equations F - P = RT In (P/atm.) = R T in (P/760 mm.) = R T In (P/1.0132 X 10adynes cm.?) P i s to be expressed, respectively, in atm., mm. (of mercury) and in dynes cm.-*. Method (2) also enables one t o retain physical magnitudes for equilibrium products (among other things), a procedure adopted by many eminent author^^^^^^^' because it has considerable value, especially in teaching. Thus, for the equilibrium '/*N, (gl
+ a/JL(g)
=
NHa (g)
at P(atm.) and T(deg.), it is possible to write
mole-
To the Editor: I would like to add some notes to the article by James E. Boggs entitled "The Logarithm of 'Ten Apples.' "' He gives no explanation of the rule that one may not take the logarithm of a physical magnitude, but only the logarithm of a number. The Napierian logarithm of the number 10 is defined2 as the "area" under the curve y = 1/x from x = 1 to x = 10, that is, as the integral
It might, therefore, be thought that the Napierian logarithm of "10 apples" could be defined as the dimensionless "area" under the curve y = (l/x) "reciprocal apples" from x = 1 apple to x = 10 apples, so that in "10 apples" e 2.303. That this leads to an inconsistency can be shown by its application to an important example, namely p H = -loglo [H+]. Let [H+] = 10 mole I.-' By adapting the definition just described for Napierian logarithms to define Briggsian logarithms, it follows that pH = -log,, "10 mole 1.-"' = -1 and that pH is a number. Then = 10 is a number, whereas it is also the [Hf] = physical magnitude 10 mole I.-' This inconsistency is removed by insisting that one must use only numbers and not physical magnitudes as exponents of numbers (e or 10 as a rule). There are at least two methods, in addition t o the primed symbol method used by James E. Boggs,' to indicate that a symbol stands for the numerical measure of a physical magnitude in certain units. Examples are: (1) I n the equation pH = -log,, [Hf], [H+] is to be taken to mean the numerical measure of the couceutration in mole I.-', so that if [Hf] = 10 mole I.-', this measure is 10 and p H = -log,, 10 = -1. Objections to this procedure have been given by Sir Harold J e f f r e y ~ . ~(2) The method dealt with a t length by E. A. Guggenheim4 may be used, accortl'hg to which pH = log,^ ([H+]/mole I.-'), the symbol [H+], like other symbols in general, being used to represent a physical magnitude, say [H+] = 10 mole I.-', in which case pH = -log,o 10 = -1. This method has the great advantage of indicating the units in which the physical magnitudes in an equation are to be expressed. For example, in the
' BOGGS,J. E., J. CHEM.EDUC.,35, 30 (1958). R., AND H. ROBBINS, "What is Mathematics?" 'COURANT, Oxford University Press, London, 1941 443. Cambridge, at the JEFFREYS,H,, "Scientific Infer&:>' University Press, 1957, p. 88.
and K,
=
Kc (RT)-1
=
K.P-1, A F " = -RT In (K, stm.)
so this notation also indicates, in the general case, the quantity of reaction to which its symbols refer. For the specific rate constant of a first order reaction it is likewise possible to write k (see.) = Ae-"IRT (see.), since here lc and A are frequencies in see.-' The exponent E/RT is a number and requires no modifying symbols to be attached to it. The equation denoting the equilibrium states through which one mole of an ideal gas (which is assumed to have constant heat capacities Cv and Ca) passes during a quasi-static adiabatic process may be written in the form
(&,)(A) Cn/C. =
a constant number
for pressure measured in atmospheres and volume in liters per mole; the exponent C,/C, is clearly a number. Finally, following Professor Guggenheim again,4 it is desirable to insist that graph coordinates should always represent numbers, not physical magnitudes. This explains the anomalous result obtained above, by the customary graphical method, that In "10 apples" N 2.303. The graph should have been one of thr 1 dimensionless number y = ;/apples against x/apples,
-
where x means a number of apples: there is theu obtained the result In (10 apples/apples)
=
Sl1O:
-dx= 2.303
"
and this is not in any way different from In 10 2.303. I n general this means that "areas" represented on graphs will also be dimensionless numbers. For example, in the familiar method for determining the increase in entropy (AS) due to rise in temperature at constant pressure from information concerning the
' GUGGENHEIM, E. A,, AND J. E. PRUE,"Physiochemical Calaulations," North-Holland Publishing Co., Amsterdam, 1955, p. l ; An Advanced Treab GUGGENKEIM, E. A,, 'rThe~m~dynamic~: ment for Chemists and Physicists," North-Holland Publishing Ca., Amsterdam, 1957, p. 1. 6 DENBIGH, K., ('The Principles of Chemical Equilibrium," Cambridge, s t the University Press, 1955, Chap. 4. 6 ZEXANSKY, M. W., "Heat and Thermodynamics," MoGrawHill Book Co.. Inc.. New York. 1957. Chm. 18. ''
Press, London, 1957, Chap. XX.
