Instrumental end point detection technique more widely applied.
Plotting two straight line graphs from the experimental data and determining the point of their intersection solve a number of problems in analytical chemistry (i.e., potentiometric and conductometric titrations, the composition of metal-chelate complexes and binding interactions as ligand-protein). The relation between conductometric titration and the volume of titrant added lead to segmented linear titration curves, the endpoint being defined by the intersection of the two straight line segments. The estimation of the statistical uncertainty of the end point of intersecting straight lines is a topic scarcely treated in detail in a textbook or specialized analytical monographs. For this reason, a detailed treatment with that purpose in mind is addressed in this chapter. The theoretical basis of a variety of methods such as first-order propagation of variance (random error propagation law), Fieller’s theorem and two approaches based on intersecting confidence bands are explained in detail. Several experimental systems described in the literature are the subject of study, with the aim of gaining knowledge and experience in the application of the possible methods of uncertainty estimation. Finally, the developed theory has been applied to the conductivity measurements in triplicate in the titration of a mixture of hydrochloric acid and acetic acid with potassium hydroxide.
- straight lines
Titrimetry is one of the older analytical methods , and it is still found [2–4] in a developing way. It plays an important role in various fields as in 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). Independent values of chemical quantities expressed in SI units are obtained through gravimetry and titrimetry (classical analysis).
In titrimetry, the quantity of tested components of a sample is assessed through the use of a solution of known concentration added to the sample which reacts in a definite proportion. To identify the stoichiometric point, where equal amounts of titrant react with equal amounts of analyte, indicators are used in many cases to point out the end of the chemical reaction by a color change.
Information on reaction parameters is usually obtained from an analysis of the shape of the titration curve, whose shapes depend on some factors such as the reaction of titration, the monitored specie (indicator, titrant, analyte or formed product), as well as the chosen [10, 11] instrumental technique (Table 1); that is, spectrophotometry, conductimetry or potentiometry, for instance. The importance of titrimetric analysis has increased with the advance of the instrumental method of end point detection.
|Conductimetric titrations||Electrical conductivity|
|Potentiometric titrations||Potential of an indicating electrode|
|Amperometric titrations||Diffusion current at a polarizable indicating (dropping mercury or rotating platinum) electrode|
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.
|Breakpoint||Jones and Molitoris ; Shanubhogue et al. |
|Changepoint problem||Csörgo and Horváth ; Krishanaiah and Miao |
|Common intersection point||Rukhin |
|Hockey stick regression||Yanagimoto and Yamamoto |
|Intersystem crossing||Kita et al. |
|Piecewise linear regression||Vieth |
|Segmented regression||Piegorsch |
|Transition||Bacon and Watts |
|Two phase linear regression||Christensen ; Lee et al. ,; Seber [26, 27]; Shaban ; Sprent |
At the point of intersection (xI), the two lines have the same ordinate. The estimation of statistical uncertainty of end points obtained from linear segmented titration is the subject of this chapter. The topic is scarcely treated in [36, 37] analytical monographs. The method of least squares is the most common choice and the most appropriate when the relative statistical uncertainties of the 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 methods bases on intersecting hyperbolic confidence bands as weighted averages [57, 58] of the abscissas of the confidence hyperbolas at the ordinate of intersection will be dealt in detail in this chapter. In addition, several experimental systems will be the subject of study, with the aims of gaining knowledge and experience in the application of these methods to uncertainty estimation.
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 N1 observations on the first line
and N2 the observations on the second
where a1, b1, a2, b2 are the usual least squares estimates of the kth line (k = 1, 2), respectively. As it is stated in the introduction section when the relative statistical uncertainties of the x data are negligible compared to the y data, the use of the least squares method is the alternative most common. The ordinate variance can be considered on a priori grounds to vary systematically as a function of the position along the curve, so that weighted least squares analysis is appropriate. Formulae for calculating the intercept a, the slope b and their standard errors by weighted linear regression  are given in Table 3, where the analogy with simple linear regression (i.e., wi = 1), is evident.
Note that in summation (1) and (2) by dividing by N1 and N2, respectively, we get
At the point of intersection, the lines (1) and (2) have the same ordinate and the abscissa of intersection (denotes by ) is given by
Random error in the points produce uncertainty in the slopes and intercepts of the lines and therefore in the point of intersection. The probability that a confidence interval contains the true value is equal to the confidence level (e.g., 95%).
