Unified Approximations: A New Approach for Monoprotic Weak Acid

Research: Science and Education

Unified Approximations: A New Approach for Monoprotic Weak Acid–Base Equilibria Harry L. Pardue,* Ihab N. Odeh, and Teweldemedhin M. Tesfai Department of Chemistry, Purdue University, West Lafayette, IN 47907-2084; *[email protected]

Few subjects included in chemistry curricula have inspired as much interest as acid–base chemistry. A search of the JCE Online index suggested during the review of this manuscript identified some 700 articles published on the subject. In addition to the primary references (1–9), the literature cited includes a chronological listing of selected articles (10–98) published in this Journal since 1936. This article focuses on a limited but important aspect of acid–base equilibria. As is the case with introductory treatments in most general and analytical texts, this article focuses on calculations of hydrogen ion concentration for situations involving monoprotic weak acids and weak bases in otherwise pure water. Unless noted, it is assumed throughout that analytical concentrations of the weak acids or weak bases are larger than about 2 × 10᎑6 M and that approximation errors of ±5% or less are satisfactory. Classical approaches to equilibrium calculations (1–3) were developed before the advent of electronic calculators and computers. Because the only tools students had to solve numerical problems were log tables and slide rules, procedures were designed to give the simplest possible mathematical solution for each problem. More than 40 years of observing students struggle with the classical approach to acid–base equilibria has convinced the senior author that there must be better options. The need to master a variety of different situations resulting from classical approximations frustrates students (4), increases the possibility of conceptual errors (5), and, most importantly, detracts from time and effort that could be devoted to other equally important aspects of acidbase chemistry. More recently, graphical methods have been developed (6–8) that help students identify simplified relationships analogous to those resulting from classical procedures. Spreadsheet procedures based on the use of pointer functions have also been used for computer-assisted solutions of equilibrium problems (6). Still more recent articles have described exact equations for quantities of titrant added versus some monitored response such as pH or electrochemical potential (9). Despite these developments, introductory treatments in most general and analytical texts tend to focus on approximate calculations of hydrogen ion concentrations. This article describes a new approach for such approximations, identified herein as unified approximations. A simple decision criterion is used to determine if each situation should be treated as a deprotonation reaction or a protonation reaction. Depending on the results of this decision, one of two similar procedures is to develop and solve a quadratic equation that gives hydrogen ion concentrations within 5% of the correct value for all conjugate acid–base concentrations larger than about 2 × 10᎑6 M. As is shown later, the unified approximations give satisfactory results for a range of situations for which 10 different sets of decisions, simplifying assumptions, mathematical www.JCE.DivCHED.org

•

procedures, and equations are needed using classical procedures. Advantages of the unified approach relative to classical (1–3) and graphical (6–8) methods are that it can be mastered in a small fraction of the time required for these options and it includes a decision criterion that virtually eliminates conceptual errors. These features improve student morale and leave more time for other equally important topics. Mathematical Description Given that it is relatively easy to use known values of the hydrogen ion concentration with fractional equations to calculate equilibrium concentrations of conjugate acids and bases, this article focuses on calculations of hydrogen ion concentrations. Although activity effects are ignored for sake of convenience, the same principles apply when activity effects are taken into account.

Generalized Problem It will be convenient to use the following generalized form of a typical problem involving a monoprotic conjugate acid–base pair: Given a monoprotic weak acid, BH, with a deprotonation constant, Ka, calculate the hydrogen ion concentration in an aqueous solution containing analytical concentrations, CBH ≥ 0 and CB ≥ 0, of the conjugate acid–base pair.

Treat as a Deprotonation or Protonation Reaction? The first step is to decide if a situation should be treated as a deprotonation or protonation reaction. An initial reaction quotient is used to develop a decision criterion for this purpose. Writing the ratio of the initial reaction quotient to the deprotonation constant, Q 0Ka, in terms of initial concentrations and recognizing that the initial hydrogen ion concentration in pure water at 25 ⬚C will be CH = 1 × 10᎑7 M, the following relationship can be obtained: −7 Q0 1 C HC B 1 1 × 10 C B = = Ka K a C BH Ka C BH

(1a)

If the ratio of the initial reaction quotient to the deprotonation constant is less than unity, that is, if Q 0Ka < 1, it will be necessary for the reaction to increase the hydrogen ion concentration above 1 × 10᎑7 M to reach equilibrium. Setting the right-most part of eq 1a to be less than 1, the resulting inequality is easily rearranged into the following form, −7 1 1 × 10 CB < 1 Ka C BH

rearrange

⇒

Vol. 81 No. 9 September 2004

•

DC D /P ≡ K a

(1b) C BH > 1 × 10 −7 M CB

Journal of Chemical Education

1367


equilibrium-constant relationships. The following set of reactions would be used for situations in which KaCBHCB > 1 × 10᎑7 M.

in which DCD/P is a decision criterion to determine whether a situation should be treated as a deprotonation or protonation reaction. Any combination of a deprotonation constant and analytical concentrations that satisfies this inequality will require a deprotonation reaction to increase the hydrogen ion concentration above 1 × 10᎑7 M thereby making the solution acidic. Any combination giving an inequality less than 1 × 10᎑7 M will require a protonation reaction to decrease the hydrogen ion concentration, thereby making the solution basic. Examples illustrating the benefits of this decision criterion are discussed later.

