Applying Le ChÃ¢telier's Principle To Model Strong ... - ACS Publications

Article Cite This: J. Chem. Educ. 2018, 95, 521−527

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Applying Le Châtelier’s Principle To Model Strong Acid−Strong Base Titration Philippe H. Mercier* Department of Natural Sciences, LaGuardia Community College, CUNY, 31-10 Thomson Avenue, Long Island City, New York 11101, United States S Supporting Information *

ABSTRACT: Le Châtelier’s principle is used as a basis to derive an expression to calculate solution pH during titrations that involve only strong acids and strong bases. Central to this model are the reaction quotient and a reequilibration term that represents the extent of neutralization during a titration. Unlike the method traditionally taught to students to determine the pH of these acid−base mixtures, the expression obtained is exact and applies throughout the entire range of a titration curve without the need for modification or segmentation. The expression is the same as one that is obtainable using charge balance; however, the present derivation applies Le Châtelier’s principle instead, and therefore remains within the scope of a general chemistry curriculum. Instructors are then able to reinforce and deepen understanding of previously learned concepts in acid−base chemistry while connecting them to chemical equilibrium and titration curves. KEYWORDS: First Year Undergraduate/General, Second Year Undergraduate, Upper-Division Undergraduate, Analytical Chemistry, Curriculum, Problem Solving/Decision Making, Acids/Bases, Equilibrium, pH, Titration/Volumetric Analysis

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INTRODUCTION Acid−base titration is an essential topic in the general chemistry curriculum because it is important in many areas of chemical research. The present work was motivated by an extra credit activity and a series of questions posed to students in one of the author’s General Chemistry 2 laboratory sessions on acid−base titration. They were asked to construct a theoretical titration curve for a strong acid titrated by a strong base and to compare that curve to the one they obtained with their experimental data. As part of the class discussion, the chemistry of neutralization and its connection to the autoionization of water was examined. Developing students’ abilities to calculate the pH of solutions during acid−base titrations is an exercise in integrative learning that calls on them to demonstrate high competency in many areas. They must understand the nature and properties of acids and bases;1,2 they must know how to perform stoichiometric calculations,3−6 and know how to distinguish between changes in moles and changes in molarities.7 Misconceptions or deficiencies in any of these areas can create severe barriers to student learning, particularly in relation to new material that builds on these concepts. Stoichiometry, simple acid−base properties, and solution chemistry are often taught in the first semester of a year-long general chemistry sequence; students have to merge that knowledge synergistically with concepts taught in the second semester such as chemical equilibrium and additional acid−base chemistry, including titration. Additionally, students are expected to be comfortable with algebraic, exponential, and logarithmic manipulations; they must also learn several expressions related to hydronium ion concentration and pH determinations, all of which are applicable only under specific conditions. Table 1 is a list of some of these © 2018 American Chemical Society and Division of Chemical Education, Inc.

expressions along with a description of their applicability. With so many topics and expressions for students to learn to properly calculate pH for even the simplest kind of titration, that which involves only strong acids and bases, serious questions have been raised over the wisdom of teaching students how to construct titration curves.8,9 Considered separately, however, there is no debate as to the appropriateness of teaching students topics such as simple acid−base properties, stoichiometry, or solution chemistry, as they are well within the abilities of first-year students to grasp. It is when these concepts are brought together, and a more sophisticated analysis and appreciation of the material is needed to understand a more difficult topic such as acid−base titration, that students’ minds are challenged in new and meaningful ways. In section 5.3 of the revised Guidelines and Evaluation Procedures for Bachelor’s Degree Programs published in 2015 by the American Chemical Society Committee on Professional Training, it is made clear that, upon completion of foundation courses, the courses that are taken after general chemistry, students are expected to be capable of pursuing in-depth study in those areas of chemistry.10 In section 5.2, general chemistry guidelines are addressed, and in them it is stated that, as an introductory course, its purpose is mainly to prepare students for foundation courses. The guidelines broadly outline the topics that should be covered in general chemistry in order to give institutions the flexibility to tailor their programs. This means that programs have much discretion in deciding what topics to emphasize and de-emphasize depending on each Received: July 28, 2017 Revised: February 19, 2018 Published: March 1, 2018 521

