[membrane science and technology] ion exchange membranes - fundamentals and applications volume 12...

Irreversible Thermodynamics

Chapter 5

5.1. PHENOMENOLOGICAL EQUATION AND

PHENOMENOLOGICAL COEFFICIENT

Many theoretical approaches to the transport phenomena in a solution arebased on the Nernst–Planck equation (Planck, 1890):

Ji ¼ �Ciuidmidx

(5.1)

where Ji is the ionic flux, Ci the ionic concentration, ui the ionic mobility and mithe electrochemical potential.

Equation (5.1) is equivalent to

J ¼ gX (5.2)

where X is the driving force and g is the proportionality constant.In the discussion of ionic fluxes in a solution dissolving two kinds of ions,

we have to pay attention to the fact that both ions influence each other even iftheir charges are the same or different. Further, we have to notice that the ionicfluxes affect solvent fluxes. These phenomena mean that the fluxes and drivingforces are not independent and they are coupled together. The Nernst–Planckequation does not concern with these mutual effects (Kimizuka, 1988). Theirreversible thermodynamics expressed these mutual effects by the followingphenomenological equation (Kedem and Katchalsky, 1961):

Ji ¼ Li1X 1 þ Li2X 2 þ � � � þ LinXn ¼Xnk¼1

LikXk ði ¼ 1; 2; . . . ; nÞ (5.3)

where Xi is the force, Ji the flux, Lik the phenomenological coefficient and i and k

are components. The matrix composed of the phenomenological coefficients issymmetrical as suggested by the following Onsager’s (1931) reciprocal theorem:

Lik ¼ Lki ði; k ¼ 1; 2; . . . ; nÞ (5.4)

Equation (5.3) is introduced on the assumption that the thermodynamic functionshold in a fine irreversible space. This suggestion is termed ‘‘the assumption ofpartial equilibrium’’ and holds more strictly in the circumstance of being more closeto equilibrium states. The actual electrodialysis system is not formed in the equili-brium process so that the irreversible thermodynamics exhibit only approximatedmeaning in the electrodialysis system. However, the irreversible thermodynamics is

DOI: 10.1016/S0927-5193(07)12005-2

dx.doi.org/10.1016/S0927-5193(07)12005-2.3d

Figure 5.1 Two-cell membrane system in which electrolyte solutions are partitioned byan ion exchange membrane.

Ion Exchange Membranes: Fundamentals and Applications68

considered to be applicable in the circumstances being apart to some extent fromequilibrium states (Dunlop, 1957; Dunlop and Gosting, 1959).

Fig. 5.1 shows a two-cell apparatus, in which electrolyte solutions arepartitioned by an ion exchange membrane and the following system is realized:

(1)
The ion exchange membrane is placed in a boundary between cell I andcell II and forms a continuous interface.
(2)
The total system is closed, and there is no energy transfer between theclosed system and the external one.
(3)
There is no appearance and disappearance of material caused by achemical reaction.
(4)
The system is isothermal.
In the circumstance described above, the entropy production s in the system isrepresented by the following equation (Staverman, 1952):

s ¼dS

dt¼

Xi

JiX i (5.5)

where Ji and Xi are the force and flux of component i indicated as:

Ji ¼ �dMI

i

dt¼

dMIIi

dt(5.6)

Xi ¼zi Dcþ vi DPþ Dmi

T¼

DmiT

(5.7)

where dMIi and dMII

i are the quantity changes of component i during dt in cellsI and II, respectively. Dmið¼ m0i � m00i Þ and Dmið¼ m0i � m00i Þ are an electrochemicalpotential difference and a chemical potential difference between the cells,respectively. DPð¼ P0 � P00Þ and Dcð¼ c0

� c00Þ are the pressure difference and

potential difference between the cells. zi and vi are an electric charge number and

Irreversible Thermodynamics 69

partial molar volume of component i, respectively. The following dissipationfunction is introduced from Eqs. (5.5)–(5.7):

Ts ¼Xi

Ji Dmi ¼Xi

Jiðzi Dcþ vi DPþ DmiÞ

¼ I Dcþ J DPþXi

Ji Dmi ð5:8Þ

where I, J and Ji are an electric current, volume flow of a solution and a massflux of component i, respectively.

