**Acknowledgements**

The research was supported by University of Muthanna, Iraq. The authors acknowledge the Ministry of Sciences and Technology in Iraq for providing them with the isolates of arbuscular mycorrhizal (AM) fungi to complete their research.


**About the Editors**

**J. A. Tenreiro Machado** was born in 1957, graduated with 'Licenciatura' (1980), PhD (1989) and 'Habilitation' (1995) degrees in Electrical and Computer Engineering at the University of Porto. From 1980 to 1998, he worked as a professor at the Department of Electrical and Computer Engineering, University of Porto. Since 1998, he has been a coordinator professor at the Institute of Engineering of the Polytechnic Institute of Porto, Department of Electrical Engineering. His main research interests are: nonlinear dynamics, modeling, fractional calculus, evolutionary computing, control, and robotics. He is the author of more than 680 papers and his Scopus h-index is 54.

**Antonio ´ M. Lopes** obtained his PhD and 'Habilitation' in Mechanical Engineering in 2000 and 2018, respectively, both at the Faculty of Engineering of the University of Porto (FEUP), Portugal. Currently, he works at the Department of Mechanical Engineering (DEMec) and the Associated Laboratory for Energy, Transports and Aeronautics (LAETA/INEGI). Antonio ´ M. Lopes has been involved in teaching activities for the master's degree in Mechanical Engineering, the master's degree in Industrial Engineering and Management, and the doctoral program in Mechanical Engineering, all at FEUP. He participated in several research projects in the area of Mechanical Engineering and Automation. His research interests include: complex systems, fractional calculus, nonlinear dynamics, automation, robotics, control, systems modelling, and simulation. Antonio ´ M. Lopes has published about 130 papers in international journals with high impact factors, and nearly 100 book chapters, plus conference papers. His Scopus h-index is 23. He currently participates on the Editorial Board of eight international journals, and has been a guest editor of several journal Special Issues.


*2.2. Energy Framework Analysis*

Energy frameworks were constructed in *CrystalExplorer* using the CE-B3LYP model (basis set: 6-31G(d,p)), taking into account only intermolecular interactions within the radius of 3.8 Å from the centroid on the central molecule [30]. All frameworks were scaled equally so as to facilitate comparisons.

### *2.3. Polymorph Assessment through Hydrogen Bond Propensity Models*

Initially, the definition of a hydrogen bond was set as established earlier; however, this resulted in an excessive number of contacts within each form, some of which were chemically incorrect. Therefore, the bond angle was modified to be larger than 133◦ (the lower limit of the hydrogen bonds detected in both forms). All hydrogen bond acceptors and donors were selected, with the latter list including carbon. Systems with errors or disorder were excluded from the study, together with organometallic compounds and entries with an R-factor > 0.075.

A set of data containing entries whose chemistry was relevant to ganciclovir was generated so as to build the statistical model for prediction. Potential bias or error was decreased by ensuring that each functional group (5 substructures; Please refer to Supplementary Materials) was represented by an adequate number of hits. After analysis, donor or acceptor candidates with a low number of relevant hits were omitted to avoid regression failures. A logistic regression model was fitted on the data, producing its corresponding area under the receiver operating characteristic (ROC) curve value as a measure of the extent of correct predictions. This procedure was performed separately for both polymorphs.

### **3. Results and Discussion**


**3. Conclusion**

It is usually assumed that a certain level of genetic diversity is necessary for the long-term prosperity of a species. Indeed, different genotypes confer different levels of resistance to various environmental stresses, and consequently, the greater the diversity of the genotypes in a population, the more effective its ability to withstand unfavorable conditions will be. Therefore, the existence of a low level of genetic diversity is often considered to represent a crucial stage for the survival of a species or even a sign of its extinction. However, despite several attempts to determine the average levels of genetic variation in different categories of plants (e.g., Gitzendanner & Soltis, 2000; Nybom, 2004), it is still unknown what level of polymorphism should be considered critical for the existence of a certain species. Thus, some rare endemic species exhibit higher levels of genetic diversity than the average values found for this category of plants, such as *Oxytropis chankaensis* (Table). At the same time, very low levels, or even an absence of genetic variability has been found in other narrowly endemic plants (e.g., *Bensoniella oregona*, Soltis et al., 1992). It is impossible to predict the fate of a species on the basis of its genetic diversity alone. Here, we have attempted to address some species existing under extreme conditions in the context of a variety of environmental, biological, evolutionary and other influences.

