Research: Science and Education
Representing Fractional Distributions in Chemistry Addison Ault Department of Chemistry, Cornell College, Mount Vernon, IA 52314;
[email protected] Many phenomena in chemistry involve fractional distributions. Sometimes it is a chemical substance that is fractionally distributed. Other times it is a chemical process that is fractionally distributed. An example of the fractional distribution of a substance is the distribution of members of a conjugate pair between acidic forms and basic forms. An example of the fractional distribution of a process is the distribution of a reaction among alternative paths. The purpose of this paper is to present a common approach to the representation of all fractional distributions. Using a common approach provides three advantages. First, it is simple. Second, it is easy to understand, and third, a common approach can reveal otherwise hidden relationships between phenomena. A General Representation of a Distribution Suppose a substance can exist in several forms. We can represent the total as a sum of fractional contributions to the total. total forms = A + B + C + D + … We can then represent the various fractional contributions in this way:
fraction as A:
A
fraction as B:
B
fraction as C:
C
fraction as D:
D
D 1432
< A>
1 1 1
< C>
Suppose we want to know the conditions under which each of the members of a conjugate pair of acid and base will predominate. Let us represent the acidic member of the pair by 0, the basic member by 1.
…
We can then represent the fractional concentration of the acidic member of the pair, A−H, by α0, and the fractional concentration of the basic member of the pair, A:–, by α1. We can now write the distribution equations in this way.
B1
B 1 1 B0
B0 B1
1 B0 1 B1
(1)
HA
H
K1 A Replacing the ratios in the denominators by the functions that give these ratios we get the two distribution fractions.
1 …
1
For the ionization of a weak acid in water the ratio of the members of the conjugate pair is given by the ratio of the hydrogen ion concentration and the acid ionization constant, as indicated here. A B1 K1 B0 HA H (2)
B0
…
A:– form 1 the basic form
A—H form 0 the acidic form
…
1 < A > 1 …
< A>
A Simple Example
B0
…
1 A B C < > < > … 1 < A > < A > < A>
C
Whenever there is a distribution among forms, the question of interest is—Under what conditions will the value of a particular fraction approach 1? Or—Under what conditions will a particular form predominate? The answer to the question is found by evaluating the terms in the denominator of each of the expressions for fractional concentration.
…
Dividing each fraction top and bottom by the numerator converts each fraction to the following form.
B
The Question of Interest
1 1
K1 H
B1
1
H K1
1
(3)
These two distribution fractions indicate the relative concentrations of the members of a conjugate pair as a function of the pKa of the pair and the pH of the solution. When the hydronium ion concentration, [H+], of the solution equals the ionization constant of the pair, K1, (when the pH of the solution equals the pKa of the pair) the fractional concentration of each member of the pair will be 1/2; the members of the pair will be present at the same concentration.
Journal of Chemical Education • Vol. 85 No. 10 October 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
Research: Science and Education
When the hydronium ion concentration exceeds the ionization constant of the pair, the acidic member will predominate, and when the hydronium ion concentration is less than the ionization constant of the pair, the basic member will predominate. Another way to say this is that when the pH is on the acidic side of the pKa the acidic member will predominate, and when the pH is on the basic side of the pKa the basic member will predominate. Furthermore, when the hydronium ion concentration and the Ka are known, we can calculate the exact values of the fractions. Extension to Examples with More Than Two Forms The example just presented was that of a monoprotic weak acid. A monoprotic acid, with a single ionization constant, can be present in an aqueous solution in two forms, the “acidic” form and the “basic” form. These forms were represented in the example by 0 and 1, and their fractional concentrations were represented by α0 and α1. A Diprotic Weak Acid We can consider a diprotic weak acid in a similar way. For a diprotic acid there are two ionization constants, K1 and K2, three possible forms, 0, 1, and 2, and, correspondingly, three fractional concentrations, α0, α1, and α2. (The equations and the example are presented in full in the online supplement. See also ref 1.) A Triprotic Weak Acid For a triprotic acid there will be three ionization constants, four possible forms, and four fractional concentrations. (The equations and examples are presented in full in the online supplement.) Cooperativity Sometimes when there are two equivalent acidic or basic sites in a molecule the presence of one will affect the other. Thus, for example, pK1 and pK2 for malonic acid are 2.86 and 5.70, as shown in Scheme 1. Because the pKa of acetic acid is 4.76, we see that the first proton of malonic acid is lost more easily than that of acetic acid, and that the second proton of malonic acid is lost less easily than that of acetic acid. This type of “cooperativity” in which the pKa values for successive proton transfers are farther apart than expected is called “negative cooperativity”. This is the kind of “cooperativity” to which we are accustomed. We are, therefore, surprised to learn of “positive” cooperativity, cases in which the second proton transfer takes place as easily or, amazingly, even more easily than the first! As is so often the case this surprising chemical phenomenon was first observed in biochemical systems. Apparently when
O HO
O OH
malonic acid: pK1 = 2.86; pK2 = 5.70 Scheme 1. A diprotic weak acid.
