E. A. GUGGENHEIM University of Reading, England
No
FEWER than five different kinds of notation have been used in physics and chemistry. These will be described in turn. The first is the best; the second is correct but restrictive; the third is cumbrous; the fourth is intolerable; the fifth is even worse.
I The characteristic feature of this notation, in contrast t o the other four, is that each symbol denotes a physical quantity, not its measure i n terms of particular units. It is therefore sometimes called "quantity calculus." It has also been called "equations of quantities." The present author was taught it as a school boy by Alfred Lodge who had described it in an article entitled "The multiplication and division of concrete quantities.'" The advantages of this notation have been emphasized by Henderson2 who attributed it to Stroud. This notation has been used by some of the greatest theoretical physicists, in particular Planck and Sommerfeld. Its use is spreading, but surprisingly slowly. Most people who understand this notation like it and use it. The power of quantity calculus will be illustrated by several applications later on. A t this stage the notationa will be illustrated by a single example, the equation of state of a perfect gas PV = RT. Here P denotes the pressure, V the molar volume, R the gas constant, T the absolute temperature. For the pressure we might write, for example, P = 1.2atm. = 91.2 cm. He = 0.912 m. ~i = 1.216 X 106 dyne cm.? = 0.1216 J. om.? = 1.216 X 106g. cm.-'s.-P
or alternatively for the measure of this pressure in atmospheres we write P/atm. = 1.2 but under no circumstances may P, in quantity calculus, be equated to a pure number. Likewise we may write of the gas constant : R
= = = =
8.31 J. deg.? mole-I 8.31 X 10' erg deg.? mole-' 1.98 e d . deg.-I mole-' 0.082 stm. 1. deg.-' mole-'
or alternatively R/J. deg.-I mole-' = 8.31 but under no circumstances may R, in quantity calculus, be equated t o a pure number.
' LODGE,A,, Aiatwe, 38, 2 8 1 3 (1888).
J. B., Math. Gaz., 12, 99-104 (1924). HENDERSON, The followine abbreviations will be used: J. ioule: 8.. second: C, coulumb, a, ohm; and the familiar g., cm., etc.
I1 I n this notation each symbol denotes the measure of a quantity in a single system of units, no other units ever being used. The chosen system may be, for example, centimeter-gram-second-electrostatic-chargedegree or meter-kilogram-second-coulomb-degree, hut once chosen it must be strictly adhered to. With the former choice the gas constant R = 8.31 X 10'and with the latter choice R = 8.31. This notation is favored by electrical engineers who find the latter choice of units convenient for all their purposes. Most physical chemists would, however, find such a choice too restrictive since it excludes the use of liters, atmospheres, calories, faradays, etc. The former choice would further exclude the use of amperes, volts, etc. To sum up, this notation is correct but too restrictive for many physicists and chemists.
m I n this notation each symbol denotes the measure of a quantity i n specified units. Of the three tolerable notations this one has been most used in the past. This system, used correctly, implies tmo rules from which there is no escape: (a) The unit in which each quantity is measured must be stated. ( b ) Change of unit requires change of symbol denoting the measure.
This notation may be applied to the equation of state of a perfect gas as follows: Let P denote the pressure measured in dyne cm.? P' P" V V' V" R R' R" R"'
denote the pressure measured in J. m . 3 denote the pressure measured in atm. denote the molar volume measured in c m g denote the molar volume mertsured in denote the molar volume measured in 1. denote the gas constant measured in erg deg.? mole-' denote the gas constant measured in J. d e g . 3 mole-' denote the gas constant measured in 1. atm. deg.-'mole-' denote the gas constant measured in cal. deg.? mole-'
Then R R' R" R"'
= 8.31 X = 8.31 = 0.082 = 1.98
10'
We have the equations PV P'V' P"V"
= = =
RT = 4.16 X 1O7R"'T R'T = 4.16 R"'T RUT = 0.041 Rr"T
JOURNAL OF CHEMICAL EDUCATION
It is clear that this notation is extravagant in the use of symbols. Moreover there is great danger of numerical errors unless the unit in which each quautity is measured is displayed close to each formula.
