Concerning Units Robert L. Wadlinger The Pennsylvania State University, The Capitol Campus, Middletown, PA 17057 Clear scientific description of our physical world requires use of proper units. Units come in two distinct types: fundamental and descriptiue. Fundamental SI units of measurement, from which others are derivable, include kilogram, meter, second, and mole. Descriptive units, including atom, molecule, electron, and waue (cycle), serve to relate which physical entity is being measured. Omission of critical units of both varieties from definitions and laws of science has caused confusion as well as stagnation in understanding the nature of elementary i articles and their roles in elemenGy mechanisms. For example, omission of the descriptiue unit light waue (cycle) from the photon-defining relation r = hu has confused that law's meaning for over 75 years. The primary purpose of this paper is to re-introduce that unit, and to examine the clearly defined photon model which is deduced. To do this, some proper and improper usages of the descriptive units atom and molecule from historical cases will first be presented. Omission of a critical fundamental unit will also be examined. The familiar rotational unit radian has been termed dimensionless ( 1 ) and unitless (2). I t will he shown that this is erroneous, and that the radian is as sound a fundamental unit as the mole, since neither can be derivable from other fundamental units. The absence of radian and waue (cycle) from such "unitless" constants as the fine structure constant could be preventing physical insights into an elementary rotational mechanism intrinsic to the atom. Atoms and Molecules John Dalton expressed weight percent in terms of weights of atoms in deducing a method of arriving a t the relative weights of atoms, the chemist's invaluable atomic weight scale. Let us follow the presentation of Grunwald and Johnsen (3). We will use mass for Dalton's weight. Dalton let the definition of % hold forth, stating that 88.88 g of oxygen in 100 g water must equal the mass of one oxygen atom times the number of oxygen atoms in 100 g of water. Analogously, the same relation holds for hydrogen. In unit form, we express these statements as 88.88 g oxygen - # g oxygen X # oxygen atoms (1) 100 g water 1oxygen atom 100 g water 11.12 g hydrogen # g hydrogen # hydrogen atoms X (4 100 g water 1hydrogen atom 100 g water The units on the left-hand side (1.h.s.) of the equals sign must be the same as those on the right-hand side (r.h.s.1 of any equation; hence, only with the units expressed as in (1)and (2) can this he true. The unit oxygen atom cancels across the multiplication sign in (11, and the unit hydrogen atom cancels likewise in (2). Dividing (1) by (2), Dalton concluded 8 g oxygen - mass of 1oxygen atom 1 g hydrogen mass af 1hydrogen atom # of oxygen atoms X (3) # of hydrogen atoms Presuming the last ratio on the right of eqn. (3) equals unity, Dalton found that mass of 1oxygen atom =8 (4) mass of 1hydrogen atom 942
Journal of Chemical Education
Without considering the atom as a unit, Dalton could not have achieved this monumental masterpiece of deduction. H,O Molecule Mass
Full development of Dalton's idea led to science's ability to calculate the mass of a single atom or molecule, though one could not weigh either directly. Today, every first course in chemistry includes such a calculation, an example being 18 g HzO!mole -- 3 x 10P" Hz0 (5) 6 X 102bmaleculesHzOImole 1HzO molecule This result is corroborated beyond reasonable doubt by mass spectrometer measurements. The molecule imagined by Dalton and others proved to be precisely what one finds in nature. The value of the use of the descriptive units atom and molecule is thus significant. IUPAC omits all descriptive units from scientific constants, definitions, and laws based on the presumption that elementary entities such as atom and molecule are understood to be present. This presumption can lead to misunderstandings, exemplified by the following. The Boltrmann Constant Max Planck claimed to he the first to show that division of the ideal nas law constant R hv Avoeadro's number N ~rovides
R JouleIKelvin-mole Joule =k N molecule!mole Kelvin-molecule
(6)
Physics texts, chemistry texts, and handbooks shun the unit molecule when presenting the units for k . In so doing, k is stripped of its basic physical meaning: the energy per (degree) Keluin per single ideal gas molecule. This stripping starts with use of IUPAC's units for Avogadro's number, N mole-', in relation (6), yielding k Joule-keluin-'. The reader then is supposed to comprehend the explicit elementary entity pertinent. If some physical meaning is attributed to k other than energy-Kelvin-'-molecule-', then that must he explicitly expressed as a new definition. However, this has not been the case historically. For example, the foundation-stone of statistical thermodynamics defines the entropy S as the entropy per ensemble, and 12 as the number of complexions in the ensemble (51, by S=klnR
