Some Opportunities for Reinforcement of Learning Among the Subdisciplines of Chemistry A. L. Underwood Emory University, Atlanta, GA 30322 I t is a common lament among physical chemists that students, even at the graduate level, are inadequately prepared in mathematics. The problem resides less in the lack of formal courses than in a failure to develop a "feel" for expressing chemical situations in mathematical language. A related problem is the tendency of students to compartmentalize their studies into quarter or semester packages; much of what was "learned" seems unavailable in later courses. Students frequently express resentment when asked to apply simple acid-base theory from the first-year course to estimation of the charge on a protein molecule in a biochemistry course. Much of this may he inevitable, hut the faculty must not encourage it; we must search continually for ways to utilize material from an earlier course in a later one and to reward the students who are successful in carrying over material from one course to another. There are many examples of topics taught in different contexts using different symbols hut where the mathematical formalism is the same. One which bas been frequently noted involves the Bouguer-Beer law in analytical chemistry and the first-order rate equation in physical ( I ) . Presented here are some additional cases of equations of similar form which occur in quite different situations and which have been less widely discussed. Pointing out the similarity, which may not otherwise occur to the student, may assist in his effort to relate physical situations to mathematical form. Hyperbolic Approach Toward Saturation Michaelis-Menten Kinetics. For many enzymic reactions, the graph of initial velocity, u, versus substrate concentration [S], at constant enzyme level is a hyperbola as shown in Figure 1.This curve is the locus of the Michaelis-Menten equation, which is generally seen ( 2 ) in the form
is preferable, of which the commonest (although not necessarily the best) is the Lineweaver-Burk version ( 3 )shown in Figure 2. The equation of this graph is ohtained by taking reciprocals of both sides of the Michaelis-Menten equation:
I t is seen that the slope is KMIV,,, and the intercept on the 110axis is l/Vma* Solving eqn. (1) for [S] and taking logarithms, one readily obtains pS = ~ K -Mlog-
U
vmax -u
(3)
The analogy both to the Henderson-Hasselbalch equation, [acid] pH = pK, - log [conjugatebase] and, for that matter, to the Nernst equation for a redox couple Ox ne- = Red, is obvious RT [Red] E=EQ--1"nF [Ox] Of course, where u = V,,,/2, pS = ~ K Mjust , as pH = pK. when a weak acid has been half converted into the coniueate base. The mathematics when a base is half protonated & an oxidant half reduced is the same as when an enzyme has been half wn\.erted into enzyme-suhsrrate cumplcx. 'Ch~ihas, in Tart, heen wintrd out in >I book t.11which t w undc.mraduates are likrly t u mcuunter. On page 56 oi the samv I ~ w (k1 I is thi* stntemcmt regarding the h\ l,erbdic. curve: "Such a result is ohtained whenevera prockSs depends upon a simple dissociation; if, for a dissociation XY = X Y, [Y] is held constant, plotting IXY] against [XI will give [a hyperbola].'' The same writers note (p. 59) the similarity to Langmuir's isotherm.
+
+
The asymptote, V,,, is the limiting velocity when [S] = m , i.e., when all catalytic sites of the enzyme are occupied by substrate molecules. The Michaelis constant, KM,is equal to [S] when u = V,,,/2. To evaluate V,. and KM,a linear plot
Figure 1. Graph of eqn. ( I ) for Michaelis-Menten kinetics and eqn. (8)far the Solvent extraction of a weak acid.