3. First-order propagation of variance for
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 . Some authors [61, 62] evaluate the uncertainty in on this way. First-order propagation of variance retains only first derivatives in the Taylor expansions and this procedure leads to
Then, the standard error estimate of is as follows
The end point depends on four least squares parameters a1, a2, b1, b2 that are random variables. Segment one parameters depend only on measurements made along segment one and these are statistically independent of the measurements along 2. However, Δa, and Δb are correlated random variables because each involves b1 and b2. Note that, a1 and a2 are related to b1 and b2 by means of Eqs. (1) and (2).
The variances of Δa and Δb are given by
and for the covariance between Δa and Δb, we get 
It should be noted that in the calculations, the variance regression estimates from both line segments are pooled into a single sp2, by using the following formula which weights each contribution according to the corresponding [64–67] degrees of freedom
The standard deviation in Eq. (13) is calculated on the assumption that the s(y/x) values for the two lines are sufficiently similar to be pooled.
From expression in Table 3 for the variance of the intercept (sa2 = V[a]), we may derive
in which s2 is s(y/x) in Table 3; ∑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
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, , 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 . This method depends on the distribution function of the random variable . 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 . However, , even is regarded as the ratio of two normally distributed variables, is not normally distributed and, indeed, becomes more and more skewed  as the variance levels increase. For sufficiently small variance though, , is approximately normally distributed, and under these circumstances, confidence intervals may be calculated from the standard deviation of 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
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+N2 − 4) inherent in the standard deviation of , could be misleading. Note that because involves the ratio of random variables first-order propagation of variance is not exact . Evidently, is a random variable not normally distributed unless is small enough. When the variances of the responses are not necessarily small, one a solution to this problem is to apply the called Fieller’s theorem [38–43]. Another point of view focuses the problem in the calculation of the limits of the confidence intervals by using the confidence bands for the two segmented branches.
5. The Fieller’s theorem
Any linear combination z of normally distributed random variables is itself normally distributed.
If the standardized variable is distributed as N(0, 1), then z is distributed as t.
Consider now any pair of individual line segments written as a difference z as follows
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 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, b1 and b2 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
and therefore, is distributed as N(0, 1) and according to (ii)
which on rearrangement leads to
which may be factored as
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 N1 + N2 − 4 degrees of freedom. Note that in Eqs. (21), (23), (24) and (25), the corresponding values of V[Δa], V[Δb] and Cov[Δa, Δb] are given by Eqs. (16), (17) and (18), respectively, as in the first-order propagation of variance for .
The first and last group of symbols enclosed in braces in Eq. (25) has the form of hypothesis tests, that is, two-tailed tests, for significant difference of intercepts and significant difference of slopes, respectively. When the hypothesis test for different slope fails, the coefficient of xI2 becomes negative finding two complex roots , so Fieller confidence interval embraces the entire x-axis (the lower and upper limits should strictly be set to −∞ and ∞, respectively) at the chosen level of confidence.
This method has been extensively in some other contexts in analytical and chemical literature (Table 4).
|Arrhenius plot||Cook and Charnock ; Han ; Puterman et al. |
|Calibration curves||Baxter ; Bonate ; Mandel y Linning ; Schwartz [50–52]|
|Estimation of safe doses||Yanagimoto and Yanamoto |
|Estimation of uncertainty in binding constants||Almansa López et al. |
|Models for biologic half-life data||Lee et al. |
|Position and confidence limits of an extremum||Asuero and Recamales ; Heilbronner |
|Standard addition method||Franke et al. |
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 result by the projection on the abscissa of the surface between the four hyperbolic arcs .
Because a confidence band, bounded by two hyperbolic arcs, is associated with each regression line, it is obvious that the point of intersection, xI, is only a mean value, with which a certain confidence interval is associated. If the signal values both before and after the point of intersection are normally distributed around the line with a constant standard deviation, the point of intersection and its statistical confidence interval will be estimated by the projection of the intersection onto the abscissa. The confidence interval (xl, xu) for the true value of the equivalence point is given by the projection on the abscissa of the common surface delimited by the four hyperbolic arcs.
