\_2 \hat{\mathbf{x}}\_{II} V[b\_2]}{\Delta b\_1 \Delta b\_2} \right) \tag{49}$$

$$\mathcal{Cov}(\Delta a\_2, \Delta b\_1) = -2 \left( \frac{\overline{\pi}\_2 \hat{\pi}\_1 V[b\_2]}{\Delta b\_1 \Delta b\_2} \right) \tag{50}$$

$$\text{Cov}(\Delta b\_1, \Delta b\_2) = 2 \left( \frac{\hat{\mathbf{x}}\_I \hat{\mathbf{x}}\_{II} V[b\_2]}{\Delta b\_1 \Delta b\_2} \right) \tag{51}$$

The standard error estimate is given by

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

<sup>1</sup>Covða1, b1Þ

whose roots give the two values of ðxIÞ1. Since the point of intersection xI belongs to one of the

Similarly, if it is regarded as belonging to the other response function, there is another confi-

<sup>2</sup>Covða2, b2Þ

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.

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 xI is the volume at which the straight lines one and two intersect and xII the volume at which the two and three lines intersect, the difference xII–xI

By multiplying Δx by the molarity of titrant, we have the amount in millimoles of the second

V½Δx� ¼ V½x^I� þ V½x^II� þ CovðΔa1,Δa2Þ þ CovðΔa1,Δb2Þ þ CovðΔa2,Δb1Þ þ CovðΔb1,Δb2Þ ð47Þ

Δb<sup>2</sup> þ Δa<sup>1</sup> Δb<sup>1</sup>

<sup>V</sup>½y2� þ <sup>x</sup><sup>2</sup>

Δb1Δb<sup>2</sup> � �

<sup>2</sup>V½b2�

<sup>Δ</sup><sup>x</sup> <sup>¼</sup> <sup>x</sup>^II � <sup>x</sup>^<sup>I</sup> <sup>¼</sup> �Δa<sup>2</sup>

First-order propagation of variance applied to Δx leads to [65] the following expression

CovðΔa1,Δa2Þ ¼ 2

�

�

ðxIÞ<sup>1</sup> þ ðyI � a1Þ

ðxIÞ<sup>2</sup> þ ðyI � a2Þ

<sup>2</sup> � <sup>t</sup> 2

<sup>2</sup> � <sup>t</sup> 2

<sup>1</sup>V½a1� ¼ 0 ð44Þ

<sup>2</sup>V½a2� ¼ 0 ð45Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2F<sup>α</sup> 2,n�2

should be

ð46Þ

ð48Þ

q

2

2

b1ðyI � a1Þ þ t

response functions, then a certain confidence interval is associated with it.

b2ðyI � a2Þ þ t

If we desire the confidence bands for the entire line, the critical constant

7. Statistical uncertainty of endpoint differences

ðb2 <sup>1</sup> � t 2

70 Advances in Titration Techniques

ðb2 <sup>2</sup> � t 2

<sup>1</sup>V½b1�ÞðxIÞ

dence interval associated with it

<sup>2</sup>V½b2�ÞðxIÞ

2 <sup>1</sup> � 2 �

2 <sup>2</sup> � 2 �

substituted for tα/2, originating wider bands.

acid, that is, acetic acid, in the reaction mixture.

denoted as Δx, is given by

where

$$s\_{\Delta \mathbf{x}} = \sqrt{V[\hat{\mathbf{x}}\_I] + V[\hat{\mathbf{x}}\_{II}] + \sum\_4 \mathbf{Cov}} \tag{52}$$

where ∑<sup>4</sup> is the sum of Eqs. (48)–(51).

Attempts to derive confidence limits for Δx as we get in the previous Fieller's theorem section fails because the quantity analogous to z of 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 t distribution. Because in this case the exact confidence limits cannot be calculated, we use the small variance confidence interval

$$\mathbb{C}.I. = \mathbf{2}t\_{\alpha/2}\mathbf{s}\_{\Delta \mathbf{x}} \tag{53}$$

#### 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 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.

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

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 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 + xi) 2 , being the weighting factor (100 + xi) <sup>2</sup> (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 3 show the straight line segments with the corresponding selected points.

Figure 1. Conductometric titration of amixture of perchloric and acetic acids with potassium hydroxide (data shown inTable 5).



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

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

, being the weighting factor (100 + xi)

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 3 show the

Figure 1. Conductometric titration of amixture of perchloric and acetic acids with potassium hydroxide (data shown inTable 5).

