the elusive chemical potential
TRANSCRIPT
The elusive chemical potentialRalph Baierlein Citation: Am. J. Phys. 69, 423 (2001); doi: 10.1119/1.1336839 View online: http://dx.doi.org/10.1119/1.1336839 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v69/i4 Published by the American Association of Physics Teachers Related ArticlesEARTH SATELLITES Phys. Teach. 51, 202 (2013) The Chain of Command Phys. Teach. 51, 248 (2013) Building a multifocal lens Phys. Teach. 51, 250 (2013) Revisiting the Least Force Required to Keep a Block from Sliding Phys. Teach. 51, 220 (2013) Teaching Electron—Positron—Photon Interactions with Hands-on Feynman Diagrams Phys. Teach. 51, 232 (2013) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html
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The elusive chemical potentialRalph BaierleinDepartment of Physics and Astronomy, Northern Arizona University, Flagstaff, Arizona 86011-6010
~Received 12 April 2000; accepted 10 October 2000!
This paper offers some qualitative understanding of the chemical potential, a topic that studentsinvariably find difficult. Three ‘‘meanings’’ for the chemical potential are stated and then supportedby analytical development. Two substantial applications—depression of the melting point andbatteries—illustrate the chemical potential in action. The origin of the term ‘‘chemical potential’’has its surprises, and a sketch of the history concludes the paper. ©2001 American Association of Physics
Teachers.
@DOI: 10.1119/1.1336839#
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I. INTRODUCTION
It was the semester’s end in spring 1997, and I hadfinished teaching a course in thermal physics. One ofstudents said to me, ‘‘Now I really understand temperatubut what is the meaning of the chemical potential?’’ Cleara topic needed greater emphasis, both in class and in thetext. I vowed to do better the next year, and—altogetherspent several years looking into responses to the questio
The present article first describes three meanings ofchemical potential, next develops them analytically, andnally gives two substantial examples of how the chemipotential is used. Some observations are interleaved, andpaper concludes with a short history.
For whom is this paper intended? I wrote primarily fsomeone—instructor or student—who already knows abthe chemical potential but would like to understand it bettSome portions of the paper are original, but much of it cosists of material that is common knowledge among textbwriters. I have gathered together interpretations, insights,examples to construct a kind of tutorial or review articlethe chemical potential.
II. MEANINGS
Any response to the question, ‘‘What is the meaning ofchemical potential?,’’ is necessarily subjective. What safies one person may be wholly inadequate for another. HI offer three characterizations of the chemical potential~de-noted bym! that capture diverse aspects of its manifold nture.
~1! Tendency to diffuse. As a function of position, thechemical potential measures the tendency of particlediffuse.
~2! Rate of change. When a reaction may occur, an extrmum of some thermodynamic function determines eqlibrium. The chemical potential measures the contribtion ~per particle and for an individual species! to thefunction’s rate of change.
~3! Characteristic energy. The chemical potential providescharacteristic energy: (]E/]N)S,V , that is, the change inenergy when one particle is added to the system at cstant entropy~and constant volume!.
423 Am. J. Phys.69 ~4!, April 2001 http://ojps.aip.org/aj
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These three assertions need to be qualified by contexconditions, as follows.
~a! Statement~1! captures an essence, especially whentemperatureT is uniform. When the temperature variespatially, diffusion is somewhat more complex anddiscussed briefly under the rubric ‘‘Further commentin Sec. IV.
~b! Statement~2! is valid if the temperature is uniform anfixed. If, instead, the total energy is fixed and the teperature may vary from place to place, thenm/T mea-sures the contribution. When one looks for conditiothat describe chemicalequilibrium, one may focus oneach locality separately, and then the division by teperature is inconsequential.
~c! The system’s ‘‘external parameters’’ are the macscopic environmental parameters~such as externamagnetic field or container volume! that appear in theenergy operator or the energy eigenvalues. All exterparameters are to be held constant when the derivain statement~3! is formed. The subscriptV for volumeillustrates merely the most common situation. Note thpressure does not appear in the eigenvalues, and sothe present usage—pressure is not an external paeter.
These contextual conditions will be justified later. Thnext section studies diffusive equilibrium in a familiar cotext, ‘‘discovers’’ the chemical potential, and establishescharacterization in statement~1!.
III. THE TENDENCY TO DIFFUSE
The density of the Earth’s atmosphere decreases wheight. The concentration gradient—a greater concentralower down—tends to make molecules diffuse upwaGravity, however, pulls on the molecules, tending to mathem diffuse downward. The two effects are in balance, cceling each other, at least on an average over short timesmall volumes. Succinctly stated, the atmosphere is in elibrium with respect to diffusion.
In general, how does thermal physics describe such afusive equilibrium? In this section, we consider an ideal isthermal atmosphere and calculate how gas in thermal elibrium is distributed in height. Certain derivatives emer
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Figure 1 sets the scene. Two volumes, vertically thincomparison with their horizontal extent, are separatedheight by a distanceH. A narrow tube connects the uppevolumeVu to the lower volumeVl . A total numberNtotal ofhelium atoms are in thermal equilibrium at temperatureT;we treat them as forming an ideal gas. What value shouldanticipate for the numberNu of atoms in the upper volumeespecially in comparison with the numberNl in the lowervolume?
We need the probabilityP(Nl ,Nu) that there areNl atomsin the lower volume andNu in the upper. The canonicaprobability distribution gives us that probability as a suover the corresponding energy eigenstatesC j :
P~Nl ,Nu!5 (statesC j with Nl in Vl
and Nu in Vu
exp~2Ej /kT!
Z
[Z~Nl ,Nu!
Z. ~1!
The symbolZ denotes the partition function for the entisystem, andk is Boltzmann’s constant. The second equalmerely defines the symbolZ(Nl ,Nu) as the sum of the appropriate Boltzmann factors.
In the present context, an energy eigenvalueEj splits natu-rally into two independent pieces, one for the particles inlower volume, the other for the particles in the upper vume. Apply that split to the energy in the Boltzmann factImagine holding the state of theNu particles in the uppervolume fixed and sum the Boltzmann factor exp(2Ej /kT)over the states of theNl particles in the lower volume. Thastep generates the partition functionZl(Nl) for the lowervolume times a Boltzmann factor with just the energy of tparticles in the upper volume. Now sum over the states ofNu particles in the upper volume. The outcome is the tform
Z~Nl ,Nu!5Zl~Nl !3Zu~Nu!. ~2!
~More detail and intermediate steps are provided in Chaof Ref. 1.!
Fig. 1. The context. The narrow tube allows atoms to diffuse from oregion to the other, but otherwise we may ignore it.
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Common experience suggests that, given our specificmacroscopic system, the probability distributionP(Nl ,Nu)will have a single peak and a sharp one at that. An efficiway to find the maximum inP(Nl ,Nu) is to focus on thelogarithm of the numerator in Eq.~1! and find its maximum.Thus one need only differentiate the logarithm of the righand side of Eq.~2!. Note that increasingNl by one unitentails decreasingNu by one unit; via the chain rule, thaintroduces a minus sign. Thus the maximum in the probaity distribution arises whenNl andNu have values such tha
] ln Zl
]Nl5
] ln Zu
]Nu. ~3!
The equality of two derivatives provides the criterion fthe most probable situation. This is the key result. The flowing paragraphs reformulate the criterion, explore its iplications, and generalize it.
First, why reformulate? To connect with functions that adefined in thermodynamics as well as in statistical mechics. A system’s entropyS can be expressed in terms of lnZ,T, and the estimated energy^E&:
S5^E&T
1k ln Z. ~4!
Thus the Helmholtz free energyF provides a good alternative expression for lnZ:
F[^E&2TS52kT ln Z. ~5!