The entropy change dS in a closed system is expressed by the followingCarnot–Clausius equation based on the second law of the thermodynamics:

dS ¼ deS þ d iS ¼dQ

Tþ d iS (5.9)

Here, dQ is quantity of heat supplied to the system from the surroundings anddeS the entropy change in the system caused by the heat supply indicating thereversible process. diS corresponds to the irreversible entropy change arising in thesystem and it is equivalent to the entropy production s defined by Eq. (5.5). TdiSis equivalent to the dissipation function Ts (Eq. (5.8)). Accordingly, Eq. (5.8)expresses that the entropy production in the system is arisen by the electric currentdue to electromotive force, the mechanical flow due to pressure difference and thediffusion due to chemical potential difference. Based on the irreversible thermo-dynamics, the phenomenological equation in the ion exchange membrane systemis introduced starting from (5.8) as follows (Kedem and Katchalsky, 1963):

I ¼ LE Dcþ LEP DPþXi

LEi Dmi (5.10)

J ¼ LPE Dcþ LP DPþXi

LPi Dmi (5.11)

Ji ¼ LiE Dcþ LiP DPþXi

Lik Dmi (5.12)

Further, the phenomenological equation is also introduced as follows fromEq. (5.8):

Ji ¼Xi

ðzkLik Dcþ vkLik DPþ Lik DmkÞ (5.13)

where zk and vk are an electric charge number and partial volume of component k.The phenomenological equation introduced above is equivalent to Eq. (5.3) and theOnsager’s reciprocal theorem is realized between each phenomenological coeffi-cient. In the above ion exchange membrane system including cations i+ and anionsi�, the coupling of ionic fluxes with fluxes presented in the phenomenologicalequation (Eqs. (5.10)–(5.13)) is depicted in the model of Fig. 5.2 (Kimizuka, 1988).

J i+Ji−

J

P

Figure 5.2 Fluxes, driving forces and membrane phenomena (Kimizuka, 1988, p. 162).


Further, the membrane phenomena are expressed as follows based on the phen-omenological equation (Staverman, 1952; Sakai and Seiyama, 1956; Yamabe andSeno, 1964):

(1)
Permeability(a) Electric conductivity (permeability)
Putting DP ¼ 0 and Dm ¼ 0 in Eqs. (5.10) and (5.13)

LE ¼I

Dc

� �DP¼0;Dm¼0

¼1

E

XziJi ¼

Xi

Xk

Likzizk

(5.14)

(b) Hydraulic (mechanical) permeabilityPutting Dc ¼ 0 and Dmk ¼ 0; in Eqs. (5.11) and (5.13)

LP ¼J

DP

� �Dc¼0;Dm¼0

¼1

P

XviJi ¼

Xi

Xk

Likvivk (5.15)


(c) Electric transport number of component iPutting DP ¼ 0 and Dm ¼ 0 in Eqs. (5.10) and (5.13)

ti ¼ziJiPziJi

� �DP¼0;Dm¼0

¼ziJi

I¼

ziPk

Likzk

LE

¼

ziPk

LikzkPi

Pk

Likzizk

(5.16)

(d) Mechanical (hydraulic) transport number of component iPutting Dc ¼ 0 and Dm ¼ 0 in Eqs. (5.11) and (5.13)

ti ¼viJiPviJi

� �Dc¼0;Dm¼0

¼viJi

J¼

viPk

Likvk

LP

¼

viPk

LikvkPi

Pk

Likvivk

(5.17)

(2)
Electrokinetic phenomenaThese are the phenomena appearing in a flowing solution under an
applied potential difference. When the concentrations on both sides of themembrane are the same (Dmi ¼ 0), the following equations are introducedfrom Eqs. (5.10) and (5.11):

