Stanley J. Gill University of Colorado Boulder
I I
The
Chemical ~ o t e n t i a ~
The fundamental concept of the chemical potential ( f i ) was introduced by Gibbs in his classical paper On the Equilibrium of Heterogeneous Substances (1). Using this concept he was able to derive precise criteria for chemical and phase equilibria. Its importance is reflected in part by the many experimental studies of solutions and chemical reactions as interpreted by means of the chemical potential (g). I n teaching the concept of the chemical potential one still follows, to a large measure, the writings of Gibbs and Lewis (1,$). The concept is introduced in a formal manner by partial derivatives of the energy or some other thermodynamic function such as the free energy. Although this provides a precise definition, it does not give one an immediate intuitive meaning for the chemical potential. A certain physical vagueness is often associated with the term. Suppose the concept of temperature was first introduced to a student as a partial derivative of energy with respect to entropy under the conditions of constant volume and material content of the object. This too provides a precise definition, but it does not use any of the familiar physical properties associated with temperature. If such an approach were generally used in introducing temperature, most natural intuition would he thrown aside. Since the ideas of temperature can be developed without the formalism of derivative coefficients as part of the definition, it would seem that a similar approach to the chemical potential would assist in gaining a physical understanding of this important idea. A wggestion along these lines has been proposed by Brdnsted through the use of a generalized concept of work (5,4).Some of the advantages of his suggestions have been discussed by LaMer (5-7), MacRae (8,0), and Morrison (10). In this article we wish to illustrate some of the possibilities of presenting the chemical potential concept from the Br#nsted point of view. We shall first review the introduction of the chemical potential as Gibbs presented this idea in treating the variation of energy of open systems. This will provide a means for comparison with the definition of the chemical potential which Rr@nsteddeveloped from the idea of the chemical work process. A brief summary of the generalized work concept shows the parallel between the ideas of temperature, pressnre, chemical potential, and other thermodynamic potentials. A use of the chemical work process is illustrated by a derivation of the Gibhs-Duhem equation and by a presentation of the chemical reaction Presented as pert of the Symposium on the Teaching of Thermodynamica at the 141st Meeting of the American Chemical Society, Washington, D. C., March, 1962.
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work process. The real power of Br@nstedlsmethods lies in the logical parallelism between the various processes that form the subject of thermodynamics. The full advantage of these parallels is realized with the concept of the chemical potential. The Chemical Potential-Gibbs
Gihbs recognized the need to describe changes of energy within a system in terms of molar or mass content. The basic idea is contained in the expression for an open system: d E = T d S - pdV
+ xpjdni
(1)
j
where the summation is taken over all components in the system and M, is the chemical potential of the jth component. The energy, a function of the state of the system, is defined by the entropy, volume, and molar content of various species within the system: E = E(S, V , n,, m, . . . ) (2) I t follows that the chemical potential of species j is where the subscripts on the partial derivative indicate the conditions of constant entropy, volume, and moles of components other than j. By the definition of other thermodynamic functions, such as the Helmholtz free energy A = E - TS, and the Gibbs free energy G = H - TS, other definitions of the chemical potential can be derived: P, =
( a A / b n , ) T . ~ , n=, (aG/bn,)T.$.ni
(4)
These have the advantage of simplifying the constant experimental conditions that must be met to constant temperature and volume or constant temperature and pressure along with the constant molar requirement. However, one is still faced with the precise measurement of the functions A and G to do this. Although these definitions of the chemical potential are perfectly correct, one wonld hesitate to use an analogous procedure to define terms like temperature or pressure. In those instances we would have T
=
(aElbs)v,,
- p = (aEl3V)a.n
(5)
which as mentioned before hardly give us the intuitive ideas which are already so familiar for these concepts. We can imagine that Gibbs had an intuitive feeling for the idea of the chemical potential as the measure used to indicate the equilibrium state of chemical material throughout a system and that he was simply defining these terms in a precise fashion. A more physical meaning of this idea can be found by considering the similarity of various potentials as defined in basic work processes.
