Irreversible Thermodynamics Applied to Electrolyte Transport

J. Non-Equilib. Thermodyn.Vol. 4 (1979), pages 75-92

Irreversible Thermodynamics Applied to Electrolyte Transport

S. K. RatkjeLaboratory of Physical Chemistry, Norwegian Institute of Technology, University of Trondheim,Trondheim, Norway

Received 27 September 1977Registration Number 75Key Number 23 01 127

AbstractElectrolyte transport processes are described by means of the principles of irreversiblethermodynamics. It is shown for a given process that the physical interpretations ofthe transport parameters based on descriptions using electrochemical potentials, aredifferent from those obtained by a method using only thermodynamic variables whichare operationally defined [1]. This suggests that the method of Ftfrland, Ftfrland andRatkje [ 1 ] which is further developed in this paper, may give interpretations whichare valuable alternatives to the ones obtained from the traditional ionic description.

Introduction

When a description of isothermal electrolyte transport processes has been sought byapplication of irreversible thermodynamics, most frequently the gradients of theelectrochemical potentials have been considered as the basic forces giving rise tofluxes of charge and mass [2—6]. The electro-chemical potential is, however, notoperationally defined in thermodynamics. A thermodynamic state variable cannotbe changed by varying the amount of one ion only in the system. According to Cole-man and Truesdell [7] the choice of electrochemical potential gradients as forces ina system therefore has a special bearing in irreversible thermodynamics. A shortrepetition of their work will be given.The fundamental property of a transport system, as described by irreversible thermo-dynamics, is the entropy production dS/dt which is given by

The fluxes, Jj, are the rate of change of n parameters ai5 which give a necessary andsufficient thermodynamic description of the system:

(2)

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76 S. K. Ratkje

The conjugate forces, Xis are defined by > t

U- l , -n i*i- (3)

The fluxes are linearly independent according to the definition of a£. Each flux isassumed to be a linear, homogeneous function of all the forces:

J i= Ó L i kXk , i = l , . . . n . (4)k=L

According to Onsager [8], the phenomenological coefficients Lik obey the symmetryrelation

Lik = Lki , i, k = 1 ... ç , (5)

provided eqs. (1—4) are valid. It is important to note that eqs. (1) and (4) alone donot imply that eq. (5) is valid, as pointed out by Coleman and Truesdell [7]. Whenthe electrochemical potential gradients are chosen,as the basic forces giving rise toelectrolyte transport, eqs. (2) and (3) are not valid because ionic masses cannot beused as the parameters aie

Thus, by keeping strictly to the statistical mechanical analysis by Onsager, thereciprocal relationship (5) may not hold for the ionic description [7].From experiments it is, however, known that the Onsager relationship is fulfilledfor the ionic description [4]. This fact may, however, be regarded as a pure mathe-matical consequence of the transformation properties of two parameter sets [7].By choosing a parameter set containing the neutral components of the system andthe charge, eqs. (2) and (3) and thus (5) are valid. This set may be transformed intoa set containing ionic components through a similarity transformation, and thereforeeq. (5) is also valid for a description using the ionic components [7].We regard the first of these parameter sets as a fundamental set because it fulfillseqs. (2) and (3). Therefore we prefer to use these variables in a first description oftransport systems and only this description is called rigorous by us. One purposeof this paper has been to present further aspects of the rigorous description, whichwas first presented by Fprland, Fprland and Ratkje [ 1 ]. This will be done in the twosubsequent parts.Attention has not been focused on the physical interpretation of transport experi-ments in terms of the basic variables before and it is also the purpose of this paperto do so. By examples from electrolyte transport theory it will be shown in thethird section, that such interpretations may be different from those obtained fromthe ionic description. This stems from the fact that the variables in the neutral com-ponent description are directly related to experiments, while the ionic descriptionhas only indirect relations to experimental situations. It is hoped that the considera-tion of alternatives will provide different insights into the transport systems, and asa whole make possible a more reliable analysis.