l l ( S ) or ll(H') Figure 2. Graph af eqn. (2)for Michaelis-Menten kinetics and eqn. (9) lor the solvent extraction af a weak acid. Volume 61 Number 2
February 1984
143
Solvent Extraction of a Weak Acid For a weak acid, HB, which partitions between an aqueous phase and an organic solvent, it is easily shown that
where KnHRis the distribution or partition coefficient for rhe = IHR ,,,: IHRlnq, and D is il distrihlsprcies HR (I.+:.. tion ratio expressed using consentmtlrms summed over all relevant species:
Equation (6), found in treatises on solvent extraction (5)and in elementary quantitative analysis texts (61,is modified easily to accommodate other equilibria such as dimerization in the organic solvent. What has not, to our knowledge, been pointed out is that eqn. (6) can he written
It is seen that eqn. (8) looks exactly !ike eqn. (1) and that Figure 1can he labelled to represent either equation. Just as . u approaches V,. when more and more enzyme molecules are working, so D approaches KD,, as more and more B- ions are nrotonated to form the extractable snecies HB. This is scarcely a scientific breakthrough, hut it can he pointed out to undergraduates that the two equations both describe the approach to saturation for the simple case X Y = XY, and that the approach is hyperbolic. I t is seen that when D = KD,$~, [H+] = KWB,analogousto the Michaelis-Menten case where [S] = K Mwhen u = Vm.,/2. Because Lineweaver and Burk were biochemists, the double reciprocal plot has not appeared in analytical texts, hut
.
~~
~~
+
1 - K w s + [H'lw --
=
E
K.,.
(
+
1 -
for the volume of a gas under standard conditions adsorbed by a unit mass of solid as a function of the pressure, p , a t constant temperature. A graph of u versus p is, of course, is the limitingvalue hyperbolic. Further, c = u,,,b, where u, of u corresponding to complete surface coverage. The assumotions in Lanrmuir's derivation, althouah in- orobahlv . valid for most realsurfaces, are approximately correct fo; the bindina of protons to a base in solution. of substrate molecules to ma& enzymes, of oxygen to myoglobin, etc.: binding at definite sites, all sites equivalent, no interactions between sites or between hound ligands. Langmuir sounds like a protein chemist! On page 339 of reference (7) is found the following: "To test whether [the Langmuir isotherm] fits a given set of data, we take the reciprocal of each side to give llu = 11 (u,.,bp) llu,.,. A plot of l l u versus l l p ought to give a straight line if the Langmuir isotherm is obeyed." Historically, the Michaelis-Menten equation ( 8 ) slightly predates Langmuir's (9).
+
Successive Formation Constants
We close with an equation which is well-known in two different areas and which was derived, apparently independentlv, hv two workers who orohahlv never heard of each k derivation rhgs true, with not the slightest other: ~ h second hint of plagiarism. The teachers who use the eauation seem not to talk-to each other, nor do students cart; it from one course to another. The hemoglobin molecule, Hb, can bind four molecules of oxygen. For each step, a macroscopic equilibrium constant may he written:
(9)
Kn&+l., KD, IH I KD,, Thus a plot of 1/D versus l/[H+] yields a straight line of slope K.,,IKD,, and intercept llK~,,, as shown in Figure 2.