Equation~3! indicates that the rate of change ofF with par-ticle number is the decisive quantity; so define thechemicalpotentialm by the relation2
m~T,V,N![S ]F
]NDT,V
5F~T,V,N!2F~T,V,N21!. ~6!
In the language of the chemical potential, the criterionthe most probable situation, Eq.~3!, becomes
m l~T,Vl ,Nl !5mu~T,Vu ,Nu!. ~7!
The chemical potentials for atoms in the lower and upvolumes are equal.
But what about the less probable situations? And theproach to the most probable situation? Some detail will hhere. The partition functionZu(Nu) for the upper volume hasthe explicit form
Zu~Nu!5@~Vu /l th
3 !e2mgH/kT#Nu
Nu!, ~8!
where l th[h/A2pmkT defines the thermal de Brogliwavelength and wherem denotes an atom’s rest mass. Tchemical potential for the atoms in the upper volume is th
mu5mgH1kT lnS Nu
Vul th
3 D . ~9!
For the atoms in the lower volume,m l has a similar structurebut the gravitational potential energy is zero.
The explicit forms form l and mu enable one to plot thechemical potentials as functions ofNl at fixed total numberof atoms. Figure 2 displays the graphs. Suppose we fothe gaseous system with the numberNl significantly belowits ‘‘equilibrium’’ or most probable value. Almost surely atoms would diffuse through the connecting tube fromVu to
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Vl and would increaseNl toward (Nl)most probable. Atomswould diffuse from a region where the chemical potentiamu to a place where it ism l , that is, they would diffusetoward smaller chemical potential. In Newtonian mechana literal force pushes a particle in the direction of smapotential energy. In thermal physics, diffusion ‘‘pusheparticles toward smaller chemical potential.
The details of this example justify the first characterizatof the chemical potential: as a function of position, tchemical potential measures the tendency of particles tofuse. In general, in the most probable situation, which isonly situation that thermodynamics considers, the chempotential is uniform in space. Before the most probable sation becomes established, particles diffuse toward lochemical potential.
Alternative derivations of these conclusions aavailable.3,4 Their great generality complements the derivtion given here, whose specificity provides an opportunityexplore the ‘‘tendency to diffuse’’ in graphic detail. Moreover, if one uses Eq.~9! and its analog form l in Eq. ~7!, thenone finds that the number densityN/V drops exponentiallywith height. Recovering the ‘‘isothermal atmosphere’’ prvides students with a welcome sense of confidence informalism.
Just as temperature determines the diffusion of energy~bythermal conduction and by radiation!, so the chemical potential determines the diffusion of particles. This parallel is prfound, has been noted by many authors,5 and goes back aleast as far as Maxwell in 1876, as Sec. IX will display.
A. Uniformity
Martin Bailyn remarks sagely, ‘‘The@chemical potentialsfor various particle species# ...will be uniform in equilibriumstates. ... In this respect,@they# supplant density as the parameter that is uniform in material equilibrium.’’6
That uniformity, moreover, holds good when two or mophases coexist, such as liquid and solid water, or liqusolid, and vapor. The equality of chemical potentials acrosphase boundary can serve as the basis for derivingClausius–Clapeyron equation~which gives the slope of thephase boundary in a pressure-temperature plane!.
If a particle species is restricted to two disjoint regionhowever, then the chemical potential may have different v
Fig. 2. Graphs of the two chemical potentials as functions ofNl and Nu
5(Ntotal2Nl). The arrows symbolize the direction of particle diffusion reltive to the graphed values of the chemical potentials~andnot relative to thelocal vertical!. WhenNl is less than its most probable value, particles dfuse toward the lower volume and its smaller chemical potential; whenNl isgreater than its most probable value, diffusion is toward the upper voland its smaller chemical potential.
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ues in each region. For example, suppose the vertical tubFig. 1 were provided with a valve, closed initially, and ththe lower volume were initially empty. If the valve were theopened, atoms would diffuse downward, but the valve cobe closed again before the chemical potentials in the loand upper regions reached equality. Equilibrium couldachieved in each region separately—but with unequal checal potentials.
Later, when we discuss batteries, we will find such a sation. When a battery is on open circuit, the conduction eltrons in the two~disjoint! metal terminals generally havdifferent chemical potentials.
IV. EXTREMA
Now we focus on the connection between the chempotential and extrema in certain thermodynamic function
Return to the canonical probability distribution in Sec.and its full spectrum of values forNl andNu . With the aid ofEqs. ~5! and ~2!, we can form a composite Helmholtz freenergy by writing
F~Nl ,Nu
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from equilibrium: the numberNi of molecular species Bichanges byDNi , which equals the stoichiometric coefficiebi . Then the change in the Helmholtz free energy is
DF5(i
S ]F
]NiD
T,V, other N’s
DNi
5(i
m ibi50. ~14!
The partial derivatives are precisely the chemical potentiand the zero follows because the imagined step is away fthe minimum. Equilibrium for the chemical reaction impliea constraint among the various chemical potentials:
(i
m ibi50. ~15!
The constraint provides the key equation in derivingchemists’ law of mass action.
Equation~14! provides the initial justification for the second characterization of the chemical potential: when a retion may occur, the chemical potential measures the conbution ~per particle! to the rate of change of the functiowhose extremum determines equilibrium.
Note that the rate of change isnot a rate of change withtime. Rather, it is a rate of change as the reaction procestep by step.
B. Other extrema
In different circumstances, other thermodynamic functioattain extrema. For example, under conditions of fixedergy and fixed external parameters, the entropy attainmaximum. Does the second characterization continuehold?
To see how the extrema inSand in the Gibbs free energG are related to the chemical potential, we need equivaexpressions form, ones that correspond to the variables thare held constant while the extrema are attained. To dethose expressions, start with the change inF when the sys-tem moves from one equilibrium state to a nearby equirium state. The very definition ofF implies
DF5DE2TDS2SDT. ~16!
~Here E is the thermodynamic equivalent of the estimatenergy^E& in statistical mechanics.!
A formal first-order expansion implies
DF5S ]F
]TDV,N
DT1S ]F
]VDT,N
DV1S ]F
]NDT,V
DN. ~17!
The last coefficient is, of course, the chemical potentialm.The first two coefficients, which are derivatives at constN, can be evaluated by invoking energy conservation. Wthe system moves from an equilibrium state to anothernearby, energy conservation asserts that
TDS5DE1PDV, ~18!
providedN is constant. Use Eq.~18! to eliminateTDS in Eq.~16! and then read off the coefficients that occur in Eq.~17!as 2S and 2P, respectively. Insert these values into E~17!; equate the right-hand sides of Eqs.~16! and ~17!; andthen solve forDE:
DE5TDS2PDV1mDN. ~19!
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This equation expresses energy conservation under thegen-eralization that the numberN of particles may change bualso under therestriction that all changes are from one equlibrium state to a nearby equilibrium state.8 The involutedsteps in the present derivation are required because, astart, we knew an expression form only in terms of theHelmholtz free energy.
Now, however, from Eq.~19! we can read off that
m5S ]E
]NDS,V
. ~20!
Next, divide Eq.~19! by T, solve forDS, and read off therelation
2m
T5S ]S
]NDE,V
. ~21!
Finally, addD(PV) to both sides of Eq.~19!, rearrange tohaveDG[D(E2TS1PV) on the left-hand side, and reaoff that
m5S ]G
]NDT,P
. ~22!
If a reaction comes to equilibrium at fixed temperature apressure, the Gibbs free energy attains a minimum. Equa~22! shows that the chemical potential then plays the sarole that it did under conditions of fixed temperature avolume. Gibbs often considered the case of minimum eneat fixed entropy and external parameters.9 Now Eq. ~20!shows that the chemical potential retains the same role.
