Membrane Transport

CHAPTER

MEMBRANE TRANSPORT

10

10.1 INTRODUCTION The conventional approach to analyzing membrane transport is based on the specific models of the classical laws of electrostatics (Poisson’s equation) and hydrodynamics (NaviereStokes equations) combined with Fick’s diffusion equation. The NernstePlanck equation is used for the transport of charged particles in an electrical field. Specific Fick or NernstePlanck equations treat all flows as independent, requiring mobility being subject to gradients of electrostatic and chemical potentials, which may not always be fulfilled, especially in concentrated solutions. However, interactions (coupling) are a common phenomenon in membrane transport. One of the trends in such analysis is the approach of linear nonequilibrium thermodynamics and phenomenological equations to incorporate the coupled phenomena into membrane transport. Since the interactions between the permeant and membrane may be complex, it may be useful to describe the problem with the phenomenological approach without the need for detailed examination of the mechanism of transport and interactions. The nonequilibrium thermodynamics approach may identify the coupling between the flow of a substance and a reaction, or between two flows. This chapter summarizes the formulations for passive, facilitated, and active membrane transport problems.

10.2 MEMBRANE TRANSPORT Membranes may have various physical and chemical structures, and hence are able to restrict transport processes by having different permeabilities for different substrates. Determining properties of equilibrium across and within a membrane may help in understanding the transport phenomena through membranes.

10.2.1 MEMBRANE EQUILIBRIUM Besides thermal equilibrium, the principle of electroneutrality is also satisfied in membrane transport. Because of the absence of mechanical equilibrium, a pressure difference known as osmotic pressure exists between subsystems separated by the membrane. In the case of substrates in ion form, both nonpermeating and permeating ions create an electrical potential difference known as membrane potential across the membrane. For the separated parts of A and B, electrochemical equilibrium for permeating species k is Nonequilibrium Thermodynamics. https://doi.org/10.1016/B978-0-444-64112-0.00010-1 Copyright © 2019 Elsevier B.V. All rights reserved.

453

454

CHAPTER 10 MEMBRANE TRANSPORT    
o    
o ek þ RT ln ak aok þ zk Fj B m ek þ RT ln ak aok þ zk Fj A ¼ m

(10.1)

where a is the activity, j is electric potential, F is Faraday constant, and z is number of valence electron. For dilute aqueous solutions, we assume that the standard activity for each species and the electrochemical standard potential for each species are the same in each phase at the same temperature and pressure aokA ¼ aokB

and

wo m kA

wo

¼ m kB

at ðT; PÞA ¼ ðT; PÞB

(10.2)

The equations above Pare based on the integration of the partial molar Gibbs energy equation ei dNi for component k at constant T and volume Vk of pure component. dG ¼ SdT þ Vk dP þ i m Substituting Eq. (10.2) into Eq. (10.1), we can formulate the isothermal equilibrium condition dG ¼ 0 as   akB (10.3) þ zk FðjB  jA Þ ¼ 0 Vk ðPB  PA Þ þ RT ln akA If PA ¼ PB, we have the Nernst equation

  RT akB ln DJ ¼ jB  jA ¼  zk F akA

Under the condition of electroneutrality, Eq. (10.1) describes the Donnan equilibrium across a membrane, which separates solutions containing nonpermeating ions. With the Donnan equilibrium, differences of pressure and electrical potential will appear. If the nonpermeating components are electrically neutral, only the pressure difference occurs. In biological systems with dilute aqueous solutions, the last term in Eq. (10.3) disappears, since zk ¼ 0 and the activity of the species determines the osmotic pressure (P). For water, we have     RT awB DP ¼ PB  PA ¼  ln (10.4) Vw awA We may introduce the following approximations. Firstly, for ideal solutions, the activity coefficients are unity gk ¼ 1, and concentrations are equal to mole fractions: ak ¼ xk . Secondly, using the definitions xwA ¼ 1 

m X

xkA ;

Vw ¼ V=Nw ;

ckA ¼

k¼2

NkA V

and the approximation ln(1  x) z x for x  1, we have  X  X   m X 1 Nw m Nw m NkA y xkA ¼  ckA ln xwA y  Vw V k¼2 V k¼2 N k¼2

(10.5)

where m is the number of solutes and Nw ¼ N for dilute solutions. Substituting Eq. (10.5) into Eq. (10.4), we find the approximation called the van ’t Hoff equation ! m m X X ckB  ckA DP ¼ PB  PA yRT k¼2

k¼2

10.2 MEMBRANE TRANSPORT

455

In a system with one single nonpermeating component dissolved in water, the equation above becomes DP ¼ PB  PA yRTðc2B  c2A Þ If component 2 is present in phase B only, we obtain DP ¼ PB  PA yRTc2B

(10.6)

Consider a solution of 0.1 mol per liter of sucrose in water (phase B). It is at equilibrium with pure water (phase A) across a membrane, which is impermeable to sucrose. When the temperature is 37  C in both phases, we can determine the osmotic pressure by assuming that equilibrium exists and the solution is ideal. For this system, the electrical potential difference is zero jB  jA ¼ 0 and the solutions are ideal. The water is the only permeating species and from Eq. (10.6), we have DP ¼ PB  PA yRTc2B ¼ [8.314 (310) (0.1) (1000 l/m3)] ¼ 257.7 kPa. This simple example shows that the osmotic pressure may be considerable depending on the concentration of the permeating species. When a membrane system has two phases, m number of permeating components and zk ionic valences, the thermodynamic state of the composite system is determined uniquely by T, PA, PB, mole fraction xk in the two phases, and the electrical potential difference jB  jA across the membrane. These all add up to 1 þ 2þ2m þ 1 ¼ 4 þ 2m variables. These variables are restricted by m equilibrium relations (Eq. 10.1), so that the degrees of freedom are 4 þ m. This is a special form of the Gibbs phase rule for electrochemical or chemical membrane equilibrium.

EXAMPLE 10.1 MEMBRANE EQUILIBRIUM An aqueous solution (phase A) of 100 mmol/l of NaCl is in equilibrium across a protein-tight membrane with an aqueous solution (phase B) of NaCl and protein. The protein concentration is 5 mmol/l with a negative ionic valency of 10. Determine the difference in electrical potential and hydrostatic pressure across the membrane when both solutions are assumed to be ideal and the temperature is 25  C. Solution: Fig. 10.1 shows the membrane system with the phases A and B. There are four components, water (w), sodium ions (Naþ), chloride ions (Cl), and protein (P), and the first three are permeating. After neglecting the pressure difference, the equilibrium conditions for sodium and chloride ions from Eq. (10.3) are !   Naþ B (a) RT ln þ z Naþ FðjB  jA Þ ¼ 0 Naþ A

Membrane Phase A Water +Na+ +ClΨA PA

FIGURE 10.1 Equilibrium between two phases A and B separated by membrane.

Phase B Water +Na+ +Cl+Protein ΨB PB

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CHAPTER 10 MEMBRANE TRANSPORT

RT ln

Cl B Cl A

!

  þ z Cl FðjB  jA Þ ¼ 0

For water, where z ¼ 0, we obtain the equilibrium condition     RT xwB ln yRT PB  PA ¼  Vw xwA

m X k¼2

ckB 

m X

(b)

! ckA

(c)

k¼2

With z(Naþ) ¼ þ1, z(Cl) ¼ 1, and z(P) ¼ 10, electroneutrality in phase B yields ½Naþ  Cl  10P B ¼ 0. Therefore, the concentration of sodium and chloride ions cannot be the same in phase B. The condition of electroneutrality in phase A is Naþ ¼ Cl ¼ 100 mmol/l. The relations above with the given data determine the differences in electrical potential and pressure across the membrane, and the concentration of sodium and chloride ions in phase B. With the concentrations in phase A, we have
þ  Na Cl B ¼ ð100 mmol=lÞ2 Using this relation with the electroneutrality condition above, we obtain  Naþ B ¼ 128 mmol=l and ClB ¼ 78 mmol=l

The electrical potential difference is now obtained from Eqs. (a) or (b):   8:314ð298Þ 128 ln ¼ 6.4 mV jB  jA ¼  ð1Þð96500Þ 100 This shows that the algebraic sign of the potential difference is determined by the algebraic sign of the charge of the nonpermeating ion. The osmotic pressure difference is calculated from Eq. (c): DP ¼ PB  PA y8:314ð298Þ½ð128 þ 78 þ 5Þ  ð100 þ 100Þ ¼ 27.3 kPa As expected, this shows that the pressure is largest on the side of the membrane having the largest concentration of nonpermeating components. In the absence of NaCl, the potential difference disappears, and the pressure difference decreases (Garby and Larsen, 1995).

10.2.2 GAS PERMEATION The rate of diffusion for different substances is affected by membrane permeability. Since membranes are permeable for only some substances, different molecules will have different rates of diffusion. In gas permeation, a gas species is separated based mainly on its permeability in hollow fiber and spiral wound membranes. The hollow fiber systems can have an inside diameter up to 200 microns and hence very large surface-to-volume ratios, but high pressure drops inside the tubes. The basic flow for a species i Ji is pi Ji ¼ DPi (10.7) Dx where pi is the permeability of the membrane for species i, Dx is the thickness of the membrane, and DPi is the partial pressure difference that is the driving force. The permeability is the product of the solubility of the gas in the membrane and the diffusivity of the gas in the membrane. In terms of total pressures and mole fractions, Eq. (10.7) reduces to  pi  (10.8) PH yH;i  Pp yp;i Ji ¼ Dx

10.2 MEMBRANE TRANSPORT

457

where Pp is the total permeate pressure, PH is the total pressure on the high pressure side, yH,i is the mole fraction of species i on the high pressure side, while yp,i is the mole fraction of i in the permeate. The selectivity is the ratio of permeabilities  aij ¼ pi pj The selectivity should be greater than 20 to accomplish significant separation of species i from species j. For a completely mixed mixture on both sides of the membrane system, the external mass balance yields Jin yin ¼ Jout yout  Jp yp or yin ¼ ð1  qÞyout þ qyp where q ¼ Jp =Jin . If q is specified as a design parameter, we can estimate the yout by yin  qyp (10.9) 1q For a binary gas mixture, inserting Eq. (10.8) for species i and j into the external mass balance leads yout ¼

to  pi ari  PH yout  PL yp Dx   pj arj
  PH ð1  yout Þ  PL 1  yp Jp 1  yp ¼ Dx Jp yp ¼

(10.10) (10.11)

where r is the molar density at the permeate conditions, PL is the total pressure at the low-pressure side, and a is the area for mass transfer. After dividing Eq. (10.10) by Eq. (10.11) and substituting Eq. (10.9), we have 0 1 yin  qyp PL  yp y r 1q PH C  p  ¼ aij i B (10.12) @ A yin  qyp PL  rj 1  yp 1  yp  1 1q PH This equation reduces to the following quadratic form once the fractions are eliminated x1 y2p  x2 yp þ x3 ¼ 0 where

!  q PL ri þ aij  1 x1 ¼ 1  q PH rj   ri q PL yin 1 þ þ x2 ¼ ð1  aij Þ  1q r j 1  q PH 1  q r yin

x3 ¼ aij i rj 1  q 

The root needed is positive and between zero and one.

