Journal of Chemistry

Volume 2015 (2015), Article ID 540368, 13 pages

http://dx.doi.org/10.1155/2015/540368

## An Exact Method to Determine the Conductivity of Aqueous Solutions in Acid-Base Titrations

^{1}Laboratorio R-105/R-107, Área de Química Analítica, Departamento de Química, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Avenida San Rafael Atlixco 186, Colonia Vicentina, 09340 México, DF, Mexico^{2}Área Ingeniería de Materiales, Departamento de Materiales, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Azcapotzalco, Avenida San Pablo 180, Colonia Reynosa-Tamaulipas, 02200 México, DF, Mexico

Received 6 November 2014; Accepted 8 January 2015

Academic Editor: Patricia Valentao

Copyright © 2015 Norma Rodríguez-Laguna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Several works in the literature show that it is possible to establish the analytic equations to estimate the volume of a strong base or a strong acid ( and , resp.) being added to a solution of a substance or a mix of substances during an acid-base titration, as well as the equations to estimate the first derivative of the titration plot , and algebraic expressions to determine the buffer capacity with dilution . This treatment allows establishing the conditions of thermodynamic equilibria for all species within a system containing a mix of species from one or from various polyacid systems. The present work shows that it is possible to determine exactly the electric conductivity of aqueous solutions for these Brønsted acid-base titrations, because the functional relation between this property and the composition of the system in equilibrium is well known; this is achieved using the equivalent conductivity values of each of the ions present in a given system. The model employed for the present work confirms the experimental outcomes with the H_{2}SO_{4}, B(OH)_{3}, CH_{3}COOH, and H_{3}PO_{4} aqueous solutions’ titration.

#### 1. Introduction

It is well known that electrolyte solutions frequently obey Ohm’s Law, where the conductance (, reciprocal of the resistance ()) is the electrical property of the solution that establishes readily the inherent facility with which it transports electric charges, under the influence of an electric field. The conductance depends on the temperature, the solution’s composition, and the geometry of the electric field applied in reference to the solution. However, the electrical conductivity () of the system can also be defined, which does not depend on the said geometry [1–4].

The functional relation between the electrical conductivity and the conductance is established with reference to a geometric empirical factor (, known as cell constant with units of the length reciprocal), in agreement with the following:

On another respect, Kolhrausch’s law, or law of ions’ independent migration, correlates the electric conductivity with the solution composition, this being a linear law of concentrations, as shown by the following: where is the absolute value of the charge, is the limiting equivalent conductivity (at infinite dilution), and is the molar concentration of the th ion, whereas the sum concerns all ions comprised.

During the course of volume titration, changes occur on the number and type of ions present in the solution to be titrated, also with an increase of the overall solution volume. Thus, in order to represent the conductometry plots obtained from the said titrations, it is recommended to define the corrected electric conductivity through dilution () for the measured value of the conductivity [4–7] in agreement with the following expression: is the initial volume of the titrated system and the titrant added volume.

The literature states that there are linear relations (per segments) of the with the titrant’s volume, with different slopes at each side of any of the equivalence points, provided the titrating reactions are quantitative. The plots of deviate from linearity in the vicinity of the equivalence point when the reactions are less quantitative, which generally occurs when the solution becomes more diluted. Thus, the linear behaviors of the said plots are observed only at points far from the equivalence points [5–7]. Therefore, in systems where there are several titration reactions with relatively small slope changes it becomes difficult to decide what points can be considered within the linear trends of the plot.

Several works, reported in the literature, describe robust models to predict and interpret pH plots as a function of the titrant added, obtained during acid-base titrations. Among them, documents can be found that describe the functions where the titrant volume is resolved as a function of pH [8–12] up to more extensive works, such as that excellent revision by Asuero and Michalowski [13] that, apart from describing much more robust models to predict acid-base titration plots, describe other functions related with them, such as the pH derivatives with respect to or the buffer capacity of various systems.

In previous works we have presented explicit titrant volume expressions as a function of pH () for titrations of mixtures of species of the same proton polydonor system with a strong acid or base [14, 15]. Also, these papers presented the exact equations for the pH derivative with respect to the and the buffer capacity with dilution () of those systems. In order to construct the plots obtained from the said equations, several spread sheets were built at Excel,* Microsoft*.

The objective of this work is to present a robust model that allows estimation of the conductometric titration plots in complicated mixtures, without the need to find the linear regions of the plots before and after each equivalence point, with the aim to define exactly and precisely the volume of each of those equivalence points.

Thus, the present paper shows how to apply advantageously these equations in order to build the conductometric plots () without approximations, using nonlinear functions that remain implicit over the spread sheets used to build the graphs of the acid-base titrations.

#### 2. Materials and Methods

##### 2.1. Methodology

###### 2.1.1. Acid-Base Titrations of Mixtures of Species of the Same Proton Polydonor System:

In order to establish a model that allows determination of the system’s equilibrium conditions during an acid-base titration and to estimate the pH-metric plot, namely, , it seems adequate to solve the pH as a function of the volume (as established in the previous function), using the necessary equations to express the equilibrium condition in terms of the two variables. However, apart from the titrations of strong acid with strong base or those of strong base with strong acid, the equation to solve, at least, is a 3rd order polynomial equation [16].

