In the Classroom edited by
Applications and Analogies
Ron DeLorenzo Middle Georgia College Cochran, GA 31014
A Closer Look at the Addition of Equations and Reactions Damon Diemente Trinity School, 101 West 91st Street, New York, NY 10024
Chemists occasionally find it convenient or even necessary to express an overall reaction as the sum of two or more component reactions. In high-school courses, students are likely to encounter the addition of reactions in thermochemistry, electrochemistry, and kinetics. This article looks into a few examples drawn from these three branches of chemistry. Subtle and surprising discoveries await us. To introduce the discussion, we consider the atmospheric corrosion of calcium: Ca(s) + 1/2 O2(g) + CO 2(g) → CaCO 3(s) which can be thought of as the sum of these components: Ca(s) + 1/2 O2(g) → CaO(s) CaO(s) + CO2(g) → CaCO3(s) I have used customary chemical parlance here to identify one reaction as the sum of two others. Some may challenge this choice of words, asking: are we really adding reactions here, or are we merely adding equations? This is not an idle question, because reactions and equations are distinct things, and the word “addition” takes on special meanings when applied to each of them. Chemical reactions are real events of nature taking place at the molecular level. A set of reactions are said to be “added” when one follows the other with at least one product in each prior reaction serving as a reactant in a later one. Chemical equations, on the other hand, are conventional symbolic representations of reactions. The addition of chemical equations is closely analogous to the addition of algebraic equations, with the arrow replacing the equal sign. Any set of balanced equations can be “added”, even if none represents a reaction that anyone has ever observed. All one need do is go through the process, vain though it may be. Now in the example above, there is a close correspondence between reactions and equations. After all, calcium powder exposed to dry air really does react with oxygen and carbon dioxide to yield calcium carbonate. And calcium exposed to oxygen alone really does form the oxide, which, subsequently exposed to carbon dioxide, yields the carbonate. So the overall equation reasonably summarizes a performable reaction, and the component equations reasonably summarize a performable sequence of two reactions. Therefore we can rightly claim to have added both the equations and the reactions represented by the equations. But as the rest of this article is meant to show, the situation is not always so simple. Sometimes the overall equation is a good representation of a real reaction actually happening among the molecules but the component equations are not; other times the reverse is true. We begin with a look at two thermochemical examples.
The thermochemical application centers around Hess’s law. Here, the point of the addition is the calculation of a desired enthalpy change or heat effect. For example, these equations 2 HCl(aq) + MgO(s) → MgCl2(aq) + H2O(l)
∆H1
2 HCl(aq) + Mg(s) → MgCl2(aq) + H2(g)
∆H2
2 H2(g) + O2(g) → 2 H2O(l)
∆H3
can be added (with one reversal and two doublings of coefficients), and by Hess’s law we can add their heat effects as well (also appropriately adjusted), to obtain as an overall result, the combustion of magnesium: 2 Mg(s) + O2(g) → 2 MgO(s)
∆H4 = 2 ∆H2 – 2 ∆H1 + ∆H3
Note that in this case, we are adding equations but are not adding reactions. When a piece of burning magnesium is under consideration, the first three equations of this set cannot possibly represent reactions because hydrochloric acid, water, magnesium chloride, and hydrogen have absolutely nothing to do with the combustion of magnesium. Hess’s law is simply a convenience: if by some means we can determine ∆H1, ∆H2, and ∆H3 and then invoke the law of conservation of energy, we can calculate ∆H4. And it is a safe bet that this problem is solved in dozens of high-school chemistry labs across the country every school year. This is done by performing the reactions represented by the equations. The first two reactions of this set are easily run in familiar Styrofoam-cup calorimeters, making ∆H1 and ∆H 2 experimentally accessible. The third reaction is of course an explosion. It may be performed as a demonstration, but ∆H 3 has to be looked up in a handbook or some other compilation. Then we use Hess’s law to calculate the heat of combustion of magnesium. The burning of magnesium ribbon is another popular demonstration not performable in a Styrofoam cup. Hess’s law gives access to ∆H4 without having to attempt this vigorously exothermic reaction in the usually available high-school laboratory equipment. That’s the whole point. That’s the convenience Hess’s law provides. We turn now to a second thermochemical illustration, the seemingly straightforward reaction between the elements sodium and chlorine whose heat effect is the heat of formation of sodium chloride: Na(s) + 1/2 Cl2(g) → NaCl(s)
