The Dimensions of Logarithmic Quantities Implications for the "Hidden" Concentration and Pressure Units in pH Values, Acidity Constants, Standard Thermodynamic Functions, and Standard Electrode Potentials Philip Molyneux Macrophile Associates, 53,Crestway. Roehampton. London SW15 5DB. U.K.
In this paper the chemical implications of the propriety of taking the logarithm of a dimensioned quantity are discussed. By using quantity calculus, it is shown that this is a proper procedure, with the consequent units also being logarithmic, and additive rather than multiplicative. This allows equilihrium constants to have units rather than, as some present approaches require, having to he forced into a dimensionless form, while i t also clarifies the units of a number of derived quantities in physical chemistry such as pH, acidity constants (i.e., pK.), standard thermodynamic functions, and standard electrode potentials.
proper procedure, hut that the result has logarithmic units which are additive rather than the more usual multiplicative ones.
Logarnhms In Physlcal Chemistry
Inasmuch as operations on equations involving physical quantities, numerical values and units follow the ordinary rules of algebra (9, IO), then taking the decadic logarithm of eq 1 gives log m = 1+ log gram (2)
Mathematical functions and expressions that contain the logarithm of a quantity are commonplace in the physical sciences. Where the quantity involved is dimensionless, then clearly taking the logarithm leads to a dimensionless quantity. However, in physical chemistry, frequently the starting quantity is dimensioned. This applies, for example, to the pH of a solution; to the pK,valueof an acid; to the derivation of standard thermodynamic functions (AG*, AH*, AS*) from the equilibrium constant (and its temperature dependence) of a physical process or a chemical reaction; and to the derivation of standard electrode potential from the Nernst equation. However, despite the wide usage of the examples cited, there are some doubts about the dimensions of the derived quantity, or indeed whether taking logarithms in these cases is a proper procedure (1-8). There seems to he two schools of thought on the subject. The proponents of the first school hold that taking the logarithm of a dimensioned quantity is a perfectly acceptable procedure, which leads to a dimensionless result regardless of the dimensions of the original quantity. One apparent justification of this latter conclusion is that since the differential of In x is dxlx, which is dimensionless, then In x must be dimensionless (and likewise for log x). (Throughout this paper, "In" and "log"are used as abbreviated forms of "log," and "loglo", respectively.) The fallacy of this argument will be shown later. However, it is clear that the numerical value of In x (and logx) must depend on the units of x itself, so that in this sense in x (and log x) is dimensionally sensitive. The proponents of the opposing school of thought hold that taking the logarithm of a dimensioned quantity is an improper and even a meaningless procedure, and that the quantity must be converted into a dimensionless form before the logarithm is taken. Against this viewpoint is the fact that such derived logarithmic quantities are used widely and successfully, as illustrated by the examples cited above and further discussed below. Nevertheless, in such usage, as for example in the case of pH, there is always the rather curious circumstance that the units (i.e., in this case, of the hydrogen ion concentration) "disappear" when the logarithm is taken, only to "reappear" when the antilogarithm (i.e., the exponential) is taken. In the present paper a third viewpoint is propounded, which is that taking logarithms in these cases is a perfectly
Some Formalism
From quantity calculus (9, lo), the value of adimensioned physical quantity is equal to the product of a numerical value (a pure number) and a unit. Thus, for example, when we say that the mass, m, of an ohject is 10 grams, this means that m = 10 X gram
(1)
The units of log m are thus "log gram", but they are additive t o the numerical value, rather than being multiplicative. The additive character of the logarithmic units explains the observation mentioned previously, that the differential of In x (or of log x ) is dimensionless, for clearly on differentiation the additive constant of the logarithmic units will disappear. Similarly, for natural logarithms, the equation corresponding to 2 is In rn = 2.303.. . + In gram (3) In more complex situations, such as the examples from physical chemistry discussed below, the logarithmic unit is itself multiplied by numerical factors and units, and i t is therefore useful to have amore compact notation to indicate the presence of these logarithmic units. The suggested notation, applied to eqs 2 and 3, is and
Log m = l(gram)
(4)
In m = 2.303.. . (gram1
(5)
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. Appllcatlons
