Student exploration of the Henderson-Hasselbach equation and pH readings to determine the pK _a value of 4,4′-trimethylenedipiperidine (TMDP)

1 Introduction

In our laboratory, the determining of pK _a of weak bases or acids using a pH meter is in sequence of preparation a certain molar concentration. Students preformed several pH reading for different concentration of a weak acid or base during the four-hour laboratory period. Students submit their results for the identified compound regarding their search in the literature and handbook. The name of the compound and its reported pK _a in the handbook are given to students, and they discuss theory and experimental procedure, assumptions, and sources of error. During our laboratory course, student results with a direct pH reading were in good agreement with handbook values of pK _a for monoprotic base and acids despite of approximations made in the theory and the values were within 2–5% of handbook value.

A more impressive experiment was planned for our laboratory course consists of determining of the acid and base association constant (K _a and K _b ) of a commercially available bifunctional secondary amine, i.e., 4,4′-trimethylenedipiperidine (TMDP) with unknown pK _a , using a pH meter and statistical data processing (Figure 1).

Figure 1:

The chemical structure of TMDP (ball and stick).

The determination of pK _a enhances students’ understanding of acid–base equilibria in an aqueous solution. Many different techniques have been developed for determining the acid and base dissociation constant, including potentiometry, conductometry, voltammetry, calorimetry, nuclear magnetic resonance (NMR) spectroscopy, electrophoresis, high performance liquid chromatography (HPLC), solubility, spectrometry, fluorimetry, polarimetry, kinetic, and computational. They have some merits and drawbacks based on their applications in modern chemistry (Reijenga et al., 2013). From 1907 to 1916, the works of Henderson, 1908, Sörenson, 1909, and Hasselbalch, 1916 led to a famous equation (Eq. (1)), known as Henderson-Hasselbalch equation. The equation relates three variables, including pH (negative logarithm of [H⁺]), molar concentration of dissociated acid [A⁻], and non-dissociated acid [HA], to the negative logarithm of the acid dissociation constant (pK _a ).

It should be kept in mind that the Henderson-Hasselbalch equation can only be employed for dilute aqueous solutions, and pK _a and pK _w values depend on temperature, ionic strength, and the dielectric constant of the solvent.

In addition, there is a relation between negative logarithm [H⁺] and [HO⁻] and water ionization constant value (K _w  = 1.01 × 10⁻¹⁴ at 298 K) as:

In 1928, Hammett, 1928, developed a new equation and defined the strength of an acid, the relation between the [H⁺] activity (α) and the activity coefficients (γ). Due to the difficulty in accurate determination of the parameters in Hammett’s acidity equation, the Henderson-Hasselbalch model is commonly employed in analytical chemistry.

“Education through research” has attracted great attention as an extensive and effective strategy for undergraduate learning about science. It emphasizes on engaging undergraduate students in original or applied research (Doyle, 2000). The determination of the pK _a value for a compound with unknown pK _a by a pH meter provides an opportunity to acquaint the students with thermodynamic acid or base equilibria in an aqueous solution, calculate the acid and base dissociation constant (K _a and K _b ), and the relationship between the concentration and strength of acid and base (Jeffrey & Bada, 1969). They also get familiar with mathematical and statistical analysis of data using excel software. As an interesting experiment for this lab experiment, the determination of the pK _a value of a bifunctional secondary amine, i.e., TMDP, was chosen. There is no experimentally determined report of pK _a value for TMDP in the literature and only a pK _a value of 10.99 ± 0.01 predicted by ACD/Labs software is found in literature. ChemAxon software predicted pK _a1 and pK _a2 values of 10.66 and 10.05 for TMDP. The application of TMDP as a safer and greener alternative for piperidine has attracted attention in organic chemistry. This bifunctional base was employed as the catalyst and/or solvent in some organic transformations (Gorjian & Khaligh, 2022a; Zaharani et al., 2021a). In addition, excellent catalytic activity and unique thermal behavior have been reported for its ionic liquid and molten salt derivatives (Khaligh et al., 2019; Suzaimi et al., 2022; Zaharani et al., 2021b, 2022). Therefore, students will encourage to analyze and calculate the pK _a value based on their collected data in this lab experiment. In addition, they can compare the reported pK _a value for TMDP and other similar organic secondary six-membered cyclic amines, such as piperidine, morpholine, and piperazine in the literature.

Several pedagogic goals can be achieved by this simple experiment with mathematical and statistical analyses. The students will be able to (1) prepare the solutions with a certain concentration (molarity), collect data using a pH meter, (2) understand the importance of calibration of the pH meter and calibrate the pH meter before their experiments, (3) employ Henderson-Hasselbalch equation to determine the pK _a value of an aqueous solution, and (4) describe the usefulness of the pH-metric technique for calculating acidity and basicity of aqueous organic solutions. Ultimately, through exploring pH changes in four different concentrations, students calculate the pK _a value for TMDP, as an organic base with an unknown pK _a value.