DOI: 10.1021/acs.jchemed.7b00575 J. Chem. Educ. 2018, 95, 521−527

Journal of Chemical Education

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Table 1. Expressions Used To Determine pH and Hydronium or Hydroxide Ion Concentration in Acid−Base Solutions Equation +

[H ] =

[HA]VHA − [MOH]VMOH VHA + VMOH

pH = −log[HA]0

[OH−] =

[MOH]VMOH − [HA]VHA VMOH + VHA

pH = pKw − pOH

pH ≅ − log [OH−] ≅

K a[HA]0 Kb[A−] [A −] [HA]

( )

pH = pK a + log

Applicability The net hydronium ion concentration determined by taking the difference in moles of strong acid, HA, and moles of strong base, MOH, and dividing by the total volume when there is excess acid. The initial pH of a strong acid, HA. The net hydroxide ion concentration determined by taking the difference in moles of strong base, MOH, and moles of strong acid, HA, and dividing by the total volume when there is excess base. pH of a basic solution. Approximate initial pH of weak acid, HA, if the percent ionization is less than 5%. Approximate hydroxide ion concentration of a weak base or of an originally weak acid solution upon reaching the equivalence point, if the percent ionization is less than 5%. The Henderson−Hasselbalch equation used to calculate the pH of buffers within ±1 of the pKa of the weak acid.

using charge balance.14,16,19 In this article, the same expression will be derived simply by applying Le Châtelier’s principle, and for comparison, the charge balance method will be examined as well. Central to the derivation will be the reaction quotient, and a re-equilibration term that will represent the extent of neutralization. The derivation and resulting equation are not intended to replace the traditional method students learn to calculate the pH of these acid−base mixtures; rather, they are intended to serve as complements that provide substantial additional context to the subject. Although it is quite underutilized in this regard, it will be shown that Le Châtelier’s principle can be an effective pedagogical tool to link previously learned material to chemical equilibrium and titrations of strong acids and bases.

program’s educational goals. Nevertheless, as the course that lays the groundwork for future chemistry instruction, topics taught in general chemistry that foundation courses subsequently build upon should be covered as robustly as possible. Not only do foundation courses contain more material on a topic than that found in general chemistry, but also the treatment is more intellectually demanding. It is therefore this author’s opinion that, in order for students to be properly prepared for foundational and in-depth coursework, the standards set for scope and breadth for topics taught in general chemistry should principally consider the long-term curricular and even cognitive needs of chemistry majors. As challenging a subject as titration is for students, pedagogically it is a unifying topic, and the broader chemical principles and kinds of stoichiometric operations that must be mastered to perform simple acid−base titration calculations are de rigueur in analytical chemistry. Although students may initially struggle, the cognitive skill-building that occurs as a result of the added intellectual challenge these calculations present to general chemistry students is ultimately worthwhile. Qualitatively, acid−base titration can be described in a straightforward way; it is a progressive neutralization reaction of an aqueous solution whose acidity or basicity can be determined by knowing which ion, hydronium or hydroxide, is present in the greater quantity at any moment. For simplicity, all activity coefficients will be assumed to be unity, and given its ubiquity in general chemistry textbooks and familiarity to students, hydronium ion will be the term used to refer to the hydrated proton irrespective of its actual configuration in water.11,12 The first two pairs of equations in Table 1 summarize how students are traditionally taught to calculate the pH of mixtures of strong acids and bases. The difference in moles of hydronium and hydroxide ions is divided by the total volume of solution in order to determine the concentration of whichever of the two ions is in excess. These equations produce accurate results so long as the effects of the autoionization of water can be ignored, and this is typically the case in problems given to students. As the solution pH approaches the equivalence point, however, the autoionization of water becomes increasingly important. When mixtures of strong acid and base produce solutions where hydronium and hydroxide ion concentrations are both at submicromolar levels, gross errors can arise if students use either the first or third equation in Table 1 in pH calculations. Ideally, the entirety of acid−base titration could be summarized simply and in as few expressions as possible; in fact, much work has been done and continues to be done to that end.13−23 In the case of titrations involving only strong acids and bases, an exact expression has already been derived

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GENERAL THEORY When strong acids and bases react and neutralize each other, the products are water and a neutral salt as shown in eq 1 where HA represents a strong acid, MOH represents a strong base, and MA represents the neutral salt formed in the reaction. Equation 2 is the total ionic representation of eq 1 and shows that the salt is composed of spectator ions that can be eliminated from the chemical equation. HA(aq) + MOH(aq) → MA(aq) + H 2O(l)