If entropy, however, is the function that attains an extmum, then the pattern is broken. Equation~21! shows thatthe quotientm/T takes the place ofm alone. Merely on di-mensional grounds, some alteration was necessary.10
All in all, this section supports the second characterizatand shows how the entropy extremum departs from the tcal pattern.
C. Further comments
The sum in Eq.~14! may be split into a difference osubsums over products and reactants, respectively. Insubsum, the chemical potentials are weighted by the cosponding stoichiometric coefficients~all taken positivelyhere!. If the system has not yet reached thermodynamic eqlibrium, the entire sum in Eq.~14! will be nonzero. Evolutionis toward lower Helmholtz free energy. That implies evoltion toward products if their weighted sum of chemical ptentials is smaller than that of the reactants. Here is the alog of particle diffusion in real space toward lower chemicpotential: a chemically reactive system evolves towardside—products or reactants—that has the lower weighsum of chemical potentials.
One may generalize the term ‘‘reaction’’ to apply to cexistence of phases. One need only interpret the termmean ‘‘transfer of a particle from one phase to anotheThereby one recaptures the property that, at thermodynaequilibrium, the chemical potential is spatially uniform~ineach connected region to which diffusion can carry the pticles!.
The present paragraph redeems the promise in Sec.return to diffusion. In an isothermal context, diffusion is dtermined by the gradient of the chemical potential, gradm, as
426Ralph Baierlein
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developed in Sec. III. In a more general context, one canto the first-order theory of relaxation toward equilibriumAmong the theory’s most appropriate variables are dertives of the entropy density with respect to the number dsity and the energy density. These derivatives are2m/T @byEq. ~21!# and 1/T. Thus particle diffusion is determined bylinear combination of grad (m/T) and grad (1/T); each termenters the sum with an appropriate~temperature-dependen!coefficient.11
When a system—initially not in thermal equilibrium—evolves toward equilibrium, the process can be compsometimes purely diffusive~in the random-walk sense!,other times hydrodynamic. James McLennan’sIntroductionto Nonequilibrium Statistical Mechanics, cited in Ref. 11,provides an excellent survey. Because of the diversityprocesses, I use the word ‘‘diffusion’’ broadly in this papeIts meaning ranges from a strict random walk to mer‘‘spreads out.’’
V. CHARACTERISTIC ENERGY
Now we turn to the phrase ‘‘characteristic energy’’ as pviding a meaning for the chemical potential. Here the cleest expression form is the form (]E/]N)S,V : the system’senergy change when one particle is added under conditof constant entropy~and constant external parameters!. Be-cause entropy is so often a crucial quantity, an energy chaat constant entropy surely provides a characteristic eneThe question can only be this: in which contexts doescharacteristic energy play a major role?
I will not attempt a comprehensive response but ratherfocus on a single context, arguably the most significant ctext in undergraduate thermal physics. Recall that the FerDirac and Bose–Einstein distribution functions dependthe energy«a of a single-particle state~labeled by the indexa! through the difference«a2m. Why does the chemicapotential enter in this fashion?
Derivations of the distribution functions make compasons of entropies~or multiplicities12!, either directly or im-plicitly ~through the derivation of antecedent probability dtributions!. In entropy lies the key to whym appears, and tworoutes that use the key to explain the appearance commind.
~1! Additive constant. The physical context determines thset $«a% of single-particle energies only up to an arbitraadditive constant. When«a appears in a distribution function, it must do so in a way that is independent of our choof the additive constant. In short,«a must appear as a difference. With what quantity should one compare«a and formthe difference?
The energy«a describes the system’s energy change whone particle is added in single-particle statewa . Typically,such an addition induces a change in entropy. So the cparison might well be made with (]E/]N)S,V , the system’senergy change when one particle is added under conditof constant entropy~and constant external parameters!. Thederivative, of course, is another expression for the chempotential, as demonstrated in Eq.~20!.
~2! Total entropy change when a particle is added. Somederivations of the distribution functions entail computing ttotal entropy change of either the system or a reservoir wa particle is added to the system of interest.13 To avoid irrel-
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evant minus signs, focus on the system. When a particleenergy«a is added, the entropy change consists of two pa
DS5S ]S
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T. ~23!
The chemical potential enters through the relationship~21!.As explained in Ref. 10, derivatives ofS andE with respectto N are necessarily proportional to each other~at fixed ex-ternal parameters!. So, the second term must be expressiin terms ofm, which appears again as a characteristic ene
The notion of ‘‘characteristic energy’’ can be madbroader and looser. Chemists often focus on Eq.~22! andcharacterize the chemical potential as the ‘‘partial moGibbs free energy’’~provided that particle numbers are mesured in moles!. Some authors note that, when only one pticle species is present, Eq.~22! is numerically equivalent tothe relationm5G/N, and they characterize the chemical ptential as the Gibbs free energy per particle.14 For typicalphysics students, however, the Gibbs free energy remainsmost mysterious of the thermodynamic functions, andsuch characterizations are of little help to them.15
VI. NUMERICAL VALUES
This section is devoted to qualitative reasoning aboutnumerical value that the chemical potential takes on. Taim is to develop more insight and greater familiarity. Fthe most part, the reasoning is based on the form form givenin Eq. ~20!.
A. Absolute zero
In the limit as the temperature is reduced to absolute zthe system settles into its ground state, and its entropycomes zero.~For a macroscopic system, any degeneracythe ground state, if present, would be insignificant, and sodescription assumes none.! Adding a particle at constant entropy requires that the entropy remain zero. Moreover, athe addition, the system must again be in thermal equirium. Thus the system must be in the ground state of the nsystem of (N11) particles.@One could preserve the constraint ‘‘entropy50’’ by using a single state~of the entiresystem! somewhat above the ground state, but that procedwould not meet the requirement of thermal equilibrium.#
For a system of ideal fermions, which are subject toPauli exclusion principle, we construct the new ground stfrom the old by filling a new single-particle state at the Ferenergy«F . Thus the system’s energy increases by«F , andthat must be the value of the chemical potential.
Consider next bosons, such as helium atoms, that obconservation law: the number of bosons is set initially aremains constant in time~unless we explicitly add or subtracparticles!. For such bosons~when treated as a quantum idegas!, we construct the new ground state by placing anotparticle in the single-particle state of lowest energy. Tchemical potential will equal the lowest single-particle eergy,«1 .
427Ralph Baierlein
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B. Semi-classical ideal gas
For an ideal gas in the semi-classical domain16 ~such ashelium at room temperature and atmospheric pressure!, theprobability that any given single-particle state is occupiedquite small. An additional atom could go into any one ogreat many different single-particle states. Moreover,may use classical reasoning about multiplicity and entrop17
Adding an atom, which may be placed virtually anywhesurely increases the spatial part of the multiplicity and hetends to increase the entropy. To maintain the entropy cstant, as stipulated in Eq.~20!, requires that the momentumpart of the multiplicity decrease. In turn, that means lekinetic energy, and so the inequalityDE,0 holds, whichimplies that the chemical potential is negative for an idgas in the semi-classical domain.~Implicit here is the stipu-lation that the energyE is strictly kinetic energy. Neither apotential energy due to external forces nor a rest enemc2, appears.!18
In this subsection and in the preceding one, we determiDE ~or placed a bound on it! by assessing the change in thsystem’s energy as the system passed from one equilibstate to another. How the additional particle is introducand what energy it may carry with it are secondary matteWhy? Because the system is required to come to equilibragain and to retain—or regain—its original entropy valuTo satisfy these requirements, energy may need to betracted by cooling or added by heating. Such a processautomatically compensate for any excess or deficiency inenergy change that one imagines to accompany the litintroduction of the extra particle.
To summarize, one compares equilibrium states ofN andN11 particles, states that are subject to certain constrasuch as ‘‘same entropy.’’ This comparison is what detminesDE. The microscopic process by which one imaginone particle to have been introduced may be informative,it is not decisive and may be ignored.