This general formalism can he applied to a number of examples in physical chemistry. Consider firstly the case of the pH of an aqueous solution (or other liquid), which most simply is defined by where [H+] is the hydrogen ion concentration in mol dm+ ("molar") units, or more exactly its activity. Taking the specific case of pure water, or a neutral aqueous solution, at mol dm-3 (11), this gives 25 "C, where [H+] = 1.004 X Volume 68
Number 6
June 1991
487
pH = 7.00 + log (mol-I dm3)
(7)
or with the more compact notation proposed in eq 4 above When the antilo~arithmis taken, either directly or after further calculatio~sand manipulations, the otherwise "hidden" units will reappear quite naturally. Similar considerinvolved in the evaluation ations apply to related of pH for aqueous solutions of electrolytes, such as pK, (from the ionic product for water) and pK, (i.e., the acidity constant for the ionisation constant of a weak acid). Considering thermodynamic functions for physical or chemical equilibria, the standard free energy change of the reaction or process, AGO, is related to the corresponding equilibrium constant, K, by AG*= -RTln K = -2.303.. .RTlog K
(9)
If we keep to the specific example of the ionization of water, with the ionic ~ r o d u c tK . , = 1.008 X 10-l4 mo12 dm-6 at 25 OC (thermodynamic tekperature, T,= 298.15 K) (111, then with the gas constant R = 8.314 J K-' mol-' this gives or equivalently AGe = -5.706(-14.00
+ log (mo12dm-"
kJ mol-'
(11)
Using the more compact form proposed in eq 5 for the natural logarithmic form (the numerical value produced being the same with decadic logarithms) AG* = -79.9 kJ mol-I (mol' dm-61
(12)
where the normally "hidden" concentration units are indicated by the term in braces. Takiue this a staee further, consider the units for the compon&t standard enthalpy and entropy changes, AH0 and AS-. From the Gibbs-Helmholtz equation (12) AH* = RT2d(ln K) dT
(13)
which shows that the differentiation will lead to the loss of the additive logarithmic concentration term, so tbat the units remain in their conventional form, i.e., in the present specific case, AH'+ = 55.8 kJ mol-I (11). On the other hand, the entropy function is given by so that as with AG0 there will be an additive logarithmic term, and the value will be more fully expressed as AS0 = -81 J K-' mol-' {mo12dm-"). Similar considerations apply to reactions or processes involving gases. T o take a simple example, for the thermal dissociation of calcium carbonate the equilibrium constant will equal the equilibrium pressure (or more precisely, the fugacity) of the carbon dioxide (the other twisuhsta&es invdlvedeach being crystalline solids and hence a t unit activity). The values of AGO and of AS0 (but not that of AH*) will then have an extra additive term of the logarithmic pressure units, such as (atm). In the most eeneral case of a chemical eauilihrium involving gases and also components in solution as well as pure liquids and solids, the values of AGO and of ASe (hut again, not that of AH0) will have corresponding logarithmic units derived from the gas partial pressure terms and solute concentration terms in the equilibrium constant. This also applies to the corresponding parameters for physical equilibria (solid/vapor, liquidlvapor, solid/solution, liquidlsolution, In current practice, presumably because of the supposed impropriety of taking the logarithm of K if it has units, the 468
Journal of Chemical Education
accepted procedures are either to make K dimensionless by convertinr it into a "srandnrd eauilibrium ronsranr", K*, or even to d&ne it as dimensionl&s (6).However, the present discussion indicates tbat such procedures are artificial and unnecessary, and that K values with units can be converted into loparithmic forms to give the thermodynamic parameters (and vice versa) is a perfectly consistent andrational way. I t may be objected that the representation of pressure and/or concentration units in the thermodynamic parameters is superfluous because these are standard quantities, so that the units are understood since they correspond to the standard states of the comnonents. However. eiven that we do have a notation for these-units, then it seemlpreferahle to use this to indicate them specifically, especially for didactic purposes, otherwise the units seem to disappear and reapDear in an arbitrarv fashion. Also, it enables the distinction to be made betwedn cases wheretbe logarithmic units are involved (e.g., AGO and AS0) and those where they are not (e.g., AHe). Because these "hidden" logarithmic units in the thermodynamic parameters have been discussed rather indirectly (i.e., by way of the equilibrium constant), it is helpful to clarify how they arise in the parameters themselves. Even this, however, still involves equilibria and equilibrium constants. For a substance Z, the starting point is the evaluation of the standard absolute entropy, SO, of the gas (or vapor), Z(e). which involves the summation of the entrouv contrihutions for the conversion of the crystalline solid &O K (SO = 0., bv.the third law) via the five staees: (a) solid at 0 K solid at the melting point (m.p.); (b) . . fusion to l i q u ~ r nt l m p . liquid ar the hoiling point (h.p.1: 1d8 \,apurizarim.Icr vnpor at unir togncify. P C . , 1 arm: l e ) wpm at b p . and 1 arm gas nr specrfied temperature and 1 atm
-
-
.