(1)

H+(aq) + A−(aq) + M+(aq) + OH−(aq) → M+(aq) + A−(aq) + H 2O(l)

(2)

The net ionic equation that remains is in fact the autoionization equation of water written in reverse. H+(aq) + OH−(aq) ⇌ H 2O(l)

(3)

Presented in this manner the neutralization that occurs in a mixture of strong acids and bases can immediately be associated with the ion-product constant of water, Kw. K w = [H+]eq [OH−]eq (at equilibrium)

(4)

Initially, however, the amounts of hydronium and hydroxide ions in the mixture are not at equilibrium, and the ion product is instead a reaction quotient Qw. Q w = [H+][OH−] (away from equilibrium)

(5)

Le Châtelier’s principle is a simple and clear articulation of a system’s thermodynamic imperative to reach or return to equilibrium by favoring either the forward or reverse directions of a reaction. The difference between the reaction quotient and the ion-product constant of water serves as a mathematical basis 522



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presented as shown in eq 13, r will always be slightly less than the lesser of l + HA or l + MOH in order to ensure that both quantities remain positively valued.

on which to apply Le Châtelier’s principle to acid−base neutralization. The amounts of hydronium and hydroxide ions, as well as the equilibrium of water, will be represented in terms of moles rather than molarities as the representation in the former will produce a visually simpler expression. Table 2 lists the notation that will be used throughout this article.

(13)

−

(2l + HA + MOH)2 − 4((l + HA)(l + MOH) − K wVt 2) ]/2

(14)

Equation 15 is the resulting expression for the hydronium ion concentration, and eq 16 is the same equation in terms of molarities. The corresponding equations for hydroxide ion concentration are shown in eqs 17 and 18. [H+] =

[H w +] = [OH w −] =

K w (in pure water)

HA − MOH +

(HA − MOH)2 + 4K wVt 2 2Vt (15)

[H+] =

Initially, the amounts of hydronium and hydroxide ions come entirely from water, and because they are equal to each other, they will be represented by l as shown in eqs 6 and 7. It will be possible to show that the moles of hydronium and hydroxide ions contributed by water are effectively always equal to each other in mixtures of strong acids and bases.

[HA]VHA − [MOH]VMOH +

([HA]VHA − [MOH]VMOH)2 + 4K wVt 2 2Vt

(16)

[OH−] =

(6)

(MOH − HA)2 + 4K wVt 2

MOH − HA +

2Vt (17)

K w Vt

(7)

[OH−] =

Equations 8 and 9 show the amount of hydronium ions in an acidic solution in molarities and moles, respectively, as the sum of the contributions from the acid and from water. The same applies to the total amount of hydroxide ions in a basic solution; it is also a sum of the contributions from a base and from water as shown in eqs 10 and 11. [H+] = [H w +] + [HA]

(8)

H+ = l + HA

(9)

−

K wVt 2 = (l + HA − r )(l + MOH − r )

Description

Concentrations of monoprotic strong acid, strong base, [HA], [MOH], hydronium, and hydroxide ions from water, respectively. [Hw+], [OHw−] VHA, VMOH, Vt Volumes of acid, base, and total. HA, MOH Moles of hydronium and hydroxide ions contributed by the strong acid and the strong base, respectively. l Moles of hydronium and hydroxide ions contributed by water. H+, OH− Total moles of hydronium and hydroxide ions. r Re-equilibration term

l=

(12)

r = [2l + HA + MOH

Table 2. Description of Notation Notation

Q wVt 2 = (l + HA)(l + MOH)

−

[OH ] = [OH w ] + [MOH] −

OH = l + MOH

[MOH]VMOH − [HA]VHA +

([MOH]VMOH − [HA]VHA )2 + 4K wVt 2 2Vt

(18)

Alternatively, using charge balance to derive eqs 15−18, the total positive charge must equal the total negative charge in the solution. Here, the concentrations of the acid and the base will pertain to their concentrations after they are mixed together, as shown in eqs 19 and 20. The initial charge balance expression is shown in eq 21 where M+ and A− are the counterions coming from the strong base and strong acid, respectively. The concentrations of M+ and A− can then be replaced by the concentrations of MOH and HA, respectively, and the concentration of hydroxide ion can be replaced using the ion product of water, as shown in eq 22.

