N.K.Borsuk manuscript: Testing the Definition of Thermodynamic Equilibrium - part 1

1 INFINITE ENERGY • ISSUE 102 • MARCH/APRIL 2012

AbstractThis paper reports explorations into the scientific principlesof, and ramifications from, the coupling of solute-concen-tration fluctuations to an external field. Solute-concentra-tion fluctuations are a variety of spontaneous particle fluc-tuations—random vacillations in the distribution of soluteparticles within solute-solvent solutions—that have longbeen known to exist. In the study presented here, it is posit-ed that the fluctuations are also exotic solute species thatdevelop gravitational potential energy within liquid solu-tions that are at rest within earth’s gravity. To test thishypothesis, experimental results from analytical electro-chemistry, using simple two-electrode gravity cells (Au-AuCl3-Au) with 0.5 molar aqueous auric chloride elec-trolytes, were compared against numerical expectationsdeveloped from statistical mechanics. The salient technicalfindings are these: The gravitational potential energies ofsolute-concentration fluctuations exist in accord with theo-retical modeling and calculations; they (1) add to the chem-ical potential of a solute in solution, (2) participate in theequilibrium equation and (3) preclude the electrochemicalcells from a steady state of equilibrium.

1. Introduction/Overview1.1 — The Second Law and Particle FluctuationThermodynamic Equilibrium: In simple fluid systems—gasesand liquids—thermodynamic equilibrium has two facets.Static equilibrium occurs at the macroscopic level; a fluid’senergies are balanced and latent, and so its measureableproperties cannot spontaneously change. In this condition asystem can only be passive and inanimate. Dynamic equilib-rium exists at the microscopic level of atoms and moleculeswhere the fluid is constantly in flux due to the particles'incessant and random motions. But these spontaneous par-

ticle fluctuations are small in magnitude and brief in exis-tence, and they offset or balance one another; microscaleparticle fluctuations do not alter the equilibrium propertiesof the macroscopic system. The tendency for all material sys-tems to devolve to a state of thermodynamic equilibrium isomnipresent, heedless of the size, composition or complexi-ty of a system, and is one of the statements of the second lawof thermodynamics.

Solute-Concentration Fluctuation: Within a liquid solute-sol-vent solution—a volume of saltwater, for instance—a solute-concentration fluctuation is both a stochastic reaction andan ephemeral object. In the first instance it is merely achance clustering of solute particles that haphazardly aggre-gate and then disperse. In the second, it is a subvolume with-in the bulk fluid that briefly contains more or fewer soluteparticles than its surroundings; the precise value beingalways in flux due to the ceaseless random movements ofthe particles themselves.1 In all solutions a solute is differentthan the solvent, and so fluctuated subvolumes are transito-ry cloud-like objects with distinct properties of their own:solute concentration, mass density, magnetic susceptibility.These properties are significant because they distinguish thefluctuated subvolumes from the surrounding bulk fluid.2These fluctuated subvolumes are the solute-concentrationfluctuations that are the focus of study in this paper.

Spontaneous particle fluctuation is not a new concept inphysics. It started with the idea that matter is made of atomsthat are always in random motion, and it became a corner-stone in the kinetic theory of gases that was established inthe 19th century. Since 1905, from theoretical studies byEinstein and by Smoluchowski in combination with the1907-1908 experimental confirmation of their calculationsby Perrin (dealing with the erratic movement—Brownianmotion—of particulate suspended in fluids), particle fluctu-

Testing the Definition of Thermodynamic Equilibrium

Part 1: Systems in a Gravitational Field

Norman K. Borsuk*

Thermodynamic equilibrium in a liquid solution is a balancing of the steady-state potential energiesthat play on the chemical species—the solvent and solutes—within the macroscopic system. In prac-tice, a relatively simple equilibrium equation is the balance beam that mathematically equilibrates thecomponents’ energies and makes it possible to define and predict a solution’s equilibrium properties

that, ultimately, can be tested in actual experiment.

An equilibrium equation’s efficacy is dependent, in no small part, on underlying models, theories and laws thatprescribe which quantities enter the formula in the first place. Failure to include a relevant energy or chemicalcomponent can lead to incomplete and misleading definitions, erroneous predictions and, when the oversight iscorrected and all is set straight again, possibly to revision or replacement of fundamental canon. Such is theimpact of solute-concentration fluctuations and their gravitational potential energies on equilibrium theory andthe second law of thermodynamics.

MARCH/APRIL 2012 • ISSUE 102 • INFINITE ENERGY 2

ation has been recognized as fundamental phenomena. (Forboth the theories and experiments, Perrin’s Atoms2 providesin-depth explanations, references to original publicationsand invaluable historical narrative.

Exotic Nature of Particle-Concentration Fluctuation:Thermodynamically, concentration fluctuations in gases andliquid solutions involve a forward reaction in which theentropy S of a fluctuated subvolume spontaneously decreas-es (ΔS < 0). This is a ubiquitous yet exotic thermodynamicprocess as the fluctuation drives the subvolume and its sur-roundings away from equilibrium. The forward reaction is fol-lowed by a restoring reaction in which the entropy increas-es (ΔS > 0), returning the subvolume toward equilibrium (seeKondepudi and Prigogine3). Also, fluctuated subvolumes arenot solid objects; rather, they are loose groupings of parti-cles. And it can be shown that they are cloud-like non-Brownian objects that have mass, occupy volume and aredistinguishable from the bulk fluid but are effectivelymotionless in the system. Hence, they do not exert a pres-sure in the system and, consequently, an equilibrium con-stant Kpx = pn

x/px cannot be defined for any fluctuated sub-volume x with a partial pressure equal to zero (px = 0).

Generally, equilibrium particle-concentration fluctuationsare two-part phenomena that have been recognized inphysics for over one hundred years; they are random sto-chastic processes and they are transitory subvolumes with dis-tinct properties. Additionally, they are known to have passiveinteractions with energy. For example, they are understood tobe responsible, at least in part, for the atmospheric scatteringof sunlight that results in the blue color of earth’s sky.4

1.2 — Core ThesisThe questions that have prompted these studies include: Dosolute-concentration fluctuations couple to a gravitationalfield and develop gravitational potential energy? And insolutions where the solute and solvent have different mag-netic properties (such as a paramagnetic salt dissolved intodiamagnetic water): Do solute-concentration fluctuationscouple to a static inhomogeneous magnetic field and sodevelop magnetic potential energy within the fluid solution?

In this paper I focus on the first question—the gravita-tional case. The study is premised on an inclusive model thatis defined as follows: Within liquid solutions, a solute-con-centration fluctuation is simultaneously a density fluctua-tion; the mass density of the fluctuated subvolume is brieflydifferent than that of the surrounding solution. As a result,the subvolume becomes an ephemeral solute species thatcouples to an external gravitational field; i.e., it developsgravitational potential energy within the bulk solution. Thismodel is fundamentally at odds with a standard modelimplicit to modern physics and classical thermodynamicsover one issue. In the standard model, the field-potential

energies of a solution’s microscale fluctuations are unac-counted; the fluctuations are recognized, but not their ener-gies. The conceptual disparity between the inclusive and thestandard models leads to conflicting predictions for a well-known property of liquid solutions in earth’s gravity.

1.3 — Essential TaskThe standard model predicts that a column of solute-solventsolution at equilibrium and at rest in earth’s gravity willhave a permanent vertical solute-concentration gradient(analogous to the vertical gradient of air pressure and densi-ty in earth’s atmosphere). The inclusive model predicts aslightly steeper particle gradient with greater chemicalpotential energy than the standard expectation. The differ-ing forecasts are due to the fluctuations’ gravitational poten-tial energies being included in the inclusive model, butabsent in the standard model. Those differences in the parti-cle-gradient’s steepness and potential energy are differencesin magnitude that can be calculated and accurately meas-ured. The essential task in this work has been to test calcula-tions from each model against experimental results, and soprovide evidence for resolving the conflicting predictions.