BH

+ − H + B + − H + OH

H2O

The long arrow from left to right in the deprotonation reaction reflects the fact that the weak acid donates protons making the solution more acidic. The long arrow from right to left in the autoprotolysis reaction reflects the fact that, according to LeChâtelier’s principle, hydrogen ion produced by the deprotonation reaction will suppress the autoprotolysis reaction. For situations in which KaCBHCB < 1 × 10᎑7 M, an analogous set of reactions involving protonation (hydrolysis) of a weak base would be used.

Reactions and Equilibrium Constant Expressions Having decided if a given situation will involve a deprotonation (dissociation) reaction or a protonation reaction, a student would start with the appropriate set of reactions and

Table 1. Summary of Selected Relationships for Situations Involving Deprotonation and Protonation Reactions Deprotonation Reactionsa,b

If:

Ka

Protonation Reactionsb

C BH ≥ 1 × 10 −7 M CB

H+

Ka =

( 3a)c

B−

BH

BH = C BH −

H+

−

B−

H+

−

=

CB

+

Kw H

+

Kw H

+

If:

Ka

C BH < 1 × 10 −7 M CB

(4a)c

OH − BH

(3b)c

Kb =

(4b)c

(3c)c

BH = C BH +

OH − −

(3d)e

B−

OH − −

B−

= CB

−

Kw

(4c)d

OH −

Kw

(4d)f

OH −

aSee

p 195 in (1); bTerms in curved parentheses and stylized parentheses represent effects of concentration changes and autoprotolysis, respectively. The latter terms are ignored in first approximation calculations.; cCBH > 0; dCBH ≥ 0; eCB ≥ 0; fCB > 0.

Table 2. Forms of Exact Equations Suitable for Iterative Solutions Decision Criterion

If:

Ka

C BH ≥ 1 × 10 −7 M CB

C BH < 1 × 10 −7 M If: K a CB

1368

Exact Equations

− (C B + K a ) +

Then:

H

+

=

− (C BH + K b ) + OH

•

−

Ka Kw H+

(5a)

2

Then:


(C B + K a )2 + 4 K a C BH + K w +

(C BH + K b )2 + 4 K bC B + K w +

=


KbKw OH −

2

•

www.JCE.DivCHED.org

(5b)


• What is the relationship between hydrogen ion and hydroxide ion concentrations produced by autoprotolysis?

H+

H2O

H2O

eq 2a

H+

⇒

= H+

rearrange

⇒

H

eq 2b

BH

+ H+

+

=

constant

BH

=

H+ (2b)

H+

H2O

Kw

−

H+

(2c)

• How is the equilibrium concentration of the weak acid related to the analytical concentration of the acid?

BH = CBH − H +

BH

eq 2c

BH = C BH −

⇒

H+

−

Kw H+

(2d)

• How is the equilibrium concentration of the conjugate base related to the analytical concentration of the base?

B−

= CB + H +

BH

eq 2c

⇒

B−

= CB +

− 5% -10

equation for [ OH− ]

-20

H+

www.JCE.DivCHED.org

Kw

−

H+

•

1

2

3

4

5

6

Figure 1. Percentage errors versus trial number for iterative calculations of hydrogen ion and hydroxide ion concentrations. Conditions: Ka = 2.3 x 10᎑11 M; CBH = 5.0 x 10᎑3 M; CB = 5.0 x 10᎑2 M; and [H+] = 3.73 x 10᎑12 M. Plots obtained: (䊏) by using the decision criterion or (䊐) not using the decision criterion.

Kw

Kw

=

H+

0

Trial Number

• How is the total hydrogen ion concentration, [H+], related to the hydrogen ion concentrations produced by the weak acid, [H+]BH, and autoprotolysis, [H+]H2O?

H+

+ 5%

(2a)

H2O

autoprotolysis

H2O

10

-40

= OH −

= OH −

equation for [ H+ ]

-30

• What is the relationship between the total hydroxide ion concentration, [OH ᎑], and that produced by autoprotolysis, [OH᎑]H2O?

OH −

20

Error (%)

Relationships among Analytical and Equilibrium Concentrations Whereas charge-balance and mass-balance equations are commonly used to develop expressions for concentrations of the conjugate acid–base pair (1), an alternative approach that emphasizes sources of hydrogen ion and hydroxide ion concentrations is described. Students can discover the quantitative features of deprotonation and protonation reactions by answering several simple questions as illustrated below for the deprotonation and autoprotolysis reactions described above. Stylized brackets are used to highlight terms that will be ignored in first approximation calculations.

(2e)

This process not only identifies sources of hydrogen ion and hydroxide ion concentrations explicitly but also leads students quickly and easily to expressions for equilibrium concentrations of the conjugate acid–base pair. Answering similar questions for protonation reactions leads to an analogous set of relationships. Relationships obtained for situations involving both deprotonation and protonation reactions are summarized in Table 1. However the relationships in Table 1 are developed, they can be used for virtually any approach to solving problems.