All of the species described above are represented by small, fragmented, marginal populations. Each of these species is characterized by a particular level of genetic diversity, ranging from very low (*Panax ginseng*) to high (*Oxytropis chankaensis*). Despite these differences, all of these plants exhibit weak competitiveness, and none of them can be considered prosperous. These species are not dominant in their respective plant communities, and they inhabit specific (often quite narrow) ecological niches where their existence is maintained more or less successfully for a long period. Without the effects of human activity, these species could probably exist in this state indefinitely. What are the mechanisms that ensure the long-term existence of these species in small, isolated populations at the limit of their climatic and environmental tolerance?

term persistence. In contrast, the populations of *I. vorobievii* and *I. mandshurica* contain 13 and 6 haplotypes, respectively, which may result from historical gene flow, retention of ancestral polymorphisms that accumulated over a long period in more continuous ancient populations, or a putative origin from several founders. Based on cpDNA haplotypes, demographic event analyses show that populations of both species have undergone

Apparently, the main reason for the rarity of *I. mandshurica* and especially of *I. vorobievii* is the scarcity of suitable habitats for these psammirises in Primorye. As *I. vorobievii* and *I. mandshurica* represent components of steppe vegetation and grow in specific edaphic conditions, they are members of several steppe communities that are likely remnants of previously more widespread steppe vegetation (Krestov & Verkholat, 2003). The contractions of such relic communities result from climate and natural community changes (natural succession) and, in recent years, from anthropogenic habitat destruction. Their isolation from the main part of the species' ranges, limited seed dispersal and poor vegetative reproduction make these psammirises particularly vulnerable and may lead the

It is usually assumed that a certain level of genetic diversity is necessary for the long-term prosperity of a species. Indeed, different genotypes confer different levels of resistance to various environmental stresses, and consequently, the greater the diversity of the genotypes in a population, the more effective its ability to withstand unfavorable conditions will be. Therefore, the existence of a low level of genetic diversity is often considered to represent a crucial stage for the survival of a species or even a sign of its extinction. However, despite several attempts to determine the average levels of genetic variation in different categories of plants (e.g., Gitzendanner & Soltis, 2000; Nybom, 2004), it is still unknown what level of polymorphism should be considered critical for the existence of a certain species. Thus, some rare endemic species exhibit higher levels of genetic diversity than the average values found for this category of plants, such as *Oxytropis chankaensis* (Table). At the same time, very low levels, or even an absence of genetic variability has been found in other narrowly endemic plants (e.g., *Bensoniella oregona*, Soltis et al., 1992). It is impossible to predict the fate of a species on the basis of its genetic diversity alone. Here, we have attempted to address some species existing under extreme conditions in the context of a variety of environmental,

All of the species described above are represented by small, fragmented, marginal populations. Each of these species is characterized by a particular level of genetic diversity, ranging from very low (*Panax ginseng*) to high (*Oxytropis chankaensis*). Despite these differences, all of these plants exhibit weak competitiveness, and none of them can be considered prosperous. These species are not dominant in their respective plant communities, and they inhabit specific (often quite narrow) ecological niches where their existence is maintained more or less successfully for a long period. Without the effects of human activity, these species could probably exist in this state indefinitely. What are the mechanisms that ensure the long-term existence of these species in small, isolated

populations at the limit of their climatic and environmental tolerance?

bottleneck events and expansion in the past (Kozyrenko et al., 2009).

species to extinction.

biological, evolutionary and other influences.

**3. Conclusion**

For ginseng, longevity appears to be most important factor for its survival. Indeed, a lifetime of up to several hundred years is unusual for an herbaceous plant that is incapable of vegetative reproduction. Ginseng maintains the ability to produce seeds throughout its life span, and it appears that as a result of apomixes, its seed production is not particularly dependent on environmental conditions or the availability of pollinators. A large number of fully viable seeds and an extended period of embryo maturation allow revival of populations of this species, even from a small number of individuals over the course of many years.

Similar to ginseng, *Aristolochia manshuriensis* only exhibits seed-based reproduction. The life span of this species is not as long as that of other woody plants and is near the average for woody vines. This species has a very poor ability to reproduce vegetatively, and there are major limitations on its seed reproduction as a result of adaptation to pollination only by certain insects. Though adaptation to specific pollinators prevents inbreeding, pollination decreases if pollinators are absent, and the resulting fruit development is low. In this case, high seed production (a large number of seeds per one fertilization event) and adaptability to seed transfer by wind and water seems to be of primary importance for the survival of the species. Even a single successful fertilization every several years could guarantee population restoration if conditions are suitable for seed germination. Under unfavorable conditions, plantlets of this species can exist in a juvenile state for a long time, which allows a population to survive until the re-establishment of suitable conditions.