Expectations for a solution’s equilibrium solute gradientbased on the standard model are readily found in existingliterature. Expectations premised on the inclusive modelwere derived from statistical mechanics (see Section 2).Quantitative predictions from both models were comparedagainst measurements from electrochemical experimentsusing two-electrode gravity cells (see Section 3). The cellswere rudimentary composite devices consisting of a sealedcolumn of solution and an external electrode circuit (Figure1). The fluid columns were glass tubes filled with auric chlo-ride salt dissolved in water and sealed with gold-foil caps ateach end of a tube. The gold caps also served as electrodes,and a nanovoltmeter was used to measure the voltage devel-oped between the two electrodes in response to the solute’spotential energies.

Since the experimental goal was to bring a cell to, andmaintain it at, thermodynamic equilibrium, it was impor-tant to carefully keep each cell within an isolated and tem-perature-controlled environment for prolonged periods(weeks at a time).

1.4 — Key Concepts to be Presented(1) The first key concept resulting from this study is thatspontaneous solute-concentration fluctuations developgravitational potential energy within simple fluid solutionsthat are at thermodynamic equilibrium and at rest within agravitational field. Those energies are clearly indicated inelectrochemical measurements in accord with quantitativecalculations derived from the inclusive model: A fluid col-umn will come to thermodynamic equilibrium with a verti-cal solute-concentration gradient that is steeper and that hasgreater chemical potential energy than anticipated from thestandard model.(2) The second key concept is a discovered consequence ofthe first. When a cell’s fluid column is at thermodynamicequilibrium, then the external electrode circuit will be pre-cluded from equilibrium and will produce a steady-statevoltage. This result is an outcome, albeit unintuitive, of anelectrode’s electrochemically-selective nature. A cell’s non-equilibrium state prevails because the electrode circuit pro-

Figure 1. A gravity cell is a composite system comprised of two sub-systems: a fluid column and an external electrode circuit.


duces a voltage that is directly driven by the differencebetween two of the potential energies within the fluid col-umn. Those two energies are the solute’s chemical potentialthat is referred to as Ec, and the solute’s gravitational poten-tial that is referred to as Eg. Since Ec is larger than Eg due tothe contributions by the almost innumerable field-potentialenergies of the fluctuations, the electrode circuit will exhib-it a steady-state electronic potential energy (a voltage) driv-en by the imbalance. This second finding suggests that theequilibrium statement of the second law of thermodynamicsis not valid in application to the composite systems.(3) The cumulative evidence expounded in this paper sup-ports a third key concept: The second law of thermodynam-ics is not universal. The second law is not valid in definingor predicting the systems studied here because it is premisedon an incomplete accounting of the chemical componentsand potential energies that always exist within liquid solu-tions within a gravitational field.

2. Theoretical Foundations2.1 — Ideal SystemThe ideal system in this study is a column of liquid solutioncomprised of an electrically neutral solute dissolved into asolvent, at constant and uniform temperature, and at rest ina gravitational field. At equilibrium, the ideal solution devel-ops a permanent vertical solute-concentration gradient. (Theionic or nonionic nature of the solute is inconsequential tothe terminal steady-state gradient in both theory and prac-tice. Ionic solutes in aqueous electrolytes were used in theexperiments because they provided a practical means ofaccurately testing the hypothesis.) This section of the papersets forth the general equations used to predict the equilib-rium properties of the solute gradient in the ideal system asbased on each of the two competing models (the standardand the inclusive).

2.2 — Standard ModelThe equations in this subsection are taken from, or aredirectly derived from, Guggenheim’s texts.5 The equilibriumcriterion for a chemical species i in a system at uniform tem-perature and without influence from an external field is suchthat the component’s chemical potentials have the samevalue in any two phases α and β so that:

µαi = µβi . (1.1)

When the system is within earth’s gravity, then the gen-eral condition for thermodynamic equilibrium must alsotake into account the component’s gravitational potentialssuch that:

µαi + Miθα = µβi + Miθ

β (1.2.1)

where Mi is the molar mass of species i, and θα and θβ are thegravitational potentials in different phases (vertical posi-tions) α and β of the system. Rearranging Equation 1.2.1 gives:

µαi − µβi = −Mi (θα − θβ) . (1.2.2)

For the ideal solution and perfect gases, the relationshipbetween the solute’s molar chemical potentials and molar

concentrations cαi and cβi is such that:

µαi − µβi = RTlncαicβi

(1.3)

where R is the universal gas constant and T is absolute tem-perature.

Combining Equations 1.2.2 and 1.3, the equilibrium cri-terion for perfect gas a within the gravitational field is:

RTlncαacβa

= −Ma(θα − θβ) , (1.4)

and for solute i in the ideal solution within the field is:

RTlncαicβi

= −M̌i(θα − θβ) , (1.5)

where M̌i is the effective molar mass of the solute in solutionat extreme dilution such that M̌i = Mi − (V

–i/ V

–0)M0 (where M0

is the molar mass of the pure solvent, V–i is the partial molar

volume of the solute, and V–

0 is the partial molar volume ofthe pure solvent).

Guggenheim’s formula in Equation 1.5 can be simplified.Let Ec be the solute’s molar concentration-gradient potentialenergy such that:

Ec = RTlncαicβi

. (1.5a)

And let Eg be the solute’s molar gravitational potential ener-gy in solution such that:

Eg = M̌i(θα − θβ) . (1.5b)

In substituting Equations 1.5a and 1.5b into Equation 1.5,then Equation 1.5 is more clearly seen to involve only twoforms of potential energy involving the solute:

Ec = −Eg . (1.6)

Equations 1.5 and 1.6 are equivalent definitions of equi-librium for the ideal solution in a gravitational field, as basedon the standard model.

2.3 — Inclusive ModelIn this subsection the expressions for the gravitationalpotential energies of solute-concentration fluctuations andtheir connection to the solute’s chemical potential arederived. The derivation begins by stating premises and defi-nitions, including the general result from statistical mechan-ics for the fractional level of particle-number fluctuationapplicable to the ideal system. The effective density of a fluc-tuated subvolume in solution is then determined and, fromthat, the equation for its gravitational potential energy. Thederivation is completed by equating the gravitational poten-tial energies of all solute-concentration fluctuations to thesolute chemical potential.

2.3.1 — Premises and Definitions2.3.1a: Let ni be the number of solute particles of species itypically within subvolumes s and s’ in Figure 2. Subvolumes’ is distinguished from the nearby and otherwise identicalsubvolume s due to a random increase (fluctuation) in solute


particle number Δni. Both subvolumes exist within a smalllocal region of the solution in the macroscopic fluid columnbetween gravitational potentials θα and θβ.

2.3.1b: The subvolumes are open member systems within agrand canonical ensemble. The fractional level of particle-number fluctuation for these open subvolumes is under-stood1,p.643 and is of the order:

Δnini

= 1

√ni(2)

Hence, the typical maximal magnitude of a small fluctuationin particle number within a subvolume s normally contain-ing ni number of solute particles will be such that ∆ni = ni

½.

2.3.1c: Therefore subvolume s contains ni number of soluteparticles, and fluctuated subvolume s’ momentarily containsni + ni

½ solute particles.

2.3.1d: From Equation 1.5, the effective molar mass of thesolute in solution is M̌i such that M̌i = Mi − (V

–i / V

–0)M0.

2.3.1e: M̌i is constant throughout the column and in eachsubvolume such that (M̌i)s’ = (M̌i)s = (M̌i)s1 where s1 is a unitsubvolume with only one solute particle.

2.3.1f: The effective mass of a single solute particle in solu-tion is m̌ i = M̌i /NA, where NA is Avogadro’s number of parti-cles per mole. m̌ i is also constant such that (m̌ i)s’ = (m̌ i)s =(m̌ i)s1.

2.3.1g: The volume Vs’ of subvolume s’ is equal to that ofsubvolume s, and both are ni times larger than the volumecontaining a single solute particle, so that Vs’ = Vs = niVs1.