The "hidden" units then arise a t the fourth stage, (d), in the value for the standard entropy change of vaporization, AuSe; this implies, in addition, that there are no "hidden" unite for the entropy of the solid or the liquid. For this vaporization stage, there is an equilibrium involved, between the liquid and the vapor Z(1)= Z(g)
(16)
with the equilibrium constant equal to the pressure (fugacity) of Z(g), since the pure liquid is a t unit activity. Thus, as before, AuGs for this process will have additive logarithmic Dressure units. ex.. latml, which will also be the case with iu.5'0 (but not w i t h d u ~ ~Concequently, ). these logarithmic units will alsooccur in the final \.slue of S* for thr ens at the specified temperature. This means, for example, that in the case of the values of SO for the halogens a t 25 "C ( I I ) , although those for Brz(1) and Iz(s) are complete in their conventional form (152 J K-I mol-', and 116 J K-' mol-'1, the value for Clz(g) should be written more fully as 223 J K-' mol-' {atml. Similar considerations apply to a substance in solution, where the key point is the solubility equilibrium between the pure (say, solid) component and its dissolved state (say, in aqueous solution) Z(s) = Z(ad
(17)
which correspondingly leads to logarithmic units in the value of S' for Z(aq), i.e., {mole dm-" if the standard state is the substance a t 1mol dm-"concentration. These logaritbmic units also arise in electrochemistry (13). For example, in the case of a standard electrode potential.. E0.. this involves the consideration of a reaction takine place reversibly in an elecrrwhemi~:alcell where the specitied electrode svstem is wuvled to the hydrogen elerrrode. Thus, for the siiverlsilver chloride electrdde, the cell is:
where the corresponding cell reaction (equilibrium) is The standard electrode potential, EO(AgIAgC1), is then related to the equilibrium constant of this reaction by
um constants, so that it is no longer necessary to force these into a dimensionless form. The notation should ulso enable the "hidden" concentration andlor pressure units in thermodynamic and related quantities in physical chemistry to he represented unambiguously, so that they can he retrieved when numerical values of equilibrium constants, gas pressures and solute concentrations are derived from them. Llterature Clted
(where n is the number of electrons transferred in the halfcell reaction, and F is the Faraday), so the conventional representation (11)of its value a t 25 O C needs to he written more fully in the form: E* = +0.22 V (mo12 dm-6 atm-1'2). Similar considerations apply to Es for other electrode systems. Concludlng Remarks I t has thus been shown that it is a proper procedure to take the logarithm of dimensioned quantities, such as equilibri-
6. Ha&w. F.7. Cham. Educ 1982,59,103&1036. 7. Mills. I. M.J Chrm. Edur 1989.66.887-889. 8. Laidler, K. J.J. Chsm. Educ. 1990,67,88. 9. MeClsshen, M. L. Phyaico-Chernkoi Qaontifies ond Units. 2nd 4.;R.I.C. mono^ graphs hrTeachers No 16: The Royal Institute of Chemistry: London. ,971. la. M~IIS.1.: cwitss.T.; ~ a u s yN.; . ~ o m a n nK.: , ~ u c h i t s uK., . E ~ P ~. u o n t i t i r units ~. and Symbols in Physical Chemistry: IUPACiBlackweII: Oxford. 1988. 11. Aylwsrd. G.H.; Find1qv.T. J.V SIChzmicoiDolo, 2ndodition; Wiles Sydney, 1974. 12. Atkins. P. W. Phvsieoi Chernimlrv:Oxford Univerritv: Oxford. 1978. 13. P s r s o n s , R . ~ v ~ hChsm. ~ ~ ~ 1914.37.503-516. l.
Volume 68
Number 6 June 1991
469