(10.13)

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CHAPTER 10 MEMBRANE TRANSPORT

EXAMPLE 10.2 GAS PERMEATION IN A BINARY GAS MIXTURE We want to separate carbon dioxide contaminating methane using a cellulose acetate membrane. The mixture is perfectly mixed on both sides of the membrane. The methane mole fraction in the feed (high pressure gas) is y(CH4) ¼ 0.90. The permeate pressure is 1.5 atm. At 35  C and 20 atm, the permeability of the membrane is p(CO2) ¼ 15  1010 and p(CH4) ¼ 0.48  1010 cm3 (STP) cm/(cm2 s cm Hg) (Wankat, 1994). The membrane thickness is 1 mm, and q ¼ 0.4. Estimate the membrane selectivity, permeant mole fraction yp(CO2), and flows of carbon dioxide and methane J(CO2) and J(CH4). Solution: 10 P 2 The selectivity is: aCO2 CH4 ¼ PCO ¼ 15:010 0:481010 ¼ 31:25 CH 4

The permeate mole fraction yp ¼ yp(CO2) can be determined from Eq. (10.13) with the constants !     q PL r 0:4 1:5 þ þ ð31:25  1Þ ¼ 22:43 aij i  1 ¼ x1 ¼ 1  q PH 1  0:4 20 rj     r q PL yin 1 0:4 1:5 0:1 1 x2 ¼ ð1  aij Þ i þ ¼ ð 30:25Þ þ þ  ¼ 28:91  þ 1q 1  0:4 20 0:6 0:6 rj 1  q PH 1  q   r yin

0:1 ¼ 5:21 ¼ 31:25 x3 ¼ aij i 0:6 rj 1  q  Here, rCO2 rCH4 ¼ 1. With the constants above, Eq. (10.13) yields yp ¼ yCO2 ¼ 0:216. This result displays an uphill transport of carbon dioxide, since the outlet concentration is higher than the carbon dioxide mole fraction in the feed. The pressure difference across the membrane allows for this transport direction. From the mass balance of Eq. (10.9), we have yout ¼

yin  qyp 0:1  0:4ð0:216Þ ¼ 0:022 ¼ 0:6 1q

The flow of carbon dioxide is JCO2 ¼

15:0  1010 ð76 cm Hg=1atmÞ½20ð0:022Þ  1:5ð0:216Þ cm3 ðSTPÞ ¼ 1:322  104 1:0  104 cm2 s

And yCO2 ¼ yout , since the module is well mixed JCH4 ¼

0:48  1010 ð76 cm Hg=1atmÞ½20ð0:978Þ  1:5ð0:784Þ cm3 ðSTPÞ ¼ 6:706  104 4 1:0  10 cm2 s

10.2.3 TRANSPORT IN LIQUID SOLUTIONS Consider a homogeneous membrane with thickness Dx separating an outer solution 1 from an inner solution 2. The flow occurs along the x-coordinate perpendicular to the membrane surface. The zero point of x is on the surface in contact with solution 1. The electrolyte solutions are characterized by their electrochemical potentials. Within the membrane, the chemical potential is different from both m ei1 and m ei2 . However, it is widely assumed that the potentials on the membrane surface are equal to those of the solutions they are in contact with. If the forces are expressed in terms of the chemical potentials, we have

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT

Xi ¼ 

459

de mi dx

Integration of this equation yields Z

Dx 0

Xi dx ¼

Z em2 i e

1 mi



 2  de mi dx ¼  m ei  m e1i ¼ De mi dx

Across the membrane, we also identify the mechanical pressure difference (DP ¼ P2  P1), the difference between concentration of species (Dci ¼ ci2  ci1), and the drop of electrical potentials (Dj ¼ j2  j1). These differences are related to the electrochemical potential difference by De mi ¼ Vi DP þ RTDðln ci Þ þ Ii Dj where Vi is the partial molar volume, and Ii the electrical charge of species i. In passive transport, electrolytes and other substances are transported due to concentration, pressure, and electrical potential differences. We define the x-direction of the flow to be positive towards the inner solution 2, as a resultant of a pressure difference DP ¼ P2,inner  P1,outer < 0 in accordance with nonequilibrium thermodynamic conventions, whereas many authors from the membrane process field choose to define the pressure difference as DP ¼ P1,outer  P2,inner (Katchalsky and Curran, 1967, Kargol and Kargol, 2003). In that case the flow is positive when DP > 0. It results in some sign changes in the flux and dissipation expressions and parameter definitions as fluxes are involved, when comparing this chapter with the aforementioned papers.

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT For a nonelectrolyte binary dilute solution consisting of water (w) and a solute (s), we have Dmw ¼ Vw DP þ RTDðln cw Þ ¼ Vw DP  RT Dms ¼ Vs DP þ RTDðln cs Þ ¼ Vs DP þ RT

Dcs DP ¼ Vw DP  cw cw

Dcs DP ¼ Vs DP þ cs cs

where we have used D(lncw) ¼ Dcw/cw ¼ D(1 e cS)/cw ¼ eDcS/cw and DP ¼ RTDcs is the osmotic pressure difference according to Eq. (10.6). Using standard notation, we have G ¼ H e TS and recalling that for an open system with no heat exchange nor work and negligible kinetic energy changes, the first law yields DH ¼ 0, we get for an isothermal membrane process the lost work expression TDS ¼ DG The rate of entropy production per unit area is diS/dt and the rate of lost work per unit area TdiS/dt must be equal to the change in the Gibbs energy. If we consider the transport of species i across a membrane where a chemical potential change occurs,

460

CHAPTER 10 MEMBRANE TRANSPORT " # X X X di S ¼ ðDGÞt ¼  T Ji mi1  Ji mi2 ¼  Ji Dmi  0 dt i i i

In this expression, when Dmi ¼ mi2(inner) e mi1(outer) > 0, the flow is positive. It goes from solution 1 (outer) through the membrane to solution 2 (inner) according to our direction choice. Hence, TdiS/ dt > 0, as required by the second law. This is consistent with Fick’s law and with the definition of mass transport driving force in nonequilibrium thermodynamics presented in Chapters 3 and 6, typically Xi ¼ e1/T (Vmi)T,P. For the binary system water e solute, we can express the dissipation function per unit area of membrane as a sum of molar fluxes Ji times forces Xi. X di S ¼T J¼T Ji Xi ¼ ð Jw Dmw  Js Dms Þ  0 dt i Since chemical potential differences are not readily measured, is it more suitable to introduce hydrostatic and osmotic pressure differences (Hwang, 2004). With the equations above, the dissipation equation becomes     di S DP DP ¼ Jw V w DP  J¼T  Js V s DP þ ¼ Jv DP  Jd DP  0 dt cw cs where Jv is the total volume flow and Jd is the relative flow of solute versus water defined by X J i V i ¼ J w Vw þ J s Vs Jv ¼ i

Jd ¼

Js Jw  cs cw

If the solution is sufficiently dilute, we may further simplify Jv w (JwVw), cw y 1/Vw, csVs1 and the solute flow can be expressed in terms of Jv and Jd: Js ¼ ðJv þ Jd Þcs

(10.14)

For identifying conjugate flows and forces, we may use the dissipation function per unit area of membrane. Therefore, Jv and (eDP) and Jd and (eDP) are the pairs of conjugate flows and forces and the linear phenomenological equations with four phenomenological coefficients are Jv ¼ Lp DP þ Lpd DP

(10.15)

Jd ¼ Ldp DP þ Ld DP

(10.16)

Due to Onsager’s relations, Lpd ¼ Ldp, the membrane can be characterized by three parameters.

10.3.1 TRANSPORT COEFFICIENTS Since the dissipation function is always positive according to the second law of thermodynamics, Lp and Ld must be positive, while Lpd can be either positive or negative and its magnitude is restricted by the condition LpLd  (Lpd)2 > 0.

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT

461

If we consider a membrane having the same solute concentration on both sides, we have DP ¼ RTDcs ¼ 0. However, a hydrostatic pressure difference DP still exists between the two sides, and we have a flow Jv that is a linear function of DP. In Eq. (10.15), the coefficient Lp is called the mechanical filtration coefficient, which represents the velocity of the fluid per unit pressure difference between the two sides of the membrane. The cross-phenomenological coefficient Ldp is called the ultrafiltration coefficient, which is related to the coupled diffusion induced by a mechanical pressure of the solute with respect to the solvent. An osmotic pressure difference DP s 0, produces a diffusion flow characterized by the permeability coefficient described later, which indicates the movement of the solute with respect to the solvent due to the inequality of concentrations on both sides of the membrane. The cross-coefficient Lpd relates the flow Jv to DP at DP ¼ 0 and is called the coefficient of osmotic flow. Even if there is no hydrostatic pressure difference, the gradient of solute concentration induces a volume flow. The crosscoefficients are imposed by the nature of the flow in the membrane, for example, Lpd shows the selectivity. If Jv ¼ 0, we have ðDPÞJv ¼0 ¼ 

Lpd DP Lp

The ratio eLpd/Lp is called the reflection coefficient sR and is always smaller than 1. sR ¼

Lpd Lp

The value sR ¼ 1 means eLpd ¼ Lp, and then ðDPÞJv ¼0 ¼ DP. This is the condition for an ideal semipermeable membrane, which blocks the transport of solute Js ¼ 0, no matter what the values of DP and DP are. When this is not the case, the membrane allows some solute to pass Lpd =Lp < 1. When sR < 1, on the other hand, some of the solute is reflected and the rest crosses the membrane; when sR ¼ 0, the membrane is completely permeable and is not selective. If we introduce sR into Eq. (10.15), we have Jv ¼ Lp ðDP  sR DPÞ

(10.17)

At constant volume (Jv ¼ 0), the solute permeability coefficient u is defined as ! Lp Ld  L2pd   u ¼ cs;av ¼ cs;av Ld  Lp s2R Lp where cs;av is the average solute concentration. For an ideal semipermeable membrane, sR ¼ 1, and u ¼ 0. For nonselective membranes, Lpd ¼ 0, and u ¼ cs,av Ld. The permeability coefficient is a characteristic parameter both in synthetic and natural membranes. Table 10.1 shows the permeability and reflection coefficients for some membranes. The membrane can be characterized by three parameters, typically, the mechanical filtration coefficient Lp, the reflection coefficient sR and the solute permeability coefficient u.