Iterative Approach to Exact Equations After substituting eqs 3c and 3d and eqs 4c and 4d into eqs 3b and 4b, respectively, the results can be rearranged into the forms in Table 2. These equations are easily solved using an iterative process. Whichever equation is used, a first approximation is calculated by ignoring the term in the stylized parentheses ( ). Then, subsequent approximations are obtained by including the term in the stylized parentheses with each successive value of the hydrogen ion or hydroxide ion concentration. The convergence processes for eqs 5a and 5b for a situation for which the decision criterion, KaCBHCB < 1 × 10᎑7 M, are illustrated in Figure 1. Notice that eq 5b gives a satisfactory result (Error ≅ 2 × 10᎑3%) on the first iteration whereas eq 5a requires three iterations to converge to within 5% of the correct result. Reasons for this behavior as well as conditions for which it is expected to be valid are easily explained by selective treatment of the autoprotolysis terms in eqs 3c, 3d, 4c, and 4d in Table 1. Equation 3c is used here as an illustrative example. An acidic solution must always contain a finite concentration of a weak acid initially, that is, CBH > 0. Accordingly, it is reasonable to assume that there will be some minimum initial weak-acid concentration above which the autoprotolysis term in parentheses in eq 3c will contribute 5%


•


1369


or less to the final result. For situations involving pH ≤ 7, the maximum contribution of autoprotolysis to the hydrogen ion concentration will occur in neutral solution when [H+] = √KW. This relationship can be used as follows to identify conditions for which the autoprotolysis term in eq 3c will be less than 5% of the analytical concentration of the weak acid.

Kw H

+

≤ 0.05 C BH;

H+

Kw ;

=

(6)

C BH ≥ 20 K w = 2 × 10 −6 M

The autoprotolysis term will be less than 5% of the initial weak-acid concentration for all weak-acid concentrations equal to or larger than 2 × 10᎑6 M. Analogous reasoning applies for eq 4d. The autoprotolysis term will be less than 5% of the initial weak-base concentration for all weak-base concentrations equal to or larger than 2 × 10᎑6 M. The validity of these conclusions has been confirmed using an iterative spreadsheet program to solve eqs 5a and 5b.

Unified Approximations Simplified equations in Table 3 can be obtained either by dropping the terms in the stylized parentheses in eqs 5a and 5b or by dropping the terms in the stylized parentheses in eqs 3c and 4d, substituting the simplified equations into the equilibrium-constant equations, rearranging, and using the quadratic formula to obtain the final equations. If the decision criterion is used to decide between deprotonation and protonation reactions, then these simplified equations will give results valid to within 5% for all situations satisfying the criteria, CBH ≥ 2 × 10᎑6 M (eq 5a) and CB ≥ 2 × 10᎑6 M (eq 5b). Performance characteristics of the unified approximations are discussed later. However, before doing that, it will be instructive to compare the relationships based on classical approximations needed to give satisfactory results for the range of situations covered by the unified approximations. Classical Approximations The mathematical relationships based on classical approximations needed for all situations for which the two equations in Table 3 give satisfactory results are summarized in

Table 4. These relationships are obtained by dropping appropriate terms in eqs 3c, 3d, 4c, and 4d, substituting the results into the appropriate equilibrium-constant relationship, and rearranging. As noted in footnotes a and b in Table 4 for weak acids or weak bases alone, some simple decision criteria can be used to determine if concentration changes and or autoprotolysis should be ignored. As implied by footnote c, a two-step process involving a comparison of a first estimate of the hydrogen ion or hydroxide ion concentration with the smaller of the two analytical concentrations is used for mixtures of conjugate acid–base pairs. Although eqs 10a and 10c give equivalent results, it is important to use the equation giving the larger concentration to determine if the concentration change can be ignored relative to the smaller of the two analytical concentrations. As noted earlier, the classical approach requires 10 sets of assumptions, procedures, and mathematical relationships to give results equivalent to those obtained using one decision criterion and two similar sets of procedures and relationships using the unified approximations. Performance Characteristics of Unified Approximations Results obtained using an iterative program to solve the exact equations in Table 2 were used to evaluate several performance characteristics of the unified approximations. Conclusions are summarized below.

Effects of Deprotonation Constants Effects of different deprotonation constants on percentage errors obtained for equal concentrations of a conjugate acid–base pair, namely CBH = CB = 5 × 10᎑3 M are illustrated in Figure 2. Square symbols represent results obtained using the equation for hydrogen ion concentration and triangular symbols represent results obtained using the equation for hydroxide ion concentration and [H+] = Kw[OH᎑]. The intersection of the two plots is the point at which the decision criterion in eq 1b would suggest a change from one set of reactions to the other. Filled symbols and solid lines represent errors obtained by using the decision criterion to determine whether to treat each situation as a deprotonation reaction or a protonation reaction. Open symbols and dashed lines represent errors that would be obtained if each situation were treated as either a

Table 3. Unified Approximations for Situations Involving Monoprotic Acids and Bases in Otherwise Pure Water Decision Criteria

Quantitative Relationships

If:

C BH ≥ 2 × 10 −6 M C K a BH ≥ 1 × 10 −7 M CB

Then:

If:

C B ≥ 2 × 10 −6 M C K a BH < 1 × 10 −7 M CB

Then:

1370


H+

OH

•

≅

−

− (C B + K a ) +

(C B + K a )2 + 4 ( K aC BH + K w ) 2

≅

− (C BH + K b ) +

(C BH + K b ) 2 + 4 ( K bC B + K w )


2

•

www.JCE.DivCHED.org

(7a)

(7b)


deprotonation or protonation reaction throughout the full range of deprotonation constants. Clearly, use of the decision criterion avoids errors in excess of 10% that could occur if all situations were treated exclusively as deprotonation or protonation reactions. The changes in slopes that occur at large and small values of pKa result from the fact that the hydrogen ion con-

centration increases more rapidly than the absolute error in the hydrogen ion concentration in these regions.