Next, consider the explicit expression for a chemical ptential in Eq.~9!—but first delete the termmgH. The deriva-tion presumed that the thermal de Broglie wavelengthmuch smaller than the average interparticle separation. Csequently, the logarithm’s argument is less than one, andchemical potential itself is negative. Can such a negavalue be consistent with the characterization of the chempotential as measuring the tendency of particles to diffuYes, because what matters for diffusion is howm changesfrom one spatial location to another. The spatial gradient~ifany! is what determines diffusion, not the size or sign ofm atany single point.
C. Photons
Photons are bosons, but they are not conserved in numEven in a closed system at thermal equilibrium, such as akitchen oven, their number fluctuates in time. There isspecific numberN of photons, although—when the temperture and volume of a cavity have been given—one can cpute an estimated~or mean! number of photonsN&.
There are several ways to establish that the chemicaltential for photons is zero~though I find none of them to beentirely satisfactory!. The most straightforward route is tcompare the Planck spectral distribution with the Bos
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Einstein distribution, computed for conserved bosons. Tdistributions match up if one sets the chemical potentiathe latter to zero.
One can also use a version of Eq.~20!, replacingN by ^N&.Everything about a~spatially uniform! photon gas in thermaequilibrium is known if one knows the temperature and vume. Specifically, the entropy can be written in terms oTand V, and soT can be expressed in terms ofS and V.Therefore, the energyE ~usually considered a function ofTand V) can be expressed in terms ofS and V. Then anyderivative ofE at constantS andV must be zero.
For a third method, one can examine the annihilation ofelectron and a positron or their creation. In thermal equilrium, the reaction should follow the pattern set by chemireactions:
melectron1mpositron52mphoton ~24!
for the two-photon process. Provided the temperature,ume, and net electrical charge have been specified, oneconstruct the probability thatN2 electrons,N1 positrons,and any number of photons are present. The construcfollows the route outlined in Sec. III forNl andNu ~providedone treats the electrons and positrons as uncharged sclassical gases!. The ensuing probability is a function oN2 , N1 , T, andV. Looking for the maximum in its loga-rithm, one finds the criterion
melectron1mpositron50. ~25!
Comparing Eqs.~24! and ~25!, one infers that the chemicapotential for photons is zero.
Next, note that the spin-singlet state of positronium mdecay into two or four or any higher even number of phtons. Simultaneous thermal equilibrium with respect to allthese processes@in the fashion illustrated in Eq.~24! fortwo-photon decay# is possible only ifmphoton equals zero.Here one sees clearly that the absence of a conservationfor photons leads to a zero value for the chemical poten
Finally, consider the conduction electrons in the mewall of a hot oven. The electrons interact among themseland with the radiation field~as well as with the metallic ionswhich are ignored here!. One electron can scatter off anothand, in the process, emit a photon into the oven. The reacis e1e8→e91e-1g. The primes indicate different electronic states.~The reversed process, in which a photonabsorbed, is also possible.! In thermal equilibrium, an analogous equation should hold among chemical potentials: 2me
52me1mphoton. From this equality, one infers that thchemical potential for thermal radiation is zero.
If the chemical potential for photons is everywhere zerothere any meaning to the notion of ‘‘diffusion of photons’Yes, a spatial gradient in the temperature field determinesflow of radiant energy and hence the ‘‘diffusion of photons
G. Cook and R. H. Dickerson19 provide additional andalternative qualitative computations of the chemical pottial.
Now the paper turns—for illustration—to two applicationof the chemical potential. The first—depression of the meing point—was the subject of a Question20 and fourAnswers21 in the Journal’s Question and Answer section1997. The second application—batteries—seems to beduringly fascinating for most physicists.
428Ralph Baierlein
icense or copyright; see http://ajp.aapt.org/authors/copyright_permission
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VII. DEPRESSION OF THE MELTING POINT
Consider liquid water and ice in equilibrium atT5273 Kand atmospheric pressure. If alcohol or table salt is addethe liquid water and if the pressure is maintained at oatmosphere, a new equilibrium will be established at a lowtemperature. Thus, when ice is in contact with the solutiits melting point is depressed. The chemical potential pvides a succinct derivation of the effect, as follows.
For the sake of generality, replace the term ‘‘ice’’ b‘‘solid’’ and the term ‘‘liquid water’’ by ‘‘solvent.’’ The‘‘solvent’’ here is the liquid form of the solid but may contain a solute~such as alcohol or salt!.
The solvent’s chemical potential depends on the tempture, the external pressure, and the ratioh of number densi-ties ~or concentrations!, solute to solvent:
h[nsolute
nsolvent. ~26!
In the absence of solute, the solvent and solid coexistemperatureT0 and pressureP0 , and so their chemical potentials are equal then.
After the solute has been added, the new equilibriumcurs at temperatureT01DT such that
msolvent~T01DT,P0 ,h!5msolid~T01DT,P0!. ~27!
Expand the left-hand side about the arguments (T0 ,P0,0)and the right-hand side about (T0 ,P0). To evaluate the partial derivatives with respect toT, one may differentiate Eq~22! with respect toT, interchange the order of differentiation on the right-hand side, and note that (]G/]T)P,N52S52Ns(T,P), where s denotes the entropy per molecul@Alternatively, one may use the Gibbs–Duhem relation22 fora single species~because the derivatives are evaluated inpure phases!.# The first-order terms in the expansion of E~27! yield the relation
2ssolventDT1]msolvent
]hh52ssolidDT. ~28!
The remaining derivative is to be evaluated at zero soconcentration. Solve forDT:
DT51
ssolvent2ssolid3
]msolvent
]hh. ~29!
Both a theoretical model, outlined in Appendix A, anexperimental evidence from osmosis indicate t]msolvent/]h is negative. A liquid usually has an entropy pparticle higher than that of the corresponding solid phaand so the difference in entropies is usually positive. Timplication then is adepressionof the melting point that islinear in the solute concentration~when a first-order expansion suffices!.
Qualitative reasons for a depression are given in Ref.Exceptions to the positive difference of entropies arescribed in Ref. 23. The chemical potential provides a simanalytic approach to related phenomena: elevation ofboiling point, osmotic pressure, and various solubilproblems.24
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VIII. BATTERIES
The topic of batteries immediately prompts two questio
~1! Why does a potential difference arise?~2! How large is the potential difference between the term
nals?
This section addresses the questions in order.For a prelude, let me say the following. In discussing t
first question, we will see how the spatial uniformity of thchemical potential helps one to infer and describe the spabehavior of ionic concentrations and the electric potentialaddressing the second question, we will relate the potendifference to the~intrinsic! chemical potentials and stoichiometric coefficients of the particles whose reactions powerbattery. An overall difference in binding energy will emergas what sets the fundamental size of the potential differen
A. Why a potential difference?
To have an example before us, take a single cell ofautomotive lead-acid battery. The chemical reactions arefollowing.
At the pure lead terminal,
Pb1HSO42→PbSO41H112e2. ~30!
At the terminal with lead and lead oxide,
PbO21HSO4213H112e2→PbSO412H2O. ~31!
Imagine commencing with two neutral electrodes andelectrolyte; then pour in a well-stirred mixture of sulfuracid and water~and keep the cell on open circuit!.
At the pure lead terminal, reaction~30! depletes thenearby solution of HSO4
2 ions and generates H1 ions, asillustrated in Figs. 3~b! and 3~d!. The two electrons contrib-
Fig. 3. For the electrolyte in a lead-acid cell, qualitative graphs of~a! theelectric potentialw(x), ~b! the number densitynH1 of H1 ions, ~c! theintrinsic chemical potential for those ions, and~d! the number density ofHSO4
2 ions. The abscissax runs from the pure lead electrode to the terminwith lead and lead oxide. Potentials—both electric and chemical—usucontain an arbitrary additive constant. Implicitly, those constants have bchosen so that graphs~a! and ~c! lie conveniently above the origin.
