68 Journal of Chemical Education
_
Vol. 88 No. 1 January 2011
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pubs.acs.org/jchemeduc
_
r 2010 American Chemical Society and Division of Chemical Education, Inc.
each term, including the coefficients, is defined and, further, that these
are compatible under the applied operations according to rule 1.
This is a statement of the familiar expression that one cannot add,
subtract, or compare apples and oranges. Dimensional analysis is the
method used to ensure dimensional homogeneity by making explicit
the dimensional validity of an equation by ascribing, when necessary,
the correct dimensionality to constants of proportionality (2).The
application of these rules can be useful in spotti ng errors in a derived
equation. Conversely, their satisfaction, although necessary, is not
sufficient to guarantee the correctness of an equation.
Apparent Problems in Dimensional Analysis Involving
Transcendental Functions
By definition, transcendental functions such as logarithm
(to any base), exponentiation, trigonometric functions, and
hyperbolic functions act upon and deliver dimensionless numer
ical values. We now explore some common examples of mis
conceptions involving these functions and the resolution of such
misconceptions.
Logarithms
Arrhenius Equation. If the argument of the function is already
dimensionless as, for example, the argument of the exponential
function in the Arrhenius equation
k ¼ A exp 
E
a
RT
ð3Þ
then, according to rule 1 both k and the preexponential factor A
must necessarily have the same dimensions (of reciprocal time)
because the exponential factor is dimensionless. A difficulty arises
when one has to insert a quantity with dimension (expressed in a
given system of units) in the argument of a transcendental function.
The logarithm is defined as the reciprocal operation of exponentia
tion (12),thatis
y ¼ log
b
x if x ¼ b
y
ð4Þ
where b, x,andy are real numbers, b being the base of the logarithm.
This definiti on precludes the association of any physical dimension
to any of the three variables, b, x,andy, so as not to violate rule 1.
Consider the following example discussed by Molyneux (8)
logð10 gramsÞ¼logð10 gramÞ
¼ logð10Þþlogðgram Þ
¼ 1 þ logðgramÞ
ð5Þ
where the log is to the base 10. Molyneux's suggestion that the
last equality defines the meaning of log 10 gram is mistaken for
the following reason. If we follow this suggestion, then if we
apply definition in eq 4 one may ask: what is the exponent y
(a number) to which one should raise the base b, that will yield
gram(s)? Because there are no reasonable answers to this ques
tion, the quantity (log gram) is meaningless as is eq 5 as a whole.
The notion that, in logarithmic relationships, dimensions are
carried additively (15) is a misconception: dimensions are not
carried at all in a logarithmic function.
To linearize the Arrhenius equation (eq 3), it is often modified
by taking the logarithm of both sides:
ln k ¼ ln A 
E
a
RT
ð6Þ
But, in fact, it should have been written as (14)
ln
k
k
0
¼ ln
A
k
0

E
a
RT
ð7Þ
where k
0
is the unit rate constant in the chosen system of units.
In this manner, the dependence of the logarithm on the unit
used is irrelevant because the argument of the logarithm in eq 7 is
a ratio of two variables having the same unit, that is, a pure
number. The method of expressing the arguments of the
logarithms in the Arrhenius equation shown in eq 7 is recom
mended when tabulating or plotting data as it avoids potential
ambiguities. In this manner, the numerical value of a (dimensioned)
physical quantity is disengaged from its dimension, such that (11)
physical quantity=unit ¼ numerical value ð8Þ
as, for example, in ΔH
f
0
/(kJ mol
1
)=285.9. This is known as
quantity calculus, the importance of which has been underscored
some time ago in this Journal and has been adopted as the Journa l 's
conventional style (11).
The differential form of the Arrhenius equation is
E
a
¼ R
dln k
dð1=TÞ
¼ RT
2
dln k
dT
ð9Þ
which is another deceptive form that may give the appearance of
taking the logarithm of a dimensioned quantity, that is, the rate
constant k. Formulas of this form occur in abundance in physical
sciences, and the resolution of this apparent problem is in the
identity (1)
dln k
dT
¼
1
k
3
dk
dT
ð10Þ
where the dimensions of k are cancelled and which is equivalent
to dln(k/k
0
)/dT, that is, taking the logarithm of a dimensionless
number.
In addition to logarithms , it is equally meaningless to include
dimensioned quantities as the arguments of trigonometric or
hyperbolic functions because these are defined as ratios (the sine
of an angle is the ratio of the length of the opposite side to the
length of the hypotenuse, the cosi ne is the ratio of the length
of the adjacent side to the length of the hypotenuse, etc.) The
hyperbolic functions, themselves defined in terms of either expo
nential or trigonometric functions, cannot operate on quantities to
which physical dimensions are attached either.
pH, pK
a
, log IC
50
, and So Forth. Mills has emphasized that the
correct definition of pH is (7, 16, 17):
pH ¼  log
10
½H
þ
mol dm
 3
!
ð11Þ
Thus, before taking the logarithm, one has to divide the concen
tration by the unit of concentration used to obtain a dimension
less or unitless number. A similar division by appropriate units is
necessary before applying the negative of the logarithm to
quantities such as the acid dissociation constant (K
a
) or the
50% inhibitory concentrations and so forth. The argument of
the logarithm of the partition coefficient (for example, P
o/w
=
C
o
/C
w
) is already dimensionless because it is a ratio of concen
trations.