2.3.2 — Effective Density of a Fluctuated SubvolumeThe typical maximal difference in the effective massbetween fluctuated subvolume s’ and unfluctuated subvol-ume s is Δms. From premises 2.3.1c and 2.3.1f, Δms is equal to:

Δms = ms’ − ms = (ni½ + ni)m̌ i − nim̌ i = ni

½m̌ i . (3)

The difference in density between subvolumes s’ and s isΔρ̌s’, where Δρ̌s’ = Δms/Vs. From premise 2.3.1g and Equation3, then Δρ̌s is:

Δρ̌s’ = (ni

½ + ni)m̌ i−

nim̌ i

Vs’ Vs

(4)=

ni½ m̌ i

= m̌ i

niVs1ni

½Vs1

Δρ̌s is the effective density of the fluctuated subvolume. Itis small and rapidly tends to zero (proportional to 1/n1/2) asthe number of particles increases. The maximal densityoccurs in the unit subvolume with one discrete solute parti-cle, and it also tends to zero as the solute becomes identicalto the solvent.

2.3.3 — Gravitational Potential Energy of aFluctuated SubvolumeBecause a solute is not identical to a solvent, the density ofthe concentration-fluctuated subvolume will be differentfrom the average value of the local fluid. This density differ-ence is expressed in Equation 4. In this case I posit that thesubvolume will have gravitational potential energy in a grav-itational field. This conjecture is counterintuitive since anysingle fluctuated subvolume generally has very brief exis-tence. But the hypothesis may nevertheless obtain throughincessantly recurring fluctuations.

From Equation 4, the gravitational potential energy ∆Es’ ofsubvolume s’ between gravitational potentials θα and θβ is:

∆Es’ = Eαs’ − Eβ

s’ = (θα − θβ)(Δρ̌s)Vs1 = m̌ i(θα − θβ)

(5)ni

½

and expressed as a ratio is:

∆Es’=

1(6)

m̌ i(θα − θβ) ni½

The energy m̌ i(θα − θβ) is the gravitational potential energyof the discrete solute in solution and, from the premises, itis effectively a constant in the system.

Equations 5 and 6 express the expectation for the gravita-tional potential energy of a single variety of fluctuated sub-volume within the macroscopic ideal system that is at equi-librium and at rest within gravity. Equation 6 is consistentwith the order of magnitude for the fractional energy fluc-tuation that is expected for the subvolume as noted by oth-ers. For example, Tolman’s notation1,p.633 is (E – E

=)2 / (E

=)2 =

1/n. Also, Hill6 noted in his textbook that the 1/n1/2 order ofmagnitude for the relative fluctuation “. . .is the standardresult in statistical-mechanical fluctuation formulas.”Another source of support for Equation 6 is found in text-book statistics.7

2.3.4 — Impact to the Solute Chemical PotentialWe are interested in the energy change to the solute.Equation 5 gives the gravitational potential energy of a sin-gle variety of fluctuated subvolume. To find the change inenergy in terms of the solute within the fluctuated subvol-ume (not the subvolume as a unit), Equation 5 is divided by

Figure 2. Representation of solute-concentration fluctuation in theideal system. Left: A representation of an ideal fluid column at equilib-rium in gravity. Right: Two subvolumes. Subvolume s has a densityequal to the bulk fluid in the local area. Subvolume s' is representedas affected by solute-concentration fluctuation resulting in a greaterdensity distinguishing it from the surrounding fluid. Solvent fluctuationsin which the solvent concentration increases are not considered here.


ni, the typical number of solute particles within the subvol-ume:

1. ∆Es’ =

m̌ i(θα − θβ).

1 (7.1)

ni ni½ ni

which gives:

∆Es’=

m̌ i(θα − θβ) . (7.2)

ni ni3/2

Designating ∆Es’ / ni as ∆Ei,s’ and defining it as the poten-tial energy change of a solute particle in any fluctuated sub-volume s’ due to the gravitational potential energy of thesubvolume as a unit, then:

∆Ei,s’ = m̌ i(θα − θβ)

. (7.3)ni

3/2

Multiplying each side of Equation 7.3 by Avogadro’s num-ber gives the molar energy change to the solute:

NA∆Ei,s’ = M̌ i(θα − θβ)

. (8)ni

3/2

Recognizing from Equation 1.5.b that M̌ i(θα − θβ) = Eg, then:

NA∆Ei,s’ = Eg

. (8.1)ni

3/2

NA∆Ei,s’ is the change in the molar potential energy of thesolute particles in fluctuated subvolumes s’ due to the gravi-tational potential energy of the subvolumes as discrete units.The dimensions of NA∆Ei,s’ are equivalent to the molarchemical potential of the solute (energy/mole of solutespecies i), and when n = 1 then the relation Ec = − Eg inEquation 1.6 is apparent. It appears reasonable to assignNA∆Ei,s’ to the chemical potential of the solute µi,s’ suchthat:

∆µi,s’ = −NA∆Ei,s’ = - Eg

. (9)ni

3/2

where ∆µi,s’ is the difference in molar chemical potential ofsolute i due to the gravitational potential energy of fluctuat-ed subvolumes s’ between the extremes of the applied field.

2.3.5 — Impact to the Equilibrium CriterionThe total chemical potential energy of the solute in the macro-scopic system due to the gravitational potential of all fluctu-ated subvolumes is the sum of all contributions in Equation 9:

∑s’

s’=1∆µi,s’ = − ∑

n

n=1

Eg(10.1)

ni3/2

which leads to:

∆µi = −Eg

–Eg

–Eg

... –Eg

. (10.2)13/2 23/2 33/2 ni

3/2

The hyperharmonic series in Equations 10.1 and 10.2 is con-vergent. For systems containing at least 108 solute particles

the numerical value is approximately:

∆µi = − ∑∞

n=1

Eg ≈ - 2.6 Eg . (11)

ni3/2

Recognizing for the ideal system that:

∆µi = RTlncαicβi

(12)

and using the results in Equations 11 and 1.5b, we arrive at:

RTlncαicβi

= −2.6 M̌ i(θα − θβ) . (13.1)

Using the simplifying notation in Equation 1.5a, thenEquation 13.1 can be transcribed to:

Ec = −2.6 Eg (13.2)

and Equation 10.2 to:

Ec = −Eg –Eg

–Eg

... –Eg

. (13.3)23/2 33/2 n3/2

Inclusive Model: Equations 10.1 through 13.3 are equiva-lent statements of the equilibrium criterion for the solute inthe system between gravitation potentials α and β based onthe inclusive model. For comparison, recall that the standardcriterion is Ec = −Eg.

Clearly, the standard expression is contained within theinclusive equation. The inclusive model subsumes the stan-dard model by accounting for the gravitational potentialenergy of all solute-concentration fluctuation, includingthose involving only a single particle (which is the case inthe standard equation). Using parameters in Table 1,Equations 1.5 and 13.1 easily lead to numerical values forthe physical magnitude and potential energy of the equilib-rium solute gradient that can be tested against experiment.

2.4 — Electrochemically Measuring the Equilibrium GradientThere are various methods for measuring the equilibriumsolute gradient: electrochemical, optical and mechanical.The electrochemical “gravity cell” is superior to the othersfor numerous reasons. A gravity cell’s voltage—its magnitudeand polarity—is an electrochemical signal that can be used

Cell Parameters Units Value

Mi (AuCl3) kg/mol 0.3 aVi (AuCl3) cm3/mol 78 M0 (H2O) kg/mol 0.018V0 (H2O) cm3/mol 18

(θα – θβ) m2/s2 9.8 R J/mol·K 8.3 T K 301 F C/mol 96,487

aestimated

Table 1. Parameters for the model cell.


to discriminate between Ec and Eg. The voltage driven by Echas positive polarity which is opposite to the voltage drivenby Eg (negative). Additionally, the gravity cell can measurethe magnitude of Eg and combined magnitudes of Ec and Eg.Optical and mechanical methods cannot obtain these meas-urements as directly, if at all. Another reason for using elec-trochemical methods is that modern voltmeters may typi-cally have an input resistance of 1010 ohm or more so thatthere is extremely small energy influence to the fluid col-umn from the measurement equipment, whereas opticalpractices (requiring a high intensity light) and mechanical(using a densimeter) are highly intrusive. Also, it turns outthe electrochemical cell reveals ramifications of the fluctua-tions’ field-potential energies when the fluid column is asubsystem within a composite device—an important taskthat cannot be accomplished in the other methods.