(10) (11)

These relationships hold upon mixing; the net effect, however, is that either HA, MOH, or both will react completely, leaving effectively only water to contribute the remaining hydronium or hydroxide ion to the solution.

[HA]mix =

pH of a Mixture of Strong Acid and Strong Base

The strong acid and base in a mixture can be thought of as being added to a volume of pure water. In the language of Le Châtelier’s principle, the equilibrium of water is then disrupted; the addition of strong acid and base increases the amounts of both the hydronium and hydroxide ions in the solution, and makes the reaction quotient numerically larger than the ionproduct constant of water. The disruption causes the forward reaction in eq 3, the neutralization reaction, to become favored in order to restore equilibrium. Equation 12 shows the reaction quotient upon addition of acid and base; eq 13 shows the return to equilibrium through neutralization represented by r. Equation 13 has two roots, and eq 14 is the physically relevant solution to r that must be substituted into (l + HA − r) in order to determine the moles of H+. When the equilibrium of water is

[HA]VHA Vt

[MOH]mix =

(19)

[MOH]VMOH Vt

(20)

[H+] + [M+]mix = [OH−] + [A−]mix [H+] + [MOH]mix =

Kw + [HA]mix [H+]

(21)

(22)

Upon rearrangement of eq 22, eq 23 is obtained, and the equivalent expression in terms of hydroxide ion concentration is shown in eq 24. The solutions to eqs 23 and 24 are eqs 15−18. [H+]2 − [H+]([HA]mix − [MOH]mix ) − K w = 0 523

(23)



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[OH−]2 − [OH−]([MOH]mix − [HA]mix ) − K w = 0

of NaOH, the independent variable is the volume of acid that adds to the total volume as well as increases the moles of acid. Computationally, this means that once the initial concentrations and volumes are known for all species, the volume of titrant becomes the only variable that needs to be adjusted at each increment. The equivalence point is pH = 7 for all the curves, and the product of eqs 15 and 17 is Kw, as expected. In addition to eq 13, there are alternative representations of the water ion-product expression that result in different numerical values and interpretations for r; however, they can all be used to derive eqs 15−18 and are therefore functionally equivalent to each other. They are shown in eqs 25−27.

(24)

Upon comparison of the derivation of eqs 15−18 through the two methods, one of the principle differences between them is with the equation and variable that must be solved. The variables solved using the charge balance method in eqs 23 and 24 are the hydronium and hydroxide ion concentrations, respectively. Using Le Châtelier’s principle, r is the variable that must be solved in eq 13, and the solution must be substituted into (l + HA − r) and (l + MOH − r) in order to determine the moles of hydronium and hydroxide ions, respectively. The r term represents the system’s response to disruption and return to equilibrium. Additionally, the contributions to the hydronium and hydroxide ion concentrations by water and the strong acid and base are individually identifiable; this makes it possible to connect the neutralization occurring between the acid and the base to the return to equilibrium. Using charge balance, however, hydronium and hydroxide ion concentrations are always treated as aggregated quantities, and this hides the chemistry involved. Although the charge balance method is not necessarily inaccessible to first-year students, it is a topic that remains outside of the scope of the general chemistry curriculum. Le Châtelier’s principle, however, not only is part of the curriculum, but also is a bridging concept that serves to connect other concepts related to chemical equilibrium. Despite their differences, however, derivations through both methods converge at the end to produce eqs 15−18; these equations are exact, and apply without modification or segmentation for all concentrations of strong acid and base, throughout the entire range of a titration curve. Of the four, eqs 15 and 17 only involve four terms and may perhaps be less challenging for students to deconstruct and understand. The initial pH of either a strong acid or strong base can be determined by setting either MOH = 0 or HA = 0. Figure 1

K wVt 2 = (HA − MOH + l − r )(l − r )

(25)

K wVt 2 = (l − r )(MOH − HA + l − r )

(26)

K wVt 2 = (l + r − MOH)(l + r − HA)

(27)