The General Concept of Work Processes
Carnot (11) in 1834 reasoned that the work produced from a thermal engine was due to the transport of a thermal quantity, which has since become known as entropy, between two different temperature states. He even likened this process to that of the transport of water by means of a water wheel from one height to another with the consequent production of work (12). Had this simple interpretation not been obscured by the controversy of the caloric nature of heat, which as Callendar (IS) and La Mer (6-7) have argued, had little to do with Carnot's arguments, the mode of presenting thermodynamics could have been greatly simplified. BrGnsted has expanded this idea by defining a work process as a process which involves the transport of a quantity between two different potentials. When such a transport is completely controlled by coupling t o some process which has been chosen as a standard (such as a weight lifting process), then the work of the first process can be measured. The work obtained is described by an equation of the general form where a positive quantity (herewith underlined) dK is transported from an initial state with conjugate potential P, to a final state with conjugate potential P,. Once a procedure is defined for measuring the quantity dK of the particular work process and a device is inventedto couple the work process with the standard, enabling dW to be measured, then the potential difference P, - P,is accessible to experiment. Though such a procedure for investigating thermodynamic systems is unfamiliar to most of us, it provides a uniform and precise way for determining the properties of state P and K. A summary of such results is tabulated for somo' f the basic work processes in Table 1. I n order to characterize a particular work process it must involve the exclusive transport of the quantity peculiar to it. We shall see this particular point now as we discuss the purposeful omission of the chemical potential from Table 1. Table 1.
A Summary of Results for Basic Work Processes Potential P
Work process Weight transport (gravitational field) Volume transnort Surface area &ansport Charge transport Thermal transnort
gh (h, height) .n (measure) .. r (surface tension) p (electrostatic potential) T temoersture
Quantity K
changes of other quantities in the initial and final regions. Equation (7) gives us the same intuitive concept of chemical potential as Carnot used in his ideas of the thermal potential. It says that work can be obtained if matter is taken from a higher to a lower chemical potential state. Furthermore, our natural inclination is to regard all spontaneous or natural processes as proceeding in a direction which dissipates work; or, in this case, the transport of matter from a region of higher chemical potential to one of lower chemical potential. This leads to the intnitive concept that, within an equilibrated system, the potentials of temperature, pressure, and chemical potential will be uniform unless there is coupling between certain work processes, such as the coupling between volume transport and weight transport with a gravitational field to produce variation of pressure with height (14). There are other points worth noting from equation (7). One is that only a difference in chemical potential can be measured from the work produced in matter transport. This is true for all other work processes that are limited by the definition in equation (6). A resuleother than the primarily important one that dissipated work produces a quantity of entropy inversely proportional to the absolute temperature--is needed to define a potential in an absolute sense. The practical determination of a potential difference involves coupling a known or calibrated work process with the process to be investigated. The manometer couples a weight transport with a volume transport so that a pressure difference can be measured by a difference in height. The only method for measuring a chemical potential difference directly is to couple an electron work process with a chemical work process, by means of an electrochemical cell. With such a procedure the effects of concentration, temperature, pressure, and other properties on the chemical potential can be determined. For practical reasons the meamrements are usually made against a lixed or reference state. This association of an electrochemical process with measurable electronic potentials to determine chemical potential differences was realized by Gibbs (15) long ago, as MacRae has recently pointed out (16). In the electrochemical cell the chemical and electronic work processes are balanced so that the work produced by one is gained by the other:
m (mass)
V (volume) o
(area)
p (charge)
where dW,
S (entrow)
= (pi
- ry)dy
(9)
and The Chemical Potential in the Chemical Work Process
The chemical work process as suggested by BrGnsted is due to the transport of chemical matter between two different chemical states described by chemical potentials. The work obtained from this transport alone is called the chemical work and is defined by dW = (pi
- #),&
(7)
The element of matter & is taken from an initial state of chemical potential p, to a final state of chemical potential p,. This transport must occur without any
The condition coupling these two processes is that
where (n)is the stoichiometry number of electrons involved in the chemical reaction a t electrodes and 5 is Faraday's number. Substituting these conditions into equations (9) and (10) gives the basic result:
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From what has so far been said, this equation applies strictly only to simple chemical transports, but can easily be generalized to include chemical reactions. Furthermore, the electrostatic potential difference is strictly measured as an electron potential difference by means of a potentiometer ( I ?,IS). The chemical work obtained in chemical transport of a single component within a mixture applies to that particular component with all others being held constant. For the transport of component j between two different chemical states we shall describe the work process by dWi = (Pii - pi!)
&i
(12)
The Gibbs-Duhem Equation
The second step is to change the state of the n moles to match that of the final region. This can be done by adding increments of volume and entropy from the volume and entropy auxiliary reservoirs in a totally controlled manner so that the work is recorded (Figure 2b). The over-all work to produce the desired change is
where we have used the variable pressure p and temperature T to indicate the intermediate states of n during this change.
