New universal absorhrnts such as spun-bonded polvolefins and liquid neutralizers containing monoethanol amine offer advantages that should also not be overlooked. Donald D. Hedberg, President Lab Safety Supply Co. 3430 Paimer Drive P.0. BOX 1368 Janesville, WI 53547
1). nXn-'11 is obtained bv the resoective differentations and Xapproaches one so thai limit 0;" - 1)/(X - 1) = n because n(ln-1) = 1 for all n, and all of this is hecause In-' = n for all n. Proof of limit (X" l)/(A - 1) where (A" 1) = (A l)(X"-' A"-2 .. . A 1) also can be shown by mathematical induction. Case 1: (A - l)/(X - 1) when n = 1 is 1, Case 2: (X2- 1)/(X - 1) when n = 2 is 2, Case 3: (A3 - l)(h - 1) when n = 3 is 3. Now when n = k, (Ak - 1)/(X - 1) = (Ak-I hk-2 . . . X 1) = k. Finally, the case where n = k 1 is taken. The k 1term is added to both sides of the (Ak - 1)/ (A - 1) relation so that k 1 is the obtained sum. Therefore ( X k - l ) / ( h - l ) + X k = (Xk+Xb-1+Ak-2+...X1) =k+Xkis the result of addition and k Xk = k 1 because Xk = 1. Q.E.D. Q.E.D. or Quod Errat Demonstratum is a Latin phrase for "that which has been proved", but in free translation, it is "quit, enough done" or "quite easily done". The latter of the free translations applies to the above proof. ~~~~
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Although not touted as a universal spill cleanup procedure, the AC procedure can be used for safe, efficient cleanup of nearly all liquid chemical spills. The AC procedure is general, easy to learn and remember, quick and easy to carry out, very economical to implement and use, and in&insicaCy safe. Learning and practicing the AC procedure will help students and-laboratory workers understand the inherent faults of emergency cleanup procedures that emphasize direct deactivation of soilled chemicals. Note that t h ~ . A vp k d u r e ii very effective for cleanupof hvdmfluor~cacid spills. There is no visible evidence of any chemical reaction &hen sand absorbs 48%, 5 M, or 1 M solution. Any silicon tetrafluoride produced by the slow reaction of silica with the hydrofluoric acid probably reacts immediately to form fluorosilicic acid. Also, sand is a safe, efficient absorbent for hot, 70% perchloric acid, a very hazardous chemical that certainly could not be absorbed safely using "new universal absorbents such as spun-bonded polyolefins." Sand is the ideal absorbent for use with the AC orocedure. Sand is a very good absorbent for most laboratory liquids, even low-viscositv, volatile liouids. Sand is more dense than most liquids, ineit to nearly d l chemicals, readily available, and very economical to use. In short, sand is the best allaround absorbent for liquid chemicals spilled in the laboratory. Commercial absorbents and spill cleanup materials will find applications in specific cleanup situations, t u t the AC soill clc,anuu orocedure should be taueht and irn~~lemented as a progresi&e step in applied chemiial technol&y. Norman S. Nelson
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To the Editor:
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William D. Hlii Jr. NO^ Carolina Central Univenity Durham. NC 27707
Rethlnklng the Clement and Desormes Heat Capacity Ratlo To the Editor: Bv coincidence. we were orenarine a oaoer makine the 63, same point as that of Bert;& a n d - ~ i ~ o n a l[198i, d 2521. concernina the erroneous discussion and derivation of the'equation Gociated with the Clement and Desormes heat capacity . . ratio ex~eriment.The error was first Dointed out many years ago [1947,24,251], but apparently ~ k r t r a n d and McDonald are unaware of this reference. We would make two points beyond those in the more recent article: (1) There is no information whatever about the heat capacity ratio, per se, in the adiabatic expansion experiment. In the ideal gas approximation as presented by Hertrand and Mc1)onald. the thermal oronertv of the ras under study that determines the results bf t i e experiment is C,. nothine more or less. The ratio enters the discussion as a matter of convenience and depends on the assumption that C, - C, = R. which is not strictly true for real cases. (2) The e;roneous assumption of reversibility in theherivation of the commonly used ea 9, is completelv counteracted by the introductionbf an approximatik (truncation of the &arithmic expansion) rc.sulting in the pruperly derived eq 8! Edwin F. Meyer and George H. Stewart Texas Woman's University Dentan. TX 76204
Avogadro's Number: A Perverse View Comments on Euler's Theorem for Homogeneous Functions and Proofs Thereof To the Editoc The discussion of Euler's theorem with respect to thermodynamic functions by R. J. Tykodi (J.Chem. Educ. 1982,59, 557) and by Michael A. Adewurni (J. Chem. Educ. 1986, 63(7), 610) were and still are very interesting. There really should be no confusion about LCHospital's rule for any function involving a numerator and a denominator. This includes findine the limit of (An - l)/(A - 1) when X approachm 1. The deri\,'ati\.eof the n h e r a t o ; is divided hy the derivative of the dcn(miator and then evaluated to the limit. In many cases the differentations are executed many times before finding the limit, but in the case of (An- l)/(A 282
Journal of Chemical Education
To the Editor: Poskozim. Wazorick. Tiemoetoaisal. and Poskozim 11986. . . 63, I251 bring togethe; a usefui set of illustrations with a common ~ u r ~ o s e - t ohelo students (and their instructors?) compreh;ndthe size of ~ i o a d ~ r onumber. 's On reading this, I asked myself, perhaps perversely, what could he said to make Avogadro's number seem smaller than it does in such illustrations. Because 279is almost exactly 6 X loz3, if you start with one of anything and double it a mere 79 times, you have Avogadro's number of them. It's best to start with something small. Thomas A. Lehman Bethel College North Newton. KS 67117