These equations are all equivalent to eq 13 because HA and MOH can be viewed as playing one of two roles; they can be viewed either as substances that increase the moles of hydronium and hydroxide ions in solution, respectively, or as substances that reduce the moles of hydroxide and hydronium ions, respectively. As a result, the value and interpretation of r changes in each equation accordingly. In eq 13, conversely, r can clearly be interpreted as a neutralization term at all stages of a titration. In eq 25, however, r is a neutralization term in acidic solutions, and is a dissociation term in basic solutions. The opposite is true for r in eq 26. For completeness, eq 27 is included although it can be confusing and impractical to use; in it, r represents a dissociation term throughout an entire titration. In eq 13 r has a chemical interpretation that is clearly consistent with eq 3 and with Le Châtelier’s principle. In comparing eqs 13, 25, 26, and 27, r can be viewed simply as a compensatory term that ensures that the amounts of hydronium and hydroxide ions are always positively valued. Pedagogically, it is important, however, to present r in a consistent chemical context that remains aligned with the concepts students are learning. It is therefore strongly recommended for general chemistry instruction that solely eq 13 be used to express the ion-product of water. Given their mathematical equivalence, it can nonetheless still be useful in some cases to examine the alternative expressions to eq 13. With eqs 25 and 26, for instance, one can view the overall neutralization process in acid−base mixtures as two neutralizations occurring in tandem; one neutralization is represented by the difference in moles of the strong acid and the strong base; the second neutralization is the difference between l and r and represents the net contributions to both ions from water. It can then immediately be inferred from eqs 25 and 26 that water effectively contributes equally to both hydronium and hydroxide ions; such an inference is not as straightforward with eq 13. Before the equivalence point, the hydronium ion concentration contributed by water is the difference between eq 15 and the remaining concentration of strong acid, which can be calculated using the first equation in Table 1. The result is shown in eq 28, and it applies up to and including the equivalence point.

Figure 1. Simulated titration curves of 25.0 mL samples of 0.10 M (blue), 1.0 mM (red), and 1.0 μM (yellow) HCl titrated by 0.10 M, 1.0 mM, and 1.0 μM NaOH, respectively, and a 25.0 mL sample of 0.10 M (green) NaOH titrated by 0.10 M HCl using the negative log of eq 15.

shows the simulated titration curves obtained using the negative log of either eq 15 or 16 for 25.0 mL samples of 0.10 M, 1.0 mM, and 1.0 μM HCl titrated by 0.10 M, 1.0 mM, and 1.0 μM NaOH, respectively, and a 25.0 mL sample of 0.10 M NaOH titrated by 0.10 M HCl. In the titration of HCl, the volume of base is treated as the independent variable that adds to the total volume as well as increases the moles of base with each increment. In the titration

[H w +] =

(HA − MOH)2 + 4K wVt 2 − (HA − MOH) 2Vt (28)

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by the strong acid and those produced by water are, in principle, indistinguishable from one another.

Beyond the equivalence point, water contributes all the hydronium ion in solution, and those concentrations can be determined using eq 15 alone. The hydroxide ion concentration in acidic solutions comes entirely from water and can be found by dividing Kw by eq 15. Figure 2 is a graph of the net

Determination of the Equivalence Point

Unlike titrations of weak acids or bases where students are taught to determine the equivalence point mathematically, students are often given a chemical rationale to explain why the equivalence point of strong acids and bases is pH = 7 at room temperature. They are taught that the salts formed by the counterions of strong acids and strong bases are neutral. Using eq 15, the equivalence point of strong acids and bases can be quickly determined by setting HA = MOH, and this gives students a mathematical rationale to understand the concept before they consider it for weak acids and bases. [H+] =

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Figure 2. Graph of the concentration of hydronium ion contributed by water versus concentration of hydroxide ion also contributed by water for a simulated titration of 25.0 mL of 0.10 M HCl by 0.10 M NaOH in 1.0 mL increments up to 24.0 mL.

K w (at the equivalence point)

(29)