429Ralph Baierlein
icense or copyright; see http://ajp.aapt.org/authors/copyright_permission
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uted to the terminal make it negative. In the context offigure, the negative charges on the terminal and the net ptive charge density in the nearby electrolyte produceleftward-directed electric field in the electrolyte. By themselves, the concentration gradients in the ion densities woproduce diffusion that restores a uniform density. The eltric forces oppose this tendency. Specifically, acting onHSO4
2 ions, the field produces a rightward force and prevethe concentration gradient from eliminating the depletioActing on the H1 ions, the field produces a leftward forcand sustains the excess of H1 ions.
The situation is an electrical analog of air molecules inEarth’s gravitational field: the effects of a force field andconcentration gradient cancel each other.
Over an interval of a few atomic diameters, the electrolhas a positive charge density. Beyond that, the electrolytessentially neutral~when averaged over a volume that cotains a hundred molecules or so!. The combination of a positively charged interval and a negative surface charge onelectrode provides a region of leftward-directed electric fiand a positive step in electric potential~as one’s focus shiftsfrom left to right!. Figure 3~a! illustrates the step.
At the lead oxide electrode, reaction~31! depletes thenearby solution of H1 ions @as illustrated in Fig. 3~b!# andsimultaneously makes the terminal positive. The positcharges on the terminal produce a leftward-directed elecfield. The electric force on the remaining H1 ions preventsdiffusion from eliminating the depletion.
The reaction depletes HSO42 ions also, but the 3-to-1 ratio
in the reaction implies that the effect of the H1 ions domi-nates. In fact, the electric force on the remaining HSO4
2 ionspulls enough of those ions into the region to produce a loexcess of HSO4
2 ions, as illustrated in Fig. 3~d!. ~Later, aself-consistency argument will support this claim.! The elec-trolyte acquires a net negative charge density over an inteof a few atomic diameters. The combination of a negativcharged electrolyte and the positive surface charge onelectrode produces a leftward-directed electric field andother positive step in electric potential, as shown in Fig. 3~a!.
In summary, a chemical reaction at an electrode is a sor source of ions. Thus the reaction generates a concentrgradient and a separation of charges. The latter produceelectric field. In turn, the electric field opposes the tendeof diffusion to eliminate the concentration gradient andcharge separation. The electric field is preserved and, acover an interval, produces a step in electric potential.
For each ion, its chemical potential may be decompointo two terms:~1! the chemical potential that the ion wouhave in the absence of an electric potential and~2! the elec-trical potential energy of the ion due to the macroscopic etric potentialw(x). The first term was called by Gibbs th‘‘intrinsic potential,’’ and it is now called theintrinsic orinternal chemical potential. The subscript ‘‘int’’ may be readin either way. Thus an ion’s chemical potentialm has theform
m5m int1qionw, ~32!
whereqion is the ion’s charge.The intrinsic chemical potential is an increasing functi
of the ion’s concentration~provided the physical system istable!.25 Thus Fig. 3~c! displays a graph form int, H1 thatshows the same trends that the number densitynH1 does.
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At thermal equilibrium the chemical potential for eacspecies is uniform in space~in each connected region twhich diffusion can carry the particles!. For a positivelycharged ion, the concentration will decrease where the etric potential increases; a comparison of graphs~a! and ~b!illustrates this principle. For a negatively charged ion, topposite behavior occurs. Thus consistency among thegraphs in Fig. 3 requires the concentration variationsHSO4
2 ions to be opposite those for H1 ions.The gradient of Eq.~32! lends itself to a simple interpre
tation if the ions exist in adilute aqueous solution. Then thintrinsic chemical potential for a specific ion depends stially on merely the concentrationn(x) of that ion. To saythat the chemical potential is uniform in space is to say tthe gradient of the chemical potential is zero. Thus Eq.~32!implies
S ]m int
]n D dn
dx2qionEx50, ~33!
whereEx denotes thex-component of the electric field. Thelectric forceqionEx annuls the diffusive tendency of a concentration gradient,dn/dx.
B. How large is the potential difference?
To determine the potential difference when the cell hcome to equilibrium on open circuit, we start with a basprinciple: at chemical equilibrium, the Gibbs free energytains a minimum~in the context of fixed temperature anexternal pressure!. Taking into accountboth reactions~30!and ~31!, we have
DG5(i
bim i50. ~34!
The sum of stoichiometric coefficients times chemical pottials must yield zero. Note that electrons appeartwice in thissum, once when transferredfrom an electrode and agaiwhen transferredto the other electrode.
Each chemical potential has the two-term structure dplayed in Eq.~32!. Some care is required in evaluating thterm containing the electric potential. Indeed, the ions aelectrons need separate treatment. We will find that the ioelectric potential terms inDG cancel out and that the electronic contribution introduces the potential difference.
For the ions, we may use the values of the chemicaltentials at the center of the electrolyte. Although the reactuses up ions at the electrodes, those ions are replenifrom the plateau region, and that replenishment is part ofstep from one equilibrium context to another.
An alternative way to justify using the central locationto note that the chemical potential for each ion is unifothroughout the electrolyte. Thus every location givessame numerical value, but the center—because it typifiesplateau region—will ensure that, later, the ionic intrinschemical potentials are to be evaluated in the bulk regionthe electrolyte.
The number of ‘‘conduction’’ electrons is the same in treactants and the products. The electrons are merely onferent electrodes. Consequently, the net charge on the iothe same for the reactants and the products. When thosecharges are multiplied by the electric potential at the cenand then subtracted, the contributions cancel out. All that
430Ralph Baierlein
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ions and neutrals contribute to the sum in Eq.~34! is theirintrinsic chemical potentials, suitably weighted.
Quite the opposite is true for the conduction electroTwo electrons are removed from the metallic lead atpositive electrode, and two are transferred to the lead atnegative electrode. The contribution of their electric potetial terms to Eq.~34! is
2~22e!wpos1~22e!wneg52~22e!Dw, ~35!
where Dw[wpos2wneg is the positive potential differencbetween the terminals and wheree denotes the magnitude othe electronic charge. The electron’s intrinsic chemicaltentials, however, have the same numerical value in thepieces of lead,26 and so they cancel out in Eq.~34!.
The upshot is that the minimum property forG implies
DG5(iÞel
bim int,i2~22e!Dw50. ~36!
The summation includes the ions and neutrals but excluthe electrons.
Solving for the potential difference, one finds
Dw521
qch3 (
ions andneutrals
bim int,i , ~37!
whereqch denotes the magnitude of the electrons’ chargFor the ions, the intrinsic chemical potentials are to be evaated in the bulk region of the electrolyte.
The sum obviously depends on the cell’s composition.the species in the electrolyte, eachm int can be expanded todisplay a dependence on concentrations. Thus composand concentrations determine the potential difference.significantly more can be said, as follows.
For an ion in an aqueous solution, the chemical potentiadifficult to calculate explicitly. If the electrolyte were a dilutgas, computation would be much simpler. The intrinchemical potential for each ion or neutral molecule woucontain, as a term, the particle’s ground-state energy«g.s.~reckoned relative to the energy of some standard state,as the energy of free neutral atoms and free electrons atat infinite separation!.27 That is to say,m int5«g.s.1¯ . Dif-ferences in binding energywould dominate the sum in Eq~37!, would set the fundamental size of the open-circuit ptential difference, and would provide the primary enersource for a current if the circuit were closed. Qualitativethe same situation prevails in an aqueous solution.28
Appendix B describes a more common route to the rein Eq. ~37! and reconciles the two different values forDG.Dana Roberts29 and Wayne Saslow30 provide complementarytreatments of batteries. A chemist’s view of a battecouched in language that a physicist can penetrate, is gby Jerry Goodisman.31
Having seen—in two applications—how the chemical ptential is used, we can turn to a sketch of how it arosewas named.