some proteins accept a first proton they undergo a conformational change that produces or reveals a basic site even more basic than the first, and a second proton transfer then follows immediately. When this is true there are only two significant forms, form 0 and form 2; form 1 is converted to form 2 so rapidly that the concentration of form 1 is never significant. Correspondingly, when this is true, there are only two significant distribution functions, α0 and α2. These equations are presented in full in the online supplement along with Metzler’s beautiful chemical example of positive cooperativity in a small molecule from ref 2. We see that positive cooperativity, in general, is characterized by the “depletion of intermediate states”. Application to Enzyme Kinetics This approach is well suited to enzyme kinetics, as indicated in full in the online supplement and also in ref 3. Application to Reactions with More Than One Path to Product We can conceive of the existence of more than one path between starting materials and product, as demonstrated in the following two examples. Solvolysis of an Alkyl Halide There is nothing new here concerning solvolysis of an alkyl halide except, possibly, the approach (4). Acid-Catalyzed Hydrolysis of Esters For the hydrolysis of an ester in aqueous acid we can imagine three contributing paths. The first would involve catalysis specifically by hydronium ion: specific acid catalysis. The second would involve catalysis by all Brønsted acids; general acid catalysis. The third would be the uncatalyzed reaction, or “water” reaction. The total rate would be the sum of the rates of reaction along these three paths. The details of this approach are presented in full in the online supplement. See also (5). Application to the Reaction A + B → C → D Many reactions can be described as A reacting with B to give an intermediate, C, that then goes on to the product, D. The process is often represented in this way.
A B
k1
C
k2
D kļ1 The rate of the reaction can be thought of as the rate of formation of the intermediate, C, multiplied by the fraction of the intermediate that goes on to D. This approach, which is equivalent to the results of the steady-state approximation, is presented in full in the online supplement. Cooperativity Revisited Cooperativity can be seen as a “depletion of intermediate states”, a circumstance in which the only forms present at significant concentrations are 0 and n.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 10 October 2008 • Journal of Chemical Education
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Research: Science and Education
A Simple MWC Model for Allosteric Interactions Many models have been proposed for the dependence of
αn, or Y, the binding fraction, on the concentration of S that
amplifies the effect of S and thus produces cooperativity. One of the most widely accepted of these models is that of Monod, Wyman, and Changeux, the MWC model. The online supplement considers a simplified version of the MWC model that indicates the nature of the mechanism of amplification (6). Summary Many chemical phenomena that involve the fractional distribution of materials or processes over two or more forms or paths can be represented in the same way. Benefits The unified point of view provides at least three benefits. First, it is simple; it involves a pattern that is easily extended to the next example. Second, it is easy to understand. Term by term inspection reveals all significant chemical interactions. Third, the unified point of view makes it possible to see an unfamiliar phenomenon as yet another example of a common behavior. Previous insights can then be applied to the new phenomenon. Examples Examples of the fractional distributions of materials include the various possible forms of mono-, di-, tri-, and polyprotic acids, and the various possible free and combined forms of enzymes in enzymatic reactions. Examples of fractional distributions of processes include the solvolysis of an alkyl halide, the acid- and
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base-catalyzed hydrolyses of esters, and the typical reaction of A with B to form C, which then goes on to D. Cooperativity Cooperativity is a phenomenon in which “intermediate forms are depleted”. A simplified dimeric example of the MWC model is analyzed in the online supplement. Literature Cited 1. Ault, A. J. Chem. Educ. 2001, 78, 500–503. 2. Metzler, D. Biochemistry: The Chemical Reactions of Living Cells; Academic Press, Inc.: New York, 1977; pp 193–194. 3. Ault, A. J. Chem. Educ. 1974, 51, 381–386. 4. For an interesting public argument concerning this approach to rate expressions see: Breslow, R. J. Chem Educ. 1990, 67, 228–229; Northrop, D. B. J. Chem Educ. 1993, 70 , 999–1001; and Breslow, R. J. Chem Educ. 1993, 70 , 999–1000. 5. Ault, A. J. Chem. Educ. 2007, 84, 38–39. 6. Stryer, L. Biochemistry, 3rd ed.; W. H. Freeman and Company: New York, 1988; pp 239–241.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Oct/abs1432.html Abstract and keywords Full text (PDF) Links to cited JCE articles Supplement A full-length version of this manuscript, including a more complete presentation of the derivations and examples
Journal of Chemical Education • Vol. 85 No. 10 October 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