(7)
Placing k entropy units per molecule on the r.h.s. of (7) mandates those same units for S on the 1.h.s. Thus. an envarious types of ensembles of more than one molecule. However, he correctly presents the entropy per molecule as S=a+klnp
(7a)
wherein p is the number of ways in which a molecule could, over a period of time, have a given average energy ( a is shown to equal zero f o r p = 1 a t absolute zero). Later, Boltzmann's relation for a number of molecules is presented S=a
+ k In W,,,
(7bj
wherein Wm,,is the net number of distinguishable permutations among the most probable populations of molecules among the energy states. Since a and k have the same units in each ( l a ) and (Ib), S retains the same units of entropy per molecule. This is critical in the same manner as the unit molecule is critical to eqn. (5). Yet, much confusion exists when some authors present S as the entropy of the ensemble. S is simply the average entropy of one molecule in the ensemble. There is a distinct need for clarification of this m i n t for currently ambiguity prevails for the student. Next, iet us consider another example of misunderstandinr which occurs when an elementary entity is "understood" tdhe present. Waves and Dollars In our society, the lawyer ensures precise specificity in drawing up the scientist's employment contract. The salary is expressed as a pa33 frequency, for example 100,000 dollars per year, or $100,000/year or $100,000 year-'. Never is the numerator unit of the salary deleted for abbreviation purposes. To write simply 100,00O/year or 100,000 per year is to present complete ambiguity as to what monetary entity the scientist will receive (sous, yen?). Yet when the scientist expresses light frequency, the numerator physical entity is completely disregarded, though it is "understood to he present. Light Light frequency units are usually (and incorrectly) expressed as reciprocal time. For example, u = 10's waue-secand-' (or cvcles uer second. c ~ s are ) exuressed as u = 10'5 further confuses the physical meaning, resulting in ambiguity.) The wave of light, the fundamental repeating unit in the classical electromagnetic wavetrain, is as physically-real as the dollar, atom, or molecule. Indeed, analogous to the mary time standard. When the teacher tries to explain to the student the physical meaningfulness of the classical light velocity/wavelength/ frequency relationship c
meterlsecond = X meterlwave X u wavelsecond
(8)
i t is almost impossible to explain eqn. (8) without the unit wave. Yet, without major exception, the standard science texts use the abbreviated expression c
meterlsecond = h meter X u second-'
(9)
whereuoon the teacher must s u ~ ~the l vunit wave to clarify. unitsare-5 wave-meter-' using eqn. (8). T h e wave number units must involve a number of waues We must specify what is being counted in one meter, just as we must specify what is being counted in one second. The Photon The greatest impact on science of the omission of the unit waue for light. however. lies in its absence from the hoto ond r j l n i n c leu d 11hy~ics and rhcmisrry. thc t'l3nc.k-Einstein le"of relariun ( = III.I'hecntwv i n t r i n ~ ~ c t ~ ~ r h t . " i ~ a r t i c lizht s A d u is the light (the photon) is 6 , h is ~ h k ' constant, frequency. Placing proper light frequency units into this expression we have Joule-second wave r Joule = h Xu(10) wave second
wave of light. It is accepted that that action is intrinsic to the photon (7). Thus, the photon is precisely one wave long, being the "unit cell" of the classical wavetrain. Just as Dalton's use of the unit atom produced quantitative evidence that seemindv-continuous matter consisted of atoms, use of the unit &e in eqn. (10) shows that the seemingly-continuous classical wavetrain of light is composed of single-wave photons. The atom is the fundamental repeating ort ti on of an element; the ohotan is the fundamental repeating portion of the iight wavetrain. The units in relation (5) above are a mandate that so many grams of mass are intrinsic to one water molecule. The units for Planck's constant are no less a mandate: so many Joule-seconds of action are intrinsic to one waue of any light radiation. Let us prove this beyond doubt. Science has adopted the SI magnitude h = 6.626 X Joule-second as the elementary or least action in light. Calculating a value of r for a given wavelength light using this magnitude for h provides a value cA by LA