PEDAGOGICAL CONSIDERATIONS AND RECOMMENDATIONS As mentioned earlier, the derivation and the equation obtained for hydronium ion concentration are not meant to replace the traditional method to calculate the pH of acid−base mixtures. They are meant to augment students’ understanding by providing further context through concepts that are already known. Over the past academic year students at LaGuardia Community College who were enrolled in General Chemistry 2 were given a written activity on acid−base titration after conducting the laboratory experiment. They were presented with both the traditional method taught to determine titration curves of strong acids and bases, and with eq 15; they were then asked to calculate the pH of a 10.0 mL sample of 0.10 M HCl titrated by 0.10 M NaOH in 1.0 mL increments using their method of preference. In the fall semester eq 15 was provided without derivation, and in the spring semester it was derived in a manner similar to how it was presented here. Students provided feedback through a survey in which they were asked, among other questions, if they had been able to complete the activity, if they preferred one method of calculation over the other, and if they considered it appropriate and useful to teach eq 15 to future cohorts. Of note, 87% of students indicated that they found it appropriate and useful for instructors to teach eq 15 irrespective of their preference between the two methods. It was found that 60% of students found the traditional method simpler to follow, as they preferred to have a standard set of steps to solve a problem. The other 40% of students preferred having a single expression at their disposal to do a calculation, and favored using eq 15. In either case, students appreciated being taught a new way to understand acid−base titration and perform pH calculations. The majority of students, 61%, were able to complete all the calculations on their data sheet in the time available; however, it was not imperative for them to do so. At the beginning of the activity students were told it was much more important for them to learn how to perform the calculations, and that they should not feel rushed in any way. At the end of the activity only 7% of students continued to experience difficulties with the calculations. It is this author’s opinion that these results are encouraging and show that, at least at a basic level, acid−base titration calculations are well within the abilities of first-year students to grasp. For instructors who would like to introduce this material to their students outside of a formal lecture, the activity, survey, and aggregated results of the survey can be found in the Supporting Information. Alternatively, some may consider

contributions of water to hydronium and hydroxide ion concentrations relative to each other and is for the simulated titration of 25.0 mL of 0.10 M HCl by 0.10 M NaOH in 1.0 mL increments up to 24.0 mL. The data points are presented up to 24.0 mL of NaOH rather than 25.0 mL so as to not obscure the earlier data points, most of which are at sub-picomolar concentrations. Figure 3 is a complement to Figure 2 and

Figure 3. Graph of the concentration of hydronium ion contributed by water (line) and of hydroxide ion contributed by water (scatter marker) versus titrant volume, for a simulated titration of 25.0 mL of 0.10 M HCl by 0.10 M NaOH in 1.0 mL increments up to 24.0 mL.

shows the effective contributions of water to both ions as the solution nears the equivalence point. As the titration of strong acid continues, water effectively contributes an increasing proportion of both ions; at the equivalence point water effectively contributes entirely to both. Considered from another viewpoint, Figure 3 can also be used as a graphical illustration of the common ion effect. Water is a very weak electrolyte, the presence of the strong acid, a strong electrolyte, suppresses the dissociation of water because they both share the hydronium ion in common. As the strong acid is progressively neutralized, the degree of dissociation in water increases. It is evident from both Figures 2 and 3 that the effective amounts contributed by water to both ions are equal to each other at each point in a titration. For clarity, it must be reiterated that these are effective contributions. It is just as valid to imagine that the neutralization between the strong acid and strong base leaves a residual amount of both ions in order to satisfy equilibrium requirements. The net effect, however, is the same in both cases; once formed, the hydronium ions produced 525



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ORCID

introducing this derivation using the tabular method that catalogs initial, change, and equilibrium concentrations, commonly known as the ICE method, especially since it is used to teach equilibrium determinations for weak acids. It is recommended that at least two hours be set aside in a laboratory session or recitation, and that the session be highly interactive. Students are very helpful to each other in these settings, and this allows conceptual misunderstandings to surface and often be quickly addressed by peers. It is also recommended that students have calculators at their disposal that can easily accommodate long expressions on a single screen such as graphing calculators; otherwise, it is quite easy for computational errors to be made. Alternatively, if computer laboratories are available, students could perform the calculations in Excel. There is, however, a noticeable consolidating benefit to the students when they repeat these calculations for each pH value in the titration curve; using Excel makes the repetition unnecessary. Consequently, while they may complete the activity more quickly, the opportunity for knowledge consolidation through concerted practice may be lost. Although all the material used in the derivation will be familiar to students, the approach to the topic will nonetheless seem new. This should therefore engage not only students who understand the concepts well but also students who still have more to learn. The derivation of eq 15 connects many topics through Le Châtelier’s principle that are taught in general chemistry: mole and molarity, acid−base neutralization and titration, molecular and net ionic equations, the common ion effect, and chemical equilibrium. Bringing all of these concepts together can help instructors establish and reinforce a strong mathematical foundation before they introduce students to the more nuanced chemistry of weak acids and bases.