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Irreversible thermodynamics of electrolyte transport 77

1. A rigorous derivation of fluxes and forces for an isothermal systemConsider the system illustrated in Fig. 1, consisting of two elements in contact witha heat reservoir. Charge is introduced into the system through a pair of electrodes,one in each compartment. An expression for dS, the total entropy change of thesystem is sought.

dQobs

ô,

(Ð-dnn

= T,

= dni2 ( 2 )

Fig. 1: The transfer of charge and mass in an isothermal system consisting of two separate parts.The electric potential difference ÄÖï1>8 is measured by inserting one electrode in eachcompartment.

When electrolytes are transported isothermally from one side to the other, thesystem as a whole is closed. The first law of thermodynamics is accordingly

dU = dq — pdV - Ä Ö<*5 dQ . (6)

U is the internal energy of the total system. The term pdV is the mechanical workdone by the system, while the last term is the electric work done in the outer circuitwhen a charge dQ is passing across the electric potential difference Ä Ö008 betweenthe two electrodes. The term dq is the heat received by the system, at temperature T.

Each local element with volume í of the system is open. The local entropy is subjectto the condition:

ds = Ydu + ^dv+ Ó dn.· (7)

The local internal energy is denoted by u, while ni} i = 1,... k, are the mass variablesaccording to the phase rule, including the components introduced'into the systemby the electrodes. It has only been emphasized before by F0rland et al. [ 1 ] that thenumber of components chosen to describe the system should conform with the phaserule. The choice is possible because local equilibrium is a prerequisite for each sub-volume of the system. One advantage is that a choice of independent fluxes will beensured. Negative components may arise in this connection, meaning that a compo-nent is removed from the system, but a change to components which are all positivecan always be performed [9]. In eq. (7) changes in dipole energies have been neglectedin accordance with de Groot and Mazur [10].

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78 S. K. Ratkje

The total entropy change can be described by the contributions from the twoelements and the heat reservoir at temperature T

dS = ds! + ds2 - γ . (8)

By inserting the thermodynamic identities

,3s . __lJ?8^ --K j - i k (9)(3çË.í,ç, ~ Ô (8çË'ô'ç; - ô , é - 1 ... k , W)

into (7), we obtain for dS from (8)

dS = ã (du! + du2) + ã (Pl dvt + p2 dv2) -

- Ó (Mudnu + M i a d n « ) - . (10)

The subdivision into two elements yields for the total energy change

dU = dUi +du2 . (11)

The change in U from eq. (6) is

dU = — pi dv! - p2 dv2 - Ä <i>obs dQ + dq . (12)

As the local electric potential Ö is not operationally defined, the traditional splittingof the electric work term into separate parts assigned to the volume elements (seee. g. [6]), has not been performed. The reference problem for the separate parts isthus avoided [11]. The mass leaving element 1 is received by element 2. Thus, wehave

dni2 = — drin , i = 1 ... k . (13)

Eqs. (1 1 — 13) are now inserted into (10), giving

1 é k

dS=-Â<I>o b sdQ-^ Ó AMidn . (14)1 1 i=l

The difference (ìß2 — ìç) has then been replaced by Ä μ{ . Eq. (14) gives the entropyproduced in the discontinuous system. In the isothermal case there is no change inS due to the heat flow.For practical purposes a continuous description of the transport processes is wanted,to make possible integration of concentration dependent quantities. Measurements




generally give integrated quantities. A continuous description may be obtained forhomogeneous systems by considering the region with concentration gradients of theexperimental system as built up by a row of subsystems, like the one given in Fig. 1 .Within each subsystem there is a very small concentration difference, and the fluxesconsidered are constant. Thus for each subsystem, the differences A<I>obsand Δμ{ ineq. (14) may be replaced by the gradients v<J>obs and íì£. The differential entropyproduction is

dt" Ô dt Ô dt ·

The electrodes of each subsystem are connected via potentiometers to the electrodesin the neighbouring compartments. The electric potential differences in the outercircuits are additive. As the set of fluxes in the different subsystems may have differ-ent values, no stationary state conditions are assumed. The integration over a widerange of concentrations may be carried out step by step by this analysis.The mass fluxes Jj can be defined as the mass fluxes in the volume element in whichthe gradients of chemical potential are equal to