Ideally, the steady-state open-circuit cell potential U forthe gravity cell is a function of only two variables, Ec and Eg:

U = U(Ec, Eg) = −(Ec + Eg)/nF (14)

where n = 3 for the Au3+ ion and F is the Faraday constant.From the standard model, the steady-state cell potentialwould indicate zero volts at the meter since Ec = − Eg.However, from the inclusive model in which Ec = −2.6 Eg, thesteady-state cell potential would be:

U = −(−2.6 Eg + Eg)

= +1.6 Eg

. (15)nF nF

2.5 — The Steady State of the CellAccording to Equation 15, when the fluid column is at equi-librium then the cell will have a permanent non-zero volt-age. This voltage is a distinct signal indicating Ec is in excessof the standard expectation and in excess of Eg. Additionalcharacteristics (derived in Section 4) are these: The steady-state voltage will have a definite and measureable magni-tude, and its polarity will be such that the bottom electrodeis always the cell’s positive terminal (cathode)—which isopposite to the polarity of the voltage driven by Eg.

2.6 — 20th Century Gravity-Cell StudiesSpanning 47 years during the early to middle 20th century,a small number of “gravity-cell” studies reported their exper-imental findings from research of ionic transference num-bers. Those studies tested for distinct voltages from electro-chemical cells by inverting them in gravity or spinning themin a centrifuge.8-11 Roughly, the cells were fluid columnssimilar to the cells in this study. Although the experimentalprocedure was straightforward in each case, the authorsnonetheless reported an apparently common problem:unexpected or anomalous voltages suggesting excessivesolute-concentration gradients as in Equation 15.

The common task in each of the early studies was to meas-ure an initial peak voltage driven by Eg. As the particle gra-dients developed, the authors anticipated that the initialvoltage would decay as the slowly developing solute-con-centration gradients and Ec became established. Theyexpected a cell’s voltage to eventually approach zero as thecell’s fluid column came to equilibrium. These expectationsfollow the standard model and were explained by Tolman8

in 1911 in the footnote on page 122 of his paper:

The electromotive force [voltage] which is producedin the salt solutions by the action of gravity must becarefully distinguished from the actual changes in theconcentration of the solute which gravity will pro-duce. The difference in concentration between thesolution in the upper and lower ends of a verticaltube, or the central and peripheral portions of a rotat-ing tube can be calculated from simple thermody-namic considerations. It is to be expected, however,that this difference in concentration will be very slowlyestablished. The difference in potential between thetwo ends of the solution is an immediate phenome-non which occurs as soon as the tube is set up. It isevident that when the final change in the concentra-tion of a salt solution has completed itself there willno longer be any potential difference between theupper and lower portion of the solution.

He continued (pages 137-138) with a description of unex-pected signals he encountered:

It is a striking fact that upon stopping the rotator [thecentrifuge] the residual electromotive force is alwaysfound to be in the opposite direction from that pro-duced by the centrifugal force. It will also be seenfrom an examination of the data that there is a gen-eral tendency for this residual electromotive force toincrease somewhat and then in the course of a fewminutes to gradually disappear.

Essentially, Tolman’s attention was sharply drawn toanomalous voltages with polarities opposite to those pro-duced by the gravitational field (centrifugal force). He wrotethat he was fairly confident the anomalies were not due tothermal effects. And he investigated the possibility they weredue to concentration gradients created by small pockets ofsolution trapped within pores in the cell walls, that hadstrongly different solute concentration than the bulk fluid,and that dispersed into the fluid column during slowdownof the centrifuge. In the end, he was unable to determine thesource of the problem voltages.

MacInnes and Ray9 in 1949, in regards to voltage datathey obtained using a series of increasing centrifuge speeds,stated: “The same accuracy has not yet been obtained withdecreasing speeds, a phenomenon for which we can give noexplanation.” It appears they found problems related toTolman’s: cell voltages that were not in keeping with expec-tations during slowdown of the rotor.

Ray, Beeson, and Crandall10 in 1958 wrote: “. . .in somecases a residual potential of up to 10 μV remained after per-mitting the cell and contents to stand or upon carrying outcentrifugal runs.” Apparently Ray et al. also encounteredproblem voltages related to Tolman’s. Also, this groupreported a problem regarding contamination of the elec-trodes that I encountered. They wrote: “. . .experience hasshown that particular care needs to be taken to avoid oily orsurface active impurities that can contaminate the electrodesand also to avoid colloidal material which tends to concen-trate in the region of one or the other electrode.”

Grinnell and Koenig11 in 1942 tested a basic gravity celland reported: “It was found that in spite of the care taken to


exclude oxygen, the e.m.f. [voltage] of a cell in one of its ver-tical positions was not constant but always showed a slightchange with time. In the cells here studiedthis change did not exceed 0.004 micro-volt per minute.”

They went on to describe their methodof minimizing the unavoidable spuriousvoltages that gradually developed duringtheir tests: they quickly inverted the cells,taking less than 60 seconds between ameasurement at one vertical position andthe other. Although the authors attributed(without elaboration) their voltage prob-lem to oxygen contamination, the indica-tions suggest that they were observing andunconsciously attempting to overcome theeffects anticipated in the inclusive model.

In the 20th century studies, theresearchers were intent on measuring vari-ous electrolytes’ gravitational or centrifu-gal potentials—not concentration-gra-dient potentials. It is not surprising,therefore, that there are no quantita-tive results for comparison to the pres-ent work. Still, from their reports ofanomalous voltages it is a reasonablepossibility that the researchers didencounter effects from the field-poten-tial energies of solute-concentrationfluctuation, but could not compre-hend the electrochemical signals—and, in the gravity-cell study, activelytried to circumvent them—because theeffects were inexplicable without anunderlying conceptual model. At thetime, of course, the inclusive model did not exist.

3. ExperimentsRecall from Section 1.3 that the cells used in this study wereglass tubes filled with a liquid electrolyte consisting of auricchloride (AuCl3) salt dissolved in water, and sealed with goldfoil at each end of the tube. Testing a cell involved setting itupright and keeping it undisturbed for prolonged periods,often for days or weeks at a time, in an environment as freeas possible of extraneous influences, and constantly moni-toring its voltage with a nanovoltmeter. Eventually a cell-in-test was inverted (180 degree rotation). Each inversion tookabout 5-10 seconds to complete, and afterwards the cellwould again rest undisturbed until the next inversion. Thissection reports experiments involving two cells in three sep-arate tests.

3.1 — Materials and Equipment“Au-AuCl3-Au” denotes an electrochemical cell with twogold electrodes and an auric chloride electrolyte connectingthem. The two cells used in tests reported here both usedaqueous electrolytes at about 0.5 molar. The differencebetween the cells was that one was 7.5 cm tall and the otherwas 2.5 cm tall. Each cell body was a thick-walled Pyrex tube1.5 cm internal diameter with a 0.13 mm thick Teflon wash-er sealing the gold foil electrode (99.99% Au, annealed, 0.13

mm thick) at each tube-end (Figure 3). A cell was first assem-bled and then filled with electrolyte through a small diame-

ter Pyrex tube (a fill-neck) that was weldedat one end to the middle of the cell bodyand sealed at the other end with a compres-sion fitting and Teflon gasket. A small airgap (a small bubble) was left inside thecapped end of the fill-neck to mollify pres-sure changes due to expansion/contractionof the materials. The electrolyte was pre-pared by electrolysis of a dilute HCl solu-tion using gold electrodes from the samestock as the finished electrodes used in thecells. The solution was then gently heatedto drive off the acid and to concentrate thesalt solution. This process was repeated anumber of times over a period of about twoweeks. The final 0.5 molar concentration ofthe electrolyte was roughly determined byvisual comparison to the depth of color in a

set of columns of known concentra-tions of aqueous cupric sulfate at 1.0M, 0.5 M and 0.1 M, and another setcontaining aqueous nickel sulfate solu-tions.