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CHAPTER 10 MEMBRANE TRANSPORT

Table 10.1 Solute Permeability and Reflection Coefficient for Some Membranes (Katchalsky and Curran, 1967) Membrane

Solute

Solute permeability, u 1015 Mol/(dyne cm)

Reflection Coeffiecient, s

Toad skin

Acetamide Thiourea Methanol Ethanol Isopropanol Urea Urea Ethylene glycol Malonamide Methanol Urea Glucose Sucrose Urea Glucose Sucrose

0.0041 0.00057 11 11 7 0.008 17 8 0.04 122 20.8 7.2 3.9 31.6 12.2 7.7

0.89 0.98 0.50 0.44 0.40 1 0.62 0.63 0.53 e 0.013 0.123 0.163 0.0016 0.024 0.036

Nitella translucens

Human red blood cell

Visking dialysis tubing

Dupont “wet gel”

10.3.2 MEASUREMENTS OF TRANSPORT COEFFICIENTS The membrane transport coefficients may be measured based on the equations       Jv DP Js and sR ¼ and u ¼ Lp ¼ DP Jv ¼0 DP DP¼0 DP Jv ¼0 Using Eq. (10.14) and the definition of sR, the solute flow can be expressed in terms of the solute permeability Js ¼ cs;av ð1  sR ÞJv  uDP

(10.18)

The definition of u requires that Jv ¼ 0 and according to Eq. (10.15) it happens only if there is a nonzero mechanical pressure difference as DP ¼ sR DP. It implies that DPs0 and u cannot be regarded as a measure of purely diffusive permeation. Therefore, experimental measurements of u may be difficult (Narebska et al., 1995, Kargol and Kargol, 2003). To isolate the purely diffusive permeation, we can rewrite Eq. (10.18) by introducing Jv as Eq. (10.17) as 
Js ¼ cs;av ð1  sR ÞLp DP þ u  cs;av ð1  sR ÞLp sR DP

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT  We may define ud, as the purely diffusive permeability coefficient for the solute ud ¼

Jd DP

463  . DP¼0

The use of ud allows us to express the solute flow with two independent contributions of the mechanical and osmotic pressure difference. Js ¼ cs;av ð1  sR ÞLp DP þ ud DP

(10.19)

where we have used a correlation ud ¼ ð1  sR ÞLp cav Hence, we have the following relation between u and ud: ud ð1 þ sR Þ ¼ u. Some measured transport parameters are listed in Table 10.2. Parameters sR and uex were measured at Jv ¼ 0 and u and ud were computed from the relations above. They are both smaller than uex, showing that the experimental measurement overestimates the purely diffusive flow and incorporate a contribution due to a nonzero mechanical pressure difference. Also, u is larger than ud showing that u always incorporate a nondiffusive contribution. For a simplified model of the membrane having cylindrical pores of radius r and length l with Np pores per unit surface area, Poiseuille’s law expresses the volume flow Q for a pressure difference along the z direction Pz¼0ePz¼l as follows Q¼

pr 4 ðPz¼0  Pz¼l Þ 8hl

where h is the viscosity of the liquid. The flow Jv is according to our pressure difference definition expressed by   Np pr 4 Np pr 4 ðPz¼0  Pz¼l Þ ¼ Jv ¼ ðDPÞ (10.20) 8hl 8hl Using Eqs. (10.15) and (10.20), we find the filtration coefficient Lp at DP ¼ 0 Lp ¼

Np pr 4 8hl

Let us assume that DP ¼ 0, but there is a concentration difference and the solute can diffuse. If the solution is ideal, van ’t Hoff’s law states that

Table 10.2 Measured Transport Parameters (Kargol and Kargol, 2003) Membrane

Solution

cav (mol/m3)

Neprophane Neprophane Cellophane Cellophane Dialysis membrane

Ethanol Glucose Ethanol Glucose Glucose

400 200 300 150 300

u 1010 Mol/(Ns)

ud 1010 Mol/(Ns)

Lp 1012 m3/(Ns)

sR

19.9 9.95 6.68 3.31 3.21

19.5 9.35 6.55 3.01 2.84

5.0 5.0 2.23 2.23 1.09

0.025 0.065 0.02 0.1 0.13

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CHAPTER 10 MEMBRANE TRANSPORT

DP ¼ RTDcs and then, from Eq. (10.18), we have Js ¼ uDP ¼ uRTDcs;av

(10.21)

According to Fick’s law, the diffusion flow in each pore of transverse area of p r and length L is given by    J ¼ D pr 2 L Dcs;av 2

where D is the diffusion coefficient and Dcs ¼ cs,downstream  cs,upstream. With the total number of pores Np, the total flow becomes    Js ¼ Np D pr 2 L Dcs;av (10.22) Comparing Eqs. (10.21) and (10.22) yields u¼

 Np D pr 2  1  s0R RT L

where sR0 is the fraction of solute reflected. From a hydrodynamic point of view, the reflection coefficient sR may be defined as rs 2 (10.23) 1  sR ¼ 1  r where rs and r are the radius of the solute molecule and the pore, respectively. If rs > r, and sR ¼ 1, since the solute molecules do not fit into the pores, all the solute is reflected. Eq. (10.23) results from the ratio of the effective area of the pores, which is pðr  rs Þ2 , to the total area effective area ðr  rs Þ2 rs 2 ¼ ¼ 1 2 actual area r r Generally, the reflection coefficient decreases as the molecular radius increases.

EXAMPLE 10.3 TIME NECESSARY TO REACH EQUILIBRIUM IN A MEMBRANE TRANSPORT Consider two containers of volumes V1 and V2 and the initial solute concentrations of c1 and c2 (c1 > c2) separated by a permeable membrane. Estimate the time necessary to reach the equilibrium at which c1 ¼ c2. Solution: The changes in the number of moles of solute in containers 1 N1 and 2 N2 are calculated as a function of the area a of the membrane and the solute flow Js dN1 dN2 ¼ Js a ; ¼ Js a dt dt We know that Ni ¼ Vi ci ; Js ¼ uRTðc1  c2 Þ, and then we have V1

dc1 ¼ auRTðc1  c2 Þ dt

(a)

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT

V2

dc2 ¼ auRTðc1  c2 Þ dt

465

(b)

After subtracting Eq. (b) from Eq. (a), we obtain

  dðc1  c2 Þ 1 1 ¼ auRT þ ðc1  c2 Þ dt V1 V2   Integration of this equation yields: DcðtÞ ¼ ðc1  c2 Þ ¼ Dcð0Þexp tto

(c)

where to is defined using Eq. (c) to ¼

V1 V2 ðAuRTÞðV1 þ V2 Þ

If we assume that V1 [ V2, we have a simplified approximation for to to z

V2 auRT

In practice, after a time of 4to, the concentration differences between the two containers will vanish, and to is considered the characteristic time. This approximation may be useful in cellular transport, and artificial kidneys and lungs.

10.3.3 FRICTIONAL FORCES AND RESISTANCE-TYPE PHENOMENOLOGICAL COEFFICIENTS We can express the phenomenological coefficients in terms of the frictional forces; assuming that for a steady state-flow, the thermodynamic forces X are counterbalanced by a sum of suitable frictional forces F. Thus, for a solute in an aqueous solution, we have Xs ¼ Fsw  Fsm

(10.24)

where Fsw is the frictional force between solute and water, and Fsm is the corresponding friction with the membrane. Similarly, the force on water is Xw ¼ Fws  Fwm

(10.25)

The terms Fsm and Fwm comprise complex hydrodynamic interactions within the membrane matrix and should be regarded as macroscopic averages. For sufficiently swollen membranes, however, Fsw which indicates the interactions of solute and solvent, may approach to free diffusion. The individual frictional forces Fij are assumed to be linearly proportional to the relative velocity vij ¼ vi  vj , and because the proportionality factor fij is the frictional coefficient per mole of ith component, we have Fij ¼ fij ðvi  vj Þ

(10.26)

ci fij ¼ cj fji

(10.27)

The fij obey the reciprocity relations

where ci is the concentration of species i in the membrane. Introducing Eqs. (10.27) and (10.26) into Eqs. (10.24), and (10.25), we obtain

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CHAPTER 10 MEMBRANE TRANSPORT

Xs ¼ fsw ðvs  vw Þ þ fsm ðvs  vm Þ cs Xw ¼ fsw ðvw  vs Þ þ fwm ðvw  vm Þ cw Choosing the membrane as the frame of reference and for vm ¼ 0, we have Xs ¼ vs ðfsw þ fsm Þ  vw fsw   cs fsw cs Xw ¼ vs þ vw fwm þ fsw cw cw Since the flows Js and Jw are expressed as Js ¼ csvs and Jw ¼ cwvw, we may express these equations in terms of the flows Xs ¼ Xw ¼ 

fsw þ fsm fsw Js  Jw cs cw

fsw cw fwm þ cs fsw Js þ Jw cw ðcw Þ2

(10.28) (10.29)

These equations represent the resistance type of formulations Xs ¼ Ks Js þ Ksw Jw Xw ¼ Kws Js þ Kw Jw The resistance phenomenological coefficients are given by Ks ¼

fsw þ fsm fsw cw fwm þ cs fsw ; Kws ¼ Ksw ¼  ; Kw ¼ cs cw ðcw Þ2

10.3.4 KIRKWOOD’S PROCEDURE Integration of force Xi yields Z Dx 0

Z Xi dx ¼

mIIi

mIi



  dmi dx ¼  mIIi  mIi ¼ Dmi dx

The chemical potential difference can be written for the solute and water as Dms ¼ Vs DP þ RT