Effects of Conjugate-Base Concentration The effects of changes in the conjugate-base concentration for fixed values of the deprotonation constant, Ka = 1 × 10᎑7 M, and initial weak-acid concentration, 1 × 10᎑5 M,

Table 4. Combinations of Initial Concentrations and Deprotonation Constants Corresponding to 10 Situations Resulting from Classical Approximations Row

Ignore Concentration Change

Ignore Autoprotolysis

Approximate Equations (Error < 5%; CBH ≥ 2 x 10᎑6 M or CB ≥ 2 x 10᎑6 M) Weak Acids Alone: CBH > 0 M, CB = 0 M

1

Yesa

Yesb

H+

≅

2

No

Yesb

H+

≅

3

Yesa

No

H+

≅

K a C BH

−K a +

(8a)

K a 2 + 4K a C BH 2

K a C BH + K w

(8b)

(8c)

Weak Bases Alone: CBH = 0 M, CB > 0 M 4

Yesa

Yesb

OH −

≅

5

No

Yesb

OH −

≅

6

Yesa

No

OH −

≅

K bC B

−K b +

(9a)

K b 2 + 4 K bC B 2

K bC B + K w

(9b)

(9c)

Mixture of Conjugate Acid–Base Pairs: KaCBH/CB > 1 x 10᎑7 M 7

Yes

Yes

H+

8

Noc

Yes

H+

1

≅ Ka

≅

C BH CB

(10a)

(C B + K a )2 + 4 K a C BH

− (C B + K a ) +

2

(10b)

Mixture of Conjugate Acid–Base Pairs: KaCBH/CB < 1 x 10᎑7 M 9

Yes

Yes

OH −

10

Noc

Yes

OH −

a

If C/K ≥ 100.

b

If KC ≥ 1 x 10᎑13 M2.

c

1

≅ Kb

≅

CB C BH

(10c)

− (C BH + K b ) +

(C BH + K b )2 + 4 K bC B 2

(10d)

If the first estimate, [H+]1 or [OH᎑]1 is larger than 5% of the smaller of CBH and CB .

www.JCE.DivCHED.org

•


•


1371


102

b'

a' 10

100 10-1 -2

10

10-3

equation for [OHⴚ ]

equation for [H + ]

10-4 10-5 10-6

a

10-7

b

Unsigned Relative Error (%)

Unsigned Relative Error (%)

10

b'

a'

5% error

1

2

5% error

100

a 10-2

equation for [OHⴚ ] ⴙ

equation for [H ] 10-4

10-6

b

10-8 10-8

0

2

4

DCD/P > 1 × 10

6

8

−7

10

12

DCD/P < 1 × 10

14

1

2

3

4

5

6

7

pC B

−7

pK a Figure 2. Effects of the deprotonation constant on relative errors for equal concentrations of conjugate acid–base pairs (CBH = CB = 5 x 10᎑3 M): [a, b (䊏, 䉱)] results obtained using the decision criterion and [a’, b’ (䊐, 䉭)] results that would be obtained if reactions were treated as deprotonation or protonation reactions throughout.

Figure 3. Effects of conjugate-base concentration on relative errors for fixed weak-acid concentration and deprotonation constant (CBH = 1 x 10᎑5 M; pKa = 7.0): [a, b (䊏, 䉱)] results obtained using the decision criterion and [a’, b’ (䊐, 䉭)] results that would be obtained if situations were treated as either deprotonation or protonation reactions throughout.

are illustrated in Figure 3. The plotting format is the same as that described for Figure 2, with solid points and lines representing results obtained by the proposed procedure and the intersection of the two plots representing the change-over point between deprotonation and protonation reactions. As with Figure 2, use of the decision criterion to guide the correct process leads to errors less than 5% for all situations and avoids errors in excess of 100% that could occur for some situations if the decision criterion were not used.

As an example, the conjugate-base concentration corresponding to the largest error for a deprotonation constant of Ka = 1.8 × 10᎑5 M and weak-acid concentration of CBH = 1 × 10᎑3 M would be CB,max = 0.18 M.

Conditions for Maximum Error Two important features that can be generalized to situations involving weak-acid concentrations and deprotonation constants larger than 1 × 10᎑7 M are also illustrated in Figure 3. First, errors for acidic solutions increase with increasing weak-base concentrations. Second, the error is a maximum at the point at which the decision criterion, Q0Ka, is 1 × 10᎑7 M. Given that the conjugate-base concentration, CB,max, corresponding to the maximum error in acidic solutions occurs when the decision criterion is 1 × 10᎑7 M, the decision criterion can be rewritten and rearranged as follows: Ka

C BH = 1 × 10 −7 M C B, max rearrange

⇒

(11)

7

C B, max = 10 K a C BH

Summary of Performance Characteristics Some consequences of the above observations are summarized in Figure 4. The dashed line near the bottom of the figure corresponds to the minimum weak-acid or conjugatebase concentration, namely 2 × 10᎑6 M, to ensure errors of 5% or less. Square points and the associated line for pKa < 7 represent the maximum conjugate-base concentrations for which an error of 5% can be assured at each value of pKa assuming a weak-acid concentration of 2 × 10᎑6 M. Results were calculated using eq 11 in the following form: C B,max = 10 7K a C BH, min

(


•

(12)