IX. SOME HISTORY
J. Willard Gibbs introduced the chemical potential in hgreat paper, ‘‘On the Equilibrium of Heterogeneous Sustances,’’ published in two parts, in 1876 and 1878. In thdays, Gibbs was doing thermodynamics, not statisticalchanics, and so he differentiated the system’s energy w
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respect to the macroscopic mass~denotedmi by him! of thesubstance denoted by the subscripti. Entropy and volumewere to be held constant. Thus Gibbs introduced a mascopic version of Eq.~20!. He had denoted the system’s eergy and entropy by the lower case Greek letters« and h,respectively; so, presumably, he chose the letterm to providea mnemonic for a derivative with respect to mass.
Early in the paper, Gibbs called his derivative merely t‘‘potential.’’ 32 Later, he found it necessary to distinguish hderivative from the electric potential and gravitational potetial. He introduced the term ‘‘intrinsic potential’’ for a derivative that is ‘‘entirely determined at any point in a maby the nature and state of the mass about that point.’’33 Inshort, the intrinsic potential is a local quantity, dependentthe local mass~or number! density, say, but not on fieldcreated by distant charges or masses. For example, Gwould have called the second term on the right-hand sideEq. ~9! the intrinsic potential for the atoms in the upper voume.
Nowhere in his major paper does Gibbs use the te‘‘chemical potential.’’ That coinage seems to be have beintroduced by Wilder Dwight Bancroft.34 A Ph.D. student ofWilhelm Ostwald, Bancroft was a physical chemist at Conell and the founder of theJournal of Physical Chemistry~in1896!. Yale’s Beinecke Library preserves five letters froBancroft to Gibbs,35 dated 1898–1899. In the earliest letteBancroft adheres to Gibbs’s usage. In his second letter, d18 March 1899, however, Bancroft says he is trying to fitime to write a book on electrochemistry and uses the phr‘‘the chemical potentialm.’’ He employs the phrase nonchalantly, as though it were a familiar or natural phrase. Molikely, Bancroft found a need to distinguish between telectric potential and Gibbs’s~intrinsic! potential. The term‘‘chemical potential’’ for the latter would make the distinction clear.~Altogether, Bancroft uses the new term againtwo more of the later letters.!
The fourth letter, dated 4 June 1899, has some wondelines. Bancroft mentions that ‘‘it has taken me seven yehard work to find out how your equation should be appliedactual cases’’ and comments, ‘‘The chemical potential is sa mere phrase to everyone although Ostwald uses it wicertain specious glibness.’’ Then he launches into his pertion:
If we can once get used to writing and thinkingin terms of the chemical potential for the com-paratively simple case of electromotive forces, itwill not be so difficult to take the next step and tothink in terms of the chemical potentials whenwe are dealing with systems which cannot betransposed to form a voltaic cell. So far as I cansee at present, our only hope of converting or-ganic chemistry, for instance, into a rational sci-ence lies in the development and application ofthe idea of the chemical potential.
Widespread understanding of how to use the chemical potial was slow in coming.
In a reply to Bancroft, Gibbs acceded to the coinage, wing about the need ‘‘to evaluate the~intrinsic or chemical!potentials involved’’ in a working theory of galvanic cells.36
Electrochemists remain true to Bancroft’s usage. In refring to Eq.~32!, they would call the termm int the ‘‘chemicalpotential’’ and would cite the entire right-hand side as t‘‘electrochemical potential.’’37
431Ralph Baierlein
icense or copyright; see http://ajp.aapt.org/authors/copyright_permission
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Physicists sometimes split the right-hand side of Eq.~32!into ‘‘internal’’ and ‘‘external’’ chemical potentials and denote the sum as the ‘‘total’’ chemical potential.38
Most often, however, physicists accept whatever emerwhen one forms the derivative indicated in Eqs.~6!, ~20!,and ~22! and call the result the chemical potential. In thpaper, I adopted that usage.
Nonetheless, the splitting of the chemical potential invarious terms warrants further comment. Gibbs notedthe zeroes of energy and entropy are arbitrary; consequethe chemical potential may be shifted arbitrarily in valuea term of the form39 (constant)2T3~another constant!. Thisresult is seen most easily by regarding the chemical poteas a derivative of the Helmholtz free energy, as in Eq.~6!.
Today, the Third Law of Thermodynamics gives usnatural zero for entropy. The zero of energy, however, mseem to remain arbitrary. So long as the number of partiof speciesi remains constant in whatever process one csiders, the zero for the energy of those particles remairrelevant. If, however, particles of speciesi are created orannihilated, then one must include the rest energy,mic
2,wheremi denotes the rest mass, in the explicit expressionthe chemical potential. The rest energy term is absoluessential in a description of the early universe and mother aspects of high-temperature astrophysics.40 In short,special relativity theory provides a natural zero for the eergy of ~free! particles.
Now this section returns to history. James Clerk Maxwwas fascinated by Thomas Andrews’ experiments on cardioxide: the coexistence of liquid and vapor, the criticpoint, trajectories in the pressure-temperature plane,mixtures of CO2 with other gases. Moreover, Maxwell habeen impressed by the geometric methods that Gibbslined in his first two papers on thermodynamics. In this cotext, Maxwell developed—independently of Gibbs—his owversion of the chemical potential and some associated rtionships. When Gibbs’s comprehensive paper appeaMaxwell dropped his own formalism and enthusiasticarecommended Gibbs’s methods.41
Speaking to a conference of British chemists in 18Maxwell distinguished between what we would today c‘‘extensive’’ and ‘‘intensive’’ thermodynamic propertiesThe former scale with the size of the system. The latterMaxwell’s words, ‘‘denote the intensity of certain physicproperties of the substance.’’ Then Maxwell went on, eplaining that ‘‘the pressure is the intensity of the tendencythe body to expand, the temperature is the intensity oftendency to part with heat; and the@chemical# potential ofany component is the intensity with which it tends to expthat substance from its mass.’’42 The idea that the chemicapotential measures the tendency of particles to diffuse isdeed an old one.
Maxwell drew an analogy between temperature andchemical potential. The parallel was noted already nearend of Sec. III, and it is developed further in Appendixwhich explores the question, why does this paper offer thcharacterizations of the chemical potential, rather than jusingle comprehensive characterization?
ACKNOWLEDGMENTS
Professor Martin J. Klein provided incisive help in tracing down Bancroft’s contribution. Professor Martin Bailyand Professor Harvey Leff read preliminary versions and
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APPENDIX A: HOW THE SOLVENT’S CHEMICALPOTENTIAL VARIES
This appendix focuses on howmsolvent varies with the~relative! concentration of solute, denotedh. A verbal argu-ment comes first; then a derivation with equations will suport it.
As a prelude, however, let us note that a liquid is largincompressible. To a good approximation, its volume istermined by the number of molecules and the temperatThe volume need not be considered an external param~unlike the situation with a gas!. I will adopt this ‘‘incom-pressible’’ approximation here.
To begin the verbal argument, recall that one may thinkmsolventas the change inF total, the Helmholtz free energy fothe system of solvent and solute, when one solvent moleis added to the liquid~at constant temperature and constanumber of solute molecules!.
Addition of a solvent molecule increases the volume ofliquid. Thussolutemolecules have more space in whichmove. Their entropy increases, and soF total undergoes asupplemental change, which is a decrease. Therefomsolvent(T,P,h) is less thanmsolvent(T,P,0).