Dcs 4s DP þ DP ¼ cs cs

Dmw ¼ Vw DP  RT

Dcs 4w DP  DP ¼ cw cw

where f ¼ ciVi is the mean volume fraction of component i in the external solution. On the other hand, integration of the right sides of Eqs. (10.28) and (10.29) requires the concentration profiles in the membrane and the dependence of the fij’s on X. For a homogeneous membrane, we may assume that the fij’s are constant, and instead of integrating over the concentration, we can use the mean values given by

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT

467

ðcs Þ0 þ ðcs ÞDx 4 ; cw ¼ w 2 Vw

cs;av ¼

where fw is the mean volume fraction of water in the membrane. Assuming the equality of the chemical potential on the surface of the membrane and in the surrounding solution, we may write ðcs Þo ¼ KcIs ;

ðcs ÞDx ¼ KcIIs ;

cs;av ¼ K

cI s þ cII s 2

where K is the distribution coefficient of solute between the membrane and the solution in equilibrium with it whereas cIs and cIIs are the concentration in solution. The K is taken as a concentrationindependent parameter uniform throughout the membrane in an ideal behavior. These assumptions permit the integration of the right of Eqs. (10.28) and (10.29) over the mean concentration. Since in dilute solution fw ¼ cwVw w 1, and fs is usually sufficiently small to make fsDP < DP, Eqs. (10.28) and (10.29) yield the following expressions DP ¼ Dx

fsw þ fsm fsw Vw Js  cs;av Dx Jw K 4w

!   DP fsw fwm cs fsw  DP  þK ¼ Dx Js  Dx Jw 4w 4w 4w Þ ð4w Þ2

(10.30)

(10.31)

where Dx denotes the thickness of the membrane. The permeability coefficient of the solute flow is determined at zero volume flow (Eq. 10.18) ð Js ÞJv ¼0 ¼ uDP In this case, the solute flow at Jw ¼ 0 is practically the same as at Jv ¼ 0. So, Eq. (10.30) at Jw ¼ 0 leads to DP ¼ Dx

fsw þ fsm Js K

and therefore, u¼

K Dxðfsw þ fsm Þ

(10.32)

In the ideal membrane transport, K and the frictional coefficients are concentration independent, and u becomes a constant characterizing the mobility of the solute in the membrane. Eq. (10.32) shows the characteristics of the permeability coefficient u. It is obvious that the total frictional resistance is larger in a membrane than in free solution. Although fsw is approximately equal to fsw of the solute diffusing freely in the solvent, we must consider the additional friction fsm due to interactions of the solute with the membrane. If the membrane, for example, has a porous structure and the penetrating molecules are sufficiently large, then fsm may be very large and u w 0. In this case, the system will behave as an ideal semipermeable membrane. If the solute penetrates by dissolving in the lipoid components of the membrane, then a large friction between the solute and membrane will develop, and the value of fsm will be large. However, if the attraction between the solute and lipoid is very large, K may increase to a larger value than fsm.

468

CHAPTER 10 MEMBRANE TRANSPORT

The reflection coefficient sR is also defined at Jv ¼ 0 using the relation   DP sR ¼ DP Jv ¼0 However, the coefficient s0 at Jw ¼ 0 is not exactly equal to the coefficient s at Jv ¼ 0. The relationship between sR and sR0 is sR ¼ s0R 

uVs Lp

(10.33)

The coefficient sR0 is readily obtained from Eqs. (10.30) and (10.31) at Jw ¼ 0 s0R ¼

1 Kfsw  4w 4w ðfsw þ fsm Þ

In view of Eq. (10.33), we have sR ¼

1 Kfsw uVs   4w 4w ðfsw þ fsm Þ Lp

(10.34)

Substituting Eq. (10.29) in the equation above, we get sR ¼

1 uDxfsw uVs   4w 4w Lp

(10.35)

Eqs. (10.33) and (10.34) help to understand the physical meaning of the reflection coefficient. Ideal semipermeable membranes prevent the permeation of the solute; hence, u ¼ 0 and sR ¼ 1 if we make use of fw ¼ cwVw w 1. Generally, an increase in fsm (due to large molecular weight) decreases u (Eq. (10.32), and hence increases sR. On the other hand, if K increases simultaneously with fsm, then u may increase, while sR decreases. In some cases, the increase in K is so strong that u assumes large values, which make sR negative. This causes negative anomalous osmosis, since DPð ¼ sDPÞ becomes negative. In coarse nonselective membranes, we have uvs uDxfsw þ ¼1 Lp fw If we assume that the solute penetrates only through the water-filled channels in the membrane (with volume fraction fw), then the solute-water frictional coefficient is close to that of free diffusion Do o fsw ¼ fsw ¼

RT Do

(10.36)

and the solute flow will be given by Js ¼ D4w

DP=RT Dx

where D is the diffusion coefficient in the membrane. The relation between u and D is given by u¼

Dfw RTDx

(10.37)

10.3 PHENOMENOLOGICAL ANALYSIS OF MEMBRANE TRANSPORT

469

Using Eqs. (10.36) and (10.37), we obtain for sR uvs D  o D Lp

sR ¼ 1 

(10.38)

EXAMPLE 10.4 DIFFUSION CELL WITH ELECTROLYTES The diffusion cell shown in Fig. 10.2 has an aqueous solution of NaCl with a concentration of 100 mmol/l. Later, 0.1 mmol radioactive Na with a specific activity of 1  108 units is added to chamber A, which has a volume of 1.0 l and is stirred continuously. Measurements show that the radioactivity in reservoir A decreases at a rate of 14 units per minute. The process is at steady state. Estimate the flow of sodium ions, the diffusion coefficient, and the mobility at 298.15K and in a transfer area of 100 mm2. Solution: At the time of addition, the specific activity is ( 1108 ) (0.1). Initially, the relative decrease in the concentration of radioactive Naþ in reservoir A is !  dcþ 1 14 A ¼  7 ¼ 1:4  106 1=min (a) þ 10 dt cA The initial concentration of labeled Naþ is 0.1 mmol/l. Since there is no chemical reaction, the following reactiondiffusion equation   X X X dni ¼ n  n ¼ n i Jr  aJ out i in i out i dt yields dnþ A ¼ aJ þ ; dt

þ nþ A ¼ cA V

(b)

From Eqs. (a) and (b), we find  3     þ    dcA V 10 1:4  106 8 2 ¼ ¼ 2:33  10 mol=ðm s ð0:1Þ Jþ ¼  a dt 104 60 þ

þ

L The diffusion coefficient is estimated from: D ¼ cþJcLþ ¼ cJþ ð0Þ A

B

A

x CiA ψA TA PA

Ji

CiB ψB TB PB

L

FIGURE 10.2 Diffusion cell with uniform concentration and electrical potential chambers connected with a barrier with length L.

470

CHAPTER 10 MEMBRANE TRANSPORT

This equation assumes a quasi-steady state during the measurement of the initial decrease in radioactivity, and the concentration of labeled Naþ is negligible in reservoir B. If we assume that L ¼ 0.005 m, we obtain        1:4  106 0:005  103 1 dcþ VL A ¼ D¼ þ ¼ 1:17  109 m2 =s a 60 104 cA dt 9

Di 1:1710 ¼ 4:7  1013 mol m2/(Js) The mobility is obtained from: bi ¼ RT ¼ 8:314ð298Þ

EXAMPLE 10.5 DIFFUSION CELL AND TRANSFERENCE NUMBERS The diffusion cell shown in Fig. 10.2 has NaCl mixtures in the two chambers with concentrations c1A ¼ 100 mmol/l and c1B ¼ 10 mmol/l. The mobilities of Naþ and Cl ions are different and their ratio yields their transference numbers bþ/b ¼ tþ/t ¼ 0.39/0.61 (NaCl). The transference number t for an ion equals the fraction of the total electrical current carried by the ion when the mixture is subjected to an electrical potential gradient. For monovalent ions, we have tþ/t ¼ 1. Estimate the diffusion potential of the cell at steady-state conditions at 298K. Assume that activity coefficients are equal in the two reservoirs. Solution: Due to electroneutrality, the local concentrations of Naþ and Cl ions are the same, and are denoted by c1(x). The flows of ions will be the same and directed from A to B. The NernstePlanck equation describes the diffusive flow of ions at constant T and P   dci ci zi F dj þ Ji ¼ bi ðRTÞ RT dx dx Using the equation above and eliminating fluxes (flows), we have       dj bþ  b RT 1 dc1 ¼ þ þ dx F c1 dx b z  b z After integrating this equation over the diffusion cell, we obtain the diffusion potential     bþ  b RT c1A ln Dj ¼ jA  jB ¼  þ þ   F b z b z c1B With the relations tþþt ¼ 1, and zþ ¼ z ¼ 1, we have the difference of diffusion potential     RT c1A ln Dj ¼ 1  2tþ F c1B With the data given, the potential difference is Dj ¼ ½1  2ð0:39Þ

  8:314ð298Þ 100 ln ¼ 13mV 96500 10

The molar flows of the ions and NaCl are all the same. In that case, we estimate the flows for Naþ and for Cl, and divide them by bþ and b, respectively. By adding these results, we obtain   2bþ b dc1 J1 ¼  þ RT b þ b dx Comparing this equation with Fick’s law for diffusion Ji ¼ Di ðdci =dxÞ, the diffusion coefficient for NaCl is obtained by D1 ¼

  2bþ b ðRTÞ ¼ 2 1  tþ bþ ðRTÞ bþ þ b

The mobility of NaCl is obtained directly from Di ¼ bi RT

(a)

10.4 COMPOSITE MEMBRANES

b1 ¼

471

  2bþ b ¼ 2  t þ bþ bþ þ b

For an electrolyte with arbitrary ionic valences, Eq. (a) may be generalized by D1 ¼

2bþ b ðzþ  z Þ ðRTÞ bþ zþ  b z

EXAMPLE 10.6 ESTIMATION OF FLOW IN A DIFFUSION CELL Each chamber of the diffusion cell shown in Fig. 10.2 has an aqueous solution of NaCl with concentrations c1A ¼ c1B ¼ 100 mmol/l at 298K. An electrical potential difference of 100 mV is established between the two chambers. Estimate the diffusion flow of NaCl and its direction if D1 ¼ 1:48  109 m2/s. Solution: Here, the externally established potential difference causes a concentration gradient and hence a flow, provided that the mobilities of the ions are different. Due to electroneutrality, the local concentrations of Naþ and Cl ions are the same and are denoted by c1(x). Also, the flows of negative and positive ions are the same J þ ¼ J  . The flows of ions are oppositely directed. Using the NernstePlanck equation at constant T and P for Naþ and Cl ions and eliminating the flows yields dc1 c1 ðbþ  b Þ F dj ¼ þ b þ b RT dx dx

(a)

This equation shows the magnitude of an externally induced concentration gradient. The gradient is zero if bþ ¼ b, which is nearly true for KCl. The flow of NaCl can be obtained by substituting Eq. (a) into Fick’s law    þ  b  b F dj J 1 ¼ D1 c 1 þ b þ b RT dx With bþ/b ¼ tþ/t ¼ 0.39/0.61 (NaCl), L ¼ 0.005 m, and D1 ¼ 1:48  109 m2/s, the above equation yields      1  0:39=0:61 96500 0:1 ¼ 2:54  105 mol=ðm2 s J1 ¼ ð1:48  109 Þð100Þ 1 þ 0:39=0:61 8:314ð298Þ 0:005 For arbitrary ionic valences, the flow may be obtained from J1 ¼ D1 c1

ðzþ Þ2 bþ  ðz Þ2 b zþ bþ  z b

!