IV This is a hybrid notation specially favored in many German texts. Each symbol is used t o denote the measure in units not always speciJied and the same symbol i s retained when the units are changed. The writer presumably knows what units are implied, but the reader is left guessing. I n this notation R = 1.98 in one place and R = 0.082 in another which is ahsurd. This notation is intolerable.
v The last notation t o be mentioned is a confused and capricious mixture of the first and third. Some symbols denote the physical quantity, regardless of units; other symbols denote the measure in particular units according to the writer's whim. This confused notation is often accompanied by confused terminology such as "let P denote the force per square centimeter." This statement describes a hybrid of notation I "let P denote the force per unit area" and notation I11 "let P denote dynes per square centimeter." Such notation and terminology is even worse than that described under IV. EXAMPLES OF "QUANTITY CALCULUS"
The remainder of this article is devoted to examples of notation I or "quantity calculus" so that the reader may see and evaluate its power. It is perhaps useful to repeat that there is no objection whatever to notation I1 as an alternative provided the user i s willing to restrict himself completely to only a single system o f units, but how many chemists would agree t o do so? The first example is taken from Lodge's article which is unsurpassed as an exposition of quantity calculus. This illustrates the power of quantity calculus to deal with a mixture of units. Two masses M placed a distance d apart attract each other with a force equal t o the weight of a mass m. Assuming that the earth's radius R = 4000 miles and that M = 1 ton, d = 1 yard, m = grain, calculate the mass E of the earth. We have
so that E = Ma R2 m dz =: 1 ton X
The next example illustrates the self-checking power of quantity calculus. Let us use Stokes' law u = W/VTa
t o calculate the terminal velocity u of a sphere of radius a and weight (corrected for buoyancy) W through a medium of viscosity 7. We take as our example W = 2mg. = 4 g. c m . 3 8 . 3 a = 0.5 mm.
n
We obtain 2 mg. 6 s X 4 g. om.-' 8 . 3 X 0.5 mm. - 6r X 4 g. em.?2 Xs.-'lo-'X g.5 X 10-a cm.
U =
and without going any further it is obvious that something is wrong. The mistake is of course that we have substituted for W the mass instead of the weight. The correct calculation is: 2 mg. X 9.8 X 10%cm. s.? 6 s X 4 g. cm.-LB.-IX 0.5 mm. g. X 9.8 X lo1 cm. s.? 2X = 6 r X 4 g. cm.? s.? X 5 X lo-' cm. - 2 X lo-' X 9.8 X lo2 em, s.-l 6 r X 4 X 5 X lo-" = 0.52 cm. 6 . 3
U =
The above examples are all rather simple. The following one is more difficult. The molar couductance A of the magnesium ion in water at 25'C. extrapolated t o infinite dilution is 106.1 W 1mole-' ~ m and . ~the viscosity of water a t 25'C. is 8.95 X g.cm.-I s.?. Assuming Stokes' law calculate the effective radius b of the magnesium ion from the formula =
zaF' 6snLb
where z = 2 is the charge number of the magnesium ion, F is the faraday equal t o 0.965 X 105C mole-' and L is Avogadro's constant. equal t o 0.602 X l W 4 mole-'. We have z2F' b = &",LA
- 3.72 X 10'0 1.077 X 10P6g. cm. 1 ton
= 3.45 X 10-'6
-
= 1 ton X (8 X 2240 X 7000) X (4OOO X 1760)P = 6 X loP'ton
Another example of mixed units is the following. Calculate the resistance R of a cable of length 1000 yd.. of cross section 0.20 in.2 and of resistivity 1.2 X 10" S l cm. 1OOO yd. X 1.2 X 10-50 em. 0.20 in.' - 1000 X 36 X 2.54 em. X 1.2 X 10-'R em. 0 . 2 0 X (2.54 cm.)' = 1000 X 36 X 2.54 X 1.2 X 0.20 X 2.54 X 2.54 = 0.92 R
R =
VOLUME 35, NO. 12, DECEMBER, 1958
= 3.45
X 10-8
5
J. 8.'
--
g. cm. erg. s . ~
g. cm.
= 3.45 lo-' cm. = 3.45 A.
The notation of quantity calculus is especially clear and tidy for labeling the axes of a graph, e.g.:
