Chemical Education Today
Letters DG, ∂G/∂j, and the Physical Meaning of a Derivative Some papers published in this Journal and elsewhere (1– 3) deal with ∆G, the difference of Gibbs energy of a system in which a reaction occurs once, and the change in Gibbs , , mG, energy due to the chemical reaction symbolized as ∆G or ∆rG, defined as ∆rG = (∂G/∂ξ)T,P . A discussion is proposed for the conditions where these two quantities are equal (2). Nevertheless, these papers deal mainly with the mathematical aspect of these two quantities and not enough with their physical meaning. If we go back to the latter, not only do we put the finger on the correspondence of these two aspects but we also point out the particularity of each of them. Let us consider ∆rG = (∂G/∂ξ)T,P . By definition, ∆rG = lim (∆G/∆ξ)T,P ∆ξ →0
Mathematically this limit can be imagined easily. In abstraction it is possible to imagine the difference ∆ξ tending to zero. But physically, this extrapolation doesn’t have any significance and the corresponding physical concept of ∆ξ → 0 is that ξ varies by an amount so small that it could be neglected when compared to its original value. ξ is defined as ξ = (ni – ni°)/νi and ∆ξ equals (ni′ – ni)/νi where ni and ni′ are the amounts of constituent i at successive times. The change corresponding to ∆ξ → 0 can be ∆ξ = 1 if the original value of ξ is very large, that is, if the system has a large size and the reaction occurred previously many times until reaching a certain value of ξ. So ∆rG can be defined as the change in Gibbs energy of a closed large size system when the reaction occurs once after having run a large number of times, at given T and P. ∆rG depends on T, P, and ξ . If we want to point out the relationship between the mathematical and physical aspects of (∂ G/ ∂ξ ), the variation domain of ξ on the abscissa must be much larger than indicated in the previous papers, so the chemical potentials of the products and reactants don’t change significantly when the transformation occurs once and the drawing of G vs ξ for ∆ξ = 1 (Fig. 1 in ref 1 and Fig. 4 in ref 3) can be assumed to be a straight line, the slope of which equals the difference ∆G = Gf – Gi and thus, considering the physical meaning of the derivative (∂ G/ ∂ξ ), leads to the equality of ∆G and ∆rG. “Mathematics offers a wonderful shorthand of precise formulation of well-standardized ideas…[but] the formal severity of a mathematical treatment has its desadvantages” (4 ). That severity shouldn’t keep us away from the physical meaning of the concepts. Literature Cited 1. 2. 3. 4.
Spencer, J. N. J. Chem. Educ. 1974, 51, 577. Pauleau, Y. L’actualité Chimique 1978, Sep, 45. MacDonald, J. J. J. Chem. Educ. 1990, 67, 380. Lewis, G. N.; Randall, M. Thermodynamics; McGraw-Hill: New York, 1961; p 23.
The author replies: Twenty-five years have passed since I wrote the paper cited in Jemal’s letter (1). I have reviewed an average of about a paper per year on this subject since then and have seen many other published papers on this topic. This seems to confirm the difficulty and confusion relating to changes in thermodynamic quantities for chemical reactions and in particular to the change in Gibbs energy. In fact, if it were not for a dimensional inconsistency encountered when relating ∆G° and the equilibrium constant, quite possibly the distinction between ∆G and ∆rG would never have produced much interest. A check of my bookshelf of six current texts showed that all used the unit kJ for ∆rH, ∆rS, and ∆rG for chemical reactions. Each text had a different explanation for why, when using the relation ∆G° = {RT ln K, ∆G° must be given units of kJ mol {1. There are two precepts that must be adhered to in any discussion concerning the meaning of changes in thermodynamic parameters for a chemical reaction. First, stoichiometric coefficients are unitless; and second, the advancement of a reaction is described in terms of intensive thermodynamic quantities having units of kJ mol {1, not kJ (2). As Bent, before me, pointed out, the units on ∆G and ∆rG are different. This is because ∆G and ∆rG refer to different processes. ∆G (= Gf – Gi) is a finite difference and refers to the process whereby stoichiometric quantities of reactants (e.g., one mole of A in the reaction A 2B) are converted to stoichiometric quantities of products (e.g., 2 moles of B in the preceding reaction). ∆G and ∆H and ∆S calculated according to this process do have units of kJ. But this introduces a further complication. Having found ∆H to be, say, 100 kJ for a reaction involving a mole of A, how do students find ∆H for the formation of 2 moles of A? Again a dimensional inconsistency arises. The multiplication of tabulated intensive thermodynamic parameters having units of kJ mol {1 by moles gives units of kJ for the reaction parameter in question. ∆rG is a derivative, not a finite difference, and gives the instantaneous values of the Gibbs energy for products and reactants. Thus ∆rG has units of kJ mol {1, as made clear in the relation between ∆rG° and ln K. It is not ∆G° but ∆rG° that allows the direct calculation of the equilibrium constant. In general ∆G and ∆rG differ because one is a rate and one is not. If we choose to accept the very narrow definition of ∆G as a rate then we have the problem of what symbol we should use for a finite difference. Literature Cited 1. Spencer, J. N. J. Chem. Educ. 1974, 51, 577. 2. Craig, N. C. J. Chem. Educ. 1987, 64, 668. J. N. Spencer Department of Chemistry Franklin and Marshall College Lancaster, PA 17604-3003
Mohamed Jemal Faculty of Science Chemistry Department 1060 Tunis, Tunisia