The ability for vegetative propagation is a pathway for the survival of some species (*Oplopanax elatus*, irises and *Microbiota decussata*). This pathway also allows for the rehabilitation of a species, even after a significant reduction in its population, and it effectively allows clonal colonies to reoccupy their habitats, where suitable conditions for the species are associated with an absence of competitors. The flexibility of the reproductive system of *O. elatus*, combined with its different modes of reproduction allows it to renew heterozygous genotypes by clonal growth and to contribute additional variability resources through sporadic seed reproduction. At the same time, some features of the biology of this species (the long life of a single clone, overlapping generations and the ability to crosspollinate) also help to maintain a certain level of polymorphism.

For *I. mandshurica* and *I. vorobievii,* vegetative reproduction (even if poor) seems to be the only way to survive under unfavorable conditions. Rare reproduction through seeds allows for the maintenance of a certain level of genetic diversity in its populations. However, the spread of both species beyond the borders of their existing populations is unlikely as a result of the scarcity of suitable habitats. Poor vegetative reproduction and limited seed dispersal as well as isolation from the main part of the species range (*I. mandshurica)* or its occurrence in a single locality (*I. vorobievii)* make populations of both species vulnerable. In the case of *I. vorobievii*, this vulnerability may lead to rapid species extinction.

The pattern of genetic diversity and the structure of populations in the endemic species *M. decussata* are congruent with the leading edge model of colonization. It could be proposed that extant southern populations represent the putative species range center. Populations that expanded southward during dry and cool periods at the Oligocene/Miocene boundary have become completely extirpated following climate changes in the Quaternary. The distribution area of *M. decussata* became fragmented quite some time ago through the displacement of the species toward mountain peaks and the extinction of stands in the adjoining territories. The remnant *M. decussata* populations have the potential for survival under ongoing climate changes and global warming because of this species' physiological and ecological range of tolerance, reproduction by layering and other life history traits.

In the case of endemic or rare species, one must distinguish between relics left by the extinction of related populations and newly evolved taxa. All relics that survived the repeated periods of Pleistocene climate cooling have apparently experienced a severe genetic bottleneck, not only as a result of genetic drift associated with the reduction of populations, but also because of selection acting in a rapidly changing climate. Because of this selection, the fittest individuals have survived. Among the representatives of the ancient tropical floras (*P. ginseng, O. elatus, A. manshuriensis*) only a small number of genotypes are likely to have possessed such fitness. By acquiring a mechanism for survival in harsh environments, these species have lost much of their genetic diversity.

In the case of the narrowly endemic species *O. chankaensis*, the situation is different. This species possesses adaptive mechanisms that enable it not only to successfully renew its populations in the coastal zone, which are exposed to frequent flooding and other adverse factors, but also to maintain the high level of recombination responsible for the survival of the species in a changing environment. This is a relatively young species, and its biological characteristics may promote its prosperity and wide distribution; the only obstacle to this is its high habitat specificity; it only inhabits the sandy shore of a large lake where there is intense insolation and high air humidity. However, there are no such habitats nearby, and the small number of plants in some populations makes this species vulnerable.

The patterns observed for various rare species often do not fully correspond to the general idea of survival at the edge of their range. Plant species exhibit tremendous variation in life history traits that may help them survive in harsh or changeable environments. The cause of a plant's rarity depends on the effects of different historical, biological and genetic factors. In all cases, different compensatory mechanisms, such as increased longevity and fertility, the formation of soil seed banks and vegetative reproduction, are involved. The adaptations of these species are not always successful because the historically established balance between reproduction and dispersal can be disturbed. However, in the absence of destructive human activities, many rare species could exist for an indefinite time.

### **4. Acknowledgements**

We are grateful to Dr. E.V. Sundukova and Dr. V.N. Makarkin for help with the manuscript preparation. This work was supported by the Grant "Molecular and Cell Biology" of the Russian Academy of Sciences (No 09-I-P22-03), by the Grant "Biological Diversity" of the Russian Academy of Sciences (No 09-I-P23-06), by the Grant "Biological Resources of Russia: Estimation and Fundamental Basis for Monitoring" (No 09-I-ОBN-02), by the Grant No 11- III-В-06-094 of FEBRAS and by Grant "Leading Schools of Thought" from the President of Russian Federation.