<sup>2</sup> (see Table 5).

the third (branch C) to the excess of potassium hydroxide.

2

straight line segments with the corresponding selected points.

different variance V(100 + xi)

72 Advances in Titration Techniques

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).

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 endpoint, corresponding to the neutralization of perchloric acid (Figure 2), in order to follow the procedures previously detailed. The first endpoint is located at 16.367 mL and therefore 1.637 mmol of HClO4. 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 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.


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 procedures previously detailed. The first endpoint is located at 16.367 mL and therefore

(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

The second endpoint, corresponding to the complete neutralization of both perchloric and

corresponds to acetic acid in the sample, 17.831 mL. If the above methodology is used for lines,

propagation of variance, is 0.039 mL. The confidence limits are calculated using

x ( I

(II)] equal to 34.197

of 16.455 and 16.279 mL, identical to the first ones indicated.

Figure 3. Conductometric titration of acetic acid in the mixture (branches B and C).

) the volume in which lines B and C intersect, the difference

acetic acids, is located at 34.197 mL. If

x (II ) s d [ x

4. The estimated standard error at the endpoint, using the first-order

) is the volume in which lines A and B intersect, and

) (34.1971

0.0478, and 34.305 and 34.089 mL for the

x (II ) – x ( I t = 2.262

–16.3665 mL)

1.637 mmol of HClO

74 Advances in Titration Techniques

B and C (Figure 3) give

confidence limits.

x (II Intersecting Straight Lines: Titrimetric Applications http://dx.doi.org/10.5772/intechopen.68827 75 However, as it is indicated in the section on "statistical uncertainty of endpoint differences," the statistical uncertainty of Δx is 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 Δx, is not applicable since the magnitude analogous to z in 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 t distribution. The application of (first-order) propagation of the variance is nonetheless feasible, leading this procedure to an expression for the standard error of Δx of 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 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.]<sup>I</sup> = 0.156 mL. Second equivalence point (mixture of 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,s[Δx] = 0.040 mL; t s[Δx] = 0.086 mL, [IC]Δ<sup>x</sup> = 0.172 mL. The confidence interval, as expected, is higher than that found for x(II), despite decreasing the value of Student's t by increasing the number of degrees of freedom: N<sup>1</sup> + N<sup>2</sup> N<sup>3</sup> 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.


However, as it is indicated in the section on

this case, the product of random variables.

calculated in terms of the Student

x (

the second point (acetic acid):

<sup>I</sup>) = 16.358 mL,

equivalence point (mixture of perchloric and acetic acids):

Δ

x of the same type as Eq. (9) for a single endpoint.

s [ x (

x = 17.887 mL,

The confidence interval, as expected, is higher than that found for

the statistical uncertainty of

76 Advances in Titration Techniques

error of

[ x Δ

(perchloric acid):

value of Student's

branch (A), volumes of 4

–44 mL, 6 points.

35

"statistical uncertainty of endpoint differences,

t distribution. The application of (first-order) propagation

<sup>I</sup>)] = 0.078 mL, [I.C.]

[ Δ

(II) = 34.244 mL,

x

N 1 + N<sup>2</sup> N<sup>3</sup> 2

s [ x

–34 mL, 8 points. Third branch (C), volumes of

x] = 0.086 mL, [IC]

x ( I ) and x (II ). The

<sup>I</sup> = 0.156 mL. Second

Δ

(II), despite decreasing the

(II)] = 0.027 mL, t s

<sup>x</sup> = 0.172 mL.

3 = 13).

x is not a simple combination of uncertainties for

[ x (

x] = 0.040 mL; t s

x

attempt to deduce equations analogous to Eqs. (22) and (25) in order to calculate the confidence limits for Δx, is not applicable since the magnitude analogous to z in Eq. (20) implies, in

This quantity is not normally distributed, and therefore, no exact confidence limits can be

of the variance is nonetheless feasible, leading this procedure to an expression for the standard

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

(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

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

–12 mL, 5 points. Second branch (B), volumes of 22

<sup>I</sup>)] = 0.035 mL, t s

s [ Δ

t by increasing the number of degrees of freedom:

"

Intersecting Straight Lines: Titrimetric Applications http://dx.doi.org/10.5772/intechopen.68827 77


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].