Ji = -fa , i = 1 ... k . (16)

The positive charge flowing in the outer circuit, Q, can be varied freely by externalmeans. We define the current, I, by

dQ

The current inside the system is identical to I when no charge is assumed to accumu-late in the system. This is the same as assuming local electroneutrality. Still· accumula-tion of mass may occur. The forces conjugate to Ji are from eqs. (3, 15 and 16)

and the force conjugate to the electric current is similarly

= — - ycE>obsi ô ·

It should be noted that the fluxes of the electrode components are always uniquelyrelated to the current flux. Take as an example the electrode reversible to the chlorideion, Ag(s)lAgCl(s). The components introduced into the system by the electrode areAg(s) and AgCl(s). Their fluxes are

JAg = I and JAgci^"1· (2°)



80 S. K. Ratkje

The relations (20) make possible an exclusion of the electrode components from thelast term of eq. (15). In the chosen example, (15) is rearranged to

ï -VMAg - VMAgC1) · É - E VMi · Ji , (21)

where n is equal to k minus the number of electrode components.The effect is that the observed electric potential gradient must be corrected for thegradients VÂg and íìÁâÏ1 to get the force conjugate to I, when the electrode com-ponents are excluded among the mass fluxes. The chemical potential gradients ofAg(s) and AgCl(s) are different from zero if a pressure gradient exists, giving thecorrected force for the chosen example:

... nn = - Ô íÖ = - Ô t7*0"* + (VA* - VA*CI) VP) · (22)

From now on the symbol íÖ will be used in the expression for the force conjugateto I. Fluxes of electrode components are then not regarded. According to Prigogine[12] one flux, Jn, can be chosen as the zero reference for the other fluxes at mechani-cal equilibrium. The reference component can thus be omitted from the ç components.We are then left with (n — 1) mass fluxes and one charge flux in a rigorous, continu-OUS description of the isothermal coupled transport in homogeneous systems:

n-lJi = - Ó Ly VMj - Lin 7Ö , i = 1, ... ç - 1 , (23)

n-lI = - Ó Lnj VMj - Lnn 7Ö . · (24)

j=i

The term LJJ is a phenomenological coefficient which fulfills the reciprocity relation(5), because the fluxes and forces are defined according to (1—4). The transportcoefficients Ly to be regarded as local variables of an element with volume dv.Eqs. (23—24) conveniently may be rearranged to [ 1 ]:

n-lJ i = — Ó 1̂ VMj + t i l , i= l , . . . n - l . . (25)

j = i

The coefficients lij are electrode independent because they can be obtained from(25) for 1 = 0. The relation between the diffusion coefficients ly and the phenomen-ological coefficients Lu is furthermore [ 1 ]:

- l . (26)

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Further, tA is the transference number of component i, defined by

Lini = 1 ... n— l . (27)

The frame of reference for ti is then the n'th component with flux Jn. If the n'thcomponent is the solvent, ti is the Hittorf transference number.A characteristic property of the phenomenological eqs. (23—24) is that the fluxesare independent variables for any value of I. Among the forces, only the gradientsin the chemical potentials of all the n components depend on each other throughthe Gibbs-Duhem's equation, but this does not affect the Onsager relationship, (5),[13]. All parameters involved are operationally defined in thermodynamics anda continuous description is obtained without specific assumptions. In equilibriumthermodynamics, structure models should not fail to describe the macroscopic thermo-dynamic variables. Parallel to this, we can say that transport models should not failto describe the parameters given in eqs. (23—24).