J
=
6.626 X
J-s wave xvwave S
(11)
Now, for the same wauelength light, let the frequency he expressed in megawaues. J-s megawave €A J = h X lo6 X u X 10F (12) megawave S We see that the magnitude of h increases hy a factor of 106 in this case. Obviously, choice of any other frequency units yields a different value for h for a given EA and corresponding wavelength light. The point is this: choice of h = 6.626 X 10F4 J-s for the elementary action in light radiation requires a one wave light segment, and only that, to he pertinent. (Exactly one basic unit of action is intrinsic to one wavetrain segment and to one alone.) If that action is intrinsic to the photon, as textbooks indicate, then the photon is one wave long., and the wnvcpartirlc dudlitv paradox d l i g h t is resolved. 1< he primary tvhlence rur thv panicle nature oilixht mvrg\. is tlrz t . x r r ~ ~ i n ~ l \ ~ rimc i h ~r ~ i ~r w f h ~ inehsorutirm rd .~~h,therrleral -~~~~~ of that light energy, preceding the ejection of the electrons in the nhotoelectric effect. One nhoton eiects one electron as if the classical wavetrain were concentrated into an extremely short length (hence the paradox). The one-wave photon model presented here is compatible with this evidence, since, for examde, femtoseconds (10-%s) are intrinsic to one wave of visiblk light. Other evidence includes the Comoton Effect, wherehv collisions between photons and electroLs were ex~lainedin terms of the articulate nature of the ~hotons. he one-wave photon readily explains this evidence, since the extension of the single wave in the direction of its travel is extremely short. Compton studied X-rays, which lie at the extremely short wavelength end of the electromagnetic spectrum, hence each X-ray photon would behave just like a particle under the given experimental conditions. BUT, the waue nature of this "particle" would also be demonstrated under other appropriate experimental conditions. So, both wave and particle attributes would he shown, yet no paradox is involved. The literature makes various claims for the wavetrain length pertinent to single photons, even suggesting wave packets. The conclusions drawn from logical analysis of units are scientifically more sound than conjectural speculation. Laws of science are a urecise auantitative exoression of ohserved behavior, and their units are part of t i e laws. This demonstrates the traredv which can accomDanv the presumption that scientists &llLunderstand" the prop& elementary entity to be present in a constant, definition, or law of science. The wave-particle light paradox can he resolved by use of proper descriptive units in the light particle defining equation, even though it has seemed unresolvable since about 1906.
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Volume GO
Number 11 November 1983
943
Light and Matter Planck's constant is the central quantum focus for both light and matter particles, the equations expressed for a photon and an electron being, respectively, Joule-second cxr=h (7 = 11") (13) wave and Joule-second rnuh=h wave Equation (14) is the de Broglie matter wave equation. Realize that proper wavelength units meter-waue-1 are used in this relation, yielding the units h Joule-second-wave-', wherein the pertinent matter wave is meant. Of significance is the realization that the least action h pertains to both light and matter waves. That is, h Joule-second is intrinsic to one wave of light as well as to one wave of matter. Use of eqn. (13) permits accurate knowledge of the energy and time period simultaneously intrinsic to the photon. The relation r = Alc is accurately determinable, then c is accurately calculable, since h is accurately known. Likewise, for a n electron, A is calculable as accurately as the velocity is measurable, since m and h are very accurately known. I t is critical that, when inspecting any equation or constant which contains Planck's constant, one considers which wave is pertinent. We shall shortly consider the fine structure constant, given by
The Fine Structure Constant or Relativistic corrections to the Bohr H atom theory permitted scientists to account for the ultra-fine spectral lines into which certain main lines resolved as better optical instruments were used. The constant a appeared in the equations pertinent. However, even in the Bohr non-relativistic theory, one can find a , when one divides the light velocity c into the product of the principal quantum number n and the electron velocity in state n.
fine structure constant.