∆Hf (NaCl)
We can establish a Born–Haber cycle for this reaction and use Hess’s law to express ∆H f(NaCl) in terms of a variety of heat effects from truly elementary reactions, viz.: Na(s) → Na(g)
∆Hsubl (Na)
Na(g) → e + Na (g) ᎑
+
JChemEd.chem.wisc.edu • Vol. 75 No. 3 March 1998 • Journal of Chemical Education
IP(Na) 319
In the Classroom
/2 Cl2(g) → Cl(g)
1
1
Cl(g) + e → Cl (g) ᎑
᎑
Na (g) + Cl (g) → NaCl(s) +
᎑
/2 BCl᎑Cl EA(Cl)
∆H latt(NaCl)
which add as written to give: Na(s) + 1/2 Cl2(g) → NaCl(s)
∆H f(NaCl)
We see that ∆H f (NaCl) is the sum of the heat of sublimation of sodium, one half of the chlorine–chlorine bond energy, the ionization potential of sodium, the electron affinity of chlorine, and the lattice energy of sodium chloride. These elementary heat effects are typically presented in lessons on chemical periodicity, covalent bonding, the solid state, etc. Hess’s law and the addition of equations allow us to pull a lot of information together and gain deep insight into the formation of a binary compound from its elements. But whatever insights we gain, and convenient though Hess’s law may be for purposes of calculation, it is once again clear that we are adding equations but not adding reactions. If solid sodium and gaseous chlorine really do produce solid salt, the overall reaction has nothing to do with gas-phase ions. If sodium vapor (which is not the standard state of the element) is involved in the reaction, the resultant enthalpy change is not properly called a heat of formation. This is a general result in thermochemistry: overall reactions are usually meant to be taken literally, whereas the set of added equations is treated almost as a chemical allegory. The situation is very different in electrochemistry. Here addition is routinely used to compose complete cell potentials from half-cell potentials as in this example (1): cathode: H2O2 + 2 H+ + 2 e᎑ → 2 H2O anode: 2 VO
2+
Ᏹ1° = 1.77 V
+ 2 H2O → 2 VO2+ + 4 H+ + 2 e-
cell: 2 VO + H2O2 → 2 VO2 + 2 H 2+
+
+
Ᏹ2° = ᎑1.00 V Ᏹ3° = 0.77 V
Note that when the half-reactions are added, some species are cancelled out to avoid apparent redundancies in the cell equation. But this cancellation of apparently redundant reagents in electrochemical reactions calls for some sober reflection, because electrochemical half-reactions are more than just a convenience for purposes of calculation. The halfreactions really do happen. In fact, in a cell employing a salt bridge the half-reactions run in separate beakers, and the reactants in the full cell reaction never actually come into physical contact with each other. Clearly, then, if anything really happens, it must be the half-reactions. Furthermore, note that the cell equation gives no clue that the cathode solution has to be acidic in order for the reduction half-reaction to run. Note too that the reduction product, water, has completely vanished from the cell equation. We have to pay close attention to the half-reactions, and not to their sum, if we want to keep track of what is really going on at the molecular level as the cell draws current. In fact, it is not clear that the cell equation has much significance beyond the stoichiometric; that is, we can use it to calculate the theoretical yield of VO2+ ion from given quantities of VO2+ and H 2O2. Of course, we might also wish to calculate the standard potential of the cell, but no overall chemical equation is needed to do that. We have just to add the standard half-cell potentials. So in electrochemistry, overall equation is the allegory, and the separate half-reactions are real. 320
In kinetics applications, too, reactions as well as equations are regularly added. Experimental rate laws are interpreted in terms of a mechanism that may or may not be completely sequential, but that does in an important sense add up to an overall equation. Consider for instance the gas-phase reaction between nitrogen dioxide and carbon monoxide (2, 3): NO2 + CO → NO + CO2 At high temperature (ca. 200 °C), this reaction is first order in each reactant; at low temperature (ca. 100 °C), it is second order with respect to NO2 and zeroth order in CO. This behavior is inexplicable without the postulation of separate mechanisms at each temperature together with the basic principle that overall rate is determined by the first slow step in the mechanism. At high temperature we evidently have (4) slow
fast
NO2 + CO → O–N–O–C–O → NO + CO2 and at low temperature (5) slow
NO2 + NO2 → NO3 + NO fast