Triangular points and the associated line for pKa > 7 represent the maximum weak-acid concentrations for which an error of 5% can be assured at each value of pKa assuming a weak-base concentration of 2 × 10᎑6 M. Results were calculated using an equation analogous to eq 11 as follows:

C BH,max = 10 7K a C B, min

(


)

= 10 7K b 2 × 10 −6 M = 20 K b

for C BH, K a ≥ 1 × 10 −7 M

1372

)

= 10 7K a 2 × 10 −6 M = 20 K a

•

www.JCE.DivCHED.org

(13)


Errors ≤ 5% Are Not Assured Region 5 on Figure 4 corresponds to analytical concentrations of the conjugate acid–base pair less than 2 × 10᎑6 M. The unified approximations, like classical approximations, will fail for some very low weak-acid–weak-base concentrations. Although additional approximate equations can be developed based on assumptions that acids or bases are fully protonated, these equations also fail for many situations. Given that it is very difficult to identify situations for which these additional equations will and will not give satisfactory results, it probably is best to use iterative solutions to exact equations for situations in this region. Discussion Although classical approximations decrease the mathematical complexity of monoprotic acid–base equilibria, they increase the conceptual complexity by increasing the number of different situations that students must master. Graphical methods combined with proton balance equations (6–8) guide students to the most significant species and simplified mathematical relationships. However, these procedures require a multi-plot graph for each situation (8) and are included in only one undergraduate text (7) with which the authors are familiar. The unified approximations reduce the conceptual complexity by combining solutions for a relatively large number of different situations into just two similar sets of processes. A simple decision criterion is used to differentiate between the two processes. The two processes yield results for some types of situations not included in existing texts. Processes used to solve problems by either the unified or classical ap-

www.JCE.DivCHED.org

•

CBH = 2 × 10

ⴚ6

CB = 2 × 10

ⴚ6

axis axis

101

101

b

a

Region 2 basic

10ⴚ1

100

Region 3 acidic

C B−,max

10ⴚ1

CBH,max

10ⴚ2

10ⴚ2

10ⴚ3

10ⴚ3

Region 1 acidic

10ⴚ4 10ⴚ5 10

ⴚ6

2 × 10

Region 4 basic

CBH,min

ⴚ6

CB,min

10ⴚ4 10ⴚ5

2 × 10

ⴚ6

Region 5 exact equation

10ⴚ7

0

2

4

6

8

CBH / (mol/L)

100

CB / (mol/L)

The solid lines in Figure 4 represent not only maximum weak-acid or conjugate-base concentrations to ensure errors ≤ 5% but also the points at which the decision criterion would differentiate between acidic and basic solutions. Accordingly, the region above the horizontal dashed line at C = 2 × 10᎑6 M can be grouped into four regions. Regions 1 and 3 correspond to situations that would be treated as deprotonation reactions and regions 2 and 4 correspond to situations that would be treated as protonation reactions. Approximation errors will be 5% at all points on the solid lines and less than 5% for all other points above the horizontal dashed line at C = 2 × 10᎑6 M. One reason the equations fail at lower concentrations is related to the autoprotolysis terms that are ignored in eqs 3c and 4d. For example, at low concentrations of a weak acid, hydrogen ion from autoprotolysis tends to suppress deprotonation of the acid. Because this effect is reflected in the autoprotolysis term that is ignored in simplifying eq 3c, the unified approximations tend to give larger errors at lower weak-acid or weak-base concentrations. For deprotonation reactions, the only situations for which it has been found that the usual classical approximations give smaller errors than the unified approximations are those for which CB = 0 and KaCBH < 1 × 10᎑14 M2. For such situations, eq 8c tends to overestimate the hydrogen ion concentration by a smaller percentage than eq 7a underestimates the concentration. Analogous observations apply for protonation reactions.

10

ⴚ6

10ⴚ7

10

12

14

pK a Figure 4. Conditions to assure approximation errors of 5 % or less: [a (䊏)] maximum conjugate-base concentrations for errors ≤ 5%; [b (䉱)] maximum conjugate-acid concentrations for errors ≤ 5%; regions 1 and 3—use equation for hydrogen ion concentration; regions 2 and 4—use equation for hydroxide ion concentration; and region 5—use exact equations.

proximations require similar degrees of understanding of the underlying chemical processes. As mentioned earlier, the reduced time required for students to master the unified approximations frees more time for other topics, including a subjective understanding of acid–base equilibria. One disadvantage of the unified approximations is that they require solution of a quadratic equation for each situation. This disadvantage is offset by the decreased instruction time required for this option, the reduced opportunities for conceptual errors and the fact that the quadratic equations that must be solved are just slightly more complex than those required for many situations using classical approximations. Moreover, terms that are ignored in classical approximations tend to be negligibly small in the unified approximations. As examples, the (C + K)2 terms and the Kw terms in eqs 7a and 7b will frequently be negligibly small relative to the KC terms. Although the unified approximations were developed for monoprotic acids and bases, they can also be used when monoprotic approximations are applied to polyprotic acids and bases. Also, for most titrations of monoprotic weak acids or weak bases, one equation can be used to describe all parts of a titration curve from the beginning to the equivalence point. Regarding weak-acid or weak-base titrations, the unified approximations are particularly useful for calculations of pH versus titrant volume up to and including the equivalence point. However, an alternative procedure (9) is more useful for calculations of titrant volume for specified values of pH throughout a titration curve, including regions after the equivalence point.


