LETTERS
To the Editor: Recently1- three different methods were suggested to rrrover the solution of the differential equation
well known to neighbors of Brooklyn Polytech. would do well to read this most delightful bit of scientific journal prose.
from the solution of the differential equation
These methods involve series expansions, use of the L'Hospital Theorem and the limit of the series (1 x/n)". However, it is instructive to note that by a slight change of the form of the solution of eqn. (2), one can recover the solution of eqn. (1) without recourse to any mathematical theorem. To see that, we write the solution of (2) in the usual and rearranged forms
+
In b - In a b-a
+ ln(b - zb) --aIn(a - x)
(3)
Now the limit of the right hand side of (3), when b -c a is, by definition, the derivative of the logarithmic functions In b and ln(b - x), i.e.,
which is the solution of eqn. (1).
To the Editor: Your February 1966 issue mas of particular interest because of the article on Dipole Uxnents. Because of the considerable influence which attaches to an article in the JOURNAL of CHEMICAL EDUCATION, and the brief reference on page 67 to some of my work [J. Phys. Chem, 60, 1336 (1956); not J. Chem. Phys. as actually cited], I would like to make a few comments. My main thesis was not that accurate density measurements are unnecessary, but rather that no densities of solutions need be determined when v2 (the partial specific volume of the solute in solution) can be evaluated by other means, of which a number are available. The quantity v2 appears in the equation for the specific orientation polarization of the solute at infinite dilution according to the exact equation [cf. also: eqn. 7, J . Phys. Chem., 61, 376 (1957)J:
S. BAER HEBREW UNIVERSITY
JERUSALEM, ISR~EL
A. BEN-NAIM OF NEWYORKAT STOXY BROOK SWTEUNIVERSITY STONY BROOK, L. I., NEWYORK
A. S., J. CBEM.EDUC., 35,251 (1957). SAID, LOWELL, S., J. CHEX. EDUC., 42.552 (1965).
To the Editor: I wonder sometimes about the significance of some of the newer 'Lwords" that appear in textbooks. I recently found that the solution to which a titrant is added is 'alled the "titree"! I doubt it! This is hardly in the category of thesuggestion by Giauque, et al. [J. Chem. Phys., 42,8 (1965) ] that "pernt" he used to refer to approximate temperatures in connection with gradual transitions. Incidentally, readers who are not familiar with this West Coast advocacy of a word 680
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Although the idea that density measurements could be obviated occurred to me independently, a literature search showed that Treiber and Porod [Maatsh., 80,481 (1949)l suggested this much earlier. Therefore, my paper was in effectwritten to substant,iatethe validity of such obviation. Having been successfully applied to fifty cases, and failing in none, I feel the method is sufficiently substantiated. I am convinced that much time has been wasted (including my own) and is still being wasted in determining unnecessary densities.
To the Editor: Dr. Estok's proposal has great potential, and he has gathered very convincing evidence for it. Ap-
parently it has not received the attention it deserves, and thus has not (at least yet) been examined in practice by many other leading workers. While I thought it worth notice, I could not recommend it for this reason. It would certainly be very helpful if interested parties would calculate moments by Estok's procedure as well as by conventional methods. I would like to take this opportunity to call attention to the very promising recent work of J. R. Weaver and R. W. Parry [Inorg. Chem., 5,703 (1966)l on calculation, of dipole moments. This is an examination, both theoretical and experimental, of certain procedures not following Debye's approach, and should be particularly interesting to those who wish to determine moments in pure liquids or polar solvents.
To the Editor: Dawber, Brown, and Reed in "Acid Catalyzed Hydrolysis of Sucrose [THIS JOURNAL, 43, 34 (1966)l focused attention on a reaction which has a teaching potential that is often overlooked. Generally sucrose inversion is used as a convenient reaction to introduce polarimetry or to illustrate firs& order kinetics. Dawber, Brown, and Reed pointed out how the reaction could be used to illustrate the Hammett acidity function. I suggest that all of these a3pects-and more--can be conveniently covered if the reaction is utilized in an "open end" type of investigation. I have found the following sequence to be both practical for the professor and satisfying for the students. Students are told that the phenomenon of inversion occurs, and they are asked to investigate the effect which changing the sucrose concentration (at constant pH) has on the rate of inversion. Each student (or pair of students) prepares a solution and observes the changing rotation for one hour. By plotting the ratio of the original sucrose concentration to the concentration remaining at time (t) vs. time, the class obtains a family of curves. These data serve as an excellent basis for the introduction of an application of statistics (Are the differences in the curves significant?) with the fringe benefit of a conclusion that surprises many. Tentative conclusions about the order of the reaction can also be proposed. A student may spot the exponential appearance of the curves. A plot of the log of the ratio of the coocentrations vs. time confirms the sound foundation for his suspicion. Further, because of the relationship that the slope of the line of this plot has to the reaction rate constant, the expression for the rate constant of first-order reactions can be derived directly without using differential equations. A11 investigation of the rate of sucrose inversion with changing pH is a reasonable followup to the previous work. Such an investigation yields, ultimately, the family of straight lines of differing slopes which confirm
the firseorder character of the reaction. With a little prodding, the students recognize that this also provides an insight into the hydrolysis nature of the reaction. This leads to a full discussion of the reaction inechanism. Thoughtful students notice that the slopcs of the lines do not appear to increase proportionately with changing pH. A plot of the reaction rate vs. acid molarity gives a curve such as Dawber, Brown, and Reed present (Fig. 4). The unexpected implicatioiis drawn from this curve prepares students for the discussion of the Hammett acidity function which terminates the investigation. The collection of the data and the discussions take five laboratory periods. The data obtained even by beginning students are very satisfactory, if reasonable care is taken. I shall be glad to furnish details of appropriate concentrations and specific techniques to interested readers. Obviously this particular sequence does not exhaust the instructional possibilities of the approach. However, whether this approach is used or not, the combination of the intrinsic simplicity of the reaction and the quantitation possible with a polarinieter merits more attention as a teaching tool.
To the Editor: Dr. Pode's critical reflections on the CBA and CHI311 Study programs [THISJOURNAL, 43, 98 (10(i(i)]werc most interesting. I was particularly happy that he did not pit one program against the other or align himself with those who would merge these studies into the summum bonum-the perfect high school chcmistry course. As a former member of the CHEM Study staff, and as one who is using the course materials for the fifth successive year, I admit to considerable bias ~vhiclr was born of much sweat but few tears. The impression that the CBA teacher's guide "leans more toward helping with teaching" while that for CHElI Study "indulges in more background discussion" may appeal to the CBA teacher hut I feel that CHEM Study teachers will demur. Of course, these discussions are helpful, but they represent only one (perhaps minor) pillar of support to the inexperienced teachcr. The emphasis in the CHEM Study (T.G.) is on intent and approach, development of ideas, the class schedule, lab hints, and the solutions to the exercises and problems. I wonder if the claim that CHEhl Study is guilty of "introducing ideas without adequate discussion" is shared by practicing teachers? I t has been my esperience that the most useful textbook is characterized by what it omits rather than by what it includes. Undoubtedly, more could have been done in the areas Dr. Pode describes, but how far do me go with the average high school youngster? Volume 43, Number 7 2, December 7 966
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