The most problematic disturbanceto a cell-in-test came from vacillationsin the ambient temperature. To mini-mize this problem, the experimentalmethod and auxiliary equipment wereas follows.

During test, a cell was housed in awatertight canister and was main-tained for weeks at a time (14 weeks inone experiment) within a 76 liter tem-

perature-controlled water tank. The tank was set on a wood-en table, and the tank and table together were insulated withpolyester batting 5 cm thick with a large dead-air gap(around 10 cm) between the tank and batting, and then thebatting was covered with a layer of aluminum coated Mylarfilm (BoPET film).

The canister was made of heavy-walled polyvinylchloride(PVC, 16.8 cm outside diameter, 14.4 cm internal diameter,15 cm length). It was fitted with a copper jacket (14.5 cmlengths of 1.3 cm diameter copper cable) lining the insidewall to help stabilize the cell against small temperaturechanges and gradients imposed from the water bath.Additionally, copper strips 2.5 cm wide by 1 mm thick weremounted close to the cell to further aid thermal uniformityand stability (Figure 4).

The tank’s water was stirred by a propeller driven at about48 revolutions per minute by an electric motor that was free-standing and connected to an acrylic propeller shaft with arubber coupling (to minimize conductance of heat andmechanical vibration). The water was heated by four electricheaters (50 ohm, 10 watt resistors) that had been sealedinside brass tubing. A 104 ohm thermistor that constitutedthe fourth leg of a Wheatstone bridge was used as the tem-perature probe, and a reference voltage to the bridge wassupplied by a precision 9 V dc voltage regulator. The bridgevoltage was sensed by a low-drift preamplifier that exportedan analog signal to a Fuji proportional-integral-derivative

Figure 3. 7.5 cm and 2.5 cm cells withbasic materials: a Pyrex tube, a Teflonsealing washer and a finished elec-trode cut from gold ribbon.

Figure 4. A 7.5 cm cell mounted into the canister.


temperature controller (model PXG4) that controlled thecurrent to the heaters. Using this control system, theabsolute temperature of the water (around 22°C) varied lessthan 0.002°C from setpoint for the great majority of time inany test period. The heaters typically controlled with lessthan 5 watts. The temperature drift of the control systemand the heat input from the stirring are unknown.

Each cell was monitored with a Keithley 2182-A nano-voltmeter with an accuracy around ± 0.02 μV. The inputimpedance of this meter is specified at greater than 1010

ohm, and so the cell potential was monitored in very nearlyan open-circuit state. Two thin copper wires (about 0.13 mmin diameter) connected a cell’s electrodes to the meter, andwere installed inside copper tubing (about 3 mm internaldiameter) with about 1.5 m of the tubing coiled andimmersed in the water tank to help thermally isolate the cellfrom the outside air temperature and to bring the wires tothe bath temperature.

Even with the aforementioned precautions against tem-perature intrusions from the outside air to the cell, it seemedthat a cell in test could still be affected by room-air temper-ature vacillations that were more than about 2°C from thetank temperature. The room’s daily temperature vacillationswere reduced by installing a thermostat-controlled coolerand heater that kept the room air within 2°C cooler than thetank temperature. Future set-ups should include at least 3 to

5 meters of connection wire (withinthe copper tubing) submerged in thewater tank. Inverting a cell wouldmechanically stress the metal elec-trodes and cause a temporary volt-age of not more than 0.04 μV (usingthe 7.5 cm cell). However, these spu-rious metal-stress voltages were briefand very low level, and wereinsignificant to the test results.

A small strip of Teflon (“S”-shapeTeflon strip about 2 mm wide by 10mm length, in Figure 5) and an inertgas float (the small air bubble from

the fill neck; about 2 mm diameter) were included in the cellcolumns in two tests, and just the Teflon strip in the other.These were included because during inversion they wouldstir the electrolyte and mechanically clear contaminationfrom the electrode surfaces. This surface contamination thatI suspected of interfering with the cells was similar to thecontamination problem reported by Ray et al. After eachinversion, a cell would again rest undisturbed for as long as22 days until the next inversion.

3.2 — Cell Voltage Driven by EgMeasuring the magnitude and polarity of cell voltage inresponse to the solute’s gravitational potential energy (Eg)was accomplished in test of a 7.5 cm cell (Figure 6). This cellhad been built and sealed (and used in various experiments)about 14 months before the start of the experiments report-ed here. In this test, the cell was inverted about 2-5 minutesafter each previous inversion. This rapid-inversion methodkept the fluid column reasonably well mixed since the cellhad a Teflon strip in the core to facilitate mixing of the elec-trolyte during each rotation of the device. Sharp voltagechanges (vertical spikes in Figure 6) were produced immedi-

ately upon each inversion (13 trials total), and indicate thesolute’s gravitational potential. The average magnitude ofthese voltage spikes was −0.83 μV, and divided by the 7.5 cmheight of the cell implies a value of −0.11 μV/cm with thebottom electrode having a negative-going polarity with eachinversion. The calculated expectation (using parametersfrom Table 1) for the voltage produced from the solute’sgravitational potential is −0.08 μV/cm.

Measuring the cell voltage driven by Eg (Figure 6) wasespecially important because both Eg and the voltage itdrives are constant parameters that are identical in both thestandard and inclusive models. The test provided importantresults:

1) an experimentally-derived value (−0.11 μV/cm) to com-pare against the expected value using tabulated parameters(−0.08 μV/cm);2) confidence in the calibration of the meter and its useful-ness in measuring a cell’s voltage throughout the experi-ments;3) confidence in the experimental design, equipment,method of test and the value of the acquired test results.

In Figure 6, the gradual voltage changes that followedeach spike reflected the gradual formation of the concentra-tion gradient with the bottom electrode having a positive-going polarity. The cell voltage that developed from the con-centration gradient following the very last inversion contin-ued well past the expected equilibrium value or common off-set for the cell. This type of gradual voltage excursion match-es the observations reported by Grinnell and Koenig in theirgravity-cell studies: “The e.m.f. [voltage] of a cell in one of itsvertical positions was not constant but always showed aslight change with time.” Also, the voltage excursion appearsrelated to the “residual potential” reported by Tolman. Inthis particular experiment it remains uncertain whether thevoltage excursion would have eventually subsided andreturned to a steady state at or near the common offset.

3.3 — Cell Voltage Driven by Ec + EgResults from two long-term experiments are presented inFigures 7 and 8. Together they give concise evidence for thephysical magnitude and potential energy of the solute’s con-centration gradient within a fluid column at equilibrium inearth’s gravity.

Figure 6. Rapid inversions to measure the cell voltage driven by Eg:–0.11 μV/cm.

Figure 5. Teflon strip in acell.


The first experiment tested a 2.5 cm cell over a 14½ weekperiod (Figure 7), while the second tested the taller 7.5 cmcell over 5½ weeks (Figure 8). Both cells had a Teflon stripand a small gas float in the core as described earlier. In bothexperiments, the movements of the Teflon strip and, pre-dominantly, the float caused electrokinetic effects12 (distur-bances of the double-layer interface between the solutionand the Pyrex cell wall) that resulted in the spurious voltagespikes at each inversion. In Figures 7 and 8, the top chartgives a long-term view of each test record, while the bottomchart gives a detailed comparison (an overlay) of two voltagetrajectories in each of the long-term experiments.

3.4 — Key Experimental FindingsVoltage-Polarity Inversion with Cell Inversion: The overlaidtraces reveal that the long-term cell potentials did not con-verge. When a cell’s working electrode was at rest at the bot-tom position (labeled “(a)”) then each trace indicated a morepositive steady-state cell voltage than when the cell wasinverted so that the working electrode was at rest at the topposition (labeled “(b)”). In other words, the cells’ voltages“flipped” or inverted in each trial (each cell inversion), andthe electrode brought to rest at the bottom of the columnalways came to a more positive electrical potential than theelectrode at the top.