#### **5. References**

Adams, C.A., Baskin, J.M. & Baskin, C.C. (2005). Trait Stasis Versus Adaptation in Disjunct Relict Species: Evolutionary Changes in Seed Dormancy–Breaking and

adjoining territories. The remnant *M. decussata* populations have the potential for survival under ongoing climate changes and global warming because of this species' physiological and ecological range of tolerance, reproduction by layering and other life history traits.

In the case of endemic or rare species, one must distinguish between relics left by the extinction of related populations and newly evolved taxa. All relics that survived the repeated periods of Pleistocene climate cooling have apparently experienced a severe genetic bottleneck, not only as a result of genetic drift associated with the reduction of populations, but also because of selection acting in a rapidly changing climate. Because of this selection, the fittest individuals have survived. Among the representatives of the ancient tropical floras (*P. ginseng, O. elatus, A. manshuriensis*) only a small number of genotypes are likely to have possessed such fitness. By acquiring a mechanism for survival in harsh

In the case of the narrowly endemic species *O. chankaensis*, the situation is different. This species possesses adaptive mechanisms that enable it not only to successfully renew its populations in the coastal zone, which are exposed to frequent flooding and other adverse factors, but also to maintain the high level of recombination responsible for the survival of the species in a changing environment. This is a relatively young species, and its biological characteristics may promote its prosperity and wide distribution; the only obstacle to this is its high habitat specificity; it only inhabits the sandy shore of a large lake where there is intense insolation and high air humidity. However, there are no such habitats nearby, and

The patterns observed for various rare species often do not fully correspond to the general idea of survival at the edge of their range. Plant species exhibit tremendous variation in life history traits that may help them survive in harsh or changeable environments. The cause of a plant's rarity depends on the effects of different historical, biological and genetic factors. In all cases, different compensatory mechanisms, such as increased longevity and fertility, the formation of soil seed banks and vegetative reproduction, are involved. The adaptations of these species are not always successful because the historically established balance between reproduction and dispersal can be disturbed. However, in the absence of destructive human

We are grateful to Dr. E.V. Sundukova and Dr. V.N. Makarkin for help with the manuscript preparation. This work was supported by the Grant "Molecular and Cell Biology" of the Russian Academy of Sciences (No 09-I-P22-03), by the Grant "Biological Diversity" of the Russian Academy of Sciences (No 09-I-P23-06), by the Grant "Biological Resources of Russia: Estimation and Fundamental Basis for Monitoring" (No 09-I-ОBN-02), by the Grant No 11- III-В-06-094 of FEBRAS and by Grant "Leading Schools of Thought" from the President of

Adams, C.A., Baskin, J.M. & Baskin, C.C. (2005). Trait Stasis Versus Adaptation in Disjunct

Relict Species: Evolutionary Changes in Seed Dormancy–Breaking and

environments, these species have lost much of their genetic diversity.

the small number of plants in some populations makes this species vulnerable.

activities, many rare species could exist for an indefinite time.

**4. Acknowledgements**

Russian Federation.

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5. Criterion for the quantitative titration and the influence of side-reactions

The principle of redox titrations is that the solution of a reducing agent is titrated with a solution of an oxidizing agent (or vice versa)

On the Titration Curves and Titration Errors in Donor Acceptor Titrations of Displacement and Electronic… http://dx.doi.org/10.5772/intechopen.68750 137

Table 4. Titration curves of V(II) with MnO4 �.

$$y\text{Oxr} + z\text{Reds} \leftrightarrow y\text{Redr} + z\text{Oxs} \tag{59}$$