2. The electrode dependency of the phenomenological coefficientsThe phenomenological coefficients Ly in eqs. (23—24) are electrode dependent be-cause the electrode reactions affect the composition of the electrolytes. Thisdependency will be analysed in the subsequent section. The equations which willbe derived are valid for a given set of intensive functions in a local element.Consider as a specific example the isothermal transport of the electrolytes HC1 andNaCl in H2O in a volume element with the gradients íìÇÏ1 and íìÍ3(:é. The systemis illustrated in Fig. 2. There is no restriction on the mass transport. According tothe phase rule, the components of the electrolyte are the neutral components HC1,NaCl and H2O. The flux of H2O is chosen as the zero frame of reference.

ÄÖobs

Solution

HCl (Ci)— É êé**ç/Ã·«Ë

Solution

HCl

NaCI(C2)

Fig. 2: A liquid junction separating two solutions of HCl and NaCl of different composition. Theelectric potential difference ÄÖ008 over the system is defined when the electrodes arespecified.

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82 S. K. Ratkje

The gradients in eqs. (23—24) are between two electrodes which afe included in thesystem.Consider electrodes reversible to the anion, QT, or to the cation, H+. íÖ is ob-tained from measurements with anion reversible electrodes, while 7Ö' is obtainedwith cation reversible electrodes. For the first case eqs. (23—24) give:

JHCI

Î

LII L12 L13

L2i L22 1-23

Lai L32 1-33 íÖ(28)

When the electrode reactions involve the transfer of Cl , the flux of NaCl may bedefined as the flux of Na+. This will also be the case when the electrodes are revers-ible to H+. The flux of the component NaCl is thus unaffected by the electrodereactions. The current can be controlled externally. The only change in the fluxesdue to the change of electrodes is therefore: JHci ~* JHCI- For this reason HC1 willbe denoted "the key component", and is numbered as the first component. For H+

reversible electrodes we thus have:

JHCI

IÔ 'JL21

L32 L32 33 -íÖ'(29)

The relationship between Ly and Ly will now be developed. Because the diffusioncoefficients are electrode independent, we have

l i ,=ly U = 1 , 2 . (30)

The coefficient L33, which is the electrolytic conductivity of the system, must beequal to L33:

33

For components other than the key component, we have

tj = tj i = 2, ... ç — 1 ,

which in this example means that t2 = t2 or, from eq. (27) and (31), that

L23 = L23 .

Eqs. (30, 31) and (33) may be inserted into (26) yielding

(31)

(32)

(33)

(34)




and

(L21 L21) L33 — (L13 — L13) L23 (35)

Eqs. (31, 33, 34) imply that the submatrix:

L22 L23

L32 L33

is unchanged by an alternation of the electrodes, i. e.

Ly == LJJ , i, j = 2, 3 . (36)

The two different expressions for the current can be used to develop the relationshipbetween LH and Ljj. From eqs. (28) we have

= - L31 íìé — L32 - L33 (37)

From chemical considerations, [ 1 ] the electric potential gradients over the differentsets of electrodes are related by

(38)7Ö' = íÖ + 7ì! .

Eq. (38) is inserted into (29) yielding

= - (L'31 - - Lf VM - L'32 33 (39)

As the forces are independent variables, the coefficients of eqs. (37) and (39) shouldbe identical, yielding as a result (31), (33) and

(40)

Eq. (40) is rearranged and the Onsager relationship is used. Then we get

(41)

This will reduce eq. (35) to

-21 (42)

For i = j eqs. (26, 30) and (42) give the relation

13 (43)



84 S. K. Ratkje

When the positive charge carried per mole of the key component.» z, eqs. (36) and(41—43) may be generalized to

L y = L j j , U = 2, ...n, (44)z(Lu - Lii) = Lin , i = 2, ... ç , (45)

z(L n -L ' u )=L l n -4-Li n . (46)

Here, the Onsager reciprocity relationship again has been used.A consistency check will be made on the eqs. (44—46). A generalization of eq. (38)yields

' = íö + -1 íì! . (47)