•


1373


The authors’ approach to teaching the different options is to begin by describing some of the simpler results of classical approximations, to provide an overview of the full range of classical approximations, and then to describe the unified approximations. All students to whom the options have been presented, including chemistry, health-science, and food-science majors, have been unanimous in their preferences for the unified approximations. Literature Cited 1. Harris, D. C. Quantitative Chemical Analysis, 6th ed.; W. H. Freeman and Company: New York, 2003; pp 181–199. 2. Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 7th ed.; Saunders College Publishing: Philadelphia, PA, 1996; pp 137–146. 3. Christian, G. D. Analytical Chemistry, 5th ed.; John Wiley & Sons: New York, 1994; pp 182–196. 4. Adcock, J. L. Teaching Brønsted-Lowry Acid–Base Theory in a Direct Comprehensive Way. J. Chem. Educ. 2001, 78, 1495– 1496. 5. Po, H. N.; Senozan, N. M. Henderson–Hasselbalch Equation: Its History and Limitations. J. Chem. Educ. 2001, 78, 1499– 1503. 6. Freiser, H. Concepts and Calculations in Analytical Chemistry, A Spreadsheet Approach; CRC Press: Boca Raton, FL, 1992; pp 68–74. 7. de Levie, R. Principles of Quantitative Chemical Analysis; McGraw Hill: New York, 1997; pp 51–74. 8. de Levie, R. Aqueous Acid–Base Equilibria and Titrations; Oxford Press: Oxford, 1999; pp 6–22. 9. de Levie, R. Explicit Expressions of the General Form of the Titration Curve in Terms of Concentration: Writing a Single Closed-Form Expression for the Titration Curve for a Variety of Titrations Without Using Approximations or Segmentation. J. Chem. Educ. 1993, 70, 209–217. 10. Glascoe, P. M. A Chapter in Teaching Acids, Bases, and Salts. J. Chem. Educ. 1936, 13, 68. 11. Foster, L. S. Bibliography of Elementary Discussions of (1) the Interionic Attraction Theory of Electrolytes, and (2) Modern Theory of Acids, Bases, and Salts. J. Chem. Educ. 1936, 13, 445. 12. Briscoe, H. T. Teaching the New Concepts of Acids and Bases in General Chemistry. J. Chem. Educ. 1940, 17, 128. 13. Hammett, L. P. The Theory of Acids and Bases in Analytical Chemistry. J. Chem. Educ. 1940, 17, 131. 14. Johnson, W. C. The Advantages of the Older Methods (Theories of Acids and Bases). J. Chem. Educ. 1940, 17, 132. 15. Hazlehurst, T. H. Pictures of Acid–Base Reactions. J. Chem. Educ. 1940, 17, 374. 16. Porges, N.; Clark, T. F. Graphical Correlation between pH Values, Molarities, and Dissociation Constants of Weak Acids. J. Chem. Educ. 1940, 17, 571. 17. Ginell, R. Acids and Bases: A Critical Reevaluation. J. Chem. Educ. 1943, 20, 250. 18. Logan, T. S. The Presentation of Acids and Bases in Textbooks. J. Chem. Educ. 1949, 26, 149. 19. De Ford, D. D. The Bronsted Concept in Calculations Involving Acid–Base Equilibria. J. Chem. Educ. 1950, 27, 554. 20. Park, B. J. Chem. Educ. 1953, 30, 257. 21. Devor, A. W. The Bronsted Theory Applied to Acid–Base Bal-