I ¼ LE Dcþ LEP DP (5.18)

J ¼ LPE Dcþ LP DP (5.19)

From Eqs. (5.18) and (5.19), the electrokinetic phenomena areexpressed as follows:(a) Electroosmosis

J

I

� �DP¼0

¼LPE

LE

ð5:20Þ

(b) Streaming potential

DcDP

� �I¼0

¼ �LEP

LE

ð5:21Þ


Equation (5.20) shows a solution flux caused by an electric current,and Eq. (5.21) shows a potential difference caused by a pressuredifference. They are different at a first glance, but these equationsshow the same phenomenon from the opposite view points. Thissuggestion is demonstrated by the Onsager’s reciprocal theorem,LEP ¼ LPE, and we have:

DcDP

� �I¼0

¼ �J

I

� �DP¼0

¼ �

Pi

Pk

LikzivkPi

Pk

Likzizk(5.22)

(c) Electroosmotic pressure

DPDc

� �J¼0

¼ �LPE

LP

(5.23)

(d) Streaming current

I

J

� �DF¼0

¼LEP

LP

(5.24)

Equation (5.22) shows a pressure difference caused by a potentialdifference, and Eq. (5.23) shows an electric current caused by asolution flux. They show the same phenomenon because of thesimilar reason described above, so we have:

DPDc

� �J¼0

¼ �I

J

� �Dc¼0

¼ �

Pi

Pk

LikzivkPi

Pk

Likvivk(5.25)

(3)
Diffusion potentialPutting I ¼ 0 and DP ¼ 0 in Eq. (5.10) and expressing the diffusion
potential as Dc ¼ DcDiff,

I ¼ LE DcDiff þXi

EEi Dmi ¼ 0 (5.26)


Accordingly, taking account of Eq. (5.16)

DcDiff ¼ �1

LE

Xi

LEi Dmi ¼ �1

LE

Xi

Xk

Likzk Dmi

¼ �Xi

ti

zi

� �Dmi ð5:27Þ

Equation (5.27) is a general expression of the Nernst equation (cf. Eq.(3.37)).

5.2. REFLECTION COEFFICIENT

Assuming that an electrolyte solution is placed in cells I and II ðC0i ¼

C00i ¼ Ci;DCi ¼ 0Þ in Fig. 5.1 and that the solution in cell I is pressurized

through a piston, then the water and solute i in cell I are transferred toward cellII. The fluxes of water and solutes through the membrane in this system areexpressed by Jwater and Jsolute. If the membrane in this system does not permeatethe solute i at all (Jsolute ¼ 0) and permeates only water, the solute concentrationin cell II becomes C00

i � DC00i ¼ ðC00

i V00Þ=ðV 00 þ DV 00Þ: Here, DC00

i is solute con-centration decrease and DV00 the volume increase in cell II. On the other hand, ifJwater is equivalent to Jsolute, C

00i does not change ðDC

00i ¼ 0Þ: The permselectivity

of solutes against water in this system is defined by the following reflectioncoefficient si in the irreversible thermodynamics (Schultz, 1980):

si ¼ 1�Cfiltrate

i

Cfiltrandi

(5.28)

Here, we define the permselectivity coefficient of solutes against water T solutewater by

the following equation, which is commonly applied in electrodialysis:

T solutewater ¼

Jsolute=Jwater

C0solute=C

0water

(5.29)

The relationships between Jwater, Jsolute, si and T solutewater are shown as follows:

Jsolute ¼ Jwater; si ¼ 0; T solutewater ¼ 1

JsoluteoJwater; 0osio1; T solutewater40

Jsolute ¼ 0; si ¼ 1; T solutewater ¼ 0

(5.30)


In Fig. 5.1, the osmotic pressure p developed between cell I and cell II is rep-resented by the following Van’t Hoff equation:

p ¼ RTðC0i � C00

i Þ ¼ RT DCi (5.31)