Magnitude of Steady-State Voltages: The 2.5 cm cell devel-oped a separation of 0.8 µV between the working electrodeat rest in the top position against being at rest at the bottom

position. This is a difference of 0.4 µV from a common mid-point (offset), and divided by the 2.5 centimeter height ofthe cell implies a factor of about +0.16 µV/cm from the off-set. The taller cell developed a separation of about 2.8 µVbetween the inverted positions, or about 1.4 µV from the off-set, and divided by 7.5 cm implies a factor of about +0.19µV/cm.

Comparing Test Results to the Two Models: The observedsteady-state cell potentials compare well with the expectedrange based on the inclusive model: +0.13 to +0.18 µV/cm.The cells’ voltages were not consistent with the standardmodel which predicted zero μV/cm. (Detailed calculationsare in Section 4.)

Other Considerations: The time constants for the relaxationof the concentration profiles (L2/D, where L is the length ofthe cell in centimeters and D ≈ 10-5 cm2/s) are about 7 daysfor the 2.5 cm cell and 65 days for the taller 7.5 cm cell. Theshorter cell appears to have roughly followed the expectedrelaxation period, whereas the taller cell’s gradients appearto have decayed more rapidly. Although the movements ofthe Teflon strip and gas float caused the immediate electro-kinetic spikes with each inversion, when the objects werestationary they were apparently inconsequential to the long-term potentials.

Key Significance: After days or weeks with a cell sittingundisturbed in one or the other vertical positions, the volt-ages stopped changing and came to relatively static values—but they did not converge. The steady-state traces indicate

Figure 8. 5½ week test of a 7.5 cm cell. Top: Record of test duringdays 21–40. Bottom: Overlay of voltage traces from inversions onday 26 and day 30. Trace (a): the cell’s working electrode at the bot-tom position (starting on day 26). Trace (b): the cell is inverted so thatthe working electrode is at the top (starting on day 30). The 2.8 µVseparation between the inverted positions implies +0.19 µV/cm froma common midpoint or offset.

Figure 7. 14½ week test of a 2.5 cm cell. Top: Record of test duringdays 23–103. Bottom: Overlay of voltage traces following inversionson day 63 and day 83. Trace (a): the cell’s working electrode at thebottom position (starting on day 63). Trace (b): the cell is inverted sothat the working electrode is at the top (starting on day 83). The 0.8µV separation between the inverted positions implies +0.16 µV/cmfrom a common midpoint or offset.


that Ec was greater than Eg since, in every trial, the bottomelectrode always came to a more positive electrical potentialthan the top electrode. According to the standard model thetraces should have converged to a common offset with noseparation between them since, in the standard model, Ecand Eg are equivalent in magnitude. However, the traces—their polarities and magnitudes of separation—were inaccord with calculated expectations based on the inclusivemodel: the fluid columns attained equilibrium and, concur-rently, the external electrode circuit did not.

4. Comparing Calculations to Test ResultsThe objectives in this section are twofold: (1) to derivenumerical values for cell voltages driven by Ec and Eg basedon the standard model in Equations 1.6, and on the inclu-sive model in Equations 13.1, and then (2) to compare thosecalculated expectations to the experimental results.

The molar gravitational potential energy of the solute insolution (Eg) is the basic parameter since it is identical inboth models. We calculate Eg using Guggenheim’s formula(Equation 1.5) with values for the parameters supplied fromTable 1.

4.1 — Calculating Eg

Eg = (Mi –V–i

M0) (θα − θβ)/m =V–0

(0.3kg

–78

(0.018 kg

))(9.8 m2

)/mmol 18 mol s2

= (0.3kg

– 4.3 (0.018 kg

))(9.8 m2

)/m (16)mol mol s2

= (0.22kg )(9.8

m2)/m

mol s2

Eg = (2.18

Jmol)

m

Expressed in centimeters (of vertical separation between thetop and bottom of the fluid column), the molar gravitation-al potential energy of the auric chloride solute within thefluid column at rest within earth’s gravity is:

Eg = 0.0218 J

. (17)mol cm

4.2 — Expectation for the Cell Voltage Driven by EgThe general formula for the open-circuit steady-state cellpotential is:

UEc,g=

−(Ec + Eg). (18)

nF

The expected cell potential UEgthat is developed in

response to only Eg (presuming the fluid column is wellmixed so that there is no concentration gradient and so Ec =0) is the same in both the standard and inclusive models:

UEg=

−Eg. (18.1)

nF

Inserting the value for Eg from Equation 17 into Equation18.1, and using n = 3 (for the Au+3 ion) and F = 96,487 C/mol(where F is one Faraday and C is one Coulomb), then:

UEg=

−(0.0218 J

mol)/cm

3 x 96,487 C/mol

UEg= −7.5 x 10-8 V/cm

UEg= −7.5 x 10-2 µV/cm

Rounding off the value gives the calculated expectation(using tabulated parameters) for the cell voltage in responseto just the gravitational potential energy of the solute with-in a fluid column one centimeter tall:

UEg= −0.08 µV/cm (tabulated expectation). (18.2)

The experimental value found in test (Figure 6) was:

UEg= −0.11 µV/cm (experimental value). (18.3)

The experimental value was about 38% greater than thetabulated value. I presume that the “true” cell voltage devel-oped in response to the gravitational potential energy of thesolute in solution would likely range between the tabulatedand experimental values:

−0.08 µV/cm and −0.11 µV/cm (range of UEg). (19)

4.3 — Expectations for the Steady-State Cell VoltageWhen the fluid column is at equilibrium and the value of Egis predetermined, the expectation for a cell’s steady-statevoltage in Equation 18 will depend on the model used toanticipate Eg.

Standard model expectation: Since in the standard model Ec= −Eg then the steady-state cell voltage when the fluid col-umn is at equilibrium would be zero:

UEc,g=

−(Ec + Eg)=

−(−Eg + Eg) = 0 µV/cm . (20)

nF nF

Inclusive model expectation: In the inclusive model Ec = −2.6Eg when the fluid column is at equilibrium (from Equation13.2). In this case the steady-state cell voltage would be anonzero value:

UEc,g=

−(Ec + Eg)=

−(−2.6Eg + Eg) =

+1.6Eg . (21)

nF nF nF

After inserting the range of values for Eg/nF from Equation19 (where, from Equation 18.1, Eg/nF = −UEg

), the expectedsteady-state cell voltage according to the inclusive modelwould likely range between:

+0.13 µV/cm and +0.18 µV/cm. (22)

4.4 — Recalling the Tests ResultsExperimental findings for the steady-state cell voltages(Figures 7 and 8) were:

+0.16 µV/cm from the short cell (23)


and

+0.19 µV/cm from the tall cell. (24)

5. DiscussionThe test results (Equations 23 and 24) were entirely incon-sistent with expectations from the standard model inEquation 20. But the results were signature of the inclusivemodel in Equation 22. These two outcomes are discussed inmore detail in this section.

5.1 — Shortcomings5.1a: Theoretical: Equation 4 is consistent with the 1

√n orderof magnitude expected for the typical maximal density of afluctuated subvolume, but it is 20% greater thanSmoluchowski’s formula √ 2

π ∙ 1√n (Ref. 2, p. 134). This differ-

ence does not radically alter the magnitude of the calculatedexpectations based on the inclusive model, and the matter isnot further examined here.

5.1b: Experimental: The inversion voltages (Equation 19)were obscured by the spurious electrokinetic voltages duringinversion of the cells. It would have been helpful to haveobserved those reference signals in each trial, and to havehad a greater number of experiments and control tests.Unfortunately, the cells are not useful in testing other elec-trolytes (for control tests) because the gold electrodes are notreactive to most other metal ions in this application. And tri-als using the same cells but with aqueous sodium chlorideelectrolytes demonstrated the experimental barrier: Cellsusing aqueous NaCl electrolytes in contact with gold elec-trodes did not give sufficient conductance to reliably operatethe meter. The Johnson noise level (the limit to measure-ments due to erratic voltages produced by the thermal ener-gy of charged particles in the test circuit) due to the sourceresistance of the Au-NaCl-Au cells far exceeded the nanovoltresolution required to test them. It appears that differenttests using other cells are necessary (there are numerous vari-eties that can be tested).