1374


•

ance and Respiration. J. Chem. Educ. 1954, 31, 425. 22. Thomas, E. B. Graphical Treatment Simplifies Acid–Base Equilibria Calculations. J. Chem. Educ. 1963, 40, 70. 23. Freiser, H.; Fernando, Q. Teaching Ionic Equilibrium: Use of Log Chart Transparencies. J. Chem. Educ. 1965, 42, 35. 24. Fernando, Q; Butcher, J. Calculation of Titration Error in Precipitation Titrations: A Graphical Method. J. Chem. Educ. 1967, 44, 166. 25. Waser, J. Acid–Base Titration and Distribution Curves. J. Chem. Educ. 1967, 44, 275. 26. Stock, D. I. Dissociation of Weak Acids and Bases at Infinite Dilution. J. Chem. Educ. 1967, 44, 764. 27. Flack, A. The Acid–Base Calculator. J. Chem. Educ. 1967, 44, 466. 28. Cortelyou, W. P. Corrections for Simple Equations for Titration Curves of Monoprotic Acids. J. Chem. Educ. 1968, 45, 677. 29. Heck, L. Graphical Method for Acidity Constants. J. Chem. Educ. 1969, 46, 535. 30. Goldish, D. M. Component Concentrations in Solutions of Weak Acids. J. Chem. Educ. 1970, 47, 65. 31. de Levie, R. Ratio Diagrams: A Simple Graphical Representation of Complicated Equilibria. J. Chem. Educ. 1970, 47, 187. 32. Hamann, S. D. Titration Behavior of Monoprotic and Diprotic Acids. J. Chem. Educ. 1970, 47, 658. 33. Freiser, H. Acid–Base Reaction Parameters. J. Chem. Educ. 1970, 47, 809. 34. Freiser, H. Calculation of Hydrogen Ionic Concentrations. J. Chem. Educ. 1970, 47, 844. 35. Freiser, H. Polywater and Analytical Chemistry: A Lesson for the Future. J. Chem. Educ. 1972, 49, 445. 36. Pacer, R. A. Conjugate Acid–Base and Redox Theory. J. Chem. Educ. 1973, 50, 178. 37. Drago, R. S. A Modern Approach to Acid–Base Chemistry. J. Chem. Educ. 1974, 51, 300. 38. Ophardt, C. E. Weak and Strong Acids and Buffers. J. Chem. Educ. 1976, 53, 229. 39. Olson, V. K.; Osteryoung, J.; Christie, J. H.; Turner, J. A. Visual Aid for Acid–Base Concepts. J. Chem. Educ. 1977, 54, 220. 40. Kennedy, D. E. B. Aid To Teaching Weak Acids and Bases. J. Chem. Educ. 1977, 54, 550. 41. Staff, J. Chem. Educ. Emphasis on Acids and Bases. J. Chem. Educ. 1977, 54, 626. 42. Wiger, G. R.; de la Camp, U. Conjugate Acid–Base Mixtures in the General Chemistry Laboratory. A Comprehensive Buffer Experiment. J. Chem. Educ. 1978, 55, 401. 43. Deck, J. C. Strong and Weak Acids and Bases. J. Chem. Educ. 1979, 56, 814. 44. Ladd, M. F. C. Dissociation of a Weak Acid. J. Chem. Educ. 1980, 57, 669. 45. Clare, B. W. A Comprehensive Treatment of Acid–Base Equilibria on a Pocket Calculator. J. Chem. Educ. 1981, 58, 555. 46. Willis, C. J. Another Approach to Titration Curves; Which Is the Dependent Variable? J. Chem. Educ. 1981, 58, 651 47. Naylor, R. W. A Physical Model To Demonstrate Acid–Base Conjugate Pairs. J. Chem. Educ. 1981, 58, 1017. 48. van Lubeck, H. Significance, Concentration Calculations, Weak and Strong Acids. J. Chem. Educ. 1983, 60, 189. 49. Olivieri, A. C. Another Method for Solving Equations in Acid– Base Calculations. J. Chem. Educ. 1983, 60, 911. 50. Tchapla, A.; Fabre, C. The Value of (pH) as a Criterion for


•

www.JCE.DivCHED.org


51. 52. 53. 54. 55.

56.

57.

58. 59.

60. 61. 62.

63.

64. 65.

66.

67.

68. 69. 70. 71.

72. 73. 74.

Titrimetric Differentiation between Monobasic and Polybasic Acids in a Mixed Solvent. J. Chem. Educ. 1983, 60, 1030. Bodner, G. M. Assigning the pKa’s of Polyprotic Acids. J. Chem. Educ. 1986, 63, 246. Miller, R.; Silberman, R. An Introductory Laboratory Exercise for Acids and Bases. J. Chem. Educ. 1986, 63, 447. Treptow, R. S. The Conjugate Acid–Base Chart. J. Chem. Educ. 1986, 63, 938. Schwartz, L. M. Advances in Acid–Base Gran Plot Methodology. J. Chem. Educ. 1987, 64, 947. Thompson, R. Q. Identification of Weak Acids and Bases by Titration with Primary Standards: A Modern Version of an Old Analytical Chemistry Experiment. J. Chem. Educ. 1988, 65, 179. Leharne, S. Modelling and Analyzing Chemical Equilibria on a Spreadsheet: Acid–Base Systems. J. Chem. Educ. 1989, 66, A237. Parker, O. J.; Breneman, G. L. Numerical Solutions for Diprotic Acid equilibria Using Spreadsheet Iterations. J. Chem. Educ. 1990, 67, A5. Thompson, R. J. The Extent of Acid–Base Reactions. J. Chem. Educ. 1990, 67, 220. Cardinali, M. E.; Giomini, C.; Marrosu, G. The Hydrolysis of Salts Derived from a Weak Monoprotic Acid and a Weak Monoprotic Base. J. Chem. Educ. 1990, 67, 221. Burness, J. H. An Efficient Method for the Treatment of Weak Acid/Base Equilibria. J. Chem. Educ. 1990, 67, 224. Olivieri, A. C. Solution of Acid–Base Equilibria by Successive Approximations. J. Chem. Educ. 1990, 67, 229. Herman, D. P.; Booth, K. K.; Parker, O. J.; Breneman, G. L. The pH of Any Mixture of Monoprotic Acids and Bases. J. Chem. Educ. 1990, 67, 501. Malinowski, E. R. Methods for Calculating the pH of Aqueous Solutions of Salts of Monoprotic Acids and Bases. J. Chem. Educ. 1990, 67, 502. Nakagawa, K. A Modified Graphical Method for Acid–Base Equilibria. J. Chem. Educ. 1990, 67, 673. Campanario, J. M.; Ballesteros, R. A Short Program for the Automatic Calculation of pH in Solution Having Many Acids or Bases. J. Chem. Educ. 1990, 67, 1036. King, D. W.; Kester, D. R. A General Approach for Calculating Polyprotic Acid Speciation and Buffer Capacity. J. Chem. Educ. 1990, 67, 932. de Lange, A. M.; Potgieter, J. H. Acid and Base Dissociation Constants of Water and Its Associated Ions. J. Chem. Educ. 1991, 68, 304. Gordus, A. A. Chemical Equilibrium: IV. Weak Acids and Bases. J. Chem. Educ. 1991, 68, 397. Gordus, A. A. Chemical Equilibrium: V. Seeing an Endpoint in Acid–Base Titrations. J. Chem. Educ. 1991, 68, 566. Gordus, A. A. Chemical Equilibrium: VII. pH Approximations in Acid–Base Titrations. J. Chem. Educ. 1991, 68, 759. Amend, J. R.; Tucker, K. A.; Furstenau, R. P. Computer Interfacing: A New Look at Acid–Base Titrations. J. Chem. Educ. 1991, 68, 857. Cardinall, M. E.; Giomini, C.; Marrosu, G. More About the Extent of Acid–Base Reactions. J. Chem. Educ. 1991, 68, 989. Breneman, G. L.; Parker, O. J. Spreadsheet Titration of Diprotic Acids and Bases. J. Chem. Educ. 1992, 69, 46. Lake, M. E.; Grunow, D. A.; Su, M.-C. Graphical Presentation of Acid–Base Reactions: Using a Computer-Interfaced