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

$$\begin{aligned} \Theta &= 1.403 - 0.0637 \text{x} + 1.943 \cdot 0.01034 \sqrt{\frac{1}{8} + \frac{(\mathbf{x}\_l - 9)^2}{168}} \\ &+ 0.4908 - 0.0517 \mathbf{x}\_l + 2.353 \cdot 0.0024 \sqrt{\frac{1}{5} + \frac{(\mathbf{x}\_l - 20.2)^2}{32.8}} = 0 \end{aligned} \tag{54}$$

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

$$\begin{aligned} \Theta\_{u} &= 1.403 - 0.0637 \mathbf{x}\_{u} + 1.943 \cdot 0.01034 \sqrt{\frac{1}{8} + \frac{(\mathbf{x}\_{u} - 9)^{2}}{168}} \\ &+ 0.4908 - 0.0517 \mathbf{x}\_{u} + 2.353 \cdot 0.0024 \sqrt{\frac{1}{5} + \frac{(\mathbf{x}\_{u} - 20.2)^{2}}{32.8}} = 0 \end{aligned} \tag{55}$$

which leads to xu = 16.564 mL. Both equations θ<sup>l</sup> = 0 and θ<sup>u</sup> = 0 are resolved by successive approximations. Different values are tested for the lower and upper limits to get a change of sign in θ<sup>l</sup> and θu.

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

1/Rx

4.280

4.445

4.772

5.080

5.380

5.680

 44

N3=6

x3(mean)=

 39.022

 [

Σ

Wi]1= y3(media)=

[S(xx)]1= [S(xy)]1= [S(yy)]1=

> b3=

a3=

[R2]3=

V[y/x]3=

V(b3)=

V(a3)=

cov(a3, b3)=

V[Δa]=

V[Δb]=

cov(Δa, Δb)=

Table 7.

Evaluation

 of endpoints in the titration of a mixture of HClO4 and CH3COOH

1.294E

07

cov(Δa, Δb)= Pooled variances

1.797E

07

V[Δx]=

s[Δx]=

3.984E

 with KOH 0.100 M, optimum case (Figure 4).

02

t s[Δx]=

 0.086

 1.587E

03

t(0, 05, 13)=

 2.160

V(u)=

V(l)=

17.801

17.973

 4.528E

09

V[Δb]=

4.980E

09

 5.761E

12

V[Δa]=

6.659E

06

Cov(Δb1, Δb2)=

 2.696E

03

1.172E

07

 4.605E

06

 3.004E

09

 9.37285E

12

V[x(I)]=

V[x(II)]=

Cov(Δa1, Δa2)= Cov(Δa1, Δb2)= Cov(Δa2, Δb1)=

2.233E

03

4.674E

03

 3.871E

03

 7.262E

04

 1.201E

03

 0.99998

0.354313932

0.026637531

2.214E

06

V[x(I)]=

s[x(I)]=

 0.0269

t s[x(I)]=

 0.0610

 0.0007

t(0, 05, 9)=

 2.262

V(u)=

V(l)=

34.305

34.183

8.311E

05

3.1199E

03

 0.6851

3.103E

04

 42

 40

 38

 36

 35

 y = (1/R)(100 + x) (100 + xi)2

0.5778 0.6045 0.6585 0.7112 0.7640 0.8179

4.823E

05

V[pooled]=

 1.12632E

11

4.959E

05

5.102E

05

5.251E

05

5.407E

05

5.487E

05

Δb= Δa= x(I)=

Δx=

17.8866

34.2444

0.552936

78 Advances in Titration Techniques

0.016147

1/Rx

 y = (1/R)(100 + x) (100 + xi)2

$$10.00405(\mathbf{x}\_l)\_1^2 - 0.1332(\mathbf{x}\_l)\_1 + 1.09179 = 0\tag{56}$$

$$0.00267(\mathbf{x}\_l)\_2^2 - 0.08776(\mathbf{x}\_l)\_2 + 0.720 = \mathbf{0} \tag{57}$$

being resulted from squaring and reordering the Eqs. (44) and (45), respectively (expressed as a function of the variances of a1, b<sup>1</sup> and of the covariance between a<sup>1</sup> 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


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 in Table 8).

$$\chi\_u = \frac{(8-2)16.630 + (5-2)16.487}{8+5-4} = 16.583 \text{ mL} \tag{58}$$

$$\text{tax}\_l = \frac{(8-2)16.206 + (5-2)16.339}{8+5-4} = 16.250 \text{ mL} \tag{59}$$

#### 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

x y x y 1.265 17 0.388 1.141 18 0.441 1.028 20 0.544 0.906 22 0.644 0.777 24 0.752

12 0.641 14 0.51 16 0.372

80 Advances in Titration Techniques

N1= 8 N2= 5

[SXX]1= 168 [SXX]2= 32.8

Table 8. Hyperbolic confidence intervals for the two lines: successive approximations.