Equation (5.31) corresponds to p at Jsolute ¼ 0, si ¼ 1 and T solutewater ¼ 0 in

Eq. (5.30). Staverman (1951) defined the effective osmotic pressure peff realizedin all of the situations in Eq. (5.30) as follows:

peff ¼ sip ¼ siRT DCi (5.32)

5.3. ELECTRODIALYSIS PHENOMENA

In this section, the flux of ionic electrolytes across the membrane is dis-cussed based on the approaches of Kedem and Katchalsky (1963). For sim-plicity, the ionic electrolytes are assumed to dissociate into monovalent cationsand monovalent anions, and the phenomena are explained as follows (House,1974; Schultz, 1980).

We assume a two-cell electrodialysis system presented in Fig. 5.3 consist-ing of cells I and II and a cation exchange membrane. The concentrations ofelectrolytes i in cell I are assumed to be adjusted to C0

þ for monovalent cationsand C0

� for monovalent anions, and those in cell II are adjusted to C00þ for

monovalent cations and C00� for monovalent anions. Reversible electrodes are

placed in both cells and an electric current is passed across the cation exchangemembrane through the electrodes and electric potentials on both sides of themembrane are set at c0 and c00; respectively. The pressures in both cells areregulated by a piston to be P0 and P00, respectively. The difference of electric

Figure 5.3 Two-cell electrodialysis system in which electrolyte solutions are partitionedby a cation exchange membrane.


potential Dc, electrolyte concentration DCi, pressure DP and electrolyte chem-ical potential Dmi between both cells are defined as:

Dc ¼ c00� c0

DCi ¼ C00i � C0

i

DP ¼ P00 � P0

Dmi ¼ m00i � m0i

(5.33)

The dissipation function in the tertiary system consisting of three components,three flows and three driving forces is represented by the following equation:

T diS

dt¼ Jþ Dmþ þ J� Dm� þ JW DmW (5.34)

Dmþ ¼ vþ DPþRT DCþ

C�þ

þ F Dc (5.35)

Dm� ¼ v� DPþRT DC�

C��

� F Dc (5.36)

Here, the subscripts +, � and W mean monovalent cations, monovalent anionsand water (solvent), respectively. Dmþ and Dm� are the electrochemical potentialdifference of+ions and � ions. vþ and v� are the partial molar volumes of theseions. C�

þ and C�� are the logarithmic mean concentrations Eq. 5.50 of these ions.

From the electric neutrality in this system, we have

C0þ ¼ C0

� ¼ C0i

C00þ ¼ C00

� ¼ C00i

(5.37)

From Eqs. (5.34)–(5.37), the chemical potential difference of electrolytes i, Dmi;is introduced as:

Dmi ¼ Dmþ þ Dm� ¼ vi DPþ2RT DCi

C�i

(5.38)

vi ¼ vþ þ v� (5.39)

If the electrode is reversible to anions, electromotive force E (volt) is expressedby the following equation indicating the definite relationship between E and Dc


(Katchalsky and Curran, 1965):

E ¼�Dm�F

¼ �RT

FDlnC� þ DF (5.40)

In this electrodialysis system, the transport number of a cation exchange mem-brane t+ integrated in the apparatus (Fig. 5.3) is expressed by the followingequation because the flux of cations J+ is equivalent to the flux of electrolytes Ji:

tþ ¼FJþ

I¼

FJi

I(5.41)

I ¼ F ðJþ � J�Þ (5.42)

Substituting Eqs. (5.38) and (5.40) into Eq. (5.34), we get:

T diS

dt¼ JWmW þ Ji Dmi þ IE (5.43)

From Eq. (5.43), the following phenomenological equations are introduced.