This relation can be derived from (45) as follows. The electric potential gradientscan be expressed by one flux Ji which is not affected by the electrode reactions.Only m components are contributing to the current transport. We distinguish thesecomponents from the rest, and obtain for the electric potential gradients:

m L-· n~l l!·· 1íÖ'=- Ó -f-VMj- _Ó -JJ-v/i, --V J,', i* l, (48)

j — 1 - ^ i n j — m + 1 -Lin -"-ºé•ºÐ

m ô ç-1L-- n-1 L·· i^íìß- Ó ^VMj-r^-Ji , i =£ 1 . (49)

j = l in j = m + l Ì ç Ìç

The last two terms in these equations are identical as may be shown from eqs.(32, 44). In the first terms, only the coefficients Lu and L^ make 7Ö' different

from íÖ. However, according to eq. (45) this difference is — L|n, which by insertionægives eq. (47), as it should.New information about the water transference number may be obtained from eq.(45). Consider for that purpose the system consisting of the four components HC1(1), NaCl (2), H2O (3) and the membrane component HM (4). For a more detaileddescription of this system it is referred to Fprland et al. [1 ]. The practical zero frameof reference will be the membrane component. The electric transport of water acrossthe membrane will be defined according to (27) by

, _ _^"M^Mi-o"!^· (50)

i= 1,2,3

L44 is the electrolytic conductivity of this system. By inserting (45) into (50) and byadding and subtracting L32 = L'32 in the numerator we obtain

1 i L ' 3 ' + L ' "t3=7^- + r^-1-44 ^44 •L44

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This expression can be rearranged to yield

1 3 = Ã! tir + r2 tNa* + r' tcr · (52)

Here we introduced the ionic transference numbers which are defined by [1 ],

tH+ = L14/L44 , tNa+ = L24/L44, tcl- = — (L14 + L24)/L44.

The coefficients in (52), r1? r2 and τ are defined by

ÃÉ = L13/L14, r2 = L23/L24, τ = (L'13 + L23)/(L'14 + L24) .

They are independent of the electrodes because the coefficients t3, tH+, tNa+, tcrhave this feature.The transference number of H2O is related to the transference numbers of the ionsas can be seen from eq. (52). If t3 is a linear function of tH+, tNa+ and tcl- , thecoefficients τ1 , r2 and r' may be taken as the constant number of water moleculescarried along with the respective ion.

3. Analysis of alternative flux equations

The Nernst-Planck Flux EquationsThe phenomenological coefficients Ly and Ly may be used to evaluate approxima-tions involved in the Nernst-Planck flux equations which are used frequently in theliterature [5].Consider first the simple transport system containing the components HC1 and H2O.We have for the chloride reversible electrodes in our formalism, eqs. (23—24):

JHCI = - LII VMHCI - L12 íÖ , (53)I = - L2l VMHC, - L22 íÖ . (54)

For the transport of HC1 referred to H2O the Nernst-Planck formalism gives [5],

JH+ = ~ £H V/ZH+ = - fiH VMH* , (55)

Ja- = - ci V/iqi- = ~ «a vAcr - (56)

Eqs. (55) and (56) imply that H+ and Cl" move independent of each other. Theelectrochemical potential Ai of an ion i is defined by

Mi =Mi +zri// , (57)

where μ{ is the chemical potential of the ion, and φ is the socalled local electricpotential of the ion in solution. When chloride reversible electrodes are used, wehave

íÖ = í ø — íìáé- , (58)



86 S. K. Ratkje

and similarly for hydrogen reversible electrodes , é

íÖ' = íø + VMH+ . (59)

Eq. (57) is valid for all values of φ. Hence we have

MHCI = MHCI = Air + MCI- Î ÌÇ+ + MCI- · (6°)

By inserting (57—60) into (55) and (56), we may rewrite these equations usingmeasurable quantities:

JH+ = — CH(^MHCI + v<^) ··· Chloride reversible electrodes, (61)Q- = — #CI(VMHCI ~~~ íÖ') ... Hydrogen reversible electrodes. (62)

