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The Membrane Equilibrium with Chemical Reactions
CURRENT TOPICS IN MEMBRANES AND TRANSPORT,VOLUME 19
The Membrane Equilibrium with Chemical Reactions FRIEDRICH A. SAUER Max-Planck-Imtitut filr Biophysik Frankfurt, Federal Republic of Germany
I.
INTRODUCTION
Many b i o l o g i c a l membranes show a t i g h t c o u p l i n g between f l o w and chemical r e a c t i o n s , which o c c u r i n o r a t t h e membrane. I f t h i s coupling i s complete, t h e s t o p f l o w o r s t a t i c head s t a t e of t h e membrane s y s t e m In becomes a c o n s t r a i n e d thermodynamic e q u i l i b r i u m . t h i s a r t i c l e w e w i l l a n a l y z e t h i s e q u i l i b r i u m and comp a r e it w i t h t h e r e s u l t s of n o n e q u i l i b r i u m thermodynamics. W e d i s c u s s t h e thermodynamic e q u i l i b r i u m between two p h a s e s I and 'I s e p a r a t e d by a membrane which i s r i g i d and c a n n o t move. Phases and have t h e volumes V' and v " , r e s p e c t i v e l y ( F i g . 1 ) . Without loss of g e n e r a l i t y , w e assume t h a t t h e system ( b o t h p h a s e s and t h e membrane) have t h e common t e m p e r a t u r e T. Both p h a s e s c o n t a i n t h e component water ( d e n o t e d by s u b s c r i p t w t h r o u g h o u t ) , t h e s o l v e n t f o r which t h e membrane i s permeable. There are s o l u t e conponents of t h e f o l l o w i n g kinds: components k ( k = 1, n ) , f o r which t h e
...,
21
Copyright 0 1983 by Academic Press, Inc. All nghts of reproduction in any form reserved. ISBN 0-12-1533190
22
FRIEDRICHA. SAUER
V I
V
T
T
I1
F i g . 1 . A membrane i n contact with two homogeneous phases and Selective reversible electrodes are used t o measure electrochemical potential d i f f e r e n c e s .
'
".
...,
membrane i s permeable; components j ( j = j + 1, m), f o r which t h e membrane is impermeable; and components t ( t = m + 1, r ) , f o r which t h e membrane i s i m permeable. They react a t t h e s i d e of t h e membrane f a c u ) , for i n g t h e t phase. Components s (s = r + 1, which t h e membrane i s permeable o n l y i n complete c o u p l i n g w i t h t h e chemical r e a c t i o n . According t o Gibbs ( 1 9 6 1 , pp. 83-85) o n e g e t s t h e e q u i l i b r i u m c o n d i t i o n s from t h e v a r i a t i o n s of t h e Helmholtz f r e e e n e r g y F . More p r e c i s e l y , t h e f i r s t - o r d e r v a r i a t i o n o f F = F ' + F " around e q u i l i b r i u m must be zero, when v', v " , and T a r e k e p t c o n s t a n t and t h e system a s a whole i s c l o s e € o r m a t e r i a l exchange. T h i s means t h a t
...,
6F = 6F'
+
6F"
...,
= 0
(1.1)
o b s e r v i n g t h e above-mentioned c o n s t r a i n t s . Because t h e d e f i n e d system i s t o o complex, w e d i s c u s s it s t e p by s t e p , a t f i r s t l e a v i n g o u t many o f t h e components i n t r o duced above and d i s c u s s i n g i n d e t a i l t h e d i f f e r e n c e s between e l e c t r o l y t e s and n o n e l e c t r o l y t e s . A t t h e end w e come back t o t h e g e n e r a l e q u i l i b r i u m c a s e and d i s c u s s p o s s i b l e consequences.
MEMBRANE EQUILIBRIUM
11.
