2ur meterlcircumference - 2 s radiilcircumferenee r meterlradius
rad cycle s = rad cycle s
Conclusion
Ambiguities abound in the expression of physical quantities and units ( I ) . Much attention should he paid to the actual dimension of such universal constants such as N or a. Descriptive units are most useful. In particular, one must hear in mind that physicists' language mostly considers quantities a t the atomic scale (in contrast with the "molar" chemistry) and makes universal use of the hidden descriptive units atomlparticle and cycle (or wave). Acknowledgment
I wish to thank R. L. Wadlinger for fruitful comments. Pierre Strobei C.N.R.S. Cristallographie 166x38042Grenoble. France
To the Editor:
Strohel's positive response to my article is sincerely appreciated, and his additional comments are most welcome. Especially is his electron-volt argument valuable, for the student can be confused easilv bv the ahhreviation of Dowers to 10 that takes place in thk eonversion. Why shokd the student suffer confusion for the sake of the scientists who demand ahhreviation? However, the manner in which Strohel presents his units in one equation, yet his numerical magnitudes in a separate equation, can itself he confusing. In the unit portion of his equation, Strohel actually never eliminates the descriptive unit particle from the right-hand side, so some amhiguity stillresults, since the final result must beeV1particle on each side of the equal sign-rather than the expressed kcallmole on the right-hand side (1.h.s.) I'd like to offer the reader a simpler, single-equation alternative which keeps magnitudes and units together, as per the standard method in the USA textbooks 1.6 X 10-'9J
1.0 eV -=
particle
particle
1.0 csl 4.18 J
10-~kcal 6.0 X 102sparticles 1.0 cal 1.0 mol
Thus..iust as Strohel writes 1e v l ~ a r t i c l e o nthe1.h.s. of his unit equation, the above r.h.s. hegins with the Joule equivalent . Der oarticle. then freshman chemistrv conversion factors are used (each being unity) to completely cancel all units hut kcallmol. The same magnitudes that Strohel uses then yield the correct final equation: 1.0 evlparticle = 23 kcallmol. Concerning Strohel's argument pertinent to particle density: There is a real analogy with the notion of wave numher in "Concerning Units". That is, wave numher is a linear wave density, the numher of waves fitting in one meter in the SI svstem. Just as one must not abbreviate from m ~ l - c m -to~ cm-8, because of the amhiguity which enters the physical inter~retationfor the student. likewise one must not ahhreviate from wave-m-1 to m-1, as has happened historically. Some physical entity exists per cm3, or per m, and that is the important description of reality that the student needs! Regarding pi (s): There are two ways of regarding s or 2s. One involves the simple ratio of a circle's circumference to its radius or to its diameter-a ratio of lengths per se. Ohviously, a ratio of lengths is dimensionless. So, indeed, u or 2 s may be set in the dimensionless category in this case. However, though such- dimensionlessness exists in this case, s and 2 s carry descriptive units. Setting these ratios up rigorously, we have 2rr meterlcircumference - T diameters/circumference (2) 2r meterldiameter
-
(3)
Thus, while dimensionlessness holds, unitlessness does not hold. These descriptive units provide strict definition, allowing for no amhiguity physically. I thank Charles R. Brent for this insight. When angle is pertinent, however, each a and 2 s carry the dimension anele in radians. as exnressed hvIUPAC. (When I proposed this in "Concerning Units," I was not aware that IUPAC had alreadv listed the lane a n d e dimension as the radian.) The argument of an;trigono&etric function (sin, cos, etc.) must he angle, as anyone finds out upon trying to find the values of such functions in a handbook. Either the English system is listed (degrees, minutes, and seconds), or themetric system (radian). o n e cannot take the sin, cos, etc., of a dimensionless numher, since angle is part of the definition of such functions. Aminor error is seen in the first equation of Strohel under motion". in that 2 s radians are in one the headine -"Deriodic . cycle, not the listed s radians; i.e., Strohel's pertinent equation should read
In conclusion, neither Wadlinger's paper nor Strohel's Letter would have been needed, had scientists historically taken care to retain proper descriptive units. Robert Wadlinger
4963 Creak Rd. En. LeWiSlOn, NY
14092
Different Choices To the Editor:
H. R. Kemo 11987. 64. 1911 wiselv advocates that the de treatkd as independent of values of p h y & l the units used, and he refers to the thorougb treatment by McGlashan (I).Kemp also supports, however, a sometimes inconvenient prohibition against expressions like the following (my specific example), which may he quite helpful t o a student calculating the concentration of 96% sulfuric acid having a density of 1.84 g1mL: 96-9ttcia1100-~ 98gaeib/molacid
1 . 8 4 d l m L sol" = 0.018 mol seid1mL soln (or 18 mol acidlL soh)
If we write the quantities in this case as 96 g/lOOg. 98g/mol, end 1.64 glml., as Kemp recommends, we needlessly discard information that can k e e us ~ from dividine hv 0.96. A recommended expression such as composition of soh. x density of soh. = concentration of soh. molar mass of acid
-
u
may also he required, but is it to he memorized? On the wisdom of treating, for example, the mass of acid and mass of solution as dimensionally independent when we need to do so, we may also quote McGlashan (2):
". . .the emventional assignment vfdimensions to physical quantities enables us to use 'dimenrimal analysis' to check that sume algebraic equation interrelating.. .physical quantities is a possible one, or even. . .the only one. " . . .we could. . .assign different dimensionsto lengths, breadths, and heights, and.. .to the lengths of desks, . . .of chemical bonds, and so on. ". . .Guggenheim mote. ..: 'if in the same problem. . .two authors make a different choice lfor the numher of dimensiousl. the one ehmsing the greeter number is likely to be the more competent physieiat'" (3). Even if this is a bit strong, it may srdl be worth consideration. Volume 66 Number 3 March 1989
271
