Concerning Units To the Editor:
Wadlinger1 rightly pointed out a number of traps and misunderstandings resulting from the omission of such descriptive units as atom or waue. Here are some more examples, which any chemist dealing with some physics is likely to encounter. Electron-Volt
I t is quite puzzling to read unit conversions such as 1e V = 23 kcallmol,' rememhering that 1 eV = 1.6 X 10-19 J and that kcal has the same order of magnitude as J. T o where did all these powers of 10 disappear? The answer is in the omission of the descriptive unit atom, or in this case rather particle (it could he electrons, holes, ions). The eV in the example above should obviously read evlparticle, so that eV keal mol 1-=23-X partlcle rnol particle or atom The last term corresponds t o Avogadro's constant, which contains the descriptive unit atom, and which brings in the missing powers of 10: eV eV J cal kcal mol particle J cal kcal rnol particle replacing all terms hut kcallmol by their numerical value:2
' Wadlinger, R. L. J. Chem. Educ. 1983, 60, 942.
'Equalities such as eVIJ = 1.6 lo-'* or callkcal = may surprise students, since the "number of callkcal" is obviously lo3. It should be borne in mind, however, that the last statement algebrically means 103cal/kcal= 1, and not 1 callkcal = lo3! CrawfordF. S. Berkeley Physics Course; McGraw-Hill: New Yo*. 1968; Vol. 3, pp 3-4.
270
Journal of Chemical Education
eV
- (1.6 X 10-19)(1/4.18)(10~3)(6 X loz3)kcaVmol
P -
particle
= 23 kcal/mol
~oncentratlonvs. "Density of Partlcles"
Physicists generally use the word "density" t o express a number of particles per unit volume, i.e., an atomic concentration, with the unit ~ m - ~Thus . a student may find a or-in concentration of, say. protons, given in mol Does rhis imply that mol is dimensionless? It does not, and the paradox once again comes from the omission of atom in Avogadro's constant. Perlodlcal Motion
Let us consider the fundamental equation of periodical phenomena: x = cos(wt 41, or x = cos(2avt 4). The arc unit is rad and the quantity w is correctly expressed in rad s-1. But what about the quantity Zaut? Taking v in s-' and t in s, 2avt ends up dimensionless. This illustrates the ambiguity in the definition of rad, a "dimensionless" unit (1). Now let us remember that the frequency v can also be expressed in cycles per second3 (which means that cycles has been lost when talking of v in Hz or s-I: here is another implied descriptive nnit). 2aut now has the nnit cycles instead of the wanted rad. The key is in the quantity a. We know that 2a is the number of radians per cycle, so that a should not he considered as a dimensionless number, but as a physical constant with unit radlcycle. (This is equally obvious considering the relationship v = w/2a, which yields
+
+
Finally, including the descriptive units in a and in v is necessary to obtain the correct dimension for the expression (Zavt), i.e.
2rr meterlcircumference
rad cycle s = rad cycle s
r meterlradius
Concluslon Ambiguities abound in the expression of physical quantities and units ( I ) . Much attention should be paid to the actual dimension of such universal constants such as N or a. Descriptive units are most useful. In particular, one must bear 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). Acknowledgmed I wish to thank R. L. Wadlinger for fruitful comments. Plerre Strobel C.N.R.S. Cristallographie 166X-38042 Grenoble. Francs
To the Editor Strobel'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 abbreviation of Dowers to 10 that takes place in thk eonversion. Why shoild the student suffer confusion for the sake of the scientists who demand abhreviation? However, the manner in which Strobe1 presents his units in one equation, yet his numerical magnitudes in a separate equation, can itself be confusing. In the unit portion of his equation, Strobe1 actually never eliminates the descriptive unit particle from the right-hand side, so some ambiguity still results, since the final result must be evlparticle on each side of the equal sign-rather than the expressed kcallmole on the right-hand side (1.h.s.) I'dlike to offer the reader a simpler, single-equation alternative which keeps magnitudes and units together, as per the standard method in the USA textbooks 1.0 eV -=
1.6 X 10-'9J particle
particle
1.0 cal 4.18 J
10-~kcal 6.0 X loz3particles 1.0 cal 1.0 mol
Thus...iust as Strobel writes 1eVloarticleon the1.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 (esch being unity) to com&etely cancel all units but kcallmol. The same magnitudes that Strobe1 uses then yield the correct f i a l equation: 1.0 evlparticle = 23 kcal/mol. Concerning Strobel's argument pertinent t o particle density: There is a real analogy with the notion of wave number in "Concerning Units". That is, wave number is alinear wave density, the number of waves fitting in one meter in the SI svstem. Just as one must not abbreviate from m ~ l - c m -to~ cm-3, because of the ambiguity which enters the physical interoretation for the student. likewise one must not abbreviate-from wave-m-1 to m-1, a s 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. Obviously, a ratio of lengths is dimensionless. So, indeed, r or 2s may be set in the dimensionless category in this case. However, though such dimensionlessness exists in this case, r and 2s carry descriptive units. Setting these ratios up rigorously, we have A
-- 2 s radiilcircumferenee
(3)
Thus, while dimensionlessness holds, unitlessness does not hold. These descriptive units provide strict definition, allowing for no ambiguity physically. I thank Charles R. Brent for this insight. When angle is pertinent, however, each s and 2a carry the dimension angle in radians, as expressed by IUPAC. (When I proposed this in "Concerning Units," I was not aware that IUPAC had already listed the plane angle dimension as the radian.) The argument of any trigonometric function (sin, cos, etc.) must t ~ angle, e as anyone finds out upon trying to find the values of such functions in a handhook. Either the English system is lidted (degrees, minutes, and seconds), or themetric system (radian). o n e cannot take the sin, cos, etc., of a dimensionless number, since angle is part of the definition of such functions. Aminor error is seen in the first equation of Strobel under the headine motion". in that 2a radians are in one -"oeriodic . cycle, not the listed s radians; i.e., Strobel's pertinent equation should read
In conclusion, neither Wadlinger's paper nor Strobel's Letter would have been needed, had scientists historically taken care to retain proper descriptive units. Robert Wadllnger 4963 Creek Rd. En. Lewiston, NY 14092
Different Choices To the Editor: H. R. Kemo 11987. 64. 1911 wiselv advocates that the values of p h y & l quantities de treatkd as independent of 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 be quite helpful to a student calculating the concentration of 96% sulfuric acid having a density of 1.84 g1mL: 96gttcid1100-~
9Bgaeidlmol acid
1.84&/mLsoln
= 0.018 mol aeid1mL soln
If we write the quantities in this case as 96 g1100 g, 98 glmol, and 1.84 g/mL, as Kemp recommends, we needlessly discard information that can keen us from dividine bv 0.96. A recommended expression such as composition of soh. x density of soh. = concentration of soln. molar mass of acid
-
u
may also be required, but is it to be 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 conventimd assignment of dimensionsto physical quantities enables ua to use 'dirnensimal analysis' to check that some algebraic equation interrelating.. .phy?ical 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 wrote. ..: 'if in the same problem. . .two authors make a different choice lfor the number of dimensionsl. the one ehrmaing the greater number is likely to be the more competent physicist' " (31. Even if this is a bit strong, it may still be worth consideration. Volume 66 Number 3 March 1989
271