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Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
Chemical Education Today
The Quantum Mechanical Explanation of the Periodic System I disagree with Eric Scerri’s criticism of the relations between the periodic table and quantum mechanics, in the November 1998 issue of the Journal. I think his logic is flawed both scientifically and philosophically. He attacks the classical quantum mechanical model of the orbitals of hydrogen because the closing of the shells in the quantum mechanical model does not fit the closing of the shells in the observed chemical world. But the physicists and chemists who applied this model in the 1920s and 30s knew from day one that it was only an approximate model. The amazing thing was how much of the periodic table suddenly seemed comprehensible with this inexact and very simple model. It precipitated a flood of wonderful chemistry and physics experiments over the next 50 years, which greatly enhanced our understanding of chemistry. Furthermore, in spite of a much improved understanding of the energetics of atoms and of compound formation, no one since then has been able to propose any simpler model for the energy relationships in atoms that explains any more. Scerri objects in his next-to-last paragraph that “the present semi-empirical explanation is not fully adequate.” I disagree. Where is it written that all theories must explain everything to the umpteenth decimal place? A theory, when proposed, should do two things: it should explain empirical observations better than the theory it replaces, and it should suggest further experiments that will test the limits of the theory. The aufbau principle was, on these grounds, one of the most spectacularly successful theories of the 20th century. Moreover, it was sufficiently simple that it could be understood by college students and used as a framework for organizing an immense body of chemical knowledge. No previous theory of atoms had provided anything like the organizing power of this one. Of course, there were exceptions and things that didn’t follow the rules; much of the time these were takeoff points for new and productive experiments. I think Scerri is quite right in describing his approach as “perversely rigorous” and I suggest that this particular perversion is of little value. Rigor has its uses in science, but global understanding of relationships is always more valuable than blind rigor in advancing science. Three-quarters of a century later we understand very well, in a qualitative sense, why the shells do not close as the simple hydrogen atom model suggests: electron–electron interactions are the cause of the “failure” of the simple theory. It is true that some of these deviations arise from relativistic effects, but personally I would argue very strongly against any attempt to introduce relativity in the early stages of a chemical career. The periodic table as it stands is a map of the world of chemistry; and it is a wonderful map indeed, full of complexities, unexpected relationships, and great unified regions, like organic chemistry or the first transition elements. Introductions to chemistry should be about learning to read the map, not about how to fight one’s way through the thickets of relativity. What we need in the elementary course is fewer topics, not more,
taught with a broad brush of understanding, not in a narrow, algorithmic vein. Kenneth Emerson Professor of Chemistry Emeritus Montana State University Bozeman, MT 59715
The author replies: It is rather a pity that Emerson takes my comment to be an attack on quantum mechanics, the periodic table, and current approaches in chemical education, none of which were intended. My approach is rather a philosophical assessment of the extent to which an important aspect of chemistry, the periodic system, has or has not been truly deduced from fundamental physical theory (1–4). Although it may not be written anywhere that “all theories must explain everything to the umpteenth decimal place”, it is obviously more desirable if they can. The point of my commentary was to examine the match between the present state of theoretical understanding and the explanations proffered in chemical education. Surely there can be no harm in taking a critical look at precisely what the present theory has achieved. Of course the simple orbital model began as an approximation, as Emerson states, and granted that it lends itself to teaching in broad brush strokes, but there is more to science than approximate explanations and broad strokes. The fact that at least some blemishes remain, however small, may be an indication of a deeper theory waiting to be discovered. My belief is that the numerous anomalies in the way in which electron shells are filled may call for such major revisions. In addition, as I have argued elsewhere, the frequently masked symmetry of the periodic law has yet to be fully explained (5). Contrary to Emerson’s mostly pedagogical concerns, the progress of theoretical science is always toward deeper theories which seek to explain more by drawing less and less on experimental data. For example, elementary particle physicists are not content to relate the masses of the various fundamental particles to each other but continue to strive to derive their masses from first principles. Finally, I believe that a philosophical analysis of the content of chemistry courses might reap some benefits for chemical education research, which is otherwise rather obsessed with the “learning process” and appears to regard content as being almost irrelevant. Literature Cited 1. 2. 3. 4. 5.
