1atm), and it hides the fact that the argument of the logarithm is dimensionless. By summing eq 3 for the components of a chemical reaction. multi~lvinp - . - each equation hy the stoichiometricnumber ;i for the corresponding component (1)(positive for products, negative for reactants), we obtain the equation for a chemical reaction:
Literature Cited 1. Mills, I.: Cultas, T : Homann, R Kallay. N.: Kuchitau. K Qunnaties, units o n d s y m bois in Physkal Chemistry. the IUPAC G m n Boob; Blackwell Scientific: London. 1988 (ISBN 0-632-02591-3).
2. Mills, I. M. J. Chrm. Edur
1980,66, 387-889.
Ian M. Mills University of Reading Whiteknights Reading RG6 ZAD, U. K
To the Editor: I reply to Ian Mills'comments on my paper (1)as follows: and if we put AG = 0, the condition for equilihrium, then we obtain AG" = -RT In I@ (5) where
Note that KO is always dimensionless, although K, has the dimensions of pressure raised to the power $ vi. These equations frequently appear in some form in which the quantities KO and Kp are confused, or represented by the same symbol, which is misleading and hides the fact that the argument of the logarithm is always a dimensionless ratio. It is useful to have distinct names for the dimensionless quantity @, and the quantities Kp,or Kc, or K,, etc.; the recommended names ( I , 2) are the standard equilibrium constant (or the thermodynamic equilibrium constant) for @, and the equilihrium constant on a pressure basis (K.)., or a concentration basis (K,). .. or the aciditv constant etc. Differentiating eq 5 with respect to temperature gives
k.), ?.
(1) Meaning of "log gram"
In his first paragraph, Mills has misquoted the comments that I made, a t the end of the section headed "Some Formalism", on the "meaning of 'log gram', etc". My original wording was: One general objection that may be raised to this approach is that terms such as 'log gram'have no meaning. However, this is not a serious objection from the pragmatic viewpoint so long as such terms are used in a consistent and clear fashion.
The paper indeed shows how this can be done. My approach here is that of the philosopher Ludwig Wittgenstein: "The meaning of a word is its use". I view the assignment of a use and hence a meaning to "log x" and to "log gram" as a continuation of the historical development of mathematics that led to the acceptance of previously "improper" concepts such as negative numbers, irrational numbers, imaginary numbers, and infinity. It seems to me to be more fruitful to trv to find a use and a meanine for the present concepts, &her than to assert that no such use or meaning can be found. (2) Differentiation of log x
= a In K, since @ and Kp differ by a constant Here a In factor that is temperature independent. Both a In @ and a In Kp are dimensionless (even though Kpitself may have dimensions, see eq 2). Similar results are obtained in other cases; we are always concerned with either the logarithm of a dimensionless ratio, or the difference of two logarithms (which amounts to the same thing, see eq 1).Thus the Nernst equation should be written
where m denotes molality (or more strictly activity), but the m e factors in the denominator are frequently omitted thus hiding the dimensionless nature of the argument of the logarithm. The definition of pH should he written (1)
but the mol dm4 in the denominator is frequently omitted, thus hiding the dimensionless nature of the argument of the logarithm. I would summarize the situation as follows. The problem of taking the logarithm of quantities with units never arises in science. We always require the difference of two logarithms, or the logarithm of a ratio. However because we often seem to write our equations in a sloppy way, omitting important denominators within the argument of logarithms, we confuse ourselves-not to mention our students. The moral i s self evident: do not omit t h e denominators!
In his second paragraph, Mills discusses the fact that d (log x) has no units. I showed in my paper that this arises simply from the situation that the units of log x are additive (rather than multiplicative) so that these units will disappear on differentiation. (3) IUPAC Procedure
Mills then outlines the IUPAC method (2) for dealing with standard thermodvnamic functions. etc. where there are doubts about the propriety ?f log x if x has units. My article resolves these doubts, and shows that the IUPAC procedures (that is, dividing by the appropriate standard pressures andlor concentrations) are now superfluous and should be abandoned. (4) Drawbacks of the IUPAC Procedures
From the teaching viewpoint, it is my experience that students have to be drilled to quote units (rather than just the bare numbers) a t all stages of a calculation; this applies particularly in teaching the basics of equilihrium constants and the properties of equilibrium systems. It is then most unsatisfactory, in moving on to deal with standard thermodynamic functions, to have to say that all of this was wrong and that an equilihrium constant has no units a t all- a situation attained by dividing by whatever the units happen to be! My paper shows how this schizophrenia can be avoided. (5) Tabulated Data in Handbooks M y paper also allows us to deal with the common practi-
cal situation with handbooks (see for example, 3, 4)-where the standard states of tabulated thermodynamic data are not immediately evident. The notation proposed in my paper (I)would enahle the standard states to be incorporated into the column headings of the table. Volume 72 Number 10 October 1995
955
To the Editoc
Literature Cited 1. Molyneur, P,JChom Educ, 1891,68.467-469. 2. Mills. I.; Cvitas, T.; Homann, K KsUay. N.; Kuehitsu, Eds. Quontirks, Units and Symbols inPhysicol Chamisfry; IUPACBlackwell: Oxford, 1988. 3. &lwsrd, G.H.;Rndlay, T. J. V. S I Ch~micolDoto,2nd ed.; W~ley:Sydneg 1974. 4. Wesst, R. C., Ed. Handbook ofChemisfry mnd Physies, 67th ed.: CRC Ress: Bas
Rston, FL, 198M87.