The supplemental decrease scales approximately linewith Nsolute ~becauseSsolute scales approximately linearlywith Nsolute when the solute concentration is relativesmall!. Consequently,]msolvent/]h is negative and approximately constant~for small h!.
A simple theoretical model enables one to confirm tline of reasoning. To construct the partition function for tsolution, modify the forms in Eqs.~2! and ~8! to read asfollows:
Z~T,N1 ,N2!5~N1y11N2y2!N11N2
l th,13N1N1! 3l th,2
3N2N2!
3exp@~N1«11N2«2!/kT#. ~A1!
The subscripts 1 and 2 refer to solvent and solute, resptively. Because a liquid is largely incompressible and detmines its own volume, one may replace a container voluV by N1y11N2y2 , where the constants$y1 ,y2% denote vol-umes of molecular size. An attractive force~of short range!holds together the molecules of a liquid and establishebarrier to escape. Model the effect of that force by a potenwell of depth2«, saying that a molecule in the liquid hapotential energy2« relative to a molecule in the vapophase.
In short, the model treats the liquid as a mixture of twideal gases in a volume determined by the molecules thselves~as they jostle about in virtually constant contact weach other!. The energy« provides the dominant contributioto the liquid’s latent heat of vaporization.
The solvent’s chemical potential now follows by differetiation as
msolvent52«11kT ln S l th,13
ey1D 2kTh1O~h2!. ~A2!
432Ralph Baierlein
icense or copyright; see http://ajp.aapt.org/authors/copyright_permission
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As expected,msolvent decreases approximately linearly withe ~relative! concentration of solute. Moreover, if one keetrack of where terms originate, one finds that the term2kTharises from the solute’s entropy.
Note also that the term linear inh is independentof theratio y2 /y1 . The terms of higher order inh do depend ony2 /y1 . The distinction should be borne in mind if one aswhether purported qualitative reasons for the depressiothe melting point are actually valid.
APPENDIX B: BATTERIES BY AN ALTERNATIVEROUTE
The potential difference across a battery’s terminals isten calculated by assessing the electrical work done whenchemical reaction proceeds by one step. This appendixlines that alternative route and reconciles it with the ropresented in the main text.
To determine the potential difference, we start with eneconservation in the form
TDS5DE1PDV1wel , ~B1!
wherewel denotes the~external! electrical work done by thesystem. In this alternative route, ‘‘the system’’ consiststhe ions and neutralsbut not the conduction electronsin theterminals.
In the context of fixed temperature and pressure, Eq.~B1!may be rearranged as
2D~E2TS1PV![2DG5wel . ~B2!
The electrical work equals the energy that would be requto transport the electrons literally across the potential diffence, and so Eq.~B2! becomes
2DG5qchDw, ~B3!
where~as before! qch denotes the magnitude of the electroncharges and where the potential differenceDw is positive.
The change inG is given by
DG5 (ions andneutrals
bim i , ~B4!
the sum of stoichiometric coefficients times chemical pottials. For the ions, we may use the values of the chempotentials at the cell’s center.~Justification for this step wagiven in the main text.! Because the reaction preserves eltric charge, that is, because the products have the samcharge as do the reactants, the value of the electric poteon the plateau,w~0!, cancels out in the sum. All that remainis the weighted sum ofintrinsic chemical potentials for ionsand neutrals.
Upon combining these observations with Eqs.~B3! and~B4!, we find the relationship that determines the potendifference:
Dw521
qch3 (
ions andneutrals
bim int,i . ~B5!
For the ions, the intrinsic chemical potentials are to be evaated in the bulk electrolyte.
The result forDw is, of course, the same here as in tmain text. The two routes differ in what they take to be ‘‘thsystem,’’ and so their intermediate steps necessarily dialso. The derivation in the main text takes a comprehen
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view of the cell and sees it in chemical equilibrium~when onopen circuit!. Relations internal to the system determine telectric potential difference. In this appendix, a portion of tcell ~taken to be the thermodynamic system! does work onanother portion~namely, the conduction electrons!.
APPENDIX C: WHY SO MANY?
This appendix addresses two questions.~A! Why does this paper offer three characterizations
the chemical potential, rather than just a single comprehsive characterization?
~B! Why does the chemical potential have so maequivalent expressions, which seem to make it exception
Response to Question A. Rather than immediately confronthe chemical potential, let us consider temperature andhow many characterizations it requires. One may start weither a common-sense understanding of~absolute! tempera-ture or the general expression 1/T5(]S/]E)N,V .
In characterizing temperature or listing its ‘‘meanings,’’would start with the statement
~1! Tendency of energy to diffuse. As a function of posi-tion, temperature measures the tendency of energy to dif~by thermal conduction and by radiation!.But I could not stop there. I would have to add the statem
~2! Characteristic energy. Temperature provides a charateristic energy:kT.
To be sure, I would have to qualify the second statemeOnly if classical physics and the equipartition theorem apdoes kT give the energy per particle or per mode or p‘‘degree of freedom’’~within factors of 1
2 or 32 or so!. But
even for a degenerate quantum system like the conducelectrons in a metal,kT provides a characteristic energy thone compares with the Fermi energy«F .
Thus temperature has at least two essential charactetions.
If only one species of particle is present, then the charterizations of the chemical potential can be reduced toitems: ~1! measures the tendency of particles to diffuse a~2! provides a characteristic energy. So one could seechemical potential as no more complicated than temperatand both require at least two characterizations.
But, of course, the chemical potential really comes intoown when more than one species of particle is presentreactions are possible. Then, it seems to me, I am stuckneeding to assign three characterizations.
Response to Question B. Energy, entropy, Helmholtz freeenergy, Gibbs free energy, and enthalpy commonly appeaundergraduate thermal physics. Each of these functions isoptimal function to use in some set of physical circumstances. In each case, one may contemplate adding a pato the system, and so the chemical potential can be expreas a derivative of each of the five functions.
Given the same five functions, in how many straightfoward ways can one express temperature as a derivaThree: as the derivative of entropy with respect to enerand as the derivatives of energy and enthalpy with respecentropy. The two free energies already have temperaturenatural independent variable, so there is no straightforwway to express temperature as a derivative.
We tend, however, to view the expression 1T5(]S/]E)N,V and its reciprocal as a single relationship, nas two distinct relationships. Moreover, expressing tempe
433Ralph Baierlein
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All in all, both the chemical potential and temperatuhave several equivalent expressions. In undergraduate pics, the chemical potential has two more than temperadoes, and those of temperature tend to be reduced to a siton in practice. But—as far as the number of equivalentpressions goes—the chemical potential isnot exceptional inany qualitative fashion.
1Ralph Baierlein,Thermal Physics~Cambridge U. P., New York, 1999!,pp. 148–155.
2The partition functionZ is the sum of Boltzmann factors taken overcomplete, orthogonal set of energy eigenstates. Consequently, bothZ andF depend on temperature, the number of particles of each species tpresent, and the system’s external parameters. If more than one specpresent, the chemical potentialm i for speciesi is the derivative ofF withrespect toNi , computed while the numbers of all other particles are kfixed and while temperature and all external parameters are held fixe
Equation~6! indicates that the chemical potential may be computedapplying calculus to a function ofN or by forming the finite differenceassociated with adding one particle. WhenN is large, these two methodyield results that are the same for all practical purposes. Convenialone determines the choice of method.
3Charles Kittel and Herbert Kroemer,Thermal Physics, 2nd ed.~Freeman,New York, 1980!, pp. 118–125.
4F. Mandl,Statistical Physics~Wiley, New York, 1971!, pp. 222–224.5For examples, see Ref. 1, p. 154, and Ref. 3, pp. 118–120.6Martin Bailyn, A Survey of Thermodynamics~AIP, New York, 1994!, p.213.