A comparison of (53) and (61) yields that the diffusional mobility Ln is equal tothe electric mobility L12 . For hydrogen reversible electrodes we have from (62) andthe expression for JHC1 that L'n = — L'12 . However, it should be noted that the twoassumptions follow from each other because of (46). Therefore, the description isself consistent!When an additional salt is introduced into the system, the Nernst-Planck flux equa-tions read:

JH+ = - H VMH+ = - AH vMir , (63)W = - «Na V/W = - fiNa V/iNa+ , (64)Jcr = - «ci vAci- = ~ ci V^C1" · (65)

Inserting (57—60) we get

JH+ = ~ ^H(VMHCI + v*) ··· Chloride reversible electrodes, (66)

ÉÍ&+ = — ^Na(vMNaci + v*) ··· Chloride reversible electrodes, (67)

Jci- = — #ci (VMHCI ~ íö') ··- Hydrogen reversible electrodes. (68)

The assumptions introduced may be evaluated by comparing eqs. (66—68) with (28)and (29), using the relations JH+ = JHCI» ^Na* = ^Naci? JGI- =

L12 = Ï ... Independent movement of the two salts (69)The electric mobility of an ion is equal (70)to its diffusional mobility (71)

(72)L'12+1/22=0 (73)

It may be shown that the method is selfconsistent also in this case by using eq. (46).




The method based on coupled electrochemical potentialsThe Nernst-Planck flux equations are extended by use of irreversible thermodynamics[2—7]. The coupled transport of HC1 and NaCl referred to H2O then is described by

H+

Jd-

The electrical current is

(74)

1 = JH* cr (75)

The fluxes are related by (75). The forces used in (74) have no operational definitionaccording to (3), when 1 = 0. However, a possible operational definition may befound for the ionic forces when an electric current occurs [3]. With chloride revers-ible electrodes the fluxes are related by the matrix A defined by

JHCIJNaCl

I=

"l 00 11 1

or in brief

[J] = [A] [Jion] .

o"0

- 1 Jci-(76)

(77)

From this we infer, due to the fact that the entropy production (15) in both descrip-tions must be the same, that for the forces a relation must hold as follows:

[Xion] = [AT] [X] , (78)

or by use of (38):

V/iH+ = VMHC, + íÖ =+ íÖ = VMNaC1 + VMHCI +- íö = - VMHCI -

(79)(80)(81)

The ionic forces may be given their physical interpretations through the right handside of these equations when electrical energy is introduced into the system, e. g.when íÖ or íÖ' is defined. To make the values of V/Zj accessible, electrodes haveto be defined and introduced.It was shown in the first part of the paper that our rigorous description, eqs.(23—24), fulfill the symmetry relation (5). Ionic forces and fluxes cannot beobtained in agreement with eqs. (2—3), but they may be regarded as a linear combina-tion of our basic fluxes and forces. The reciprocal relation ë9 = λ^ will be valid



88 S. K. Ratkje

because the matrix [ë] of the coefficients ë^ mathemathically is related to thesymmetric matrix [L] of the coefficients Lij by the transformation, [7]:

[X] = [A][L][AT]. (82)

For the system described by (28) or (74), we have the explicit expressions of eq.(82) from Miller [4]. In our terminology we may write:

ëç = lu + tJr L33 , (83)ë» = Ii2 + tH* tNa+ L33 , (84)ëÀ3 = In + 1À2~ÚÇ + tci- L33 , (85)ë22 = 122 + tNa+ L33 »

ë23 = 121 + 122 ~~ tNa+ tcl- L33 >

X33 = lu + 2112 + 122 + tJi- L33 . (88)

The coefficients ëõ are electrode independent quantities as they can be expressedby combinations of the parameters lu , 112,122, L33 and the ionic transferencenumbers.The physical interpretation of the coefficients ë ·̂ will, however, depend on thedescription used. As an example, take the coefficient XH. It has been interpreted asan intrinsic ion mobility, i. e. the motion the ion would have if there was no inter-action with other ions [4]. From (83), (26) and (27) we have for ëç in our termi-nology