At the equivalence point

$$p e\_{\rm eq} = \frac{1}{z+y} (\log K\_{\rm S} + \log K\_{\rm T}) \tag{60}$$

The criterion of the quantitative titration can be deduced if we consider that the substance to be determined must be oxidized (or reduced) during the titration to an extent of 99.9% [23]. This means that the amount of determinant remaining untitrated at the equivalence point should not exceed 0.1% of that originally present, i.e.,

$$\log \frac{[O\chi\_S]}{[\text{Red}\_S]} \gtrsim 3\tag{61}$$

and

T1 T2 T3 T SUM E pX 0.0001 0 0 0.0001 0.5 8.333 0.0006 0 0 0.0006 0.45 7.5 0.0038 0 0 0.0038 0.4 6.667 0.0254 0 0 0.0254 0.35 5.833 0.151 0 0 0.151 0.3 5 0.5478 0 0 0.5478 0.25 4.167 0.8919 0 0 0.8919 0.2 3.333 0.9825 0 0 0.9825 0.15 2.5 0.9974 0 0 0.9974 0.1 1.667 0.9996 0 0 0.9996 0.05 0.833 0.9999 0 0 0.9999 0 0 0 0 1 0.05 0.833 0.0001 0 1.0001 0.1 1.667 0.0008 0 1.0008 0.15 2.5 0.0052 0 1.0052 0.2 3.333 0.0343 0 1.0343 0.25 4.167 0.1947 0 1.1947 0.3 5 0.6222 0 1.6222 0.35 5.833 0.9182 0 1.9182 0.4 6.667 0.9871 0 1.9871 0.45 7.5 0.9981 0 1.9981 0.5 8.333 0.9997 0 1.9997 0.55 9.167 1 0 2 0.6 10 1 0 2 0.65 10.833 1 0 2 0.7 11.667 1 0.0001 2.0001 0.75 12.5 1 0.0004 2.0004 0.8 13.333 1 0.003 2.003 0.85 14.167 1 0.0203 2.0203 0.9 15 1 0.1238 2.1238 0.95 15.833 1 0.4904 2.4904 1 16.667 1 0.8677 2.8677 1.05 17.5 1 0.9781 2.9781 1.1 18.333 1 0.9967 2.9967 1.15 19.167 1 0.9995 2.9995 1.2 20 1 0.9999 2.9999 1.25 20.833

136 Redox - Principles and Advanced Applications

$$p e\_{\rm eq} > (p e\_{\rm eq}{}^0)\_S + 3 \frac{1}{y} \tag{62}$$

$$(\left(\mathrm{pe}\_{\mathrm{eq}}\right)\_{\mathcal{S}} = \frac{1}{y} \log K\_{\mathcal{S}} \tag{63}$$

From Eqs. (60) and (63) also we obtain

$$y \log K\_T - z \log K\_S > \Im(z+y) \tag{64}$$

If both oxidation-reduction systems in the titration involve two electrons, the difference between the log K values must be greater than 6.

In addition, if the oxidized or reduced product present in the solution containing the redox system takes part in a side-reaction, and the equilibrium position of this reaction can be kept constant, by maintaining suitable experimental conditions, the conditional oxidation-reduction constant, K', can be deduced and used similarly to those used in complex chemistry

$$K' = \frac{[\text{Red}']}{[\text{Ox}'][e^-]^\sharp} \tag{65}$$

[Red'] and [Ox'] are analytical concentrations without any respect to side-reactions. The connection between the conditional and real constants is the following one

$$K' = K \frac{\alpha\_{\text{Red}}(\text{A})}{\alpha\_{\text{Ox}}(\text{B})} \tag{66}$$

where αRed(A) and αOx(B) are the side-reaction functions, and A and B denote the substances reacting with the reduced and oxidized substance, respectively.

$$\alpha\_{\mathbb{R}\text{ ed}}(A) = 1 + [L]\beta\_1 + [L]^2 \beta\_2 + \dots \tag{67}$$

$$\alpha\_{\odot \star}(B) = 1 + [L]\beta\_1^\* + [L]\beta\_2^\* + \dots \tag{68}$$

[A] and [B] are concentrations of the species reacting with the reduced and oxidized form, respectively, β's and β\*'s are complex products or protonization constants products.

In all calculations concentration constants can be used if they are corrected to the corresponding ionic strength.

In practice, the most important side-reactions are complex formation and protonation. The oxidized and reduced form of a metal ion may form complexes of different stabilities with the complexing ligand L.

So, if the criterion of the quantitative determination is not fulfilled by a suitable pH change or by the use of a complexing agent that shifts the values of the conditional constants, the titration may be realized. For example, according to Vydra and Pribil [101], cobalt (II) can be titrated with iron (III) ions if 1,10-phenanthroline is added to the solution, and the pH is adjusted to 3 even though KCo >>> KFe.

On the other hand, if another component present in the solution has similar oxidizing or reducing properties, then interfering species can be masked, so that the conditional redox constant of the interfering system is changed to such an extent that it no longer interferes with the main reaction.