www.JCE.DivCHED.org

•

Autotitrator. J. Chem. Educ. 1992, 69, 299. 75. Baldwin, W. G.; Burchill, C. E. The Acid Equilibrium Constant Is Unity! J. Chem. Educ. 1992, 69, 514. 76. Michalowski, T. Solving Acid–Base Equilibria: Some Concerns. J. Chem. Educ. 1992, 69, 858. 77. Weltin, E. Calculating Equilibrium Concentrations for Stepwise Binding of Ligands and Polyprotic Acid–Base Systems: A General Numerical Method To Solve Multistep Equilibrium Problems (CS). J. Chem. Educ. 1993, 70, 568. 78. Kipp, J. E. PHCALC: A Computer Program for Acid/Base Equilibrium Calculations (CS). J. Chem. Educ. 1994, 71, 119. 79. Cawley, J. J. The Determination of “Apparent” pKa’s: Part II. An Experiment Using Very Weak Acids (pKa’s > 11.4).J. Chem. Educ. 1995, 72, 88. 80. DeLorenzo, R. A Dating Analogy for Acid–Base Titration Problems. J. Chem. Educ. 1995, 72, 1011. 81. Hawkes, S. J. All Positive Ions Give Acid Solutions in Water. J. Chem. Educ. 1996, 73, 516. 82. Hancock, R. D.; Martell, A. E. Hard and Soft Acid–Base Behavior in Aqueous Solution: Steric Effects Make Some Metal Ions Hard: A Quantitative Scale of Hardness–Softness for Acids and Bases. J. Chem. Educ. 1996, 73, 654. 83. García-Doménech, R.; de Julián-Ortiz, J. V.; Antón-Fos, G. M.; Alvarez, J. G. Determination of the Dissociation Constant for Monoprotic Acid by Simple pH Measurements. J. Chem. Educ. 1996, 73, 792. 84. García-Doménech, R.; de Julián-Ortiz, J. V.; Antón-Fos, G. M.; Alvarez, J. G. Dissociation Constant for a Monoprotic Acid (response re J. Chem. Educ. 1996, 74, 792). J. Chem. Educ. 1997, 74, 899. 85. Ramette, R. W. Dissociation Constant for a Monoprotic Acid. J. Chem. Educ. 1997, 74, 880. 86. Carlton, T. S. Why and How To Teach Acid–Base Reactions without Equilibrium. J. Chem. Educ. 1997, 74, 939. 87. Bonham, R. A. Determination of the Equilibrium Constants of a Weak Acid: An Experiment for Analytical or Physical Chemistry. J. Chem. Educ. 1998, 75, 631. 88. de Levie, R. Redox Buffer Strength. J. Chem. Educ. 1999, 76, 574. 89. van Lubeck, H. Why Not Replace pH and pOH by Just One Real Acidity Grade, AG? J. Chem. Educ. 1999, 76, 892. 90. Barnum, D. W. Predicting Acid–Base Titration Curves Without Calculations. J. Chem. Educ. 1999, 76, 938. 91. de Levie, R. A General Simulator for Acid–Base Titrations. J. Chem. Educ. 1999, 76, 987. 92. Silverstein, T. P. Weak vs Strong Acids and Bases: The Football Analogy. J. Chem. Educ. 2000, 77, 849. 93. Hawkes, S. J. Easy Derivation of of pH ≈ (pKa1 + pKa2)/2 Using Autoprotolysis of HA-: Doubtful Value of Supposedly More Rigorous Equation. J. Chem. Educ. 2000, 77, 1183. 94. de Vos, W.; Pilot, A. Acids and Bases in Layers: The Stratal Structure of an Ancient Topic. J. Chem. Educ. 2001, 78, 494. 95. Kooser, A. S.; Jenkins, J. L.; Welch, L. E. Acid–Base Indicators: A New Look at an Old Topic. J. Chem. Educ. 2001, 78, 1504. 96. Coleman, W. F.; Wildman, R. J. Acid–Base Equilibria in Aqueous Solutions. J. Chem. Educ. 2002, 79, 1486. 97. Hanson, R. M. Principal Species and pH in Acid–Base Solutions. J. Chem. Educ. 2002, 79, 1486. 98. de Levie, R. The Henderson–Hasselbalch Equation: Its History and Limitations. J. Chem. Educ. 2003, 80, 146.


•


1375

Unified Approximations: A New Approach for Monoprotic Weak Acid

Recommend Documents