In the Classroom
Correctly Expressing Atomic Weights Moreno Paolini* Department of Pharmacology, Biochemical Toxicology Unit, University of Bologna, via Irnerio, 48, I-40126 Bologna, Italy;
[email protected] Giovanni Cercignani and Carlo Bauer Department of Physiology and Biochemistry, Biochemistry Unit, University of Pisa, via S. Maria, 55, I-56100 Pisa, Italy
The ambiguity inherent in the reporting of atomic or molecular weights as either dimensional or dimensionless (1, 2) often leads to scientific errors. Even though both points of view have been widely expressed, and papers dealing with particular aspects of atomic and molecular weight definition have appeared in these pages over the past several decades (3–6 ), the main question remains unsettled. It is usually stated that atomic weights are dimensionless (7, 8) because they express the ratio between the weight of an atom and the weight of the atomic mass unit ( 1⁄12 of the carbon-12 isotope). If this reasoning were taken to its logical conclusion, then all physical quantities normally considered as dimensional would have to be expressed as pure numbers because they are ultimately derived from the ratio with the standard quantity chosen as the unit. However, if a table is twice the standard measure, we have no hesitation in saying that it is two meters long and this is clearly dimensional! The reason for this apparent paradox can be elucidated by the proposal of the Symbols Committee of the Royal Society of Chemistry, in agreement with the International Union of Pure and Applied Chemistry (IUPAC), the International Standards Organization (ISO), and the International Union of Pure and Applied Physics (IUPAP), whereby the value of a physical quantity is expressed by the equation (9–12) physical quantity = numerical value × unit
(1)
Considering the “atomic weight” as a physical quantity and making the suitable substitutions to eq 1, we get
atomic weight = numerical value (2) 1/12 mass of the carbon–12 isotope The written value thus obtained is clearly dimensionless because it expresses the ratio between physical quantities sharing the same dimension (atomic weight, which may be expressed in daltons, and standard atomic weight or atomic mass unit, which is exactly 1 dalton). Similarly, considering “length” as quantity, on the bases of eq 1, we have
length = numerical value 1 meter
(3)
As in eq 2, this ratio is nondimensional, since both length and standard length (length unit) are expressed in meters. To pinpoint the problem, it should be noted that the simple mathematical operation carried out in measuring a generic physical quantity establishes how many times the standard unit is contained in the quantity. This procedure must not be confused with the subsequent written or spoken expression of the measure of quantity itself.
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In fact, the theory of measure tells us that if “A” is the physical quantity, “U ” the chosen unit quantity, and “m” the ratio between them in such a way that A/U = m, then m is a pure number that expresses how many times U is contained in A. The quantity A (dimensional) is therefore A=m×U
(4)
that is, m times the unit quantity. For example, to indicate a “length” from eq 4 we obviously state that length = numerical value × meters
(5)
Similarly, to indicate an “atomic weight” on the bases of eq 4 atomic weight = numerical value × daltons
(6)
In this way the atomic mass must be considered as a dimensional entity and it clearly has to be expressed in daltons, a particular subdivision of the gram that was chosen in such a way as to avoid too many zeroes. A similar argument applies in the universally accepted definition of the Svedberg unit, 1 S = 1013 s, where the Svedberg unit is a particular subdivision of the second (s). It should be noted that in the above expressions the factor “numerical value” could be indicated as “relative value” such as “Lr = relative length”, “Mr = relative molecular weight”, etc., which are of course, pure numbers. Taken as a whole, the relative atomic (or molecular) weights (Mr) are pure numbers, but when written without subscript (i.e., M ), such weights are not dimensionless. It is also possible to obtain dimensionless relative values by giving the ratio of the dimensional absolute measure of two quantities of the same type, one of which is not necessarily the chosen measure unit, and in this way it is possible to speak of items such as relative density δ r (i.e., specific gravity), dielectric constant (ε r), etc. As an example, let us consider a typical operational definition of the molar mass of a protein, such as the Svedberg equation:
M= R×T× s D 1 –V ρ
(7)
where R is the gas constant, T is the absolute temperature, s is seconds, D is the–diffusion coefficient, ρ is the density of the solvent, and V is the partial specific volume. The dimensions in the right-hand side of eq 7 (bearing in mind that energy = force × length = mass × length2 × time2) are energy× temperature× time temperature × moles 1
length × time 2
=
mass × length2 × time1 length2 × time1 × moles
Journal of Chemical Education • Vol. 77 No. 11 November 2000 • JChemEd.chem.wisc.edu
= mass × moles1
In the Classroom
The molecular weight (or mass) of the protein is obtained by dividing its (operationally defined) molar mass by N (× moles1), and the result then clearly indicates a mass. Overall, our proposal that atomic and molecular weights be expressed as dimensional quantities (viz., daltons) is at variance with the traditional (nondimensional) interpretation of the definition of molecular weight, but is consistent with experimental reality (i.e., operational definitions). Acknowledgments We thank G. F. Pedulli (Department of Organic Chemistry) and M. Recanatini (Department of Pharmaceutical Sciences), University of Bologna; G. Moruzzi (Department of Physics), University of Pisa; and N. M. Trieff (Division of Environmental Toxicology and Community Health), University of Texas Medical Branch at Galveston, TX, USA, for critical comment on the manuscript and helpful advice. Robin M. T. Cooke helped revise the manuscript. This work was supported by CTB (Inter-Deps. Biotechnology Center).
Literature Cited 1. Akiba, T.; Toyoshima, C.; Matsunaga, T.; Kawamoto, M.; Kubota, T.; Fukuyama, K.; Namba, K.; Matsubara, H. Nat. Struct. Biol. 1996, 6, 553–561. 2. Kauranen, M.; Boutton, C.; Verbiest, T.; Teerenstra, M. N.; Clays, K.; Schouten, A. J.; Nolte, J. M.; Persoons, A. Science 1995, 270, 968–969. 3. Huff, R. B.; Evans, D. W. J. Chem. Educ. 1991, 68, 675–676. 4. Gorin, G. J. Chem Educ. 1984, 61, 1045. 5. Nelson, R. A. J. Chem. Educ. 1979, 56, 662. 6. Richardson, D. E. J. Chem. Educ. 1992, 69, 736. 7. Dawes, E. A. Quantitative Problems in Biochemistry, 6th ed.; Longman: London, 1980; p 2. 8. Commission on Atomic Weights and Isotopic Abundances, Inorganic Chemistry Division, International Union of Pure and Applied Chemistry. Pure Appl. Chem. 1980, 52, 2351. 9. Ferreira, R. Nature 1986, 324, 215–216. 10. Liebeq, C. Nature 1985, 314, 586. 11. Martinelli, L. W. Nature 1985, 316, 489. 12. Paolini, M. Nature 1986, 321, 568.
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