#### 5.1. Practical examples

(1) Calculate the pH necessary for the accurate direct titration of potassium hexacyanoferrate (III) with ascorbic acid, given that log KFe(CN)6 ¼ 6.1; the protonation constants of hexacyanoferrate(II) are log K1 ¼ 4.17, log K2 ¼ 2.22, log K3 < 1, log K4 < 1; the logarithms of all the protonation constants of hexacyanoferrate(III) are >1. The equilibrium constant of the dehydroascorbic acid-ascorbinate redox system is log KA ¼ �2.5; the protonation constants of the ascorbinate ion are log K1 ¼ 11.56 and 4.17.

The criterion for the feasibility of the titration, according to Eq. (64) is:

$$2\log K'\_{\text{Fe(CN)}\_6} - \log K'\_A > \Im(1+2) = 9\tag{69}$$

From the protonation constants of hexacyanoferrate(II), if the pH > 5.5, then αFe(CN)6(H) ¼ 1 and

$$\log K'\_{\text{Fe(CN)}\_6} = \log K\_{\text{Fe(CN)}\_6} = 6.1. \tag{70}$$

Therefore 2 x 6.1 � log K0 A > 9; or log K0 A < 3.2. If the pH ¼ 6, then αA(H) ¼ 1 þ 10�6 x 1011.57 þ 10�12 x 1015.73 ¼ 105.58

$$
\log K\_A' = \text{3.1} \tag{71}
$$

Thus, if the pH > 6, the titration can be performed with adequate accuracy.

(2) Calculate the [L] maximum for the accurate direct titration of Ceþ4 with Fe2þ, being L the organic complexant (acetylacetone) present in the solution and given that log KT ¼ 24 and log KS ¼ 11.3 (see Eqs. (28)–(30)); the global constants of Fe-(L)(II) are log β1 ¼ 5.07; log β2 ¼ 8.67.

The reactions involved would be

K0 ¼ Red0 ½ � Ox0 ½ � e

[Red'] and [Ox'] are analytical concentrations without any respect to side-reactions. The con-

where αRed(A) and αOx(B) are the side-reaction functions, and A and B denote the substances

[A] and [B] are concentrations of the species reacting with the reduced and oxidized form,

In all calculations concentration constants can be used if they are corrected to the

In practice, the most important side-reactions are complex formation and protonation. The oxidized and reduced form of a metal ion may form complexes of different stabilities with the

So, if the criterion of the quantitative determination is not fulfilled by a suitable pH change or by the use of a complexing agent that shifts the values of the conditional constants, the titration may be realized. For example, according to Vydra and Pribil [101], cobalt (II) can be titrated with iron (III) ions if 1,10-phenanthroline is added to the solution, and the pH is adjusted to 3

On the other hand, if another component present in the solution has similar oxidizing or reducing properties, then interfering species can be masked, so that the conditional redox constant of the interfering system is changed to such an extent that it no longer interferes with

(1) Calculate the pH necessary for the accurate direct titration of potassium hexacyanoferrate (III) with ascorbic acid, given that log KFe(CN)6 ¼ 6.1; the protonation constants of hexacyanoferrate(II) are log K1 ¼ 4.17, log K2 ¼ 2.22, log K3 < 1, log K4 < 1; the logarithms of all the protonation constants of hexacyanoferrate(III) are >1. The equilibrium constant of the dehydroascorbic acid-ascorbinate redox system is log KA ¼ �2.5; the protonation constants of

αR edð Þ A αOxð Þ B

1 þ ½ � L β�

K0 ¼ K

αR edð Þ A ¼ 1 þ ½ � L β1 þ ½ � L 2

αOxð Þ B ¼ 1 þ ½ � L β�

respectively, β's and β\*'s are complex products or protonization constants products.

nection between the conditional and real constants is the following one

reacting with the reduced and oxidized substance, respectively.

corresponding ionic strength.

138 Redox - Principles and Advanced Applications

complexing ligand L.

even though KCo >>> KFe.

the main reaction.

5.1. Practical examples

the ascorbinate ion are log K1 ¼ 11.56 and 4.17.