For the first line, we get:
and for the second line:
t1 and t2 are the corresponding t Student values for α/2 = 0, 05 and N1 − 2 and N2 − 2 degrees of freedom, respectively. Hence, the lower value xl of the confidence interval is obtained by solving the following equation:
The higher value xu is obtained from the equation:
and then the variance of the fitted y1 value will be given by
Note that the variance of the (weighted) mean of the one values
and that the mean y1 value and the slope b1 are uncorrelated random variables (property which was also applied in Eq. (12) without further demonstration) as shown in that follows. Taking into account that
From Eq. (31), we get for the standard error of the fitted value
Thus, the lower value, xl, for the confidence interval is obtained by solving the equation
by, for example, successive approximations with an Excel spreadsheet. The higher value xu is obtained in the same way from the equation that follows also by successive approximations
The point of view of Liteanu et al. [57, 58]. is very interesting: the authors consider the point of intersection 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 (xIl, xIu) where
The two values of the limits of the confidence interval will be given by the two solutions of the equations
As the estimation method used assumes the worst case in combining random error of the two lines, the derived confidence limits are on the pessimistic (i.e., realistic) side.
On rearrangement Eq. (40) and squaring, we have
which by simple algebra it may be ordered in powers of xl as
and taking into account the values of V[b1], Cov[a1, b1] and V[a1] (see Table 3), we get finally
whose roots give the two values of . 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 are another confidence interval associated with it
Because the intersection point belongs concomitantly to the two response functions, the two segments which together compose the confidence interval will be obtained by averaging the segments of the two separate confidence intervals, Eqs. (40) and (41). The two values of the limits of the confidence interval will be the two solutions of the second degree Eqs. (44) and (45).
The bands mentioned in this section are  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 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
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  the following expression
The standard error estimate is given by
Attempts to derive confidence limits for Δx as we get in the previous Fieller’s theorem section fails because of 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
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)  and Gzybkoski (2002) . 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 ; the identification and quantification of an unknown acid , electrolyte polymers [80, 81], acid and basic constants determinations , its use in general chemistry , microcomputer interface  and conductometric-potentiometric titrations . An accurate method of determining conductivity in acid-base reactions , the acid-base properties of weak electrolytes , and those of polybasic organic acids  have also been object of recent study.
The relation between conductometric and the volume of titrant added leads to segmented linear titration curves, the endpoint being defined by the intersection of the two straight lines segments. What follows is the application of the possible methods of uncertainty estimation of the end point of data described in the literature as well as to experimental measurements carried out in the laboratory.
8.1. Conductometric titration of 100 mL of a mixture of acids with potassium hydroxide 0.100 F
Table 5 shows the data [conductance (1/R), volume (x)] published by Carter et al. ; Schwartz and Gelb ) and corresponding to the conductometric titration of a mixture of acids, perchloric acid and acetic acid with potassium hydroxide 0.100 F as titrant agent. The points recorded belong to the three branches of the titration curve; the first (branch A) corresponds to the neutralization of perchloric acid, the second (branch B) to the neutralization of acetic acid, and the third (branch C) to the excess of potassium hydroxide.
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  select 13 points, six (volume from four to 14 mL) for branch A and seven (volume from 20 to 32 mL) for branch B. The points near to the end point of perchloric acid are deviating from linearity and are discarded in the first instance. It is also considered that the data have a different variance V⋅(100 + xi)2, being the weighting factor (100 + xi)−2(see Table 5).
In the case of acetic titration, six points (volume 35–44 mL) are selected for branch C, at first. The points of branch B near to the acetic acid end point are discarded. Figures 2 and 3 show the straight lines segments with the corresponding selected points.
Table 6 includes the intermediate results obtained in the calculation of the first end point, corresponding with the neutralization of perchloric acid (Figure 2), in order to follow the procedures previously detailed. The first end point is located at 16.367 mL and therefore 1.637 mmol of HClO4. The estimated standard error at the end point, using the first-order propagation of variance, is 0.039 mL. The confidence limits are calculated using t = 2.262 (9 degrees of freedom) and correspond to 16.455 and 16.279 mL, respectively, for the upper and lower limit, being the confidence interval equal to 0.176 mL. The application of Fieller's theorem leads to the values of 16.455 and 16.278 mL, respectively. Carter et al.  give values of 16.455 and 16.279 mL, identical to the first ones indicated.