JW ¼ LWW DmW þ LWi Dmi þ LWIE (5.44)

Ji ¼ LiW Dmi þ Lii Dmi þ LiIE (5.45)

I ¼ LIW DmW þ LIi Dmi þ LIIE (5.46)

Equations (5.44)–(5.46) contain nine coefficients (L). However, according to theOnsager’s reciprocal theorem, LWi ¼ LiW, LWI ¼ LIW and LiI ¼ LIi, so the coeffi-cients reduce to six. Equation (5.44) indicates that JW consists of the hydrodynamicterm (LWW DmW), the osmotic pressure term (LWi Dmi) and the electroosmosis term(LWIE). Equation (5.45) indicates that Ji consists of ultra-filtration term (LiW DmW),diffusion term (Lii Dmi) and electrophoresis term (LiIE). Equation (5.46) indicatesthat I consists of streaming current term (LIW DmW), diffusion stream term (LIi Dmi)and electromotive force term (LIIE).

The phenomenological equation described above includes JW (Eq. (5.44))which is difficult to measure, so we try to change this parameter here to volumeflow JV applying DmW (Eq. (5.47)) and Dmi (Eq. (5.48)) to Eq. (5.43).

DmW ¼ vWðP00 � P0Þ � vWRTðC00i � C0

iÞ ¼ vWðDP� pÞ (5.47)

Dmi ¼ vi DPþRT DCi

C�i

¼ vi DPþpC�

i

(5.48)


where

Osmotic pressure; p ¼ RT DCi (5.49)

Logarithmic mean concentration; C�i ¼

DCi

lnðC00i =C

0iÞ

(5.50)

Consequently, Eq. (5.43) is converted to

T diS

dt¼ JVðDP� RT DCiÞ þ

Jið1� C�i viÞRT DCi

C�i

þ IE (5.51)

In a diluted electrolyte solution we have

C�i � 1 (5.52)

Taking account of Eq. (5.52) in Eq. (5.51), the following dissipation function isintroduced:

T diS

dt¼ JVðDP� RT DCiÞ þ

JiRT DCi

C�i

þ IE (5.53)

Further, the following phenomenological equation is introduced from Eq.(5.53):

Ji ¼ LPð1� siÞC�i ðDP� RT DCiÞ þ oiRT DCi þ

tþI

F(5.54)

I ¼ bGðDP� RT DCiÞ þGtþRT DCi

FC�i

þ GE (5.55)

JV ¼ LP DP� siLPRT DCi þ bI (5.56)

We term the coefficients in Eqs. (5.54)–(5.56) as follows:

L is the hydraulic conductivity, s the reflection coefficient, b the electroosmotic
P i
permeability, oi the solute permeability, t+ the transport number and G theelectric conductance.

5.4. SEPARATION OF SALT AND WATER BY ELECTRODIALYSIS

A salt solution dissolving monovalent cations and monovalent anions isassumed to be partitioned by a cation exchange membrane in a two-compartmentelectrodialysis system. The concentrations of cations+and anions � in one side


compartment (cell I) and those in the other side of compartment (cell II) areassumed to be maintained to CI

þ ¼ CI� ¼ CI

S and CIIþ ¼ CII

� ¼ CIIS ; respectively.

Passing an electric current through reversible electrodes immersed in both com-partments across the membrane, cations in cell I are assumed to be transferredtoward cell II. In this electrodialysis system, the flux of electrolytes JS,K and thatof a solution JS,V transported from cell I to cell II across the cation exchangemembrane are expressed by the following phenomenological equations introducedfrom Eqs. (5.54) and (5.56):

JS;K ¼ LP;Kð1� sKÞC�SðDP� RT DCSÞ þ oKRT DCS þ

tþKi

F(5.57)

JV;K ¼ LP;K DP� sKLP;K DCS þ bKi (5.58)

where i is current density. DCSð¼ CIS � CII

S Þ and DP ( ¼ PI�PII) are the electro-

lyte concentration difference and the hydraulic pressure difference between cell Iand cell II, respectively. tþK is the transport number, oK the solute permeability, bKthe electroosmotic permeability, LP,K the hydraulic conductivity and sK the re-flection coefficient (Staverman, 1951). These are the characteristics of the cationexchange membrane incorporated in this system.