ëéé = Ln , (89)

meaning that ëð is the diffusional mobility of HC1 for V>NaC1 = 0 and íÖ'= Ï.For X22 and X23 the situation is analogous. In general, the coefficients ë^ cannot beexplained independent of the electrodes, when the electrochemical potential isdefined through a measurement involving electrodes. Thus, even if the magnitudeof Xjj is unchanged, it describes the mobility due to different types of forces fordifferent sets of electrodes.Miller, [4] observed a common limiting value for 0*ôÏÍ_0

= (ëû)ï for the same ion indifferent salt solutions, where Í is the normality of the solution. This has been takenas an evidence for XH describing the generalized mobility of one ion. In our descrip-tion the result may be regarded somewhat differently. At infinite dilution, there isno chemical coupling between the transport of HC1 and NaCl, giving L12 = 0, or inthe ionic terminology: the interaction between the fluxes of H+ and Na+ representedby X12 is zero, i. e. X12 = L12. This is reasonable in a common anion milieu. Whenthis condition is inserted into eqs. (85) and (87), we obtain:

ëÀ3 = LU - L13 , (90)ë23

= L22 — L23 . (91)

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A zero value for the right hand side of these equations corresponds to the Nernst-Einstein approximation for the components HC1 (90), and NaCl (91), (compare also(70) and (71)), which is valid at infinite dilution. This explains why ë13 —Ë23 = 0when L12 = 0. When the limit expressions of (84, 85, 87) are inserted into (83, 86,88) we find

ëð = ÚÇ* L33 , (92)ë22 = tNa

+ L33 , (93)X33 = tCi- L33 . (94)

These equations imply that in the limiting case Í -> 0, X may be interpreted as theelectric mobility of an ion. This is because electric mobility may be expressed by aconstant times the concentration, giving lim (ëð/Í)Í->0 a constant value from(92—94). Away from this limit, XH has contributions also from the diffusionalmobility of the i'th component. Thus a variation in XH may be explained by definiteand different contributions, in stead of a generalized mobility.Eqs. (83—85) and (26, 27) may be inserted into (74) to demonstrate a more importantdifference in the interpretations obtained from the ionic flux approach and ouranalysis. We obtain:

- t j L33(tH+ VMH+ + tNa+ VANa+ - tci- VAc,-) . (95)

The electrochemical potentials in (95) are expressed according to their definition(57), and after rearrangement of (74), it may be shown that the terms inside theparenthesis are equal to I/L33 (see Miller, [4] for details):

+ t3 íìáé- + Vi// . (96)

Inserting (96) into (95), the result obtained is identical to our eq. (25) which is notsurprising as the two descriptions are mutually transformable for É Φ 0. The differencein the interpretations of the result is that the ionic description, or eq. (96) impliesthe use of the concept "diffusion potential". It is defined for I = 0 as:

- FVi// = tH+ VMH+ + tNa+ íìÍ3+ - tci- VMCI- . (97)

The concept of the diffusion potential leads to physical interpretations in terms oflocal electric potentials created and maintained in the solution because of chargeseparation [4]. The emf ÄÖ01)ä of the system is interpreted as the sum of localpotentials created across the cell, (cp. (58, 59)). We describe the system withoutreference to electrical energy when the current is zero, cp. (6, 7). From (25) wehave vor 1 = 0:

(98)



90 S. K. Ratkje

The mass flux in (98) is due to differences in the chemical energy alone. The pointis that changes in dipole energies or energy due to charge separation in the systemdoes not contribute to the emf. in our approach, while this does not hold in theionic description.Thus, the entropy production given in eq. (15), may be explained in the ionic approachwith reference to local potential fluctuations, while our method neglects such contri-butions. An appropriate question in our opinion is then to ask for the physical signifi-cance of these local potentials.The emf A<J>obsof the system is proportional to the electric work done in the outercircuit. From (24) we infer that Ä Ö is equivalent to the chemical work plus resistanceloss in the cell. This equivalence gives physical interpretations of the transport proc-esses which focus on the work performed in the cell. Contributions to Ä Ö can bespecified for any volume element of the cell and we suggest that cell processes shouldalso be explained in this manner, not only by considering electrostatic interactions.For a similar analysis of ÄÖ and the possible interpretations given above, it is referredto Fprland and 0stvold [14—16]. They have given a rigorous analysis of the Donnanpotential [14—15] and the biological membrane potential [16]. In all cases they foundthat the consideration of local electric potentials in the solution leads to interpreta-tions which are different from those obtained by a rigorous analysis.