The definition of the chemical work process along with expressions for entropy and volume work transport can he used to obtain a relation of how the chemical potential depends upon the variables of temperature and pressure. We shall limit our discussion so that the properties of entropy and temperature, volume and pressure, and moles and chemical potential define the state. Consider the transport of a small number of moles u from a large region defined by potentials T,,P,, p, to another large region defined by potentials T,, p,, and p,. The transport is imagined to he conducted under complete control (revenihle) so that the work of each step is indicated in a calibrated work process. We wish to conduct this transport accordmg to the definition of the chemical work process. This means that the volume and entropy of the initial and final regions must remain constant in the course of the overall transport. This can be achieved by adding or r e moving the appropriate amounts of volume and entropy that accompany the n moles through the auxiliary actions of volume and entropy reservoirs that operate a t fixed pressure p, and temperature T,. The over-all result is depicted in Figure 1. Figure 2. Details of controlled chemical transport races. la) Removal n mole. from initiol region by expansion process into piston. (b) Changing of state of n_ mote. to match fino1 potential region. (cl Addition of nmole~toRna1region. of
The final step is much like that of the first where the n moles are forced into the final region and the entropy S, removed to the auxiliary entropy reservoir (Fig. 2c). The work is given by Figure 1. Schematic diagram for controlled trmsport of n motes from an initial to a R n d chemical rtafe. Circler represent initial a n d flnai state regions. Square with W de3ignotes work measuring process. Triongle represents entropy rourc? operating at a temperature TO.
The first step in this hypothetical transport is to remove n moles by expanding a small piston within a cylinder in contact with region i. An amount of entropy St is associated with the n moles as is a volume The pressure, temperature, and chemical potential are assumed the same as the system within t h e n moles. We remove the piston and cylinder with the moles in it and restore the entropy S, to the initial system by transport from the auxiliary entropy reservoir, see Figure 2a. The work of these first steps is given by
r,.
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ws
= (TI -
T&
- (P!
-P J ~ ,
(15)
The sum of these three steps is equivalent to what is defined as the chemical work: W
= (wi - PI)?
W = W,
+ w* + ws
(16) (17)
When the expressions given for these various steps are substituted here we find first that
The Chemical Potential of a Compound
which simplifies by use of the identity
It will be useful in considering the chemical reactions from the point of view of a work process to imagine the chemical potential of a compound of atoms to be equal to the stoichiometric sum of the chemical potentials of the atoms in the state of combination. That such a definition is reasonable can be seen by considering the work of transport of a compound A2B. The work of chemical transport in terms of the compound chemical potential is (32) dW = ( r ~ -~P qA ~ B &~ A) ~ B
and
to give
or in differential form The underlines have been retained thus far to remind us that the properties of entropy and volnme in this equation are associated with the n moles. We can now drop them once this is strictly in mind. For the case of a mixture of different components the chemical work is split into work terms for each particular component. This yields for the differential expression of chemical transport work
The same process can he visualized as a transport of elements A and B, coupled by the conditions of the compound, between the initial and final states of the compound. The work for this process is where the dashes serve as a reminder that the element is in a compound state. The stoichiometry of the com. dn ~so . that pound couples d-n ~and dn*. = 2d n a , ~and dne. =
(34)
d?~~*,s
With this substitution equation (33) rearranges to where the various n, are the moles of various components in the mixture transported of n. moles. Dropping the underline and arranging all terms on the left side we have in general
which is the familiar Gibbs-Duhem equation. The derivation here has made no particular assumptions on the nature of the system and requires only that the transport was considered as controlled. One example will show how the chemical potential might be expressed in terms of the pressure of a system held at a given temperature. Consider a pure gas which obeys the ideal gas law: pV = nRT. At fixed temperature then we have from equation (24) dr = (V/n)dp
(25)
With the ideal gas law
and upon integration from a pressure state of unit pressure where the chemical potential is p a : r=p0+RTlnp
(27)
An ideal solution which follows Raoult's law can then be treated by postulating equilibrium between vapor and solution chemical potentials: p, = pl0sl
(Raoult's Law) ( 2 8 )
For component 1:
gloL
+ RT In z,
In general we can write for a compound such as A,BoC, that The elemental chemical potentials apply specifically to a particular element within a particular compound. I n general such chemical potentials would be imagined to be different for an element in different states of chemical combination. Experimental measurements can only be made on the compound chemical potential. The Chemical Reaction Work Process
A chemical reaction involves the transformation of a set of atoms from one arrangement as a compound into another. We can think of it as a multiple transport process of atoms originally in one compound state being taken to a final different compound state. In such a case we have a number of simple chemical matter transport processes which, by the coupling requirement expressed in the stoichiometry of the reaction, are all simultaneously involved in the process of the chemical reaction. The total chemical work of a reaction is then the sum of the individual chemical transports of atoms from the reactant state to the product state. A simple example can perhaps illustrate this idea. Consider the reaction The work for a small molar transport from reactants to products is
or by redefining terms rlL =
Comparison of equation (35) with (32) suggests the definition of the compound chemical potential in terms of its constituent element chemical potentials as
(31)
for the chemical potential of a liquid obeying Raoult's law.