The basic principle of separation of salt and water by electrodialysis isexpressed by a three-compartment (cells I, II and III) electrodialysis system,which consists of a central cell (cell II) and electrode cells (cells I and II). Acation exchange membrane (K) is placed between cell I and cell II, and an anionexchange membrane is placed between cell II and cell III. Supplying an elec-trolyte solution into cells I and III, current density i is applied and an electrolytesolution being collected in cell II is taken out by an overflow extracting system,until the electrolyte concentration in cell II reaches constant. The salt accumu-lation JS,K+JS,A and solution accumulation JV,K+JV,A in cell II are given bythe following equations introduced from Eqs. (5.57) and (5.58):

JS;K þ JS;A ¼ ðtK þ tA � 1Þi

F� RT ½ðoK þ oAÞ � fLP;Kð1� sKÞ

þ LP;Að1� sAÞC�S�DC ð5:59Þ

JV;K þ JV;A ¼ ðbK þ bAÞi þ RTðsKLP;K þ sALP;AÞDC (5.60)

Here, we put DC ¼ C00�C0 ¼ �DCS, tAþ þ tA� ¼ 1; tKþ ¼ tK and tA� ¼ tA; and ne-

glect pressure difference driving force DP: DP ¼ 0. The subscript and superscriptK and A mean a cation exchange membrane and an anion exchange membrane,respectively. The superscripts 0 and 00 mean desalting side (cells I and III) andconcentrating side (cell II), respectively. C�

S is the logarithmic mean concentration


as follows:

C�S ¼

C00 � C0

lnðC00=C0Þ(5.61)

Equations (5.57)–(5.61) represent electrodialysis phenomena in a solution dis-solving monovalent ions, but they are applicable to the phenomena in the solutiondissolving multivalent ions (cf. chapter 6).

REFERENCES

Dunlop, P. J., 1957, A study of interacting flows in diffusion of the system raffinose–KCl–H2O at 251C, J. Phys. Chem., 61, 994–1000.

Dunlop, P. J., Gosting, L. J., 1959, Use of diffusion and thermodynamic data to test theOnsager reciprocal relation for isothermal diffusion in the system NaCl–KCl–H2O at251C, J. Phys. Chem., 63, 86–93.

House, C. R., 1974, Water Transport in Cells and Tissues, Edward Arnold, London,England.

Katchalsky, A., Curran, P. F., 1965, Nonequilibrium Thermodynamics in Biophysics,Harvard Univ. Press, Cambridge, USA.

Kedem, O., Katchalsky, A., 1961, A physical interpretation of the phenomenologicalcoefficients of membrane permeability, J. Gen. Phys., 45, 143–179.

Kedem, O., Katchalsky, A., 1963, Permeability of composite membranes. Part 1. Electriccurrent, volume flow and flow of solute through membranes, Trans. Faraday Soc., 59,1918–1930.

Kimizuka, H., 1988, Membrane Transport of Ions, Kyoritu Shuppan Co., Tokyo, Japan.Onsager, L., 1931, Reciprocal relations in irreversible process, Phys. Rev., 37, 405–426.Planck, M., 1890, Ueber die potentialdifferenz zwischen zwei verdunnten losungen

binarer electrolyte, Ann. Der Physik u Chemie, 40, 561–576.Sakai, W., Seiyama, T., 1956, Electrochemical studies on ion exchanger. Application of

non-equilibrium thermodynamics to ionic membrane system, J. Electr. Chem. Jpn., 24,274–279.

Schultz, S. G., 1980, Basic Principles of Membrane Transport, Cambridge UniversityPress, Cambridge, England.

Staverman, A. J., 1951, The theory of measurement of osmotic pressure, Rec. Trav.Chim., 70, 344–352.

Staverman, A. J., 1952, Non-equilibrium thermodynamics of membrane process, Trans.Faraday Soc., 48, 176–185.

Yamabe, T., Seno, M., 1964, Ion-Exchange Resin Membrane, Giho Do Co., Tokyo,Japan.

[membrane science and technology] ion exchange membranes - fundamentals and applications volume 12...

Documents