  F dj RT dx

10.4 COMPOSITE MEMBRANES Transport in membranes is mostly a complex and coupled process. It is important to understand and quantify the coupling to describe the transport in membranes. Beside kinetic studies, thermodynamics might offer a new and rigorous approach toward understanding the coupled transport without the need for detailed examination of the mechanisms of coupling. For composite membranes (with compartments) the dissipation function J in terms of flows of volume, salt, and electrical current, and the corresponding forces are (Katchalsky and Curran, 1967) J ¼ J v ðDP  DPÞ  Js Dmcs  Ij  0

(10.39)

where DPis the difference in hydrostatic pressure across the membrane, DP is the difference in the  osmotic pressure, and Dmcs ¼ DPs cs is the concentration-dependent part of the chemical potential

472

CHAPTER 10 MEMBRANE TRANSPORT

differences of the salt. DPs is the difference in the osmotic pressure due to the permeant solute (salt), and cs is the mean concentration of the salt,; Jv ¼ Jw Vw þ Js Vs , where Vw and Vs are the partial molar volumes of water and salt, respectively. j is the electric potential, and I the electrical current. Here, Jv ; Jw ; and Js represent the virtual flows. Experimentally, Jv is determined by measuring the change in volume of one or both compartments at opposite surfaces of the membranes. Eq. (10.39) yields a set of three-flow linear phenomenological equations of conductance type   DPs Jv ¼ L11 ðDP  DPÞ þ L12 þ L13 j cs   DPs Js ¼ L21 ðDP  DPÞ þ L22 (10.40) þ L23 j cs   DPs I ¼ L31 ðDP  DPÞ þ L32 þ L33 j cs As Eq. (10.39) is an appropriately derived dissipation function consisting of the conjugate flows and forces, the Onsager reciprocal rules states that Lij ¼ Lji . The set of Eq. (10.40) can also be expressed in terms of the flows using the resistance coefficients Kij, and we have the resistance-type formulation ðDP  DPÞ ¼ K11 Jv þ K12 Js þ K13 I   DPs  ¼ K21 Jv þ K22 Js þ K23 I cs

(10.41)

j ¼ K31 Jv þ K32 Js þ K33 I where the coefficients Kij are the inverse of the conductance coefficients Lij, and are symmetrical. Kij ¼ Kji .

10.4.1 TWO-FLOW SYSTEMS The set of Eq. (10.40) is related to various classical studies of electrokinetic phenomena, since the equations describe the coupled processes and yield naturally several symmetry relationships, which have been observed experimentally. Therefore, they provide a practical application of the linear nonequilibrium thermodynamic approach. For example, we may consider studies with identical solutions at each surface of the membrane, so that DP ¼ DPs ¼ 0. Then the system has only two degrees of freedom, and we have J ¼ Jv DP  Ij  0 and the linear phenomenological equations become Jv ¼ L11 DP þ L12 j

(10.42)

I ¼ L21 DP þ L22 j

(10.43)

These equations represent the basis of classical electrokinetics. For example, the magnitude of the electroosmotic volume flow per unit potential at zero pressure difference, L12 ¼ ð  Jv =jÞDP¼0, and

10.4 COMPOSITE MEMBRANES

473

the streaming current per unit pressure difference at short circuit, L21 ¼ ð  I=DPÞj¼0 , must be identical. Eqs. (10.42) and (10.43) indicate that the existence of a pressure difference will produce an electrical flow if the coupling coefficient is nonvanishing; when no pressure is applied DP ¼ 0, the action of the electrical force will cause a volume flow of water. We also observe the well-known Saxen relations between the ratios of force and flow     Jv j ¼ DP I¼0 I DP¼0     Jv j ¼ DP Jv ¼0 I j¼0 These symmetry relationships do not depend on the specific features of any given model but follow quite generally from the linear phenomenological equations of nonequilibrium thermodynamics. In Eqs. (10.40) and (10.41), a system with three degrees of freedom is characterized by the six independent phenomenological coefficients. The conductance coefficients Lij could be readily evaluated if it is possible to control the two forces, for example     Jv vJv or L12 ¼ L12 ¼ DPs =cs DPDP¼0;j¼0 vðDPs =cs Þ DPDP;j For the resistance coefficients Kij, it would be desirable to control the two flows. It may be more useful to consider alternative expressions of phenomenological equations without the need for further transformation of the dissipation function. For example, consider the set of Eq. (10.41) expressing the forces as functions of the flows. For practical purposes, it is desirable to use relations in which the independent variables are readily controlled experimentally. We may rewrite the set of Eq. (10.40) in such a way that DPs =cs becomes an independent variable, while Js becomes a dependent variable, so that we have   K11 K22  K12 K21 K12 DPs K22 K13  K12 K23 Jv þ I ðDP  DPÞ ¼ þ K22 K22 cs K22   K21 1 DPs K23 Js ¼  Jv þ I (10.44)  K22 K22 cs K22   K31 K22  K32 K21 K32 DPs K22 K33  K33 K23 Jv þ I j ¼ þ K22 K22 cs K22

10.4.2 TRANSPORT COEFFICIENTS The practical transport coefficients may be evaluated experimentally under conditions in which two of the independent variables, Jv ; DPs =cs ; and I are set equal to zero. Such a set of coefficients may be identified with six coefficients from the set of Eq. (10.44). Because of the Onsager reciprocal relations, the remaining three coefficients may be evaluated as follows

474

CHAPTER 10 MEMBRANE TRANSPORT

    Js DP  DP ¼ ¼ cs ð1  sR Þ DPs =cs Jv ;I¼0 Jv DPs ;I¼0     j Js s1  ¼ ¼ DPs =cs Jv ;I¼0 I DPs ;Jv ¼0 n1 z1 F     j DP  DP b ¼ ¼ Jv DPs ;I¼0 I Lp Jv ;DPs ¼0   where sR is the reflection coefficient and given by: sR ¼ DPDP DPs Jv ;I¼0   s1 is the transport number: s1 ¼ n1 z1 F DPjs =cs   Jv ;I¼0 b is the electroosmotic permeability: b ¼ JI v DPDP;DPs ¼0   Jv and Lp is the filtration coefficient: Lp ¼ DPDP DPs ;I¼0

With these coefficients, the set of Eq. (10.41) can be expressed in a more useful form       1 DPs b Jv  cs ð1  sR Þ  ð IÞ ðDP  DPÞ ¼ Lp Lp cs     DPs s1 Js ¼ cs ð1  sR ÞJv þ cs u þ ð IÞ (10.45) cs n1 z 1 F        b s1 DPs 1 ð IÞ Jv  þ j ¼  Lp k n1 z 1 F cs   Js and k is the electric conductance. where u is the solute permeability u ¼ DP s Jv ;I¼0   k ¼ I j Jv DPs ¼0

Here, a molecule of the salt dissociates into v1 cations of charge z1, and v2 anions of charge z2, and F is the Faraday constant. The set of Eq. (10.45) is useful for the treatment of a composite membrane consisting of compartments in series. The practical parameters above were derived long before the linear nonequilibrium thermodynamics formulations were developed. A combination of these parameters in a self-consistent formulation provides a sound basis for analysis. Table 10.2 shows some of the membrane parameters. When the permeant solute is nonelectrolyte, there is no current flow to be considered, and Eq. (10.45) can be expressed in terms of the flows as a function of the forces Jv ¼ Lp ðDP  DPÞ þ ð1  sR ÞLp DPs Js ¼ cs ð1  sR ÞJv  uDPs

10.4 COMPOSITE MEMBRANES

475

These equations correspond respectively to the set of Eq. (10.38) for the case of nonelectrolyte dilute solutions and are like Eqs. (10.17) and (10.18). The value of reflection coefficient sR must depend on the nature of both the solute and the membrane. For the case of volume flow in the absence of the concentration gradient in the permeant solute (DPs ¼ 0Þ , we see that the quantity (1sR) is a direct measure of the extent of coupling between the solute flow and the volume flow. If the membrane is completely nonselective, then sR ¼ 0; if the membrane is perfectly selective, permeable only to the solvent, sR ¼ 1. In most cases, sR will lie between 0 and 1. A diffusion flow against its conjugate gradient driven by the dissipation of another diffusional process would be called “incongruent” diffusion. For example, the flow of the ith component across a membrane may be expressed by Ji ¼ Lii Dmi þ

n X

Lik Dmk

k¼1

If Dmi ¼ 0, but Dmk s0, a flow of substance i may still be possible.