JOURNAL OF CHEMICAL EDUCATION
changes in heat capacity (Cp) with temperature, use is made of graphical integration to obtain
The Guggenheim method is to plot CP F/cal.
-
deg.?
mole-' against Tjdeg. and evaluate the "area" between the ordinates at TJdeg. and T,/deg.; if this "area" he A (a number), then A = AS/cal. deg.-' mole-', since the physical quantity which A represents has the dimensions cal. deg.-2 mole-' (those of CpjT) multiplied by deg. (that of T). Alternatively, plot C,jcal. deg.? mole-' against in (Tldeg.) and measure the "area" between the ordinates at ln(T,/deg.) and 1n(T2/deg.). If A (a number) be this "area," then A = AS/cal. deg.-' mole-', since the physical magnitude of which A is the measure has the dimensions cal. deg.-I mole-' (those of C,) multiplied by unity (those of lu(Tjdeg.)).
To the Editor: In a recent article by N. 0.SMITH(J. CHEM.EDUC., 35, 125 (1958)) the following question is posed as not having been answered to everyone's satisfaction. "Why should an intimate mixture of A and B begin to melt a t a temperature lower than the melting point of both pure components, even when their volatility is negligible?" This reader believes that a t least one answer to this intriguing question is possible if we free ourselves of the notion that if the vapor pressure of a solid is very low it is "negligible." In at least two fairly recent studies the role of the vapor phase in the eutectic fusion of dozens of binary and ternary organic systems has been investigated and found to be of considerable importance. I n one study (SORUM, C. H., AND E. A. DURAND, J. Am. Chem. Soc., 74, 1071 (1952)) it was shown that two solids can be caused to melt at their euteotic temperature even if the solids are not in physical contact. Some of the organic solids investigated had vapor pressures as low as mm. I the second investigation (PETRUCCI, R. H., '7 AND C. H. SORUM, Can. J. Chem., 34,649 (1956)) it was found that liquids obtained by this "non-contact" eutectic fusion do indeed possess the composition of the eutectic. It was further discovered that liquids can he condensed from the vapor state in equilibrium with solid mixtures even at temperatures below the eutectic temperature; in this case the liquids are actually supercooled solutions.
To the Edizor: We are victims of having lost the term "cellophane" as a trademark for our cellulose film, but nylon never was a trademark. See "Generic Names of Drugs," J. CHEM.EDUC.,34,455 (1957). The term "nylon" was coined as a new, distinctive name because the products it represents were distinctly new. However, it was decided a t the outset to dedicate it in the public domain and it was so announced by the late Dr. Charles M. A. Stine on October 27, 1938, a t the Herald Tribune Forum where the product itself was announced. Usually, reference to nylon being a lost trademark results from confusion with the famed cellophane case.
To the Editor: In the article by MYSELS(J. CHEM.EDUC.,35, 32 (1958)) the spelling of van't Hoff and le Be1 does not conform to that used by the gentlemen themselves: J. H. van't Hoffl spelled his name with a lower case v and not a capital V. Irrefutable evidence of this is provided by two signed photographs, one in E. Cohan's "J. H. van't Hoff: Sien Lehen nnd Wirken" and a second in Z. angew. Chem., March, 1911. This latter was reproduced in Volume I1 of Bugge's "Das Buch der grossen Cherniker" (Berlin: 1930). The name of le Be1 should he spelled with only one terminal 1 as in his paper (Bull. Soc. chim., 22, 337 (1874)). The use of a capital L1for le is doubtful since in his paper the whole name is set in small capitals. In the author index for the 1874 volume the name is listed as Le Be1 under L, but this is more likely to be the opinion of the indexer than the usage of the author. E. DE BARRYBARNETT 76 LONGRIDGE ROAD LONDON S.W. 5. ENGLANP 'EDITOR'S NOTE: THISJOURNAL trim to follow the custom of capitalizing particles in foreign names when they are not pro ceded by a Christian name or title, but is glad to observe any variations in personal signatures. Do readers like to see a paragraph start with a lower-case letter?
After reading the Editor's Note in proof, Dr. Barnott addressed an inquiry to The Chemical Society, London, and received the fallowing response: Dear Dr. Banetl: Practice of tho Journal has been variable in allocating capital letters when the name of a person has a particle such as de, le, or van, particularly when this particle begins a sentence; and examples of the variation can be seen also in quite recent issues. We have, however, now got a firm editorial ruling. It is to follorp the usage of tho author himself even if this means that a sentence begins with a. lower case letter; and this applies also to indexes where, far instance, we should (now at least) index van der Waals under "V" using a lower oase letter. We also now write van in full even if the author uses the abbreviation. The whole subject is complicated by Americanisation which so often loses the Continental significance.
R. S. CAHN,Editor
VOLUME 35, NO. 7, JULY, 1958