Angle in Radians Angle in a plane is defined as the extent of rotation of a ray emanating from a given point in the plane (8).In the circular method, the radian is defined as the measure of the angle of a sector of a circle that is subtended by an arc having a length equal to the radius of the circle. In making a calculation of a number of radians in a given angle, one may use any of the following equations extent of ray rotation # radians = (16) extent ofray rototionlradian angle subtended by arc # radians = (17) angle subtended by arclradion arc length suhtending angle # rodians = (18) arc length suhtending anglelradian The mathematical regulation that the same units must he present on each side of the equals sign is upheld by each of these three equations; that is, #radians = K radians
(19)
References (2) reduce eqn. (18)to the form arc length # radians = = unitless (20) arc length by erasing the unit radian from the denominator of the 1.h.s. of (la), t o make the claim of unitlessness. This is not sound scientific logic; indeed, it is false scientific logic. Relation (19) has to he enforced. Let us further show the error by analogy. 944
Journal of Chemical Education
The Mole Chemists define the number of moles of a pure substance as a ratio of masses as follows mass # moles = masslmole Never would a chemist claim that the mole is unitless since it is a ratio of masses which cancel, mentally erasing the unit mole from the denominator of the r.h.s. of (21). This would be mathematical absurdity, because the consequence would be makine one of the fundamental S I units unitless. Any statementsuch as "The radian unit is unitless" is obviously an absurdity, no different from the statement that "The mole unit is unitless."The radian is a unit which must be dealt with in constants, definitions, and laws of science like any other unit. Immediately, the question arises as to whether radian is one of the fundamental units, along with mole. Extent of rotation is not derivable from any of the other fundamental SI units, just as amount is not derivable, so this author offers that radian should be added to the list of fundamental units. The standard argument that radian can be expressed as a ratio of lengths, hence is derivable from more basic units, is false, based on the above discussion. If the argument were true, then one could state that the mole can be expressed as a ratio of masses, hence is derivable from more basic units.
In the Huhr thwry, 11
$5 3s nmsdt.rrd ~ulitle-h.St), ainrt 1111: mtiu of wlwiriez i= narurah III~IIIPS., the ..h.,., hence iiI1n lhr r.h.s., of equ. (22) has beeiconsidered unitless. This is true for the fundamental units, but not for the descr~ptweunits when the corrected units for h are applied to this expression. T h e unit waue in h causes the r.h.s. of eqn. (22) to have the unit waue as well. Thus, the principal quantum number n must bear the same unit. The constant n is the number of waves, in this case matter waves. Louis de Broglie pointed this out many years ago when he stated that n represented the number of matter waves of electron in each stationary state of the Bohr atom. He derived the mysterious Bohr postulate
mur = nh/2r
by letting n waves fit in the circular orbit by meter n waves. h -= 2rr meter wave
(23)
(24)
then placing his A = hlmu relation into equ. (24) to obtain eqn. (23). Naturally, there are routine unit changes required when one uses n waue units in the Bohr theory. The Rydberg constant proves not to have wave number units a t all, contrary to current claims. The corrected units read RH uraue3-meter-'. The Schrodineer H atom svstem and the Bohr model both produced thesame non-recativistic energy equation. The entrance of the unit wave into this eauation introduces an important new meaning simultaneou&. Agreeing with the belief of Haendler (9) that the Bohr H atom should have importance
in the mind of the student, let us show how review of one portion of this system hears fruit of a discovery type. Pauling and Wilson (10) mite the BohrlSchrodinger energy relation most simply as
I H e - w r ~ t ~eqn. l ~ g 1251,and whert.in Z is the ~ U C I P ~charge. applying f h c ;orrertt.d h u n l t ,, we ubtain hc Joule-meterlwave E, Joule = (26) (nZ/R~Z2) meterlwave Note that the numerator units force the denominator units to he those of a wavelength. This expression reduces to The reader might consider A, to he the de Broglie electron wavelength in state En, hut this is found not to he the case, ~ . the upon simple calculation of hlm v, and n 2 l R ~ Z Since unit wave in h stands for radiation wave as well as electron wave. let us consider the vossihilitv that there is intrinsic to each orwpied st:uic.nnrv state an rlectrwnay~leri~ rwlintim q,f wa\.elrneth .\,.. asswiattd u,ith eI~.ttrotlm,.t ion. L'ainr the accepted equation of Bohr
-
E, - E ,
hc
( p for photon)
=-
L"
(28)
and inserting relation (27) into (28), we obtain -h-e- = -hc
A"
X,
hc
A,
(291
Since each major term in eqn. (29) is an energy term with the product hc in the numerator, and since Ap is a wavelength of radiation, then each A, and A, must he radiation wavelengths likewise (pearslapples - pearslapples = pearslapples). A latent electromagnetic radiation exists in equilibrium with the orbiting electron in a stationary state of energy Em Most likely this is the radiation the classical scientist expected to radiate from the atom as the electron spiralled into the nucleus. Since this is thecase for the H atom then such an energy state radiation exists for all atoms. The mechanism of radiation emission can he studied further, since each c/A in (29) can he expressed as a frequency v, obtaining hu, - hv,
= hu,
(30)
whereupon, dividing through by h gives The frequency of radiation emitted equals precisely the dif-
ference in frequencies of radiation in the respective stationary states. Conclusion dtaerw; ~d t h t cit:m;t dt..rript~m of 'l'hc nta. vou~igr n ~ ~ m r phy&sl wurld that .;I wnce can 1118,vide.\ I ~ t i r u l t ~care u~ 111 uw.entin~ (teiit~i~iot~s ?nd I J U . d .ciena.t. It, rhv A11drt11 must inuolueprecise units, whether they are fundamental or descriotive. Without the descrintive units, much amhieuitvprevaiis in our physical interpretation of scientific equations, hence a return to the use of such units is advocated. Specifically, a proposal is hereby made to consider seriously placing the unit radian on the list of fundamental SI units. since extent of rotation cannot be derived from any other units. Also, and very significantly, a proposal is hereby made to change the units for Planck's constant to energy-second-wave-' from the currently-abbreviated energy-second and energyH e r t z ' . Use of these corrected units throughout quantum mechanics will produce an era of clarity in physical interpretation in that entire field. Perhaps George Gamow's suggestion that the stagnation of physics can he overcome by dimensional analysis (11) will prove to be correct. This author predicts that this suggestion will prove to he true beyond Gamow's wildest dreams. Even seemingly-impossible problems will prove to he solvable, analogous to the light wave-particle paradox. Unitlessness (dimensionlessness) may he rare and beautiful per se, hut it has no value in science, for only units permit physical interpretation. Acknowledgment The author is deeply appreciative of Dr. Henry A. Bent and Dr. James H. Lawler for critical comments which led to a clearer presentation of these ideas. Literature Cited
.
~~~~
-
(1) Bridgman, P.. "Dimensional Anslyris," 2nd Ed.. Yale University P~ess,New Haven. CT 1931.p. sears, F.. et al., ~ o l l ~hysies..ISfh e ~ ~ Ed., Addison~WesieyPub1 Co., Reading. MA, 1980, p. lfi6 Also. Rueche, F.. "Technical Physics." 2nd Ed.,Harper & Row, New York, 1981, p. 173. (31 Grunwald. %and Johnsen,R.,"Atoms.MoleculerandChemicalChange: Prentice~ Hdl. lnc.. EnglewoodCliffs. NJ, 1960, pp. 78-77. 141 I'lanek. M.,"ScientiBc Autobiobraphy and Other Paperr."Philosophical Library, New York. 1949, P. 42. (5) Castellan, G.. "Physical Chemistry: 2nd Ed., Addaon-Wesley Pub1 Co., Reading. MA. 1 9 7 1 , ~675. . (6) Adamson, A.."ATsxfbook of Physical Chemistry: Academic Press. New Y a k . 19'7% pp. 220-225.284. (7) K ~ C.,ef ~al.. "General ~ College ~ ~Chemistry." . 6th Ed., Harper & Row, New Ynrk, 1980.o.98.Also.Rrescia. F..etal.."liundamentalsofChemirtrv: 4th Ed., Academic ~ r e s s : ~ e York, w 19R0,p. 657. 181 Perlin, 1.. "T~igonnmetry."Int~mationslTexfhook Ca., Scranton, PA. 196%p. 5. (91 Huendler. R.. J. C H E M EDUC..59.874 (19821. (lor Pauline, I.., and Wilson, E.. "rntntroducliun tl, Quantum Mechance," McCrarHill Bwk Co.,NewYurk. 1 9 3 1 , ~41. . 111) Gamnw. C., "Thirty Years That Shook Physics," Dcrubleday and Cu., Inc..Garden City, NY, 1966, pp. 15'rifil.
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Volume 60 Number 11
November 1983
945