EXAMPLE 10.7 ENERGY CONVERSION IN THE ELECTROKINETIC EFFECT Discuss the energy conversion in electrokinetic systems. Solution: Electrokinetic effects are the consequence of the interaction between the flow of matter and flow of electricity through a porous membrane. The linear phenomenological equations for the simultaneous transport of matter and electricity are (Eqs. 10.42 and 10.43) Jv ¼ L11 DP þ L12 j I ¼ L21 DP þ L22 j where I is the electrical current per unit area, Jv is the volume flow of matter, Dj is the potential difference, DP is the pressure difference, and Lij are the phenomenological coefficients defined as   Jv L11 ¼ ðhydraulic permeability; filtration coefficientÞ DP j¼0   Jv L12 ¼ ðelectroosmosisÞ j DP¼0   I ðstreaming currentÞ L21 ¼ DP j¼0   I ðconductance of permeant  filled electroosmotic cellÞ L22 ¼ j DP¼0 Since L12 ¼ L21, the electrical current per unit pressure force at j ¼ 0 is equal to the volume flow Jv per unit potential difference at DP ¼ 0,     Jv I ¼ DP j¼0 j DP¼0 The efficiency of electrokinetic energy conversion for two operations modes, namely electroosmosis heo and streaming potential hsp, are expressed as

476

CHAPTER 10 MEMBRANE TRANSPORT

Jv DP output Jv DP . ¼ ¼ heo ¼  IDj input ðDjÞ2 R . ðDjÞ2 R IDj output ¼ ¼ hsp ¼  Jv DP input Jv DP where R is the resistance. The maximum values of energy conversions occur when the output forces equal half of their steady-state values. For example, heo is maximum when DP equals half the value of electroosmotic pressure 1 DP ¼  ðDPÞJv ¼0 2 The maximum energy conversion efficiency may be related to the merit b through the degree of coupling q hmax ¼

ð1 þ bÞ1=2  1 ð1 þ bÞ2 þ 1

with 1 1 1 2 q

 b¼ where q ¼

Lij ðLii Ljj Þ1=2

with Lij ¼ Lji. When the value of b is much smaller than unity, then we have hmax yb=4. Due to

Onsager’s reciprocal relations, we have hmax;eo ¼ hmax;sp Table 10.3 shows the efficiency of electrokinetic energy conversion for mixed-lipid membranes.

Table 10.3 Efficiency of Electrokinetic Energy Conversion for Various Mixed-Lipid Membranes (Caplan and Essig, 2013) Type of Membrane

  beo 3105

  bsp 3105

  hmax;eo 3105

  hmax;sp 3105

Cephalin-serine Cephalin-inositol Lecithin-serine Lecithin-inosithol Lecithin-cephalin Inositol-serine

2.108 2.235 3.687 3.946 5.855 10.044

1.714 2.255 3.686 3.942 5.852 10.156

0.527 0.558 0.921 0.986 1.463 2.511

0.428 0.563 0.920 0.985 1.463 2.539

10.5 FACILITATED AND ACTIVE TRANSPORT IN MEMBRANES Facilitated transport in a membrane involves a chemical agent as a carrier to increase the rate of transport. A chemical agent can react reversibly with a permeant, and yields high selectivity and permeability, which makes facilitated transport a very selective separation technique. The chemical agent carries the substance in the form of a carrier-bound substance; the carrier releases the substance on the other side of the membrane due to chemical conditions (mainly, pH and electric charge) and diffuses back. Usually a carrier with high association and dissociation rate constants, that are similar in magnitude is desirable. For the transport to be selective, the membrane is permeable to the flow of certain substances and impermeable to the flow of others including the carrier molecule. Therefore,

10.5 FACILITATED AND ACTIVE TRANSPORT IN MEMBRANES

477

facilitated transport is not described by Fick’s law and exhibits saturation at a higher concentration of the permeant. Various substances such as amino acids, organic acids, NaOH, NaCl, carbon dioxide, oxygen, and metals, and various ions, such as Cd(II), Cu(II), Co(II), and Fe(III), can be separated by using suitable carrier agents in liquid or in solid composite membranes.

10.5.1 LIQUID MEMBRANES Liquid membranes behave like double liquid-liquid extraction systems where the usage of organic solvent is minimized. Such devices are generally prepared as bulk liquid, emulsion liquid, and supported liquid membranes. They allow wide-scale applications in industrial separation, leading to far less expensive processes. For example, olefins, amino acids, heavy metals, gases, fatty acids, water, and inorganic salts can be separated selectively by facilitated transport. Also, solid polymer composite membranes containing a chemical agent are being tested successfully in separation technology. Some solid composite membranes are Nafion-poly(pyrrole) films with silver or sodium, activated composite membrane, and solid polymer electrode composite membranes. Composite membranes are stable and suitable for industrial applications and are usually made of a support polymer matrix of porous structure to which a chemical carrier is added. The preparation of polymer layers containing different amounts of carrier agents may require special polymerization techniques, such as interfacial polymerization. In ion separation, composite membranes utilize a chemical agent as a carrier dissolved in an organic solvent contained in a polymeric matrix or within the pores of a polymer membrane. For example, pseudocrown ethers can be used as the fixed site carriers in ion separation. Table 10.4 shows some of the applications of facilitated transport.

10.5.2 ACTIVE TRANSPORT Active transport, also known as uphill transport, requires the transport of molecules from a low concentration region to a high concentration region. Therefore, active transport must couple to another spontaneous process to transport charged or uncharged substrates against their thermodynamic forces. Some artificial membranes are used for active transport. For example, the active transport of metal ions through synthetic polymer membranes is used. Some specific examples are the cation exchange membranes made of 2,3-epithiopropyl methacrylate and 2-acrylamide-2-methyl propane sulfonic acid copolymers for the active transport of alkali and alkali earth metal ions, and some copolymer membranes for the active transport of the amino acids glycine, penylalanine, and lycine by using hydrochloric acid as the receiving solution. One possible mechanism for achieving this is that glycine and water are transferred by osmotic pressure into the membrane, and then glycine is protonated with Hþ released from sulfonic acid groups in the membrane; later, the protonated glycine is transported by means of sulfonic acid groups to the other side of the membrane regardless of the smaller electric potential difference. The transport of amino acids depends on the composition of the membranes and the structure of the amino acids. A membrane with the function of active transport can recover uranyl ions UO2þ 2 in the eluate. Uranyl ions form anion complexes with sulfate ions in sulfuric acid and can be transported against their concentration gradient through a liquid membrane with tertiary amine by using carbonate solution as the receiving solution. Polymeric anion-exchange membranes can also transport uranyl ions

478

CHAPTER 10 MEMBRANE TRANSPORT

Table 10.4 Some Facilitated Membrane Transport Systems Permeant

Carrier

Membrane

Analysis

References

Amino acids (Pamino benzoic acid) Amino acid Phenylalanine

Undeconoic acid

Organic liquid membranes ELM

R

Uglea and Zanoaga (1989) Liu and Liu (1998)

Liquid membrane Composite membrane Anion exchange membrane ELM

R

Amines and amino acids

Di(2-ethylhexyl) phosphoric acid (D2EHPA) Macrocyclic ligands

Lactic acid

Neutralization

Plyunsaturated fatty acid esters from sardine oil Binary organic acids Orthoboric acid

Agþ

Tri-n-ocytylamine Counterions

Boric acid

1,3-diols

Boric acid

Borate ions

Organic residues and inorganic salts Silver

Polymeric liquids

TR þ R

LNET TR þ R

Buschman et al. (1999) and Calzado et al. (2001) Narebska and Staniszewski (1998) Nakano & Kawamoto (1996)

SLM Ion exchange membrane SLM

TR þ R LNET

Juang et al. (1998) Selegny et al. (1997)

TR þ R

Ion exchange membrane SLM-polymeric

TR þ R

Bachelier et al. (1996) Selegny et al. (1994)

TR þ R

Ho et al. (1996)

Di(2-ethylhexyl) phosphoric acid (D2EHPA) Phenoxy compounds Cyanex-302, DeEPHA

ELM

TR þ R

Lee et al. (1996)

Bulk liquid Membrane SLM

TR þ R

Ion exchange Membrane

TR þ R

Liquid Membrane

TR þ R

Quinn et al. (1995)

Carbon dioxide

Monoprotonated ethylene diamine EDAHþ Tetramethylam monium fluoride tetrahydrate Polymers

Rankumar et al. (1999) Daoud et al. (1998) and Gumi et al. (2001) Selegny et al. (1995)

SLM

TR þ R

Carbon dioxide

Diamines

R

Ion

Water

Ion exchange Membrane Nafion 120

HC1

NaC1

Jeong and Lee (1999) Matsuyama et al. (1996) Narebska and Koter (1997) Narebska et al. (1995)

Alkali metal lons Cd(II)ions

Carbon dioxide

Carbon dioxide

Anion exchange Membrane

TR þ R

LNET LNET

10.6 BIOMEMBRANES

479

Table 10.4 Some Facilitated Membrane Transport Systemsdcont’d Permeant

Carrier

Membrane

Analysis

References

Acid

Salt

LNET

Olefins

Silver tetrafluoroborate Silver-I, Sodium-I

Anion exchange Membrane Composite (poly) ethylene oxide Nafio-poly(pyrrole)

Narebska and Warszawski (1994) Pinnau and Toy, 2001 Sungpet et al., 2001

ELM, emulsion liquid membranes; LNET, linear nonequilibrium thermodynamics; R, conventional rate equations; SLM, supported liquid membranes; TR, conventional transport equations.

selectively from the eluate of sulfuric acid containing alkali earth metal ions or cupric ions. Table 10.5 shows some applications of active transport.

10.6 BIOMEMBRANES A cell membrane is a fluid mosaic of lipids and proteins. Phosphoglycerides are the major membrane lipids that form a bilayer with their hydrophilic head groups interacting with water on both the extracellular and intracellular surfaces, and their hydrophobic fatty acyl chains in the central and hydrophobic region of the membrane. Peripheral proteins are embedded at the periphery, while integral proteins span from one side to the other. Biomembranes separate the contents of the cell from the external environment.

Table 10.5 Some Active Transport Systems Permeant

Carrier

Membrane

Analysis

References

Uranyl ions

2,3-epithiopropyl Methacrylate-dodecyl methacrylate-methacrylamide Propyl trimethyl ammonium terpolymer Oxidative Phosphorylation Reversed bienzyme

Anion exchange membranes

R

Nakano and Kawamoto (1996)

Living cell mitochondria Porous membrane

LNET

Stucki (1980)

LNET

Nigon et al. (1998) Nakano et al. (1993)

Naþ,Kþ anion Amino acids Glycine, phenylalanine, Lycine D-lactate

2,3-epithiopropyl Methacrylate-2-Acrylamide-2methyl propane sulfonic acid

Cation exchange membranes

R

(14C)-methylamine (14C)-thiocyanate

Inverted membrane Vesicles of Escherichia coli

R

LNET, linear nonequilibrium thermodynamics approach; R, Conventional rate equations.