Philip Molyneux Macrophile Associates 53 Crestway, Roehampton London SW15 5DB, U.K Reactions of Rare Earth Metals To the Editor: The recent article about the rare earth elements bv Solomon and Lee ( I ) perpetuates the widespread miscouception that the rare earth metals dissolve in acids vieldina only typical, trivalent rare earth salts. When t h i acid is hot, dilute acetic acid, the reaction yields a precipitate. This compound appears to be a mixed oxide-hydride: two hydride (or acetate) ions, bearing a total charge of 2-, replace one oxide ion, also bearing a charge of 2-, in the incompletely filled fluorite crystal lattice (2).This pheuomenon would make a n excellent research topic.
I was oleased to hear of Professor Dust's interest in our PEG moiecular weight experiment. I am very familiar with Professor dust's work in this area. As a matter of fact, in our graduate-level Polymar Chemistry Laboratory we have students determine M, bv both the titrimetric and proton NMRmethods. The e&&re to NMR is extended to include measurement of carhon-13 TIrelaxation times of PEG'S and their dimethoxy derivatives. In PEG intermolecular bydrogen-bonding restricts motion of the carbons near the hydroxyl group and decreases corresponding TI values. This effect is not observed in ethers. Although we have performed the NMR experiments for several vears. Mr. Bernier and I wanted to em~hasizethe exposure of lower level students to polymer concepts. Therefore. we chose not to mention the ex~erimentsfor advanced &dents Kathryn R. Wllllams University of Florida P.O. Box 117200 Gainesville, FL 32611-7200 Other Publications on C6Ck and CzCls To the Editor:
Literature Cited
reference8 therein.
G. L. Silver EG&GMound Applied Technologies P.O. BOX 3000 Miamisburg, OH 45343-3000
Arecent article [J. Chem. Educ. 1994,71,7041 described the formation of C6C16 and CzCl6 from the high-temperature reaction of CCL and Cr203.Readers may be interested in knowing that the formation of C6C16and C2C16from this reaction and from the decomposition of CC14was described ~reviouslvin this Journal [J. Chem. Educ. 1978.55. 52; J. Richard L. Keiter Eastern Illinois University Charleston, IL61920
Undergraduate Polymer Experiment To the Editoc I n a very thorough article K. R. Williams and U. R. Bemier described a n undergraduate polymer experiment in which students determined the number-average molecular weight, M,, of a variety of polyethylene glycol (PEG) samples [J. Chem. Educ. 1994, 71, 2651. The method emoloved was a titration-based e u d - m o u ~analv&a&ydridh sis of the b[s (i,2,4,5-henzenetetracarb~x~lic derivatives. Unknowns were DreDared . . bv mixing known masses of two commercial PEGS of nominal molecular weiehts of 200 and 400 elmol.. res~ectivelv.These molecular weights happen to fail nicely into the region of M,, that can be determined rapidly and with reasonable accuracy (8%)by proton NMRin dimethyl sulfoxide solvent (1). Consequently, a possible extension of the experiment for use in upper-level courses c a n be suggested; students could initiallv estimate the M" of their s a m ~ l e sbv 'H NMR. followed bv the derivatization and titration analysis outlined hy fllllmns and Bern~er.Such an extenswn &~ld permlt students to iudae the "fitness to purpose" ofeach analytical method. A
Literature Cited 1. Dust, J. m.; Fang,2. H.;Hsrris, J.M.Mmmmoloeulea 1990.23, 3742.
Julian M. Dust Sir Wilfred GrenfeliCollege Comer Brook, NF, Canada, A2H 6P9
956
Journal of Chemical Education
Organic Hourglass Inclusions To the Editor:
.
As an analvtical chemist in the oimnent industrv. .. I found the article "Organic Hourglass Inclusions," [J. Chem. Educ. 1994.71. 5841 verv interestine. Since BASF is one of the major s;p&iers-of aikali blue p&ment for which fuchsin is a n intermediate, I just had to try the experiment. I t worked nicely and I am looking forward to future articles on this subject (which the authors allude to on p 585"other candidates . . .are in preparation"). I found that the demo can be "s~eeded-UD" mi. bv" eoinrr cro. I placed 10 drops of sulfate on a micro slide and then added 1 d r o ~of 0.1% acid fuchsin which was mixed-in with a plastic disposable pipet. I n 30-60 min enough water had evaporated from the prep to start the crystallization process to produce the hourglass crystals. The dried prep contained a number of perfectly formed crystals that could he saved "as-is" on the microscope slide and viewed under low power (50x1 for observation. This process may work as a quick screening test for future hourglass crystal systems. James Benkn Research Associate BASF Corporation 491 Columbia Avenue Holland MI 49423 u
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