plicitly accounted for and differentiated with subscripts 1 (left)and r (right). It should be made clear to students that 1 and r refer & the electrodes of the symbolic cell, and not to the way the electrons are placed with respect to the $ symbol in the equilibrium equation, as one is used to in a thermodynamics context. One might write all or some of these three equilibrium equations the other way round. This would have no effect on the definition of the electrode potential of the Cu2+/Cuelectrode system. Leaving out the electrons or cancelling them, one would be left with the chemical equilibrium:
This equation is certainly not descriptive of the situation in the cell defined above. but would eventually become so after the cell has been shorkircuited so that ill electrons in the metal phase end up in the same thermod.mamic state and can be cancelled from the equation once zero-current conditions have been reestablished. The electrochemical and the thermodynamic IeWright conventions operate on completely different situations and should not be permitted to confuse students. A great deal of confusion is avoided if for cells in zero-current operation no reference is made to processes (reduction, oxidation) and derived concepts (cathode, anode). However, the references are there inthe literature and one would like to have a sensible argument to explain terms like reduction potential, etc. One might argue along the following lines. The IUPAC convention forces us to build a cell with the standard hydrogen electrode on the lcft. The convention also demands that with thiii cell in finite-current operation the external current must run from right to left. Therefore. in the circuit, electrons must be produced by oxidation on the IeRhand electrode (therefore called anode) and consumed hy reduction on the nght-hand electrode (therefore called cathode!. It is clear. however. that earlv introduction of the term "reduction" i i n o t likel; to be heipful to a good understandine of the electrode potential as an eauilibrium
countered redox titrations, and it seems that dedicated al-
would have to be written for these cases. -eorithms Finally, to assure "no approximations" for these redox titrations, any computational algorithm would have to take into account the changing ionic strength of the titration medium.
Donald C. Jackman Pfeiffer College Misenheimer, NC 28109 To the Editor:
In reply to Jackman's letter, I would like to make the following points.
1.Jackrnan is correct in his statement concerningdichromate titrations. In fact, our model only applies to redox systems such as
if a = b; this was pointed out at the beginning of our paper, when we set a = b = 1. We are still unable to propose a general solution for a + b. 2. Our model, contrary to Jackman's statement, can be applied to redox systems that are pH-dependent. In fact. even though we mentioned this i n t h e paper, perhaps we did not give this point the necessary emphasis. Let us consider, as a n example, the titration of Fez+with M n 0 4 in acid solution. The redox systems are
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Maarten C.A. Donkersloot Eindhoven University of Technology 5612 AZ Eindhoven The Netherlands
EO1*and E,z* need to have their values determined under the same conditions (i.e., temperature, ionic strength, auxiliary reagents, indicating, reference and auxiliary electrodes, . . .) in which the titration is supposed to be performed, but a t several pH values. A plot ofE. measured versus pH will give E,* as the intercept on the Y axis. Then, if one wants to represent the system as
for a given pH value, one will have Algorithm for Plotting Titration Curves To the Editor:
Implicit in his claim of not introducinp any approximations-in treating redox titration curves.de MO;;~ [ISSO, 67, 2261 suggests that any redox titration sy3trm may be treatpd with hls aleorithm. Indeed. his aleorithm for dotting redox titratio; curves seems to be good for syst'ems that are not dependent on pH or product concentration, but is not applicable to those situations that the student mav more likely encounter than de Moura's chosen example. Students "are much more likely to encounter the iron(1IYpermanganate and the iron(II)/dichromate titrations in their actual laboratory experiences. In the former case, the equivalence point potential is dependent not only upon the formal redox potekials of the reactants but also pH (the actual term is 4 . 0 8 pH). In the latter case, the equivalence point potential is dependent upon the formal redox potentials of the reactants, the pH, and the concentration of the chromium(II1)ion produced (the actual term is -0.118 pH - 0.017 log LC$+]). The calculational algorithm of de Moura does not treat these more commonly en-
For the system Fe3+me2+ E, = E,*
For the system Mn04-/Mn2+,at 25 'C
where x is the pH of the solution. The values ofEO1and Eo2 are the ones to be used in our model. 3. During a given titration the temperature, the ionic strength, and the pH of the solution are supposedly kept constant. 4. In the titration taken as an example in our paper both systems are pH dependent. For both of then, one has at 25 'C: E, = ED*- 0.118 p H V Consequently, the reaction will apparently become pHindependent, since the two terms (-0.118 pH) will cancel out. However, I must admit, that in order to obtain the data for Figure 1of the paper, the pH was made equal to 0, and Volume 69 Number 3 March 1992
257
the values ofE,* were taken from the literature, disregarding the ionic strength originally used. 1 don't think this invalidates the proposal the paper contains. Daniel Rodrigues de Moura Universidad Federal de Minas Gerais Belo Horizonte, MG, Brazil
Solving Quadratic Equations To the Editor:
In a recent issue, Brown [1990,67,4091 pointed out that the solution of the quadratic equation (1) ax2+&+e=0 bv the standard formula
does not work well if the absolute value of 4ac is much smaller than b2 because one of the roots then is a small difference between large numbers. For such cases he suggests a different formula. However, for 14acl