MEAN1= 9 0.83 MEAN2= 20.2 0.5538

a1= -0.06367 1.40300 =a<sup>0</sup> a1= 0.05171 -0.49081 =a<sup>0</sup> s(a1)= 0.00080 0.00806 =s(a0) s(a1)= 0.00036 0.00726 =s(a0) 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<sup>1</sup> s(y/x)1= 0.0201 t(0.05;3)= 2.353 t<sup>2</sup> s(y/x)2= 0.0048 θ DIFF-1 DIFF-2 θ DIFF-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

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

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


## 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 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.


\* 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).

8.3.4. Experimental

82 Advances in Titration Techniques

V KOH (mL) Conductance

\* Conductivity((100 + V)/100).

with potassium hydroxide (first assay).

(mS/cm)

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 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)

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

Table 9. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids

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].

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) are the graphs corresponding to the estimation of the endpoints. The points represented in the graph and then

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).

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 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 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

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

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

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

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

bands, which lead to higher confidence intervals and limits (not applied in this case).

's theorem are much stronger than those based on the use of hyperbolic confidence

0.0619 (first endpoint), 29.743

0.142 (difference). In the second case, the confidence limits cannot

's theorem leads to the same results as those obtained by the law of

0.151 (second

confidence limits: 15.334

Figure 8. Conductometric titration of acetic acid in the mixture (branches B and C).

and 14.189 mL.

following volumes

check calculations.

Fieller

The application of Fieller

endpoint), and 14.410

84 Advances in Titration Techniques

Table 10. Evaluation of endpoints in the titration of a mixture of HCl and CH3COOH with KOH 0.0992 M (data Table 9).

#### Intersecting Straight Lines: Titrimetric Applications http://dx.doi.org/10.5772/intechopen.68827 85


Table 11. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (second assay).

used in the detailed calculations are colored in the tables. The results obtained (and intermediate 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.



Table 12. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (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 Q of Dixon 0.717 and of G of Grubbs 1.110 are lower than tabulated values 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 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

used in the detailed calculations are colored in the tables. The results obtained (and intermediate 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

Table 11. Conductance and KOH volume data corresponding to the titration of a mixture of hydrochloric and acetic

14.383 0.140 mL.

21.0 2.22 2.6862

acids with potassium hydroxide (second assay).

\* Conductivity((100 + V)/100).

V KOH (mL)

86 Advances in Titration Techniques

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

V KOH (mL)

Conductance (mS/cm)

Conductance\* (mS/cm)

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).

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


discarded, the results obtained would be very close to 14.973 0.084 M and 14.421 0.053 M,

Figure 10. Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are

Figure 9. Conductometric titration of a mixture of hydrochloric and acetic acids with potassium hydroxide (data are

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

although the accuracy would improve considerably in the first case.

shown in Table 10, second assay).

88 Advances in Titration Techniques

shown in Table 9, third assay).


Table 13. Evaluation of endpoints in the titration of a mixture of HCl and CH3COOH with KOH 0.0992 M (data Table 11).

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 firstorder 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). 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 propagation law) and Fieller's theorem methods preferable to methods based on intersecting confidence bands.

## Author details

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

Table 13. Evaluation of endpoints in the titration of a mixture of HCl and CH3COOH with KOH 0.0992 M (data

Pooled variances Vol (l)= 14.345

N1= 13 [ΣWi]1= 13 N2= 15 [ΣWi]2= 15 V[Δb]= 9.902E07 V[Δb]= 1.014E06 s[Δx]= 5.583E02 t s[Δx]= 0.113 cov(Δa, Δb)= 2.893E05 cov(Δa, Δb)= 2.952E05 Vol (u)= 14.571

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 firstorder 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

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

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).

intersection abscissa in the analytical literature do not abound.

9. Final comments

90 Advances in Titration Techniques

Table 11).

to each other.

this context.

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


[26] Seber GAF. Linear regression analysis. 7.6 Two-phase linear regression. New York: Wiley, 1977. pp. 205–209

[10] Ortiz-Fernandez MC, Herrero-Gutierrez A. Regression by least median squares, a methodological contribution to titration. Chemometrics and Intelligent Laboratory Systems.

[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] Jones RH, Molitoris BA. A statistical method for determining the breakpoint of two lines.