Dung and Chen (1991)

480

CHAPTER 10 MEMBRANE TRANSPORT

Some of the proteins are involved in the transport of substances across the membrane, and some other proteins are enzymes that catalyze biochemical reactions. Proteins on the exterior surface can function as receptors and bind external ligands such as hormones and growth factors. Proteins migrate in an electric field; positively charged proteins are cations and migrate toward the cathode, while negatively charged proteins are anions and migrate toward the anode. There are several mechanisms for explaining how biological membranes can transport charged or uncharged substrates against their thermodynamic forces. It is widely accepted that cross-transports by a protein are discrete events. Biomembranes contain enzymes, pores, membrane potentials, and catalytic activities associated with the transport of substrates. It is well established that the electrostatic interactions between the membrane and a charged solute may play an important role in transport. Thus, we must establish reliable links between the membrane’s charge effect and the pore size, length distribution, and density to describe the interactions.

10.6.1 REACTION-DIFFUSION IN BIOMEMBRANES From the local conservation of mass, we obtain vci dJi ¼  þ ni Jri vt dx where n is the stoichiometric coefficient, while Jr is the chemical reaction rate (velocity). At steady state, the local concentrations do not vary with time, and we have dJi ¼ ni Jri dx

(10.46)

This equation shows that a stationary state imposes a relation between the diffusion and chemical reaction and is of special interest in isotropic membranes where the coupling coefficients vanish. For a homogeneous and isotropic medium the linear phenomenological equations expressing the forces in terms of fluxes are X dm  i¼ Kik Jk (10.47) dx k A ¼ K r Jr

(10.48)

where A is the affinity and Kr is the resistance phenomenological coefficient. Here, diffusion and chemical reaction are not coupled since the cross-coefficients between the scalar chemical reaction and vectorial mass flow vanish in an isotropic medium, according to the Curie-Prigogine principle. If the coefficients Kik are independent of position, we can differentiate Eq. (10.47) with respect to x, and insert Eq. (10.46) to obtain ! X d2 mi X dJk ¼ Kik nk Kik Jr (10.49)  2 ¼ dx dx k k After multiplying both sides by ni and summing, the equation above becomes

10.6 BIOMEMBRANES 1 0 X d 2 mi X  ni 2 ¼ @ ni nk Kik AJr dx i i;k P From the definition of affinityA ¼  ni mi , we find i

X ni d 2 m k d2 A ¼  dx2 dx2 i

481

(10.50)

! (10.51)

After substituting Jr from Eq. (10.48), we have 1 0 d2 A @X ni nk Kik A A ¼ dx2 Kr i;k ! P The expression ni nk Kik =Kr has the dimension of cm2 and will be denoted by l2 to obtain i;k

d2 A ¼ l2 A dx2 The characteristic parameter l is called the relaxation length of the coupled reaction-diffusion processes within a membrane. For a membrane thickness of Dx, dimensionless number l/Dx is closely related to the Thiele modulus used for the characterization of heterogeneous reaction columns. This dimensionless quantity is also related to the relaxation time of chemical reaction sr and the average relaxation time of diffusion processes sd as follows  2 l sr ¼ Dx 2hsd i If a reaction relaxes faster than the time necessary for diffusion across the membrane sr < hsdi, then l will be smaller than Dx, and the reaction will reach equilibrium on the surface after the reactants have diffused only a short distance within the membrane. On the other hand, when Dx is very small, which is the case in biological systems, then sr and hsdi may be of the same order of magnitude, and hence the system cannot be treated as an equilibrium state.

10.6.2 COMPARTMENTAL STRUCTURE AND DISSIPATION FUNCTION For a simple derivation of the dissipation function, consider an isothermal composite system with three compartments consisting of two external chambers, (I and II), and a membrane compartment (m) in between. The volumes of the compartments are constant (dVI ¼ dVII ¼ dVm ¼ 0). The Gibbs relations for the compartments are X dUI ¼ TdSI þ mi;I dNi;I (10.52) i

482

CHAPTER 10 MEMBRANE TRANSPORT

dUII ¼ TdSII þ

X mi;II dNi;II

(10.53)

i

dUm ¼ TdSm þ

X mi;m dNi;m

(10.54)

i

It is assumed that reactions take place only in the membrane, and the net change of the number of moles is expressed by dNi;m ¼ dNi;exch þ dNi;r

(10.55)

where dNi;exch is the number of moles of the component i exchanged with the surrounding compartments (Ni,exch ¼ Ni,I  Ni,II), and dNi;r denotes the number of moles of component i produced by the chemical reaction, which is expressed for the kth chemical reaction as follows X dNi;r ¼ n dε (10.56) k ik k where vik is the stoichiometric coefficient of the ith component in the kth extent of  reaction, and ε is the P the kth chemical reaction related to the rate of reaction as Jrk ¼ dεk dt and Ak;m ¼  i yi;k mk;m , where Ak,m is the affinity of the kth reaction within the membrane. Combining Eq. (10.54) with Eq. (10.56), we obtain X X dUm ¼ TdSm þ mi dNi;exch þ Ak;m dεk (10.57) i

k

Since the composite system is a closed system, we have dNi;I þ dNi;II þ dNi;m ¼ 0

(10.58)

dUI þ dUII þ dUm ¼ 0

(10.59)

where edNi,I is the number of moles of the ith component gained by the membrane, and dNi,II is the number of moles of component i lost by the membrane through the diffusion processes with the macroscopic driving forces of Dmi;in ¼ mi;m  mi;I and Dmi;out ¼ mi;II  mi;m . Therefore, we have X X X   mi dNi;exch ¼ (10.60) mi;m  mi;I dNi;I  mi;II  mi;m dNi;II i

i

i

The input and output flows are expressed by dNi;I dNi;II ; Ji;out ¼ dt dt The entropy contributions of the three compartments are expressed by Ji;in ¼ 

dSI þ dSII þ dSm ¼ di SI þ di SII þ di Sm ¼ di S

(10.61)

(10.62)

The contributions from the exchangeable entropy terms deS cancel one another in the equation above. By adding Eq. (10.52), (10.53) and (10.57), and using Eq. (10.60) with (10.62), we obtain X X X   mi;I  mi;m dNi;I  mi;m  mi;II dNi;II  Tdi S þ Ak;m dεk ¼ 0 i

i

k

10.6 BIOMEMBRANES

483

Dividing the equation above by dt and using Eq. (10.61), the dissipation function becomes X X di S X ¼ J¼T Ji;in ð Dmi;in Þ þ Ji;out ð Dmi;out Þ þ Ak;m Jrk  0 (10.63) dt i i k Assuming that under steady-state conditions, the difference between the output and input flows of the ith component is equal to the amount of ith component reacted in all chemical processes, and we have X Ji;out  Ji;in ¼ nik Jrk k

The last term in Eq. (10.63) can be expressed by the difference of the output and input flows ! ! X X X X X X Ak;m Jrk ¼ ni;k mi;m Jrk þ mi;m ni;k Jrk  mi;m ðJi;out  Ji;in  k

k

i

i

i

i

Introducing the equation above into Eq. (10.63), we obtain X X Ji;in mi;I  Ji;out mi;II J¼ i

i

However, only the difference in chemical potentials is measurable, and the above equation can be further transformed for practical use by introducing the affinity of the kth reaction in compartment II, Ak,II X Ak;II ¼  nik mi;II i

By introducing Ji,out from Eq. (10.61), we obtain X X X X Ji;in ðmi;I  mi;II Þ þ Jrk Ak;II ¼ Ji;in ð Dmi Þ þ Jrk Ak;II  0 J¼ i

k

i

k

With Dmi ¼ mi;II  mi;I , which is the cross-membrane difference of chemical potentials and measurable.

EXAMPLE 10.8 COUPLED SYSTEM OF FLOWS AND A CHEMICAL REACTION Discuss coupled system of flows and a first order chemical reaction. Solution: For a specific membrane, the phenomenological equations relating the flows and forces of either vectorial or scalar character may be written. Such flows and forces must be derived from an appropriate dissipation function. Consider the following dissipation function m1 Þ þ J2 ð De m2 Þ þ Jr A  0 J ¼ Jw ð Dmw Þ þ J1 ð De where the subscripts w, 1, and 2 refer to water, cation, and anion, respectively, and A and Jr refer to chemical affinity and chemical rate, respectively. Therefore, we have the corresponding linear relations in the resistance-type formulations where the forces Xj are expressed as the function of flows Jj Xw ¼ Dmw ¼ Kww Jw þ Kw1 J1 þ Kw2 J2 X1 ¼ De m1 ¼ K1w Jw þ K11 J1 þ K12 J2 þ K1r Jr ;

484

CHAPTER 10 MEMBRANE TRANSPORT

X2 ¼ De m2 ¼ K2w Jw þ K21 J1 þ K22 J2 þ K2r Jr Xr ¼ A ¼ Kr1 J1 þ Kr2 J2 þ Krr Jr In systems with several flows interacting, the resistance coefficients can reflect the extent of the interactions directly. Also, the resistance formulation utilizes the flows as independent variables, and often it is easier to measure and control the flows than the forces. The nonzero values of K1r and K2r indicate coupling between the ionic flows and the reaction. Since both Jr and A are scalars, while De mi and Ji are vectors, the coefficients Kir must be vectors. Consider that initially uncharged species M and O move into the cell, where M is reacted and transformed in to N. Some of the N flows out the cell. The transformation is mediated by the action of an enzyme confined to the interior of the cell. The component O does not take part in the transformation. However, the flow of O is coupled with the flow of M. The linear phenomenological equations are DmM ¼ KM JM þ KMO JO

(a)

DmN ¼ KN JN

(b)

DmO ¼ KMO JM þ KO JO

(c)

A ¼ Kri Jr

(d)

in

in in Here, J refers to outward fluxes, while DmM ¼ mex i  mi , and A refers to the affinity of the reaction in the cell. Eqs. (a)e(c) indicate the coupling between certain flows and lack of coupling between others. The metabolic reaction occurring in the cell is not coupled to any of the flows. After a certain time has passed the system reaches a state in which the concentrations of M and N, but not that of O become constant, so that in the stationary state we have

Jr ¼ JM ¼ JN If Aex is the affinity of the reaction measured externally, where the requisite enzyme is absent, we have    in  ex in Aex  Ain ¼ mex M  mN  mM  mN ¼ DmM  DmN

(e)

(f)

By substituting Eq. (e) into Eqs. (a)e(c), we get DmM ¼ KM Jr þ KMO JO

(g)

DmN ¼ KN Jr

(h)

DmO ¼ KMO Jr þ KO JO

(i)

A ¼ Kri Jr

(j)

in

The set of Eqs. (h)e(j) is combined with Eq. (f), and we have DmO ¼ KMO Jr þ KO JO

(k)

¼ KMO JO þ Kre Jr

(l)

ex

A

where Kre ¼ Kri  KM  KN . The only constraint is the fixing of Aex, and the system eventually reaches a state in which JO ¼ 0. In this state, an accumulation of O occurs, which is given by DmO ¼ KMO Jr ¼

KMO ex A Kre

The stationary state of the whole cell, represented by Eq. (e), yields a new dissipation function corresponding to Eqs. (k) and (l) J ¼ JO ð DmO Þ þ Jr Aex  0 In this case, a stationary-state coupling occurs between the flow of component O and the reaction. The coupling is generally a property of the membrane and is associated with enzymes that are an integral part of the membrane.