where the chemical potential in the compound state of Volume 39, Number 10, October 1962
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509
A and B is designated by PA- and p ~ . . The stoichiometry of the reaction imposes a condition between d- n ~ and dne as = &B
'/&A
=
3
(39)
where dg is called the positive increment of reaction. In thescterms the work of the chemical reaction is dW = I(2p.4
+ +B) - ( ~ D A+. !J~.)ldf
(40)
The reactant and product potential sums are given by FR =
x
Y
z
!+ =
~ Y p I L p
(46)
P
The use of these expressions in defining equilibrium conditions and the direction of spontaneous processes follows the general rules found for simpler work processes. In this way a uniform treatment of thermodynamics can be developed.
From equation (36) the chemical work of the reaction is then
Summary
As can he seen this equation has the form of other work processes. The quantity transported is the extent of reaction or reaction increment. The transport takes place between the reactant potential state to the prodnct potential state. The progress of the reaction is governed by that transport which goes to the lower reactant or product potential sum. Equilibrium is set by equality of these sums. The general case can he written from
The basic properties of work as a transport process which first let Carnot to his description of the thermal engine and which were more generally recognized by Br@nsted can he applied with conceptual advantage both in understanding the basic structure of thermodynamics and in defining the basic concepts of thermodynamic properties. The particular illustration chosen for this discussion has been the property of chemical potential, which is customarily formulated from a partial derivative of energy E or free energy G. Acknowledgment. I wish to acknowledge the valuable discussions with Dr. Melvin Hanna in preparing this paper.
where v. and v, are the stoichiometry coefficients of reactants and products and R. and P, denote various reactant and product compounds. The conditions of stoichiometry for both the compounds and the reaction lead to the requirement that where absolute values of the molar changes are indicated by the underline. From the sum of all individual chemical transports the work for the reaction transport is expressed by dMr =
(P,
- p&E
(43)
for the transport from reactants to products or by dW = (& - fin&
(44)
for the transport from products to reactants. If we use the sign convention that d is positive when reactants go to products whereas negative when products go to reactant,^ then we have dA' = ipR
- r&€
(45)
.
Literature Cited (1) GIBBS,J. W., T r a m Conn. Acad. Sci., 3, 228 (1876). (2) LEWIS, G. N., AND RANDALL, M., "Thermodynamics and
the Free Enerev of Chemical Substances." McGraw-Hill Book company, Inc., New York, 1923. ' ( 3 ) BR~NSTED, J. N., Phil. Mag., 29, 449 (1940). (4) BR~NSTED, J. N.,"Principles and Problems in Energetics," BELL,R. P., New York, Interscience Publ. Inc., 1955. ( 5 ) LAMER.V. K., FOSS,O., AND REISS,H., Ann. .I7 1'. . Acad. Sei., 51,605 (1949).
LA MER,V. K., Am. J. Phys., 22, 20 (1954). LA MER.V. K.. Am. J. Phus.. 23. 95 (19551. M A C R An., ~ , J. CHEM.ED&.; 23; 366 (194'6). MACRAE,D., J. CREM.EDUC.,32, l i 2 (1955). MORRISON, J. L., Chemistry i n Canada, July 1956. CARNOT,S., "Reflections of the Motive Power of Fire," Dover Publications, Inc.,(English translation), 1960. (12) Ibid., p. 15. (13) H. L.. Proe. Phvs. Soe. (London). 23. 153 . . CALLENDAR,
(6) (7) i8j (9) (10) (11)
(1911). (141 BR~NSTED. J. N . . Ada. Chem. Scand.. 3. 1208 (1949). . . (15j GIBBG,J. w., "dollected Works I," ~ongmans,Green, and C n ~New . York. N. Y. 1931. n. 331. (17) GUGGENHEIM, E. A,, J. C, (18) BR~NSTED, J. N., Z . phys.
Erratum In the paper "Hi3 Equilibria; the Precipitation and Solubility of Metal Sulfides" by R. H. PETRUCCI AND P. C. M o ~ w sJn., , August, 1962, p. 392, equation (12) should read as follows: K,(N - A - 8)-
[H+]=
Ks
+
I ~ { K , ( N- A - 8 ) 2(N - A )
AK.
a
+ 4K,Km(N-KsA ) (3 + A )
Also the authors wish to call to the reader's attention two other papers on the subject of equilibriumcalculations and molar solubilities byJ. N. BUTLER, J. CHEM.EDUC.,38,141,460(1961).
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