Scerri, E. R. Erkentnnis 1997, 47, 229. Melrose, M. P.; Scerri, E. R. J. Chem. Educ. 1996, 73, 498. Scerri, E. R. Br. J. Philos. Sci. 1991, 42, 309. Scerri, E. R. J. Chem. Educ. 1991, 68, 122. Scerri, E. R. Am. Sci. 1997, 85, 546. Eric R. Scerri Department of Chemistry Purdue University West Lafayette, IN 47907
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Chemical Education Today
Letters Lemon Cells Revisited The article “Lemon Cells Revisited—The LemonPowered Calculator” by Daniel J. Swartling and Charlotte Morgan ( J. Chem. Educ. 1998, 75, 181) is very interesting and useful. It certainly helps stimulate students’ enthusiasm with simple experiments in electrochemistry. I would like to make two cautionary remarks lest the electrochemical process responsible for energy transformation in the lemon cells is misunderstood. I refer to the statement in the article, “The lemon cell is peculiar in that, unlike a Daniel cell, both oxidation and reduction take place at the same electrode….The copper electrode is simply an auxiliary electrode; it acts as an electron shunt where reduction of hydrogen ions to hydrogen gas also takes place.” First, to the extent both oxidation and reduction reactions occur at the same electrode (Zn or Mg) they produce no current through the current-meter nor would they power the calculator. It is only the part of the hydrogen reduction reaction occurring at the copper or graphite electrode that contributes to the current that provides the power to the calculator and not the part of the hydrogen reduction occurring at the same electrode (Zn or Mg) at which oxidation occurs. Alas, it is this latter part which is shunted, contrary to what has been suggested by the authors. If we were able to prevent hydrogen gas evolution at the Zn or Mg electrode, we would prevent the corrosion of the electrode and would to
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that extent be saving the metal, without any loss of power supplied to the load—the calculator. In other words, in the ideal case we should have only oxidation at one electrode and only reduction at the other. Whenever both oxidation and reduction occur at the same electrode, the current gets shortcircuited (or shunted) and does not pass through the load. Second, the copper electrode is not “simply an auxiliary electrode”, it is indispensable for the operation of the cell; it is as important or as essential as the zinc or magnesium electrode is. In electrochemical literature one comes across three-electrode cells where the electrodes are designated as working electrode, counter electrode, and auxiliary electrode. The current through such a cell passes through the working electrode–electrolyte– counter electrode–load–working electrode loop; no current passes through the auxiliary electrode. In fact we can remove the auxiliary electrode and still operate the cell. In the lemon cell we have only two electrodes, which are necessary and sufficient for the operation of the cell; we can call either of them a working electrode (if we are interested in studying the process at that electrode) and the other, counter electrode. We don’t have an auxiliary electrode in lemon cells. P. Radhakrishnamurty Central Electrochemical Research Institute Council of Scientific & Industrial Research Karaikudi-630 006, Tamil Nadu, India
Journal of Chemical Education • Vol. 76 No. 9 September 1999 • JChemEd.chem.wisc.edu