NO3 + CO → NO2 + CO2 The mechanisms explain the variation of the rate law with temperature; they reveal the highly reactive and non-isolable intermediates NO3 and ONOCO, which may be confirmable by experiment once we have some idea what to look for; and they offer a truer picture than the overall equation of real molecular behavior during a chemical reaction. When the overall reaction is evaluated against the mechanism of added reactions, kinetics finds more truth in the mechanism. This is most obvious at low temperature, where there is no reaction at all between one molecule of nitrogen dioxide and one of carbon monoxide, even though that is precisely what the overall equation shows. A final and particularly arresting example. Any speculation on the origins of life on Earth must address the question of the synthesis of glycine from smaller molecules. One extensively studied reaction is the production of glycine from formaldehyde, hydrogen cyanide, and water under the influence of an electrical spark (6). The situation is meant to model possible events in Earth’s primitive atmosphere during electrical storms. The event looks simple enough; indeed it is balanced when written with unit coefficients: CH2O + HCN + H2O → NH2CH2COOH
(1)
Now this “reaction” takes place in at least two steps, and things begin to look more complicated when it is refined into a two-step sequence: CH2O + NH 3 + HCN → NH2CH2CN + H2O (2a) NH2CH2CN + 2H2O → NH2CH2COOH + NH3 (2b) In the usual way, most of us might say that eqs 2a and 2b “add” to give eq 1, and ammonia and one of the waters “cancel” in the addition process. But for many reasons this is an unacceptable simplification. Note first that (i) ammonia is required for the synthesis of glycine, (ii) two molecules of water are involved in the synthesis, and (iii) aminoacetonitrile is an intermediate product. Equation 1 reveals none of this. Clearly, eqs 2a and 2b give a more complete and accurate picture of this synthesis
Journal of Chemical Education • Vol. 75 No. 3 March 1998 • JChemEd.chem.wisc.edu
In the Classroom
of glycine than eq 1 does. And there is much more at stake here. The two-step sequence allows us to have a closer look at individual carbon, nitrogen, and oxygen atoms. When we follow individual atoms, we see with more insight the harm done when equations are too casually “added” and when seemingly redundant molecules are “cancelled” in this example. We find molecules that aren’t redundant at all and a supposed balanced equation that doesn’t even have the same atoms on each side of the arrow. To make these points clear, eqs 1 and 2 are rewritten below with distinctive fonts 1 for specific atoms. The carbon and oxygen atoms from formaldehyde are in italic. The nitrogen atom from the ammonia of eq 2a is underlined. The carbon and nitrogen atoms from hydrogen cyanide are in outline. The oxygen atoms from the water of eq 2b are in boldface. With these special tags, all of which could be experimentally emplaced with radiotracers, the two-step sequence looks like this:
bonded to nitrogen in glycine. Likewise, the carbon bonded to a different nitrogen in hydrogen cyanide is bonded in glycine to a different oxygen. (No attempt is made to keep track of the various hydrogen atoms, some of which surely exchange rapidly with the shifting hydrogen bonds.) Taken together, the examples presented here show that the process of equation/reaction addition must always be alertly approached. Sometimes it is merely and intentionally a convenience for purposes of calculation and needn’t be literally considered. But quite often close scrutiny of the addition process exposes and clarifies details of chemical behavior that might otherwise never be suspected. In these cases, we overlook a lot of chemistry if individual steps are slighted and reactions are imagined to proceed in concerted, simplified fashion.
C H2O + NH3 + H⺓⺞ → NH2C H2⺓⺞ + H2O (2a′)
1. In class, colored chalks are easier to use than unusual lettering to keep track of individual atoms.
NH2CH2⺓⺞ + 2H2O → NH2C H2⺓OOH + ⺞H3 (2b′) and these add to give: C H2O + H⺓⺞ + H2O → NH2C H2⺓OOH
(1′)
These results are rather astonishing. Equation 1, the simplified “sum” of eqs 2a and 2b, with a little “cancellation” of duplicated molecules turns out to be something of a fake. We have already recognized three significant shortcomings in eq 1, but look closely and see what else is amiss. The oxygen atom from formaldehyde and the nitrogen from hydrogen cyanide are nowhere to be seen in the glycine product. Against intuition, the carbon bonded to oxygen in formaldehyde is
Note
Literature Cited 1. Day, C.; Selbin, J. Theoretical Inorganic Chemistry; Reinhold: New York, 1962. 2. Atkins, P. W. General Chemistry; Scientific American Books: New York, 1989. 3. Oxtoby, D.; Nachtrieb, N.; Freeman, W. Chemistry, Science of Change; Saunders College Publ.: Orlando, FL, 1990. 4. Holtzclaw, H.; Robinson, W.; Odom, J. General Chemistry with Qualitative Analysis; Heath: Lexington, MA, 1991. 5. Brady, J.; John Holum, J. Fundamentals of Chemistry, 3rd ed.; Wiley: New York, 1988. 6. Orgel, L. Sci. Am. 1994, 271(10), 76.
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