In addition, students compare their pK _a value of TMDP with predicted data by different software, i.e., ACD/Labs and ChemAxon to evaluate their pH-metric procedure. This educatory experiment is suitable for second-year undergraduate students in general and analytical chemistry or upper-division organic chemistry students.

2 Materials and methods

3 Results and discussion

When TMDP is dissolved in water, it dissociates in two steps according to the reaction shown in Scheme 1. The precise value of the dissociation constants can be determined through measurements of the electromotive force; however, a fairly good value can be achieved using a pH meter. This technique can be applied in an undergraduate experiment in the general chemistry laboratory for giving students the opportunity to learn the statistical analyses of data using an Excel program.

Scheme 1:

The equilibria of ionization of TMDP in water. Note: pK _a1 = pK _w  – pK _b2 and pK _a2 = pK _w  – pK _b1

Students could find two experimental pK _a1 and pK _a2 values of piperazine, a bifunctional monocyclic secondary amine, in the literature, i.e., pK _a1 = 5.35 (pK _b2 = 8.65) and pK _a2 = 9.73 (pK _b1 = 4.27) at 298 K (∆pK _a  = 4.38) (Khalili et al., 2009). In addition, they utilized ChemAxon software to predict the pK _a values of piperazine. The predicted values of pK _a1 and pK _a2 for piperazine are 5.18 and 9.56, respectively. This good agreement between the output from the software and the experimental results (Khalili et al., 2009) gives students more confidence to use the chemical property predictors.

It is assumed that pH changes in an aqueous solution of TMDP depend on the protonation of the piperidine rings of TMDP, i.e., the first and second dissociation steps (Eqs. (3) and (4)).

The titration techniques such as potentiometric or pH metric titration can be used to determine the first dissociation constant if the pK _a1 and pK _a2 be different by more than 2 pH units, while ChemAxon predicted only a difference of 0.61. Therefore, an apparent pH is determined for TMDP and it belongs to pK _a2 = 14 – pK _b1, which named pK _a in the following paragraphs for simplicity. A diagram of species distribution of TMDP in an aqueous solution is shown in Figure 2, which was predicted by ChemAxon software with pK _a1 and pK _a2 values of 10.66 and 10.05.

Figure 2:

Predicted species distribution diagram of TMDP in an aqueous solution using ChemAxon software.

The students could exclude negligible concentration of [TMDPH₂]²⁺ according to their pH readings (Table 1, a range of pH of 11.54–12.00) and the above predicted species distribution diagram of TMDP in an aqueous solution (0.50 % and 0.10 % at pH 11.50 and 12.00, respectively). In addition, the students consider the statement: Although hydroxide anion from both dissociation steps contributes to the concentration of [HO⁻], considering the common ion effect, negligible concentration of obtained [TMDPH₂]²⁺, the effect of the hydroxide ion concentration resulting from the second dissociation of TMDP (Eq. (4)) could be ignored.

Table 1:

Description of the changes of the species concentration at equilibrium in three steps, including before, during, and at equilibrium state.

	TMDP(aq)	+	H2O(l)		[TMDPH]+ (aq)	+	[HO−] (aq)
Before equilibrium	C				0		0
During equilibrium	−x				+x		+x
At equilibrium	C−x				x		x

Therefore, the existence species at pH values of 11.5–12 included TMDP, [HO⁻], and [TMDPH]⁺. The following equilibrium can be considered dominant equilibrium (Eq. (5)), and pK _a2 = 14 – pK _b1 is named pK _a in the following paragraphs for simplicity.

(5)

The changes in the species concentration in an aqueous solution of a weak base, such as TMDP, can be studied through three steps, i.e., before, during, and at the equilibrium state, as shown in Table 1.

Henderson-Hasselbalch equation can be written by Eq. (6) for the first equilibrium:

where the α′s represent activities of the species in the equilibrium state of an aqueous solution of TMDP, and brackets and γ′s are the representative molar concentration and activity coefficient, respectively.

The activity coefficient can be calculated using the Debye-Hückel equation (Eq. (7)):

Debye-Hückel equation gives a simple form at 25 °C if the solvent is water (Eq. (8)).

where Z _i is the charge on cation and anion.

The ionic strength of the solution, i.e., I in Eqs. (7) and (8), can be obtained using Eq. (9).

where m _i is the molar concentration of cation and anion in solution.

In a dilute solution, the activity coefficient of an uncharged species is approximated to be 1.0 at all temperatures; thus, Eq. (6) can be written as Eq. (10):

At the equilibrium state, it assumed that [HO⁻] = [TMDPH⁺], and according to the charge on cation and anion, Eq. (10) can be written as Eq. (11):

A negative logarithm of the above equation gives:

The pH data collected is adequate for analysis, including four concentrations of TMDP (Sigma Aldrich, purity = 97 %, MW = 210.36 g mol⁻¹) in water. Table 2 shows data collected from 14 groups with 2–3 students over multiple days. As expected, there are slight differences in pH readings.