Although the experimental results (in Equations 23 and24) were in good accord (within 5% to 10%) with the expec-tation of +0.18 µV/cm, they were 23% - 46% greater than theexpectation using the tabulated parameters (+0.13 µV/cm).The elevated values may indicate secondary influences suchas ionic clusters in solution (for example, AuCl4-1) whichcould have influenced the test results but were not includedin the calculations. They may also indicate other influences,perhaps minor effects from equilibrium thermal fluctuation.

The experiments were delicate in that the observed volt-ages were susceptible to influence from low-level extraneoussources. For instance, deterioration of the electrolytes couldhave created subvolumes with anomalous densities thatcould have mimicked those of equilibrium fluctuation. Thecells were tested about 14 months after they had been assem-bled and sealed, and so extraneous chemical reactions wouldlikely have been exhausted after more than a year. But per-haps there were unrecognized ongoing reactions betweenthe electrolyte and, say, the Pyrex tube. Although this typeof extraneous influence is possible, if it was existent it wouldhave been very low level and highly improbable to have led

to proportional results in the different tests and also havebeen in close quantitative accord with the inclusive model.

Another possible source of experimental error is from thevery small electrical current (less than 0.5 µA) from themeter. However, it is unlikely that this current would meas-urably prejudice a cell, and coincidently match calculationsfrom theory, and then also produce voltages in close pro-portion to the height of the other cell. Did the Teflon stripor the gas float cause the long-term voltages? As with theother considerations, it is possible but unlikely. From theacquired evidence it is most likely that the objects were inert(as intended). For example, the relaxation of the gradients inthe shorter cell followed the conventional time constant andin the case of the taller cell was apparently more rapid,rebutting any significant impact from the Teflon strip andthe gas float to the long-term test results.

5.2 — The Value of the ExperimentsThe equilibrium criterion (from both models; Equations 1.6and 13.1) for a solute in the fluid column is dependent onlyon the equilibration of the “static” or steady-state potentialenergies. Diffusion, migration, convection, sedimentationand all other kinetic influences are effectively absent at ther-modynamic equilibrium for all components within themacroscopic system. In view of this and discussions above, itis not credible that superfluous influences were the source ofthe steady-state voltages; there is no tenable causal connec-tion between the steady-state voltages and spurious signalsor extraneous influences—or even the standard model.Conversely, the test results were categorically in accord withthe inclusive model.

The cells’ steady-state voltages were consistently repro-ducible. They were proportional from the tall cell to theshorter cell in separate tests within carefully controlled envi-ronments that maintained the fluid columns at, or extreme-ly close to, thermodynamic equilibrium over prolonged peri-ods. And the voltages met qualitative and quantitativeexpectations derived from the inclusive model.

6. Defining Thermodynamic EquilibriumThe inclusive model and experimental results are in accord.This is so for both the fluid columns and the composite grav-ity cells. These theoretical-experimental correspondences arethe cumulative and key technical results from this study.

6.1 — Counter Examples to the Inclusive ModelAside from the few 20th century studies discussed earlier, arethere additional known systems that may be supportingexamples of the inclusive model? Are there counter exam-ples—systems that are known to contradict the inclusivemodel? Apparently, there are only counter examples.However, the counter examples are not valid comparisonsbecause of one critical difference: Systems tested in thisstudy were carefully maintained in isolation within an envi-ronment free from an energy flux, i.e. at thermodynamicequilibrium.

Evidence: Earth’s atmosphere is an example of a familiarfluid system that seems to contradict the inclusive model.The vertical gradients in pressure and concentration of theprinciple atmospheric gases can be predicted from the stan-dard model’s equilibrium equation in Equation 1.4 (essen-


tially identical to other references13). The gradients definite-ly do not correspond to the inclusive model; they are not assteep as the inclusive model predicts.

However, earth’s atmosphere, without doubt, is not equil-ibrated (solar energy and biologically-driven chemicalchanges are examples of influences that preclude it). Yet thegas-particle gradients are undeniably consistent with numer-ical predictions from the standard model's equilibrium equa-tion. How can that be? (There are numerous similar systems,such as solutions in analytical centrifuges, which are count-er examples to the inclusive model. And they suffer the samethermodynamic shortcoming as earth’s atmosphere.Solutions in modern centrifuges are unlikely to be thermo-dynamically equilibrated for numerous practical reasons, butperhaps primarily due to the need for a remote measurementmethod, invariably optical, that projects intense electromag-netic radiation—typically, visible light—into the solutions.)

One might speculate: Perhaps when external energy influ-ences are eliminated and a system is brought ever closer tothe ideal of thermodynamic equilibrium, then measure-ments of its properties will converge, or conform with evergreater precision, to standard expectations. But the recenttests do not support this speculation. In the experimentscompleted here—experiments expressly dedicated to bring-ing a system to thermodynamic equilibrium—the conver-gence of the standard equation and actual measurementsdoes not occur. Is the problem with the measurements or theequation? The entirety of findings (the careful execution ofthe experiments, the test results, the theoretical-experimen-tal correspondences involving the inclusive model, and theproblems reported by the 20th century researchers) suggeststhe problem is with the equation. Indeed, it appears that thestandard model’s equilibrium equation completely fails, ofall places, at thermodynamic equilibrium.

Interpreting the Standard Equation: Results from this studysuggest that fluid systems such as earth’s atmosphere andsolutions in centrifuges subjected to thermal disturbancesare in a stationary nonequilibrium state. Those systemsmatch the standard model because they are neither physi-cally or theoretically close to the steady state of thermody-namic equilibrium and the standard model’s equilibrium cri-terion is not a definition of thermodynamic equilibrium—itis a statement of stationary nonequilibrium that is based onan incomplete accounting of the potential energies thatexist in gases and liquid solutions at true thermodynamicequilibrium within external fields.

6.2 — Definition of Thermodynamic EquilibriumThe distinguishing and critical experimental criterion in theexperiments here was the maintenance of the composite sys-tem (the complete electrochemical cell) as an isolated systemwithin environmental conditions highly conducive to asteady state of thermodynamic equilibrium for the fluid col-umn; to wit, an ideally quiescent and passive condition, freefrom energy and material flux with the external environ-ment, maintained for days or weeks at a time for a single test.

The Inclusive Equation: The theoretical-experimental corre-spondences developed in this study tell of the terminalsteady state of columns of liquid solute-solvent mixtures at,or extremely close to, perfect thermodynamic equilibriumwithin earth’s gravity. Under those conditions, solutions do,in fact, exhibit the impact from the gravitational potential

energies of concentration-fluctuated subvolumes. As real sys-tems approach close to the ideal of thermodynamic equilib-rium, then the inclusive equation obtains in practice:

Ec = − Eg

–Eg

–Eg

... –Eg

. 13/2 23/2 33/2 n3/2

Clearly, the standard statement, Ec = − Eg, is containedwithin the first term of the inclusive equation. A significantresult in this study is this insight: The standard model’sstatement is but a single entry—an incomplete and limitedcase—within a more general and inclusive formula.

7. SummarySpontaneous solute-concentration fluctuations are ubiqui-tous yet exotic solute species within liquid solutions, andthey participate in the balance of energy within solutions atrest within earth’s gravity. The fluctuations are not passiveobjects as previously thought—their gravitational potentialenergies have significant impact to the physics, chemistry,and thermodynamics of simple physical systems and com-posite devices. A crucial experimental practice towardsobserving those energies and their repercussions is in main-taining a system within environs that are highly conduciveto thermodynamic equilibrium.