|1/R||x||y = (1/R)(100 + x)||(100 + xi)−2||1/R||x||y = (1/R)(100 + x)||(100 + xi)−2|
|Cov(a1, b1)=||−2.168E−07||Cov(a2, b2)=||−1.445E−07|
|V[Δa]=||6.002E−06||V[Δa]=||9.640E−06||V[x(I)]=||0.0015||FIELLER||ax2 + bx + c = 0|
|cov(Δa, Δb)=||−3.613E−07||cov(Δa, Δb)=||−4.453E−07||t(0, 05, 9)=||2.262||b=||−4.953E−02||V(l)=||16.278|
|Pooled variances||t s[x(I)]=||0.0880||c=||4.053E−01|
The second end point, corresponding to the complete neutralization of both perchloric and acetic acids, is located at 34.197 mL. If x(I) is the volume in which lines A and B intersect, and x(II) the volume in which lines B and C intersect, the difference x(II)–x(I)(34.1971–16.3665mL) corresponds to acetic acid in the sample, 17.831 mL. If the above methodology is used for lines, B and C (Figure 3) give x(II) ± sd [x(II)] equal to 34.197 ± 0.0478, and 34.305 and 34.089 mL for the confidence limits.
However, as it is indicated in the section on “statistical uncertainty of endpoint differences,” the statistical uncertainty of 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.
Therefore, 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 the Eq. (9) for a single end point.
The latter methodology is applied to the optimal case detailed by Schwartz and Gelb . The corresponding data are shown in Figure 4, and the calculations necessary to locate the equivalence points, first and second, are shown in Table 7. The results obtained are: first equivalence point (perchloric acid): x(I) = 16.358 mL, s[x(I)] = 0.035 mL, t s[x(I)] = 0.078 mL, [I.C.]I = 0.156 mL. Second equivalence point (mixture perchloric and acetic acids): x(II) = 34.244 mL, s[x(II)] = 0.027 mL, t s[x(II)] = 0.061 mL, [IC]II = 0.122 mL. This latter is not correct because it does not take into account the covariances described in Section 7. If covariances are incorporated into the calculations, we get for the second point (acetic acid): x = 17.887 mL, s[Δx] = 0.040 mL; t s[Δx] = 0.086 mL, [IC]Δx = 0.172 mL. The confidence interval, as expected, is higher than that found for x(II), despite decreasing the value of Student's t by increasing the number of degrees of freedom: N1 + N2 − N3 − 2 × 3 = 13).
|1/R||x||y = (1/R)(100 + x)||(100 + xi)−2||1/R||x||y = (1/R)(100 + x)||(100 + xi)−2|
|cov(a1, b1)=||-2.373E−07||cov(a2, b2)=||-4.321E−08|
|V[Δa]=||3.356E−06||V[Δa]=||5.160E−06||V[x(I)]=||0.0012||FIELLER||ax2 + bx + c = 0|
|cov(Δa, Δb)=||−2.805E−07||cov(Δa, Δb)=||−2.463E−07||t(0, 05, 9)=||2.262||b=||−5.029E−02||V(l)=||16.280|
|Pooled variances||t s[x(I)]=||0.0784||c=||4.113E−01|
|V[x(I)]=||0.0007||t(0, 05, 9)=||2.262||V(u)=||34.305|
|cov(a3, b3)=||−1.172E−07||Cov(Δa1, Δb2)=||−4.674E−03|
|cov(Δa, Δb)=||−1.294E−07||cov(Δa, Δb)=||−1.797E−07||V[Δx]=||1.587E−03||t(0, 05, 13)=||2.160||V(u)=||17.973|
|Pooled variances||s[Δx]=||3.984E−02||t s[Δx]=||0.086||V(l)=||17.801|
Some points near to the end point appear to deviate slightly from linearity, but it is not always clear whether or not to omit these problem points in the analysis, which can be done by trial and error. The optimal point set (Figure 4) is one that minimizes, for example, the confidence interval .
The weighting factors are very similar so that the values obtained by weighted linear regression, and the simple one, become equivalent.
8.2. Conductometric titration of hydrochloric acid (0.1 M) with sodium hydroxide (0.1 M)
The data corresponding to the two branches of the conductometric titration of 0.1 M HCl with 0.1 M NaOH is shown in the upper part of Table 8 and plot in Figure 5. The cut-off point of both lines is (6.414, 0.358) [57, 58, 89].
|t(0.05;6)=||1.943||t1 s(y/x)1=||0.0201||t(0.05;3)=||2.353||t2 s(y/x)2=||0.0048|
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
And leading to xl = 16.264 mL. The highest value is obtained by solving (Eq. (37))
which leads to xu = 16.564 mL. Both equations θl = 0 and θu = 0 are resolved by successive approximations. Different values are tested for the lower and upper limits until to get a change of sign in θl and θu.