The average pH, standard deviation, standard error, and confidence interval of 95 % were calculated for pH data collected from four concentrations of TMDP using Microsoft Excel 2010 (Table 2).

As a relatively good approximation for calculating the pK _a of TMDP, the activity coefficient (γ ^±) for cation and anion can be assumed unit. These data were employed to determine the approximate the pK _a of the aqueous solution of TMDP using Eq. (12) for each series, which give a pK _a value of 11.09, 11.19, 11.20, and 11.01 for concentrations of 0.0097, 0.0194, 0.0484, and 0.0968 mol L⁻¹, respectively (Table 3).

Once the pK _a data were calculated as a function of pH and TMDP concentration, they were fit to determine the average pK _a , standard deviation, standard error, and confidence level of 95 % with the upper and lower confidence intervals of the aqueous solution of TMDP at 27.00 ± 0.04 °C (Table 4).

The determined pK _a value 11.12 ± 0.02 and 11.12 ± 0.14 with a confidence interval of 95 % was in excellent agreement with the reported pK _a of 10.99 ± 0.01 in Chemicalbook and pK _a of piperidine in the literature (11.22) (Hall, 1957).

Therefore, TMDP is only a little weaker base than piperidine by a factor of 0.8; however, TMDP and piperidine are different in the viewpoint of hydrogen bonding sites, status, multifunctionality, flexibility, physical properties, solubility, toxicity, etc. (Gorjian & Khaligh, 2022b; Khaligh et al., 2019; Suzaimi et al., 2022; Zaharani et al., 2021b, 2022).

Using the pH of each series data, students determined the pH average, standard deviation, standard error, and 95 % confidence intervals. Then, students calculate the approximate pK _a for each series, as shown in Table 3. The recorded pH values by the students probably show small differences due to temperature and concentration discrepancies.

The mean ionic activity coefficient (γ ^±) value can be calculated using Eq. (7) or Eq. (8), as shown in Tables 5 –8. Then, the constant dissociation of TMDP is determined using the obtained γ ^± (Tables 5 and 7). The pK _a values were employed to calculate the average pK _a , standard deviation, standard error, and confidence intervals (95 %) for each activity coefficient (γ ^±) (Tables 6 and 8).

Table 9 showed the pK _a and some descriptive statistics based on three different activity coefficients (γ ^±).

4 Student feedback

The outcome of this laboratory activity was in inquiry style and experimentally undetermined, as well as the instructor did not know the result. The students could experience a traditional expository laboratory experiment in approach and procedure; however, this lab experiment could be a discovery and problem-based activity (Domin, 1999).

In addition, students were given a survey of their thoughts and opinions on the determination of the pK _a value of 4,4′-trimethylenedipiperidine (TMDP). A modified learning/teaching evaluation form was prepared to evaluate the student’s performance and experience for this experiment (Council et al., 2002), available in the Supplementary Information. On the basis of initial communications with students, some changes were made to the remaining assignments with the hopes of providing a more useful and approachable learning tool. It was found that students had some questions about calibration when they did not perform the calibration themselves. Student comments in the survey suggest the importance of lectures, laboratory instruction, and prelab questions for their understanding of the acid–base concept in an aqueous solution. In addition, the determination pK _a of TMDP gave them an opportunity for some critical thinking concerning the Henderson-Hasselbach equation.

The planning and performing experiments, collecting and properly evaluating scientific data, and constructing valid arguments were key factors of this laboratory activity and the students have a good understanding of the mentioned above concepts after this experiment. Most of the students were satisfied with working to find pK _a for TMDP, a diprotic organic base with unknown pK _a in the literature, using a simple instrument in their lab. Acquaintance with the excel tools for determining statistic parameters was also an interesting experience for them.

5 Assessment

A series of postlab questions were given to assess student understanding of the lab experiment and its applications for acid–base aqueous solution systems. The postlab questions helped focus attention on the important concepts of the lab experiment, and some new questions were made to improve learning. It also encourages them to design other experimental methods to determine the dissociation constants of the conjugate acids of TMDP (K _b1 and K _b2) using the potentiometric or pH metric titration methods.

6 Conclusions

A laboratory teaching experiment for estimating TMDP basicity through corresponding pK _a value was presented. According to assumed γ ^± = 1, the pK _a value was 11.12 ± 0.14 with a 95 % confidence interval; and the pK _a values were 11.17 ± 0.21 and 11.16 ± 0.22 when the activity coefficient was calculated using Eqs. (7) and (8). All determined pK _a values were very near to the predicted pK _a for TMDP. The student groups calculated and reported their descriptive statistics which was performed by Excel formula and Excel ToolPak.