PROBLEMS

485

PROBLEMS 10.1 An aqueous solution (phase A) of 10 mmol/l of NaCl is in equilibrium across a protein-tight membrane with an aqueous solution (phase B) of NaCl and protein. The protein concentration is 1 mmol/l with a negative ionic valency of 10. Determine the difference in electrical potential and hydrostatic pressure across the membrane when both solutions are assumed to be ideal and the temperature is 25  C. 10.2 We want to separate carbon dioxide contaminating methane using a cellulose acetate membrane. The mixture is perfectly mixed on both sides of the membrane. The methane mole fraction in the feed (high pressure gas) is y(CH4) ¼ 0.70. The permeate pressure is 1.2 atm. At 35  C and 20 atm, the permeability of the membrane is p(CO2) ¼ 15.0  1010 and p(CH4) ¼ 0.48  1010 (cm3 (STP) cm)/(cm2 s cm Hg). The membrane thickness is tm ¼ 0.8 mm, and q ¼ 0.5. Estimate the membrane selectivity, permeant mole fraction yp(CO2), and flows of carbon dioxide and methane, J(CO2) and J(CH4). 10.3 We want to separate carbon dioxide contaminating methane using a cellulose acetate membrane. The mixture is perfectly mixed on both sides of the membrane. The methane mole fraction in the feed (high pressure gas) is y(CH4) ¼ 0.70. The permeate pressure is 2 atm. At 35  C and 20 atm, the permeability of me membrane is p(CO2) ¼ 15  1010 and p(CH4) ¼ 0.48  1010 [cm3 (STP) cm]/[cm2 s cm Hg]. The membrane thickness is tm ¼ 0.8 mm, and q ¼ 0.7. Estimate the membrane selectivity, permeant mole fraction yp(CO2), and flows of carbon dioxide and methane, J(CO2) and J(CH4). 10.4 A diffusion cell has an aqueous solution of NaCl with a concentration of 90 mmol/l. Later, 0.08 mmol radioactive Na with a specific activity of 1.0  108 units is added to chamber A, which has a volume of 1.0 and is stirred continuously. Measurements show that the radioactivity in reservoir A decreases at a rate of 14 units per minute. The process is at steady state. Estimate the flow of sodium ions, the diffusion coefficient, and the mobility at 298.15K and in a transfer area of 100 mm2. 10.5 The diffusion cell shown in Fig. 10.2 has NaCl mixtures in the two chambers with concentrations c1A ¼ 50 mmol/l and c1B ¼ 4 mmol/l. The mobilities of Naþ and Cl ions are different, and their ratio yields their transference numbers bþ/b ¼ tþ/t ¼ 0.39/0.61 (NaCl). The transference number t for an ion equals the fraction of the total electrical current carried by the ion when the mixture is subjected to an electrical potential gradient. For monovalent ions, we have tþ/t ¼ 1. Estimate the diffusion potential of the cell at steady-state conditions at 298K. Assume that the activity coefficients are equal in the two reservoirs. 10.6 Each chamber of the diffusion cell shown in Fig. 10.2 has an aqueous solution of NaCl with concentrations c1A ¼ c1B ¼ 80 mmol/l at 300K. An electrical potential difference of 90 mV is established between the two chambers. Estimate the diffusion flow of NaCl and its direction if D1 ¼ 1:48  109 m2/s. 10.7 A membrane, permeable to hydrogen ions but not to chloride ions, separates two aqueous solutions of HCl. The concentration of solution A is 15 mmol/l, while that of solution B is 2 mmol/l HCL. The two solutions have the same pressure (1 atm) and temperature (25  C). Estimate the electrical potential difference between the two solutions at equilibrium, as well

486

CHAPTER 10 MEMBRANE TRANSPORT

as the number of hydrogen ions that have moved across the membrane during the time taken to reach equilibrium. 10.8 The electrical potential difference between the interior and exterior of a cell is measured as 90 mV, with the cell interior negative, so that Dj ¼ jB  jA ¼ 90 mV. The cell interior and exterior ions and their concentrations are shown in Figure below. The activity coefficients for these ions are assumed to be the same in both phases. The temperature is uniform at 37  C. We want to estimate which of the three ions is closest to equilibrium. Membrane Interior; PhaseA Na+ = 12 mmol/l Cl- = 4 mmol/l K+ = 139 mmol/l

Exterior; PhaseB Na+ = 145 mmol/l Cl- = 116 mmol/l K+ = 4 mmol/l

10.9 Reconsider Example 10.9 with active transport of Naþ -ions from the intracellular to the extracellular space. Assume that 25% of the energy of hydrolysis of ATP can be used for the transport and that 38 mol of ATP are formed per mole of glucose (G) combusted. In the steady state, the extracellular concentration of Naþ is observed to be 145 mM and the intracellular concentration to be 12 mM, and the electrical potential of the intracellular fluid is determined to be 90.5 mV in relation to the extracellular space. The transcellular flux of Naþ ions is 1 mmol per minute and the temperature of the tissue is 37  C. Estimate the oxygen consumption, heat output and energy expenditure of the process when DHr;ATP ¼ 20 kJ/ mole  for hydrolysis  of ATP  and DHr;G ¼ 2867 kJ/mole for glucose combustion. Assume: q_2 q_1 ¼ 1; q_4 q_5 ¼ 4 5.    10.10 Reconsider Example 10.9 and justify the assumptions: q_2 q_1 ¼ 1; q_4 q_5 ¼ 4 5.

REFERENCES Bachelier, N., Chappey, C., Langevin, D., Metayer, M., Verchere, J.-F., 1996. J. Memb. Sci. 119, 285. Buschman, H.-J., Mutihac, L., Mutihac, R.., 1999. Sep. Sci. Tech, 34, 331. Calzado, J.A., Palet, C., Valiente, M., 2001. Anal. Chim. Acta 43, 159. Caplan, S.R., Essig, A., 2013. Bioenergetics and Linear Nonequilibrium Thermodynamics, the Steady State, second ed. Harvard University Press, Cambridge. Daoud, J.A., El-Reefy, S.A., Aly, H.F., 1998. Sep. Sci. Tech. 33, 537. Dung, H.H., Chen, C.H., 1991. J. Memb. Sci. 56, 327. Garby, L., Larsen, P.S., 1995. Bioenergetics: Its Thermodynamic Foundations. Cambridge University Press, Cambridge. Gumi, T., Oleinikova, M., Palet, C., Valiente, M., Munoz, M., 2000. Analytica Chim. Acta 408, 65. Ho, S.V., Sheridan, P.W., Krupetsky, E., 1996. J. Memb. Sci. 112, 13. Hwang, S.-T., 2004. AIChE J. 50, 862. Jeong, S.-H., Lee, K.-H., 1999. Sep. Sci. Tech. 34, 2383. Juang, R.-S., Lee, S.-H., Huang, R.-H., 1998. Sep. Sci. Tech. 33, 2379. Kargol, M., Kargol, A., 2003. Gen. Physiol. Biophys. 22, 51.

FURTHER READING

487

Katchalsky, A., Curran, P.F., 1967. Nonequilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge. Lee, S.C., Ahn, B.S., Lee, W.K., 1996. J. Memb. Sci. 114, 171. Liu, X., Liu, D., 1998. Sep. Sci. Tech. 33, 2597. Matsuyama, H., Teramoto, M., Sakakura, H., Iwai, K., 1996. J. Memb. Sci. 117, 251. Narebska, A., Koter, S., 1997. Polish J. Chem. 71, 1707. Narebska, A., Staniszewski, M., 1998. Sep. Sci. Tech. 34, 1455. Narebska, A., Warszawski, A., 1994. J. Memb. Sci. 88, 167. Narebska, A., Kujawski, W., Koter, S., Le, T.T., 1995. J. Memb. Sci. 106, 39. Nigon, C., Phalippon, J., Bonvin, C.F., Maisterrena, B., Memb, J., 1998. Sci. 144, 223. Nakano, T., Kawamoto, M., 1996. J. Memb. Sci. 113, 135. Nakano, T., Tkeda, T., Egawa, H., 1993. J. Memb. Sci. 76, 193. Pinnau, I., Toy, L.G., 2001. J. Memb. Sci. 184, 39. Quinn, R., Appleby, J.B., Pez, G.P., 1995. J. Memb. Sci. 104, 139. Rankumar, J., Maiti, B., Nayak, B., Mathur, P.K., 1999. Sep. Sci. Tech. 34, 2069. Selegny, E., Ghogoma, J.N., Roux, R., Langevin, D., Metayer, M., 1994. J. Memb. Sci. 93, 217. Selegny, E., Ghogoma, J.N., Langevin, D., Roux, R., Metayer, M., 1995. J. Memb. Sci. 108, 161. Selegny, E., Ghogoma, J.N., Langevin, D., Roux, R., Ripoll, C., 1997. J. Memb. Sci. 123, 147. Stucki, J.W., 1980. Eur. J. Biochem. 109, 269. Sungpet, A., Way, J.D., Koval, C.A., Eberhart, M.E., 2001. J. Memb. Sci. 189, 271. Uglea, C., Zanoaga, C.V., 1989. J. Memb. Sci. 47, 285. Wankat, P.C., 1994. Rate-controlled Separations. Chapman & Hall, Glasgow.

FURTHER READING Agarwal, A.K., Das, C., De, S., 2010. J. Memb. Sci. 360, 190. Ersoz, M., 2007. Adv. Colloid Interface Sci. 134, 96. Kumbasar, R.A., 2010. J. Hazard. Mat. 178, 875. Matsumoto, M., Yabushita, H., Kondo, K., 2009. Solvent Extraction Res. Dev. Jap. 16, 139. Murai, Y., Sadayuki, A., Yoshikawa, M., 2011. J. Memb. Sci. 380, 216.