D
Protonation of a Base and the Oxygenation of Myoglobin Sometimes traditional presentations obscure analogies and correlations that might he helpful to students. Since, albeit for good reasons, analytical chemists talk dissociation and pH rather than association and IH+1. . ,. manv students who are thuruughly familiar wit h sigmoidal titration curves haw nut thc faintest notio~lthnt a rraoh showine the iraction. I. of a base protonated versus [H;] is a hyperb'bla, the equation for which is
They then view the hyperbolic oxygenation curve for myoglobin (fraction saturated versus po,) as something very special in biochemistry, whereas for the simple reaction Mh + 0 2 = MhOz, no other shape makes any more sense than for the reaction RNHz + H+ = RNH3. If the hyperbolic curve is seen HS unusual, how is the student then to experi+mx=the awe which rhr siamoidnl curws for h~mualobinand the allosteric enzymes deserve? ~
~
Langmuir Adsorption Isotherm Most physical chemistry texts discuss adsorption (7), although the chapter on surface chemistry is frequently omitted from the courses where these books are used. Langmuir's model for monolayer adsorption on an idealized surface leads to an equation of the form
144
Journal of Chemical Education
The po, is tantamount to an oxygen concentration via Henry's law, and the equations above are written in the customary form. Now, Hb can be titrated with Oz,hut the K-values cannot he obtained by simple inspection of a graph of the titration data as students do in elementary courses. However, these values can, at least in principle, he extracted from the data. Knowing the initial [Hb] and the quantity of added Oz, then by measuring the partial pressure of free Ozone can calculate for that point in the titration the average number of O2 molecules hound per Hb molecule, designated F. The equation for obtaining K-values from successive determinations of F was given in 1925 (10) and is known in biochemistry as the Adair equation, which may be written as follows:
A discussion of eqn. (16) may he found in a modern hook (11). Few chemistry teachers outside the biological area have ever heard of the Adair eauation. hut inoreanic and analvtical chemists who deal with complex ions are familiar with Bjerrum's formation function. This is used to calculate successive formation constants for the binding of a ligand such as NH3 to, say, Cu(I1) ions. As presented independently in 1941 (12), the formation function, ii, which is the average number of ligands (A) hound to a metal ion, is
Bjerrum applied eqn. (17) to the stepwise addition of ammonia to metal ions, determining [NHs] by utilizing the glass electrode. Numerous adaptations have been "derived," sometimes overlooking fair acknowledgment of Bjerrum and never mentioning Adair, for obtaining successive constants by polarography, solvent extraction, spectrophotometry, and other techniques. The same sort of equation could obviously he written for the protonation of a polyfunrtional base, although students may not realize this simply because acid-base chemistry is usually treated in terms of dissociation. Summary The examples presented above could be embellished with activity coefficients and other refinements, but this would only obscure the major theme. Sometimes a simple model, stripped of frills, captures the essence of a real situation in chemistry, and sometimes, mutatis mutandis, a model with the same features approximates the truth in an entirely different chemical situation. A few students will see the basic similar-
ities even if the two situations arise in different rourses, hut those who really need our help will not. Showing less gifted students how mode$ of the same form lead toeauations of the same form takes very little extra time. ~lerdteacherswill recognize that the examples given here barely scratch the surface. Llteraiure Cited (1) Goldatein, J . H., and Day, Jr., R. A,, J. C ~ MEDuc., . 81.417 (19641. (2) Strysr, L., "Bioehemishy," 2nd ed., W. H.Freeman and Co., San Franciaca, 1981, p. 112. (8) Lineweaver. H.. and Bwk, D..J Amri. Cham. Soe.. 6 4 668 (1934). (4) Diron, M.,and Webb, E. C.,"Enzymes," 3rd d.,AcadamicPmaa. New York, 1979, p. 60. (6) S e k i i , T, and Haaegews, Y , '"Solvent Extraction Chcmiatry: FmdernsnWs and Applicstians,l. Mareel Dekkw, Ine., New York,1977, p. 117. (6) Dsy, Jr.. R. A., and Underuwd,A. L., "Quantitative Analyaia," 4thed.,Prentica.Hall, he., Engleruwd Cliffs, NJ, 1980. P. 437. (7) Levine. I. N.. "Physical Chemistry." MoGrsw-Hill Bwk Co., New York, 1978, p. 8 M#:>adis.L and h1snfsn.M L..B,rchem 2,49,3i3!1913 (9. L a q m u r . I , J Amor Cnrm Soc 3h.222 11918 1 1 Aderr.G S . J an.Cnea .6l52s IW, 11, Cantor C. R .and Schimmel. P X 'R,cphyairal Chem~atrjPen 111 ThsBahav~or ,I Rtuiwid hlarr,ao:eru.ea." B'. H brasman and C o . San Francam 19b0, p 980. (12) Bjerrum,J.,"MatelAmmineFom~titi inAqueouaSolution:Th~horyof t h e b v a r ~ i b l e StepReaetians." P. Hasse and Son, Copenhagen, 1941, p. 21.
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Volume 61 Number 2
February 1984
145