� ½ �z ð65Þ

β2 þ … ð67Þ

2 þ … ð68Þ

ð66Þ

$$\begin{aligned} \text{Ce}^{4+} + \text{Fe}^{2+} &\leftrightarrow \text{Ce}^{3+} + \text{Fe}^{3+} \\ \text{Fe}^{2+} + L &\leftrightarrow \text{Fe}L^{2+} \\ \text{Fe}L^{2+} + L &\leftrightarrow \text{Fe}L\_2^{2+} \end{aligned} \tag{72}$$

The criterion for the feasibility of the titration, according to Eq. (64) is

$$
\log K\_T - \log K\_S' > 3(1+1) = 6 \tag{73}
$$

$$
\log K\_T - 6 > \log K'\_{S'} \quad \log K'\_S < 18 \tag{74}
$$

$$K\_S' = K\_S a\_{\left[\text{FeL}\_2\right]^{+2}} \quad a\_{\left[\text{FeL}\_2\right]^{+2}} < 10^{6.7} \tag{74}$$

$$
\alpha\_{\left[\text{FeL}\_2\right]^{+2}} = 1 + \left[L\right]\beta\_1 + \left[L\right]^2 \beta\_2 \tag{75}
$$

Thus, if the [L] < 0.1 M, the titration can be performed with adequate accuracy.

(3) Calculate the [SO4 �2 ] maximum present in the solution for the accurate direct titration of Ce4þ with Tlþ, given that log KT ¼ 24 and log KS ¼ 41.6 (see Eqs. (28), (42), (43)); the global constants of Ce-SO4(IV) are log β1 ¼ 3.5; log β2 ¼ 8.0; log β3 ¼ 10.4.

The criterion for the feasibility of the titration, according to (64) is:

$$2\log K\_T' - \log K\_S > 3(1+2) = 9\tag{76}$$

$$\log K\_T' > 25.3$$

$$K\_T ' = K\_T \frac{1}{a\_{\text{Ce}-\text{SO}\_4}}, \quad a\_{\text{Ce}-\text{SO}\_4} < 10^{7.2} \tag{77}$$

$$\alpha\_{\left[\text{FeL}\_{2}\right]^{\ast 2}} = 1 + \left[\text{SO}\_{4}{}^{-2}\right] \beta\_{1} + \left[\text{SO}\_{4}{}^{-2}\right]^{2} \beta\_{2} + \left[\text{SO}\_{4}{}^{-2}\right]^{3} \beta\_{3} \tag{78}$$

Thus, [SO4 �2 ] should be < 0.39 M, to perform the titrimetry with accuracy.

#### 6. Final comments

As a matter of fact redox titrations play a prominent role in volumetric analysis of redox actives species. A systematic study of the bibliography is undertaken in order to ascertain the state of the art concerning to redox titration curves. A method for the determination of titration error in donor/acceptor titrations of displacement and electronic transference reactions has been devised; a hyperbolic sine expression being derived for the titration error, applicable to symmetrical reactions (no polynuclear species being involved in one side of a half-reaction). The hyperbolic sine expression developed is compact and allows calculating the entire titration curve without piecemeal approximations, as usually occurs by dividing the titration curve in three parts: before, in, and beyond the equivalence point regions.

The method has been applied to some experimental systems characteristics of redox titration reactions. The method proposed is also applicable to mixtures of analytes, e.g., Fe(II) þ Tl(I), as well as to multistep redox titrations, e.g., V(II)/V(III)/V(IV)/V(V) system. The forms of the redox titration curves are independent of the concentrations. However, when the concentrations involved are very low the responses of the electrodes are not appropriate. All calculations involved have been checked with the method proposed by "de Levie" [62] for the sake of comparison, and no differences were found in the numerical values obtained by both methods. A diagram for the titration error in function of the difference between the end and equivalence point (pX) is drawn in order to facilitate the graphical calculation of titration error.

Automatic titrators enable recording automatically the change with potential (E) or pH in titre during a given titration. The accuracy of the measurements can increase with the help of online microcomputer for the control and data acquisition, allowing among the possibility for curve-smoothing and differentiation.

The extension of the method to nonhomogeneous systems of the type Cr2O7 2�/Cr3þ, I2/I�, or S4O6 2�/S2O3 2� remains a challenge, this being a complex problem involving a complete reformulation of the presented equations, which implies a higher level of difficulty.

At the end of the chapter an appendix including a detailed study of the propagation of systematic and random errors on redox titration error has been carried out and spite of the complex expression obtained first on differentiation, the final expressions formulated were very compact. This topic is still under study and it will be dealt in further calculus.