Further, the equation is linear with respect to the volume even in the case of mixtures of species that belong to the same protons polydonor system, namely, . In order to simplify the nomenclature, note that, in the previous notation, the electric charges of component species have been omitted.

A common way to obtain the equation from which the titrant’s volume is solved results from establishing the electroneutrality equation in the system and substituting in it the equilibrium concentrations of the ions comprised.

Then a robust model is obtained for estimations when solving the strong base () and strong acid () volumes that were added in an acid-base titration as a function of pH [8, 13, 14].

This way, the added titrant’s volume is a function of pH as variable as well as other parameters, namely:(1)the initial volumes () of the solutions that contain each of the species of the polydonor system for titration (), considering also its counterions (when appropriate), that after mixing lead to the initial volume of the mix to be titrated (),(2)the electric charges of the species of the polydonor system and its counterions (), where represents the absolute value of the electric charge of the base of the system, ,(3)the equilibrium concentrations of the hydronium ion () and of the hydroxide ion (, where is the autoprotolysis constant of the solvent),(4)the initial molarities of the species (), that because of the balance of the component lead to (),(5)the molar fractions in the equilibrium (), defined in accordance with the mass balance for the component. Generally, these molar fractions are graphically represented through a distribution diagram because they are functions that depend only on the dissolution’s pH: where and are the global formation constants of the () species of in the system, because .

Therefore, if a volume of a mixture of species of the same proton polydonor system () with overall molarity , is titrated with a strong base or a strong acid of molarity or , respectively; it can be shown that the added titrant volume for titration with a strong base () gives rise to (5), whereas for titration with a strong acid () it leads to (6) as follows:

Now, for the same titration and considering (2) it is possible to deduce (7) to estimate the conductivity at every pH.

In (7), represents the titrant counterion and represents the counterion of the species , while represents the equivalent conductivity of the ion away from the limiting conditions that, theoretically, should be smaller than or equal to the [2]:

It must be pointed out that the second term in (7) involves the autoprotolysis equilibrium. The third term in this equation rests its greater dependence on the pH following (4), which involves the global formation equilibria of the species from the protons polydonor system. The fourth term, which comes from the spectator counterion that accompanies the acid or basic particle of the solvent as titrant, has a direct dependence with through (5) or (6), depending on the case. The last term is the one that comes from the counterions of the species of the protons polydonor system and almost does not depend on pH. The dependence of the third, fourth, and fifth terms in (7) with the pH, through the that appears in the denominator, is very small and practically null if there were little dilution during the course of titration.

Further, (7) provokes the use of a spread sheet or software to estimate titration plots using (5) and (6): with it, it becomes possible to estimate also (exactly) the electric conductivities of the system at every pH and, hence, at every added volume , provided the equivalent conductivities are given of all ions comprised in the titration. This leads then to the determination of the conductometric titration plots using (7).

The robust model also leads to obtaining many more points with very small pH increments; therefore, small increments allow estimating the derivatives through the ratio of the said increments: and .

###### 2.1.2. Acid-Base Titrations of Mixtures of Species in Solution That Come from Different Protons Polydonor Systems

When there is a mixture of species that comes from various particle polydonor systems, even when the problem of determining the titration plots of these systems can appear to be much more difficult, it turns out to be of practically equal complexity; the same methodology is followed as described in the previous section.

This is even when the polynomial to be dealt with in [H^{+}] can be of higher order every time, considering more particle polydonor systems (or more components with acid-base properties); the electroneutrality equation and the other independent equations of the system involve a only in linear terms, which allows factorizing this variable, and solve it in simpler manner.

The result is that the terms under the summation sign over the index in (5), (6), and (7) transform into a double sum: the first over the species of a polydonor system and the second over the components present in the different polydonor systems [17].

##### 2.2. Experimental

###### 2.2.1. Reagents

All reagents used were of analytical grade to prepare the dissolutions described in this paper in deionized water type I (18.2 M cm), from a Milli-Q equipment, Millipore. The NaOH (99%, Baker) used was free from carbonate and was titrated with potassium hydrogen phthalate (Merck), following the procedures described by Harris [18].

Concentrated H_{2}SO_{4} (98%, Baker) was used to prepare a nominal solution 0.025 mol·L^{−1} of sulfuric acid, which was titrated with NaOH, for the first part of this work.

CH_{3}COOH (glacial, Baker) and H_{3}BO_{3} (99%, Baker) were used to prepare the nominal solutions 0.05 mol·L^{−1} acetic acid and 0.10 mol·L^{−1} boric acid, and the mixture of 20 mL of each one was titrated with NaOH, in the second part of the present paper.

