letters Dimensions of Logarithmic Quantitites
To the Editor
I wish to comment on the article by Philip Molyneux on "The Dimensions of Logarithmic Quantities" [J. Chem. Educ. 1 9 9 1 , 68, 467-4691. Molyneux suggests that we should introduce the logarithm of units, involving expressions such a s "log (gram)", and "log (mol dm3)", into our scientific language. However, h e himself comments that such expressions have no meaning, and I am inclined to agree with him. Rather than comment further on his proposal I would like to lay out what I consider the situation to be in the many examples in physical chemistry (or science in general) where it seems that we take the logarithm of quantities that are not dimensionless. Suppose xl and x2 are two values of a quantity x. Then the quantity
ent answers according to the units we use to express the value of x. However I believe that scientific equations involving logarithms always involve the difference of two logarithms in the form of eq 1;so that in practice this problem never arises. I illustrate this below with a number of examples. Before looking a t the examples it may help to note that eq 1may be written in a differential form:
Thus d log (x) is always dimensionless, like A log (I), whether or not x is dimensionless. Notr con.iidt!r some examples drawn from physical chcmiitrx Thc wcssure deoendtmcr ofthc chemical uotenrinl of a component in a mixture of gases has the form K = @ +RTln
Herepi is the partial pressure (more properly the fugacity) of the component i, and is the value of p, when pi =pe, wherepe is some convenient standard pressure (usually 1 bar). The superscript denotes "standard"; and pe may alternatively be written h"and pa. Equation 3 often appears in the form pj =he + RT in (pi),but this is misleading and incorrect, and should be avoided; it derives from a careless neglect of the units in the equationp@ = 1bar (or
&'
is clearly dimensionless, even though the quantity x may not be dimensionless. This equation would certainly seem to imply that log (x) is itself dimensionless, whatever the dimensions ofx. But difficulties arise with log (x) whenx is a dimensioned quantity, because we do not know what to do with the units, and if we take the logarithm of the numerical value of the quantity then we appear to get differ-
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1atm), and it hides the fact that the argument of the logarithm is dimensionless. By summing eq 3 for the components of a chemical reaction. multi~lvinp - . - each equation hy the stoichiometricnumber ;i for the corresponding component (1)(positive for products, negative for reactants), we obtain the equation for a chemical reaction:
Literature Cited 1. Mills, I.: Cultas, T : Homann, R Kallay. N.: Kuchitau. K Qunnaties, units o n d s y m bois in Physkal Chemistry. the IUPAC G m n Boob; Blackwell Scientific: London. 1988 (ISBN 0-632-02591-3).
2. Mills, I. M. J. Chrm. Edur
1980,66, 387-889.
Ian M. Mills University of Reading Whiteknights Reading RG6 ZAD, U. K
To the Editor: I reply to Ian Mills'comments on my paper (1)as follows: and if we put AG = 0, the condition for equilihrium, then we obtain AG" = -RT In I@ (5) where
Note that KO is always dimensionless, although K, has the dimensions of pressure raised to the power $ vi. These equations frequently appear in some form in which the quantities KO and Kp are confused, or represented by the same symbol, which is misleading and hides the fact that the argument of the logarithm is always a dimensionless ratio. It is useful to have distinct names for the dimensionless quantity @, and the quantities Kp,or Kc, or K,, etc.; the recommended names ( I , 2) are the standard equilibrium constant (or the thermodynamic equilibrium constant) for @, and the equilihrium constant on a pressure basis (K.)., or a concentration basis (K,). .. or the aciditv constant etc. Differentiating eq 5 with respect to temperature gives
k.), ?.
(1) Meaning of "log gram"
In his first paragraph, Mills has misquoted the comments that I made, a t the end of the section headed "Some Formalism", on the "meaning of 'log gram', etc". My original wording was: One general objection that may be raised to this approach is that terms such as 'log gram'have no meaning. However, this is not a serious objection from the pragmatic viewpoint so long as such terms are used in a consistent and clear fashion.
The paper indeed shows how this can be done. My approach here is that of the philosopher Ludwig Wittgenstein: "The meaning of a word is its use". I view the assignment of a use and hence a meaning to "log x" and to "log gram" as a continuation of the historical development of mathematics that led to the acceptance of previously "improper" concepts such as negative numbers, irrational numbers, imaginary numbers, and infinity. It seems to me to be more fruitful to trv to find a use and a meanine for the present concepts, &her than to assert that no such use or meaning can be found. (2) Differentiation of log x
= a In K, since @ and Kp differ by a constant Here a In factor that is temperature independent. Both a In @ and a In Kp are dimensionless (even though Kpitself may have dimensions, see eq 2). Similar results are obtained in other cases; we are always concerned with either the logarithm of a dimensionless ratio, or the difference of two logarithms (which amounts to the same thing, see eq 1).Thus the Nernst equation should be written
where m denotes molality (or more strictly activity), but the m e factors in the denominator are frequently omitted thus hiding the dimensionless nature of the argument of the logarithm. The definition of pH should he written (1)
but the mol dm4 in the denominator is frequently omitted, thus hiding the dimensionless nature of the argument of the logarithm. I would summarize the situation as follows. The problem of taking the logarithm of quantities with units never arises in science. We always require the difference of two logarithms, or the logarithm of a ratio. However because we often seem to write our equations in a sloppy way, omitting important denominators within the argument of logarithms, we confuse ourselves-not to mention our students. The moral i s self evident: do not omit t h e denominators!
In his second paragraph, Mills discusses the fact that d (log x) has no units. I showed in my paper that this arises simply from the situation that the units of log x are additive (rather than multiplicative) so that these units will disappear on differentiation. (3) IUPAC Procedure
Mills then outlines the IUPAC method (2) for dealing with standard thermodvnamic functions. etc. where there are doubts about the propriety ?f log x if x has units. My article resolves these doubts, and shows that the IUPAC procedures (that is, dividing by the appropriate standard pressures andlor concentrations) are now superfluous and should be abandoned. (4) Drawbacks of the IUPAC Procedures
From the teaching viewpoint, it is my experience that students have to be drilled to quote units (rather than just the bare numbers) a t all stages of a calculation; this applies particularly in teaching the basics of equilihrium constants and the properties of equilibrium systems. It is then most unsatisfactory, in moving on to deal with standard thermodynamic functions, to have to say that all of this was wrong and that an equilihrium constant has no units a t all- a situation attained by dividing by whatever the units happen to be! My paper shows how this schizophrenia can be avoided. (5) Tabulated Data in Handbooks M y paper also allows us to deal with the common practi-
cal situation with handbooks (see for example, 3, 4)-where the standard states of tabulated thermodynamic data are not immediately evident. The notation proposed in my paper (I)would enahle the standard states to be incorporated into the column headings of the table. Volume 72 Number 10 October 1995
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