provocative opinion The Chemists' Delta Norman C. Craig Oberlin College, Oberlin, OH 44074
.
Colorful lahels count., esneciallv when thev .heln. draw distinctions and clarify our thinking.@ this spirit I suggest that familiar auantities such as AG. AH. and A S (which are cnmmonly written without the super bars) be called "chemists' deltas". Thev are not the simnle delta wantities that thev appear to beynor are their units ohvious:~roblems with this use of the delta svmholism have been discussed before in this Journal (1,2). ~ e n has t even suggested that the usage is a "weed in the field of thermodvnamics" that "ought to he uprooted" (1 ). As an important example, consider A€, which is defined as ~~
~
~
~
Here we have a change in the Gihbs free energy function, which is expressed in terms of stoichjometric coefficients (ui) and partial molar Gibhs energies (GJ or, what is the same, chemical potentials ( d . For the generalized chemical reaction aA+bB=eC+dD AG = cpc + dpo - WA - ~
P B
where the stoichiometric coefficients for products have positive signs and those for reactants negative signs. Despite appearances to the contrary, AG does not commonly represent the change in the Gibbs function for a process in which all of vA moles of A and of us moles of B are converted into appropriate amounts of products. Thus, AG does not equal G2 - GI as a mathematician or a student might suppose. Rather. an exnression like ea 1 is a measure of a nrocess in whit h the reactigm advnnwi to only an incrmwrital extent nccordinc" t u the reliltwe stoirhiumrtrv. svcrified in thr (le. fining chemical equation. As a consequence, no perceptible change takes place in the concentrations or in other state variables of reactants or products during such a process.
668
Journal of Chemical Education
Alternatively, we imagine a very large reactionsystem for which a finite advance by U A and us produces a AG's worth of change in G without altering any statevariables. Fundamentally, quantities like AG are derivatives not finite differences. Thus,
where 5 is the degree-of-advancement variable for the chemical reaction. The variable 5 hasunits of "moles" (vide infra). Because these quantities are derivatives (rates), Bect objected to using the delta formulation at all ( I ) . AG is a derivative, which can he pictured as the slope of a G versus 5 curve (2). A C is an intensive function with units of energy per mole. Only in the narrow sense described in the previous paragraph can AG he regarded as a delta quantity. Since usage of this symbolism is unlikely to stop, it is well to signal the peculiarity with terminology such as "chemists' delta". As Spencer has stressed (2), some distinction in the symbolism itself is needed for chemists' delta quantities. Equation 2 provides one illustration of the problem. Without the super bar in AG the same symbol C: would appear in the equation in two different roles. In AG, G is part of an intensive function, whereas, in d G ~ , p l d tG , is an extensive function.' Furthermore, there are circumstances in which one wants to use a true finite change in G, AG, in distinction from AG. Spencer and others have used a tilde, as in AG, for this purpose (2-4). But, why not use the same super bar notation for the functions for chemical change that is widely accepted
'
G = xnjp,.where the n. the moles of each species in the system, are ail positive quantities. An n, is related to a stoichiometric coeffivL, where nm is the initial number of moles of cient by n, = n, soecies i.
+
for designating individual partial molar quantities, as in Gi? This practice is exemplified in this paper. For AG itself a distinguishing name has gained some acceptance. This term is "reaction p@ential" ( 2 4 ) . Although such terminology has merit for AG, it isuse!ess i n dealing with the general concept as applied to AH, A U , AA, AS, and A € ~ The . characterization, "chemists' delta", can he used in discussine " anv - of these quantities. Another problem with chemists' delta quantities is the choice of units for these intensive functions. This is principally an issue of the units given to f . Commonly, the units of £ a r e said to be "moles" (5). However, such a designation is eonfusing tostudents who have learned to associate the mole with a quantity of some one chemical substance. In the generalized equation for a chemical reaction,
if none of the stoichiometric coefficients, a, b, c, or d, is unity, then t = 1 mol does not refer to a mole of any chemical sneeies involved in the reaction. A value of unitv for £ refers t;, one round of the ct~erniralryualion os u~rill&.~ h u sthe , use of the deiienation "mol rxn" (mole reaction) is recommended as the unit for t. This qualification also reminds evervone that F is undefined until a particular chemical equatiun is written for a reaction. As a consequence of this choice of units for 6, the units of chemists' delta quantities, surhas AC and AS,hecome"J1mol rxn"and"d/K-mol rxn", respectively. Benefits are to he found in extending further this specification of mole quantities. Although the IUPAC practice is that stoichiometric coefficients are unitless (5),such usage is in conflict with the way most chemists think and talk. We sav. "a moles of A react with b moles of B . . . ." I t seems preferable to assign "chemical species-specific" mole units to stoichiometric coefficients. Thus, the units of vc would be "mol Clmol rxn". The units of the expression that relates moles of a species to the degree of advancement of a reaction, d.