[13] Sprent P. Some hypothesis concerning two phase regression analysis. Biometrics. 1961;17

[14] Shaban SA. Change point problem and two-phase regression: An annotated bibliogra-

[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.

[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

[19] Yanagimoto T, Yamamoto E. Estimation of safe doses: Critical review of the Hockey Stick

[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-

[21] Vieth E. Fitting piecewise linear regression functions to biological responses. Journal of

[22] Piegorsch WW. Confidence intervals on the joint point in segmented regression. Biomet-

[23] Bacon DW, Watts DG, Estimating the transition between two intersecting straight lines.

[24] Christensen R. Plane answers to complex questions. The Theory of Linear Models. 4th

[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

regression method. Environmental Health Perspectives. 1979;32:193–199

1995;27:241–243

92 Advances in Titration Techniques

(4):634–645

Analytical Biochemistry. 1984;41(1):287–290

1992;25(2-3):95–112; Erratum 1994;28(1): 83

phy. International Statistical Review. 1980;48(1):83–93

Intelligent Laboratory Systems. 2008;90(2):116–122

ican Chemical Society. 1999;121(40):9265–9275

Applied Physiology. 1989;67(1):390–396

Biometrika. 1972;58(3):525–534

ed., New York: Springer, 2011

rical Unit Technical Reports BU-785-M. 1982


[59] Asuero AG, Sayago A, González AG. The correlation coefficient: an overview. Critical Reviews in Analytical Chemistry. 2006;36(1):41–59

[43] Wilkinson GN. On resolving the controversy in statistical inference. Journal of the Royal

[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.

[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

[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

[48] Bonate PL. Approximate confidence intervals in calibration using the bootstrap. Analyt-

[49] Mandel J, Linning FJ. Study of accuracy in chemical analysis using linear calibration

[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-

[52] Schwartz LM. Calibration curves with nonuniform variance. Analytical Chemistry.

[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 evaluation of acidity constants of two-step overlapping equilibria. Analytical Letters 1993;26

[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 addition method for absorption spectrometry and anodic stripping voltammetry. Analytical

[57] Liteanu C, Rica I, Liteanu V. On the confidence interval of the equivalence point in linear

[58] Liteanu C, Rica I. Statistical Theory and Methodology of Trace Analysis. Chichester: Ellis

techniques. Journal of Analytical Atomic Spectrometry. 1989;4(5):415–421

Arrhenius plots. Analytical Biochemistry. 1988;170(2):409–420

Statistical Society: Series B. 1977;39(2):119–171

ical Chemistry. 1993;65(10):1367–1372

curves. Analytical Chemistry. 1957;29(5):743–749

ments. Analytical Chemistry. 1978;50(7):980–984

1979;2(1):13–19

94 Advances in Titration Techniques

1979;51(6):723–727

(1):163–181

Chemistry. 1978;50(9):1374–1380

Horwood Ltd, 1980. pp. 166–172

titrations. Talanta. 1978;25(10):593–596


**Isothermal Titration Calorimetry**

[75] Gzybkovski W. Conductometric and potentiometric titrations. Politechnika Gdańska, Gdansk, 2002. http://fizyczna.chem.pg.edu.pl/documents/175260/14212622/chf\_epm\_lab\_1.

[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?.

[77] Donkersloot MCA. Teaching conductometry. Another perspective. Journal of Chemical

[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] Smith KC, Garza A. Using conductivity measurements to determine the identities and concentrations of unknown acids: an inquiry laboratory experiment. Journal of Chemical

[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

[81] Farris S, Mora L, Capretti G, Piergiovanni L. Charge density quantification of polyelectrolyte polysaccharides by conductimetric titration: an analytical chemistry experiment.

[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.

[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-

[85] Rosenthal LC, Nathan LC. A conductimetric-potentiometric titration for an advanced

[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

[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

in acid-base titrations. Journal of Chemistry. 2015, Article ID 540368, pp. 13

laboratory. Journal of Chemical Education. 1981;58(8):656–658

Technology. Amsterdam: Elsevier Science, 1997. p. 305

of periodic trends. Journal of Chemical Education. 2012;89(11):1442–1446

Journal of Chemical Education. 2012;89(1):121–124

pdf [accessed on August 5, 2017]

96 Advances in Titration Techniques

Education. 1991;68(2):136–137

Education. 2015;92(8):1373–1377

2011;88(5):640–642

cal Education. 1986;63(1):73–74

Educación química. 2011;22(2):166–169