Finally, CH_{3}COOH (glacial, Baker), H_{3}PO_{4} (85%, Baker), and H_{3}BO_{3} (99%, Baker) were employed to prepare the equimolar nominal solutions 0.015 mol·L^{−1} acetic acid, phosphoric acid, and boric acid. The mixture of 30 mL or each one, known as Britton-Robinson solution, was titrated with NaOH, in the third part of this contribution.

###### 2.2.2. Equipment and Materials Used for Preparation of Solutions and Titrations

Solid reagents were accurately weighed on an analytical balance OHAUS (Discovery, DV215CD), with a readability of 0.01 mg.

Liquid reagents were dispensed with plastic pipettes brand (Transferpette, adjustable) 704174 (10–100 *μ*L), 704180 (100–1000 *μ*L), or 704182 (0.5–5 mL), with accuracy better than or equal to 0.6% or glass volumetric pipettes (Pyrex, Class A) of 1, 10, or 20 mL of capacity, with accuracy better than or equal to 0.6%, were used in order to dispense the adequate amounts of the diverse liquid reagents to prepare solutions.

Glass volumetric flasks (Pyrex, Class A) of 100 or 250 mL, with tolerance better than or equal to 0.8%, were used to fill the volume to the mark.

Glass burettes (Kimax, Class B) of 10, 25, or 50 mL of capacity, with tolerance better than or equal to 0.4%, were used to dispense the titrant volumes.

###### 2.2.3. Conductometric Titrations Using Initial Large Dissolution Volumes

A laboratory instrument (PC45, Conductronic, Mexico) was used, which was equipped with a temperature sensor and a cylindrical conductivity cell of immersion (C1, 8 mL capacity: 1 cm internal diameter, 10 cm height). The cell has two platinized nickel electrodes, electric contact type mini DIN, and a cell constant cm^{−1} (readability of 1 *μ*S). The minimum volume of solution required to cover all the inner cavity of the conductivity cell was 350 mL, contained in a 600 mL beaker. The equipment was calibrated with an aqueous standard KCl 0.01 M solution (Hach) with 1.413 mS cm^{−1} conductivity at 25°C. The solution was vigorously stirred for 2 minutes after every titrant addition with a magnetic stirrer; then the stirring was suspended in order to take a stable conductivity recording.

###### 2.2.4. Conductometric Titrations Using Initial Small Dissolution Volumes

A laboratory instrument (CDM230, Radiometer, Denmark) was used, equipped with a conductivity cell CDC641T with temperature sensor and a cable A94P002. The cell has platinized nickel electrodes and a cell constant cm^{−1} (readability better than 1 *μ*S). The minimum volume of solution required to cover all the inner cavity of the conductivity cell was 30 mL, contained in a 100 mL beaker. The equipment was calibrated and used as described in the previous paragraph.

###### 2.2.5. pH-Metric Titration

A Conductronic PC45 (Mexico) laboratory instrument was used, which was equipped with a temperature sensor and a PC100 combined electrode with a BNC electric contact (). The equipment was calibrated with a * buffer* solution (Radiometer, pH 7.00), taking pH and potential difference ( in millivolts) recordings after addition of every volume of the titrant in turn; in order to correct the pH through the cell efficiency the following equation was used, as described by Islas-Martínez et al. [19]:
is an empirical parameter that relates the inverse of the cell efficiency, is the calibration pH at a point (in this case ), and is the corrected pH from the proportional systematic error to obtain the one that should have been determined for a 100% cell efficiency and 25°C temperature. The parameter is varied up to where the slope of the plot becomes equal to −59.16 mV. In this work, whenever the experimental titration graphs are presented, it is actually the that is being plotted, as explained in this section.

#### 3. Results and Discussion

In order to corroborate the equations above, the following are adequate model examples to predict the conductometric acid-base titration plots exactly.

##### 3.1. Conductometric Titrations of Aqueous Sulfuric Acid Solutions, at Different Concentrations, with NaOH

This first system constitutes a relatively simple case since sulfuric acid is a diprotic acid where the first dissociation is strong and the ampholyte (formal) of the system has a relatively low . The study was carried out at two different solute concentrations, in agreement with two procedures. The theory for this system corresponds to the first part of Section 2.1, because it corresponds to a species mixture of the same polydonor system.

In order to explain better the treatment described in Section 2.1, (9) presents the algebra of the fractions of the sulfate species as a function of pH: being the of the hydrogen sulfate ion.

Equation (10) expresses the volume of strong base added in these titrations as a function of pH:

Finally (11) expresses the conductivity as a function of pH:

###### 3.1.1. Procedure 1

20.00 mL of the 0.024 mol L^{−1} sulfuric acid aqueous solution was used to which 430.0 mL of deionized water was added (giving as a result mL and mol L^{−1}). This solution was titrated with a sodium hydroxide aqueous solution at a mol L^{−1} concentration. With these titration conditions and the parameters shown in Table 1 introduced in (9) and (10), it is possible to construct the pH-metric titration plot shown in Figure 1(a) (solid line).