Philippe H. Mercier: 0000-0002-3334-5588 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author is indebted to his many students in SCC202, the General Chemistry 2 course taught here at LaGuardia Community College, who over the years were always engaged and disposed to exploring topics discussed in class and laboratory more deeply. The author would also like to thank his colleagues Amit Aggarwal and Jennifer Vance who both helped immensely with obtaining student feedback in their sections of General Chemistry 2 and for their insights regarding acid−base chemistry. Finally, the author would like to thank the reviewers of the manuscript who offered many helpful suggestions.

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(1) Paik, S.-H. Understanding the Relationship Among Arrhenius, Brønsted-Lowry, and Lewis Theories. J. Chem. Educ. 2015, 92 (9), 1484−1489. (2) Furió-Más, C.; Calatayud, M. L.; Guisasola, J.; Furió-Gómez, C. How are the Concepts and Theories of Acid-Base Reactions Presented? Chemistry in Textbooks and as Presented by Teachers. Int. J. Sci. Educ. 2005, 27 (11), 1337−1358. (3) Nurrenbern, S. C.; Pickering, M. Concept Learning versus Problem Solving: Is There a Difference? J. Chem. Educ. 1987, 64 (6), 508−510. (4) Steiner, R. Teaching Stoichiometry. J. Chem. Educ. 1986, 63 (12), 1048. (5) Krieger, C. Stoogiometry: A Cognitive Approach to Teaching Stoichiometry. J. Chem. Educ. 1997, 74 (3), 306−309. (6) Bodner, G. The Role of Algorithms in Teaching Problem Solving. J. Chem. Educ. 1987, 64 (6), 513−514. (7) Fishel, L. Dilution Confusion: Conventions for Defining a Dilution. J. Chem. Educ. 2010, 87 (11), 1183−1185. (8) Hawkes, S. Discussion: Should Students Calculate Titration Curves? J. Chem. Educ. 2008, 85 (4), 498. (9) Harris, D. In Partial Defense of Calculating Titration Curves. J. Chem. Educ. 2008, 85 (4), 498. (10) American Chemical Society Committee on Professional Training. Undergraduate Professional Education in Chemistry: ACS Guidelines and Evaluation Procedures for Bachelor’s Degree Programs. https://www.acs.org/content/dam/acsorg/about/ governance/committees/training/2015-acs-guidelines-for-bachelorsdegree-programs.pdf (accessed February 2018). (11) Silverstein, T. P. The Aqueous Proton Is Hydrated by More Than One Water Molecule: Is the Hydronium Ion a Useful Conceit? J. Chem. Educ. 2014, 91 (4), 608−610. (12) Reed, C. A. Myths about the Proton. The Nature of H+ in Condensed Media. Acc. Chem. Res. 2013, 46 (11), 2567−2575. (13) González-Gómez, D.; Rodriguez, D. A.; Cañada-Cañada, F.; Jeong, J. S. A Comprehensive Application to Assist in Acid-Base Titration Self-Learning: An Approach for High School and Undergraduate Students. J. Chem. Educ. 2015, 92 (5), 855−863. (14) de Levie, R. Explicit Expressions of the General Form of the Titration Curve in Terms of Concentration. J. Chem. Educ. 1993, 70 (3), 209−217. (15) de Levie, R. General Expressions for Acid-Base Titrations of Arbitrary Mixtures. Anal. Chem. 1996, 68 (4), 585−590. (16) Glaister, P. A Unified Titration Formula. J. Chem. Educ. 1999, 76 (1), 132. (17) Pardue, H. L.; Odeh, I. N.; Tesfai, T. M. Unified Approximations: A New Approach for Monoprotic Weak Acid-Base Equilibria. J. Chem. Educ. 2004, 81 (9), 1367−1375.

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CONCLUSION Acid−base titration is a very important but challenging topic in a general chemistry curriculum because it draws on many other topics, each requiring considerable conceptual and mathematical understanding. Le Châtelier’s principle was used as a basis to derive an exact expression to model titrations involving strong acids and bases that applies throughout the entire range of a titration curve without modification or segmentation. It was shown that the derivation provides instructors the opportunity to review earlier topics, and to connect them to chemical equilibrium and titration curves before introducing students to the chemistry of weak acids and bases.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00575. Handout given to students on acid−base titration (PDF) Student data sheet where calculations were recorded (PDF) Fall and spring semester surveys, with aggregated survey results (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 526



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