In the weighted mean method (Table 8), the following equations are solved
Resulting from squaring and reordering the Eqs. (44) and (45), respectively, (expressed as a function of the variances of a1, b1 and of the covariance between a1 and b1). Once calculated the solutions of the Eqs. (56): 16.630 and 16.487 mL, and (57): 16.206 and 16.339 mL, we have
8.3. Experimental measurements: conductometric titration of 100 mL of a mixture of hydrochloric acid and acetic acids with potassium hydroxide 0.100 F
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%).
Analytical balance (Metler AE200) (4 decimals), conductimeter Crison (EC-Metro GLP 31), calibrated by standards of 147 μS/cm, 1413 μS/cm, 12.88 mS/cm. Burette of 50 mL (Brand) (±0.1 at 20°C).
- Hydrochloric and acetic acids mixture 0.015 M.
- Potassium hydroxide 0.1 M.
About 100 mL of hydrochloric and acetic acids mixture ≈ 0.015 M is transferred to a 250 mL volumetric flask with 100 mL of distilled water. Then, the mixture is titrated conductometrically with KOH 0.0992 ± 0.0001 M (n = 3), (previously standardized with potassium hydrogen phthalate). Table 9 shows the data [conductance, volume] as well as the product of the conductance by (100 + x)/100 to correct the dilution effect of the titrant. The data are plotted in Figure 6
|V KOH (mL)||Conductance (mS/cm)||Conductance* (mS/cm)||V KOH (mL)||Conductance (mS/cm)||Conductance* (mS/cm)|
The points recorded belong to the three branches of the titration curve; the first (branch A) corresponds to the neutralization of hydrochloric acid, the second (branch B) to the neutralization of acetic acid, and the third (branch C) to the excess of potassium hydroxide.
Figures 7 (hydrochloric acid) and 8 (hydrochloric acid + acetic acid), respectively, are the graphs corresponding to the estimation of the end points. The points represented in the graph and then used in the calculations are colored yellow (branch A), green (branch B) and blue (branch C) in Table 9, thus avoiding proximity to the breakpoints. The values obtained for the intersections of the abscissa are 15.334 mL for hydrochloric acid and 29.743 mL for the sum of hydrochloric and acetic acids. So, acetic acid corresponds to the difference, 14.410 mL. From the data of Figure 6, without discarding of points, somewhat different values are obtained: 15.383, 29.582 and 14.189 mL.
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  leads to the following volumes ± confidence limits: 15.334 ± 0.0619 (first endpoint), 29.743 ± 0.151 (second end point), and 14.410 ± 0.142 (difference). In the second case, the confidence limits cannot refer to the difference (acetic acid), since the covariates involved are not taken into account (as has been previously explained in Section 7). Three decimals were considered to compare and check calculations.
|cov(a1, b1)=||−6.572E−06||cov(a2, b2)=||−1.201E−05|
|V[Δa]=||3.279E−04||V[Δa]=||3.564E−04||V[x(I)]=||0.0007||FIELLER||ax2 + bx + c = 0|
|cov(Δa, Δb)=||−1.859E−05||cov(Δa, Δb)=||−1.932E−05||t(0, 05, 27)=||2.052||b=||−3.221E + 00||V(l)=||15.278|
|Pooled variances||t s[x(I)]=||0.0562||c=||2.469E + 01|
|V KOH (mL)||Conductance (mS/cm)||Conductance*(mS/cm)||V KOH (mL)||Conductance (mS/cm)||Conductance* (mS/cm)|
The application of Fieller’s theorem leads to the same results as those obtained by the law of propagation of errors, not being applicable to the estimation of confidence limits of the difference of volumes. The fundamentals of the first-order variance propagation method and Fieller’s theorem are much stronger than those based on the use of hyperbolic confidence bands, which lead to higher confidence intervals and limits (not applied in this case).