7Reference 1, pp. 233–234. Nuclear spin systems may be in thermallibrium at negative absolute temperatures. In such a case, the Helmfree energy attains a maximum. For more about negative absoluteperatures, see Ref. 1, pp. 343–347 and the references on pp. 352–3
8Reference 6, pp. 212–213; Ref. 1, p. 228.9J. Willard Gibbs,The Scientific Papers of J. Willard Gibbs: Vol. I, Themodynamics~Ox Bow, Woodbridge, CT, 1993!, pp. 56, 64, and 65, forexamples.
10Relation~21! can be seen as a consequence of a remarkable mathemidentity, most often met in thermodynamics: if the variables$x,y,z% aremutually dependent, then (]x/]y)z(]y/]z)x(]z/]x)y521. To understandwhy the minus sign arises, note that not all of the variables can be incring functions of the others. For a derivation, see Herbert B. Callen,Ther-modynamics~Wiley, New York, 1960!, pp. 312–313. The correspondenc$S,N,E%5$x,y,z% yields Eq. ~21!, given Eq. ~20! and the expression(]S/]E)N,V51/T for absolute temperature.
11A classic reference is S. R. De Groot and P. Mazur,Non-EquilibriumThermodynamics~North-Holland, Amsterdam, 1962!, Chaps. 3, 4 and 11James A. McLennan provides a succinct development in hisIntroductionto Nonequilibrium Statistical Mechanics~Prentice–Hall, EnglewoodCliffs, NJ, 1989!, pp. 17–25. H. B. G. Casimir develops an illuminatinexample in his article, ‘‘On Onsager’s Principle of Microscopic Reveibility,’’ Rev. Mod. Phys.17, 343–350~1945!—but beware of typographi-cal errors. A corrected~but less lucid! version is given by H. J. KreuzerNonequilibrium Thermodynamics and its Statistical Foundations~OxfordU. P., New York, 1981!, pp. 60–62.
12As used here, the term ‘‘multiplicity’’ denotes the number of microstaassociated with a given macrostate. Entropy is then Boltzmann’s contimes the logarithm of the multiplicity.
13Reference 3, pp. 134–137. Francis W. Sears and Gerhard L. SaliThermodynamics, Kinetic Theory, and Statistical Thermodynamics, 3rd ed.~Addison–Wesley, Reading, MA, 1975!, pp. 327–331.
14C. B. P. Finn,Thermal Physics~Routledge and K. Paul, Boston, 1986!, pp.190–193. Also Ref. 4, pp. 224–225.
15Here are two more characterizations that offer little help. Some autdescribe the chemical potential as a ‘‘driving force’’~e.g., Ref. 3, p. 120!.Others callm or a difference inm a ‘‘generalized force’’@e.g., Herbert B.Callen, Thermodynamics~Wiley, New York, 1960!, pp. 45–46#. Thesestrike me as unfortunate uses of the word ‘‘force.’’ In mechanics, studlearn that a ‘‘force’’ is a push or a pull. But no such push or pull drivesstatistical process of diffusion. The terms ‘‘driving force’’ and ‘‘generaized force’’ are more likely to confuse than to enlighten.
16As used in this paper, the term ‘‘semi-classical’’ means that the therma
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Broglie wavelength is much smaller than the average interparticle seption. Hence quantum mechanics has no substantial direct effect on dyics. Nonetheless, vestiges of the indistinguishability of identical particpersist~as in division byN! in the partition function!, and Planck’s con-stant remains in the most explicit expressions for entropy, partition fution, and chemical potential. A distinction between fermions and bosohowever, has become irrelevant.
The analysis in Sec. III treated the helium atoms as a semi-clasideal gas.
17See Ref. 12.18David L. Goodstein provides similar~and complementary! reasoning in his
States of Matter~Prentice-Hall, Englewood Cliffs, NJ, 1975!, p. 18.19G. Cook and R. H. Dickerson, ‘‘Understanding the chemical potentia
Am. J. Phys.63, 737–742~1995!.20Sherwood R. Paul, ‘‘Question #56. Ice cream making,’’ Am. J. Phys.65,
11 ~1997!.21The most germane of the answers is that by F. Herrmann, ‘‘Answe
Question #56,’’ Am. J. Phys.65, 1135–1136~1997!. The other answersare Allen Kropf, ibid. 65, 463 ~1997!; M. A. van Dijk, ibid. 65, 463–464~1997!; and Jonathan Mitschele,ibid. 65, 1136–1137~1997!.
22Reference 1, pp. 279–280.23Although the entropy per particle is usually higher in a liquid than in t
corresponding solid, exceptions occur for the helium isotopes3He and4Heon an interval along their melting curves. See J. Wilks and D. S. BettsAnIntroduction to Liquid Helium, 2nd ed.~Oxford U. P., New York, 1987!,pp. 15–16.
24Frank C. Andrews,Thermodynamics: Principles and Applications~Wiley,New York, 1971!, pp. 211–223. Also, Frank C. Andrews, ‘‘ColligativeProperties of Simple Solutions,’’ Science194, 567–571~1976!. For amore recent presentation, see Daniel V. Schroeder,An Introduction toThermal Physics~Addison–Wesley, Reading, MA, 2000!, pp. 200–208.
25Recall that particles diffuse towardlower chemical potential. If the con-centration increases locally, only an increase inm int will tend to restore theoriginal state and hence provide stability. For a detailed derivation,Ref. 6, pp. 232, 239, and 240.
26One might wonder, does charging the electrodes significantly alterelectrons’ intrinsic chemical potentials? To produce a potential differeof 2 V, say, on macroscopic electrodes whose separation is of millimsize requires only a relatively tiny change in the number of conductelectrons present. Thus the answer is ‘‘no.’’
27Reference 1, pp. 246–257.28Some of the energy for dc operation may come from the environmen~as
an energy flow by thermal conduction that maintains the battery at ambtemperature!. Such energy shows up formally as a termTDS in DG @seenmost easily in Eq.~B2!#. Measurements ofdDw/dT show that this contri-bution is typically 10% of the energy budget~in order of magnitude!.Gibbs was the first to point out the role of entropy changes in determincell voltage~Ref. 9, pp. 339–349!.
29Dana Roberts, ‘‘How batteries work: A gravitational analog,’’ Am.Phys.51, 829–831~1983!.
30Wayne Saslow, ‘‘Voltaic cells for physicists: Two surface pumps andinternal resistance,’’ Am. J. Phys.67, 574–583~1999!.
31Jerry Goodisman,Electrochemistry: Theoretical Foundations~Wiley,New York, 1987!.
32Reference 9, pp. 63–65 and 93–95.33Reference 9, pp. 146 and 332.34A. Ya. Kipnis, ‘‘J. W. Gibbs and Chemical Thermodynamics,’’ inTher-
modynamics: History and Philosophy, edited by K. Martina´s, L. Ropolyi,and P. Szegedi~World Scientific, Singapore, 1991!, p. 499.
35The letters are listed in L. P. Wheeler,Josiah Willard Gibbs: The Historyof a Great Mind, 2nd ed.~Yale U. P., New Haven, 1962!, pp. 230–231.
36Reference 9, p. 425.37E. A. Guggenheim introduced the phrase ‘‘electrochemical potential’’ a
made the distinction in ‘‘The conceptions of electrical potential differenbetween two phases and the individual activities of ions,’’ J. Phys. Ch33, 842–849~1929!.
38Reference 3, pp. 124–125.39Reference 9, p. 95–96.40Reference 1, pp. 262–264.41Elizabeth Garber, Stephen G. Brush, and C. W. F. Everitt,Maxwell on
Heat and Statistical Mechanics: On ‘‘Avoiding All Personal Enquiries’’ oMolecules~Lehigh U. P., Bethlehem, PA, 1995!, pp. 50 and 250–265.
42Reference 41, p. 259.
434Ralph Baierlein
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