4. SummaryAlthough the traditional ionic description of electrolyte transport processes in irre-versible thermodynamics gives a consistent analysis of the transport, the methodimplies valid special physical concepts and quantities which are not accessibleexperimentally. It has been shown above that an alternative description and inter-pretation procedure without such implications is possible. The rigorous thermo-dynamic derivation of fluxes and forces requires that the electrodes are includedin the description. A unique dependency exists, however, between different setsof electrode dependent phenomenological coefficients.

AcknowledgementsValuable discussions with Prof. T. Ftfrland and financial support from NTH's Fondare greatly acknowledged.

Bibliography[ 1 ] F^rland, K. S., F0rland, T., Ratkje, S. K., The Coefficients for Isothermal Transport. 1. Ca-

tion Exchange Membrane and Electrodes Reversible to a Common Anion, Acta Chem. ScandA31 (1977), 47.

[2] Staverman, A. J., Non-Equilibrium Thermodynamics of Membrane Processes, Trans. FaradaySoc., 48(1952), 176

[3] Kedem, O., Katchalsky, A., Permeability of Composite Membranes, Part 1, Electric Current,Volume Flow and Flow of Solute through Membranes, Trans. Faraday Soc., 59 (1963)1918.

[4] Miller, D., Applications of Irreversible Thermodynamics to Electrolyte Solutions I and IIJ. Phys. Chem., 70 (1966), 2639, ibid. 71 (1967), 616.

[5] Meares, P., Thain, J. F., Dawson, D. G., Transport across ion-exchange resin membranes, inMembranes, vol. 1., Ch. 2., G. Eisenman, Ed., Marcel Dekker, New York, 1972.




[6] Katchalsky, A., Curran, P. F., Nonequilibrium Thermodynamics in Biophysics, 2nd ed.,p. 133, Harvard University Press, Cambridge, Mass., 1967.

[7] Coleman, B., Truesdell, C., On the Reciprocal Relation of Onsager, J. Chem. Phys., 33 (1960),28.

[8] Onsager, L., Reciprocal Relations in Irreversible Processes I and II, Phys. Rev. 37 (1931),405, ibid. 38(1931), 2265.

[9] F0rland, T. in "Fused Salts", p. 82, B. R. Sundheim, Ed., McGraw-Hill, New York, 1964.[10] De Groot, S. R., Mazur, P., Nonequilibrium Thermodynamics, p. 343, North-Holland Publ.,

Amsterdam, 1962.[11] Haase, R., Thermodynamics of Irreversible Processes, p. 164, Addison-Wesley, Reading,

1969.[12] Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, 3rd edition, p. 36,

Interscience, New York, 1968.[13] De Groot, S. R., Mazur, P., Nonequilibrium Thermodynamics, p. 65, North-Holland Publ.,

Amsterdam, 1962.[14] Ftfrland, T., Qstvold, T., The Donnan Potential I, Acta. Chem. Scand., A28 (1974), 607.[15] Ftfrland, T., 0stvold, T., The Donnan Potential II, Acta. Chem. Scand. 27 (1973), 2199.[16] F0rland, T., 0stvold, T., The Biological Membrane Potential: A Thermodynamic Approach,

J. Membrane Biol., 16 (1974), 101.

Assistant professor Dr. S. K. RatkjeLaboratory of Physical ChemistryNorwegian Institute of TechnologyUniversity of TrondheimN-7034 Trondheim-NTH



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Irreversible Thermodynamics Applied to Electrolyte Transport