7.1 — Four FindingsFirst Finding: When liquid solutions are maintained inextremely quiescent environments and so attain—orapproach very close to—the ideal of perfect thermodynamicequilibrium, then the gravitational potential energies ofsolute-concentration fluctuation contribute to the totalenergy of the macroscopic fluid system and to the equilibri-um criterion. The inclusive model in Equations 13.1 - 13.3correctly anticipates the equilibrium properties of the elec-trolyte columns in the experiments.

Second Finding: When an electrolyte column is at thermo-dynamic equilibrium and is a subsystem within a compositedevice (an electrochemical gravity cell in this study), thenthe composite device can be precluded from a steady state ofequilibrium. Equations 15 and 22 correctly anticipate thenonequilibrium steady state of the electrochemical cells'external electrode circuits in the experiments. The nonequi-librium condition prevails due to two factors: (1) the gravi-tational potential energies of the solute’s concentration fluc-tuations add to the solute chemical potential Ec but not tothe solute’s gravitational potential Eg, and (2) a cell’s steady-state voltage is selectively driven by two of the solution’spotential energies: Ec and Eg. In the devices tested here, acell’s external electrode circuit remained in a nonequilibri-um steady state because Ec is 2.6 times greater than Eg.

Third Finding: The standard model’s equilibrium equationfailed to anticipate the steady-state properties of the fluidcolumns and the gravity cells because it is incomplete in itsaccounting of the potential energies within the fluidcolumns in gravity; it does not include the field-potentialenergies of the solute-concentration fluctuations except forthe single particle. Not only is the standard equation incom-plete in its accounting, it is also a misnomer; it is not a cri-terion or definition of thermodynamic equilibrium, but isinstead a description of a stationary nonequilibrium state


that is frequently observed in fluid systems.Fourth Finding: Whereas it is true that the second law of

thermodynamics is a profound and far-reaching statementabout our natural world, these studies indicate that the sec-ond law’s range is not universal; tangible exceptions exist insimple composite systems. Moreover, it appears that the sec-ond law is but one thermodynamic possibility within abroader and more general principle—alluded to by the inclu-sive model developed here—that is yet to be discovered.

The second law of thermodynamics can be generalized inthe following: Over time, energy spontaneously disperses fromhot to cold and from high potential to low toward an energy-equil-ibrated steady state. This is consistent with textbook state-ments of the second law that ply the concept of entropy, buthere the concept of energy is the key and useful quantity.

The original quantitative statement for the second lawwas derived by Clausius almost 150 years ago, and was set interms of entropy S such that: ∆S ≥ 0. Clausius’ equation con-tains two distinct sub-statements: ∆S = 0 refers to the ener-gy-equilibrated instance of the second law; ∆S > 0 refers tothe energy-dispersive instance. The “second finding” in thisstudy asserts thermodynamic processes that are exceptions tothe energy-equilibrated statement of the second law of ther-modynamics: rudimentary devices exist for which a steadystate of thermodynamic equilibrium does not obtain—infact, is precluded—when the devices are maintained in envi-ronmental conditions conducive to equilibrium.

The gravity cells’ steady-state voltages exist because a cell’sfluid column is thermodynamically equilibrated. And thevoltages indicate that the net energy available (1.6 Eg = 4.5J∙mol-1∙m-1) from a cell exceeds the energy to return the elec-tro-deposited gold metal from the bottom electrode back tothe top (1.9 J∙mol-1∙m-1), and so indefinitely continue thegravity-cell’s nonequilibrium steady state—at least ideally.(Other cell configurations, although not discussed here,appear plausible in which the steady-state voltage could occurwithout need to invert the electrodes, such as a gravity cellAu-NaOH/O2-Au where molecular oxygen is the key solute.)

7.2 — Future WorkThis research has been a first look into the scientific princi-ples and ramifications involving the coupling of solute-con-centration fluctuation to an external field. Commercially use-ful devices appear unlikely since the cells’ energies areextremely small and apparently require more material andcapital investment than any actual device could recompense.However, the general concepts are not limited to only gravi-tational or centrifugal fields, and tests of electrochemical cellscoupled to static magnetic-field gradients suggest the possi-bility of viable technological application. Investigations intothat topic are reported in Part 2 (Systems in a Magnetic Field)of this article, scheduled for the next issue of Infinite Energy.

In the magnetic-cell studies, the energy required to invertthe orientation of the electrodes to the applied field isinsignificant to a cell’s energy balance. Importantly, thestudies suggest the aim of future efforts toward devices capa-ble of demonstrating the non-universality of the energy-dis-persive instance of the second law—toward powerful devicesin which spontaneous decreases in a cell’s thermal energyand temperature are demonstrably commensurate with itsexport of electrical energy. This conversion of a cell’s latentthermal energy to electrical energy and the concurrent

decrease in the cell’s internal temperature are thermody-namic and performance characteristics that were simply tooweak to have been measured in the gravity-cell studies. Theyare worthwhile pursuits as they are important toward pro-viding evidence of the inclusive model’s practical utility anda greater understanding of thermodynamics beyond theconstraints of the second law.

AcknowledgementsMy gratitude to J.Q. Lee for financial assistance, and to bothLee and J.S. Newman for their invaluable insights that tippedthe balance and brought the experiments to life.

References1. Tolman, R.C. 1979. The Principles of Statistical Mechanics, Dover,§141, 629-649 (originally published in 1938 by Oxford Univ. Press).2. Perrin, J.B. 1916. Atoms, Translation by D.L. Hammick, Constable &Co., Chapter V, Fluctuations, 134-143, http://openlibrary.org/books/OL7164248M/Atoms.3. Kondepudi, D. and Prigogine, I. 1999. Modern Thermodynamics,Wiley, 303.4. Carey, Van P. 1999. Statistical Thermodynamics and MicroscaleThermophysics, Cambridge Univ. Press, 133-134.5. Guggenheim, E.A. 1967. Thermodynamics, Fifth Edition, North-Holland Publishing, 327-331. And in his first text: Guggenheim, E.A.1933. Modern Thermodynamics by the Methods of Willard Gibbs,Methuen & Co., Chapter XI, Gravitational Field, 153-159.6. Hill, T.L. 1986. An Introduction to Statistical Thermodynamics, Dover, 37.7. Alder, H.L. and Roessler, E.B. 1964. Introduction to Probability andStatistics, Third Edition, W.H. Freeman and Co., 100. The 1/√n orderof magnitude appears in the ratio of σ–X, the standard deviation of themean of a variable X, to the standard deviation of the distribution σsuch that: σ–X = σ/√n.8. Tolman, R.C. 1911. “The Electromotive Force Produced in SolutionsBy Centrifugal Action,” J. Am. Chem. Soc., 33, 121.9. MacInnes, D.A. and Ray, B.R. 1949. “The Effect of Centrifugal Forceon Galvanic Potentials,” J. Am. Chem. Soc., 71, 2987.10. Ray, B.R., Beeson, D.M. and Crandall, H.F. 1958. “CentrifugalElectromotive Force,” J. Am. Chem. Soc., 80, 1029.11. Grinnell, S. and Koenig, F.O. 1942. “The Determination of theTransference Numbers of Potassium Iodide from the ElectromotiveForce of Iodide-Iodine Gravity Cells,” J. Am. Chem. Soc., 64, 682.12. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. 2007. TransportPhenomena, Rev. Second Edition, John Wiley & Sons, 782.13. Resnick, R. and Halliday, D. 1966. Physics, Part I, John Wiley &Sons, 426-429 and 598.

About the AuthorNorman Borsuk is a journeyman sta-tionary engineer and an amateur physi-cist who has had an 18-year-long inter-est in the physics connecting solute-concentration fluctuation, gravity andequilibrium. He was a student at theUniversity of California at Santa Cruzfor three years, majoring in physics. Hisebook, Testing the Second Law, can be found at online pub-lishers Smashwords.com and within the online libraries ofmost manufacturers of ebook reading devices. Grants areimportant toward continuing these studies. To support thework, contact the New Energy Foundation or the author.

*Email: [email protected]

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N.K.Borsuk manuscript: Testing the Definition of Thermodynamic Equilibrium - part 1

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