The conductometric titration was carried out in triplicate, on different days, obtaining the results included in Tables 11 and 12, and also represented in Figures 9 and 10. Again, the data used in the detailed calculations are colored in the tables. The results obtained (and intermediate calculations) for the second experience are shown in Table 13: 14.913 ± 0.041 (propagation of errors and Fieller), 29.372 ± 0.120 (approximate method of propagation of errors) and 14.458 ± 0.113 (propagation of errors). In the third assessment: 15.032 ± 0.043, 29.414 ± 0.146, and 14.383 ± 0.140 mL.
|V KOH (mL)||Conductance (mS/cm)||Conductance* (mS/cm)||V KOH (mL)||Conductance (mS/cm)||Conductance* (mS/cm)|
|cov(a1, b1)=||−1.600E−06||cov(a2, b2)=||−7.774E−06|
|V[Δa]=||1.927E−04||V[Δa]=||1.590E−04||V[x(I)]=||0.0004||FIELLER||ax2 + bx + c = 0|
|cov(Δa, Δb)=||−9.374E−06||cov(Δa, Δb)=||−8.459E−06||t(0, 05, 24)=||2.064||b=||−3.420E+00|
|Pooled variances||t s[x(I)]=||0.0414||c=||2.550E+01|
|V[y/x]3=||1.792E−04||V[x(II)]=||0.0034||t(0, 05, 26)=||2.056|
|V[Δa]=||1.583E−07||V[Δa]=||9.320E−04||V[Δx]=||3.116E−03||t(0, 05, 37)=||2.026|
|cov(Δa, Δb)=||−2.893E−05||cov(Δa, Δb)=||−2.952E−05||Vol(u)=||14.571|
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 to tabulated for P = 0.05, that is, Qtab = 1.155 and Gtab = 1.15 (although the Gexp and Gtab values are practically the same). The mean ± confidence limits of the values are 15.093 ± 0.217 mL for hydrochloric acid (first end point) and 14.417 ± 0.038 mL for acetic acid (difference), which leads to molarity values of the solutions of hydrochloric and acetic acid of 0.01497 ± 0.00022 M and 0.01430 ± 0.00004 M. If the most distant value were discarded, the results obtained would be very close to 14.973 ± 0.084 M and 14.421 ± 0.053 M, although the accuracy would improve considerably in the first case.
It is worth noting the fact that when the covariance between the intercept and the slope of the straight lines obtained by the least squares method is not taken into account, the propagation of the error leads to values of much larger confidence limits, 0.429 in the example of Massart (1997) versus 0.104, or 0.648 by Liteanu and Rica  versus only 0.113, in this chapter, for the same data. As in many monographs, the covariance in the propagation of errors is not taken into account, and this is perhaps the reason why the estimates of the uncertainties of the intersection abscissa in the analytical literature do not abound.
9. Final comments
The advance of instrumental methods of end point detection increases the importance and the worth of titrimetric analysis. Physicochemical methods are intensively developed nowadays. However, in spite of this, titration continues to maintain its importance for chemical analysis. Plotting two straight lines graphs from the conductivity versus volume added experimental data and determining the corresponding intersection point of the two branches allow locating the end point in a conductometric titration. The estimation of uncertainty of end point from linear segmented titration curves may be easily carried out by first-order propagation of variance, that is, by applying random error propagation law. The weighted linear regression procedure as applied to the two branches of the conductometric titration curves leads to similar results to those obtained by the unweighted (single) linear regression procedure. The weighting factors are very similar to each other.
The covariance of measurements can be as important as the variance and both contribute significantly to the total analytical error. In particular, the strong correlation existing between the estimated slope and intercept of a straight line obtained by the least squares method must not be ignored. The inclusion of the covariance term on this respect is of vital importance, being usually of a subtractive character lowering, in this case, the confidence limits of the abscissa of the intersection point. Perhaps this omission, which leads to too greater uncertainties, may be the cause of the small number of times uncertainty is reported in this context.
The algebra associated with the Fieller’s theorem is simple, and no problem is observed with its derivation in this particular case of intersecting straight lines. However, the statistical uncertainty of endpoint differences is a complex problem. Attempt to derive the confidence limits by applying Fieller’s theorem fail in this case, being necessary to resort to first-order propagation of variance (random error propagation law). However, the algebra associated in this case though simple is cumbersome, as some terms in covariance need to be derived. As a matter of fact, greater accuracy and firmer statistical justification make first-order propagation of variance (random error propagation law) and Fieller’s theorem methods preferable to methods based on intersecting confidence bands.