would he
chemical reactions involved. In the case of the chemists' delta quantities, this common, extensive variable is [, the same degree-of-advancement variable. Thus, formally we would have for the sum of reactios 1 , 2 , and 3 to give
&&= ARl(
+
+ A&€
(4)
which is numerically equivalent to
ar;r,=AR,tARz+AR8 Although each [ in eq 3 refers to the stoichiometw of a specific chemical reaction, the ps play a common role. (In the case of the emf's the common variable is the same product of moles-of-electrons change and of [ in the two half reactions and in the overall reaction.) Thus, we are justified in makine direct use of the intensive variables of chemists' delta quantities in Hess's law.like applirations. Cunlnionls, the true nature of chemists'delta quantities is overlooked in applying them to chemical readions. Why does this simdification work? Of course, tables of thermodynamic data Eontain proper partial molar quantities_ under standard state conditions. Calculations of AGO, AHo,and AS" from such data are automatically chemists' delta quantities. The most directly accessible experimental quantity for reactions is the enthalpy change, which is measured as a true delta quantity for a process in which all of a limiting reactant is consumed. In a typical combustion experiment, a pure substance reacts with pure oxygen gas to form essentiallv " Dure . liauid water and carbon dioxide that is almost independent of the excess of oxygen gas that remains. In high precision work small corrections are made for effects of pressures and nonideal gas behavior. But, to a good approximation AH = AH for combustion of a mole of reactant. For reactions in solution, the foregoing approximation is not as good, and corrections for taking solutes to infinite dilution, where standard states are defined for enthalpies, should he made. However, a t low concentrations the enthalpy function is weakly dependent on concentration. Approimating AHo with AH for a mole of reaction is reasonable. The widely applied relationship between the change in the Gibbs function and an equilibrium constant,
mol C = (mol Clmol rxn)(mol rxn) For a partial molar quantity, such as the chemical potential, the species-specific units would also be used. Thus, rc would have units of "Jlmol C". These choices are reflected in AG for the generalized chemical equation (eq 3), for which,
builds in the proper intensive variable character for AGO. AG, which, like a true AG, is a strong function of concentrations, is rarely used, except in the guise of the Nernst equation. Thus. the intensive function nature of AG" can eo unnoticed. In summarv. .. we mav"sav.that the "chemists' delta" terminology would help in several ways. serves as a reminder of the s ~ e c i a~l r o ~ e r t i of e s a A€. a AS, etc.. for a chemical reaction. he; are intensive functions and indeed derivatives that depend on instantaneous values of the partial molar forms of various thermodynamic properties of reactants and products. They are not the simple finite differences that they appear to be. We are also led to a useful choice of units for [, u,'s, and chemists' delta quantities. In proposing this terminology routine use of wording such as "chemists' delta of the Gibbs function" in speaking about AG, for example, is not intended, The "change in the Gibbs function" is sufficient for most purposes. However, it is useful to identifv the chanee in the Gibbs function or the change in a related functio&s a chemists' delta when introducing these functions or speaking about them collectively.
Y
with the units Jlmol rxn = (mol Clmal rxn)(J/molC) + . . .
- (molBlmol rxn)(J/mol B) - ... When the intensive character of functions like AG and AR is fully appreciated, one may also wonder about the validity ofHess's law-like arguments. In such applications, we add AH;s or other chemists' delta quantities for several chemical reactions in order to secure the corresponding value for the reaction that is the algebraic sum of the equations for the several reactions. Yet, we should never add intensive variables without special consideration. For example, adding emf's, which are intensive variables, for two half reactions in order to secure an emf for another half reaction is incorrect (6). In contrast, it is accepted practice to add the emf's for two half reactions in which the electron changes cancel out. The justification for using emf's in this latter way and of using AH,'s and other chemists' delta quantities in Hess's law-like calculations is the same. Asingle value of amultiplicative variable that converts the intensive function into an extensive one for a chemical reaction is common to all of the
.
.
Chemirtry: London; pp 97-98. 6. Csstelian. G. w. Phyricol Chemistry, 3rd ed.: Addison-Wesley: Reading, MA, 1983:p 882.
Volume 64
Number 8 August 1967
669