23
NONELECTROLYTE SOLUTION WITHOUT CHEMICAL REACTION
The s y s t e m u n d e r d i s c u s s i o n c o n t a i n s w a t e r ( w ) a s t h e s o l v e n t , t h e p e r m e a n t component ( k ) , and t h e impermeant component ( j ) . A l l components a r e assumed t o b e nonelectrolytes. The v a r i a t i o n s o f F ' and F " a r e g i v e n by
where t h e p are c h e m i c a l p o t e n t i a l s and t h e n a r e t h e mole numbers of t h e components. I t i s assumed t h a t 6 T = 0;
6V'
(2.2)
= 6V" = 0
and f o r t h e components ( j ) w e have 6n
I
j
= 6n
II
j
= 0
(2.3)
Because t h e s y s t e m as a whole i s c l o s e d , one h a s t h e conditions 6n
I
W
+
(1
6 n w = 0;
6n
1
k
+
11
6nk = 0
(2.4)
I n t r o d u c i n g Eqs. ( 2 . 1 ) and ( 2 . 4 ) i n t o Eq. ( l . l ) ,o n e g e t s t h e e q u i l i b r i u m c o n d i t i o n i n t h e form
If, as w i t h A b e i n g t h e d i f f e r e n c e between I and ' I . w a s assumed, t h e components ( k ) and water a r e i n d e p e n d e n t components, one c o n c l u d e s from Eq. ( 2 . 5 ) t h a t Apk = 0
( k = 1,
...,
n);
Auw = 0
(2.6)
FRIEDRICHA. SAUER
24
The e q u i l i b r i u m c o n d i t i o n s ( 2 . 6 ) e n a b l e u s t o c a l c u l a t e p r e s s u r e and composition o f t h e 'I p h a s e , i f p r e s s u r e and c o m p o s i t i o n o f t h e I phase and t h e concen phase t r a t i o n s of t h e impermeant components ( j ) i n t h e are known. T h i s u s u a l l y l e a d s t o t h e s i t u a t i o n t h a t o n e h a s t o s o l v e a system of t r a n s c e n d e n t a l e q u a t i o n s . In case t h e s o l u t i o n i s d i l u t e , one h a s 'k
-
'kO ( T , p)
+
I n ck
RT
(2.7)
where p i s t h e p r e s s u r e and C k t h e c o n c e n t r a t i o n . Then o n e c a n f i n d a n approximate s o l u t i o n of Eqs. ( 2 . 6 ) . R e w r i t i n g Eqs. ( 2 . 6 1 , one g e t s 1
I
Pk(P'
I
Cj)
CkI
-
1
Vk(P",
I
Ck,
c
.I
J
and
= Pw(P",
II
Ck,
II
c
.I J
-
PW(PflI c
I
I
k'
c.) J
(2.9)
E q u a t i o n s ( 2 . 8 ) and ( 2 . 9 ) are of t h e form
-
(AIJ)c=c'
(2.10)
-(AP)P=pll
I n t r o d u c i n g Eq. ( 2 . 7 ) i n t o Eq. ( 2 . 8 ) and n e g l e c t i n g t h e p r e s s u r e dependence of t h e p a r t i a l molar volumes vk gives
vk
I
II
AP = - R T l n ( c k / c k )
I n a s i m i l a r way t h e l e f t - h a n d s i d e of Eq. be e x p r e s s e d and one g e t s
vw
AP =
-
(AVw)p=pll
(2.11)
( 2 . 9 ) can (2.12)
where vW i s t h e p a r t i a l molar volume o f t h e w a t e r . F o r small c o n c e n t r a t i o n d i f f e r e n c e s t h e e q u a t i o n s c a n be s i m p l i f i e d f u r t h e r . I f (2.13)
MEMBRANE EQUILIBRIUM
Eq.
25
( 2 . 1 1 ) becomes 1
Vk
AP = -RT
A ck/ c k
...,
( k = 1,
n)
(2.14)
F o r d i l u t e s o l u t i o n s w e have v k c l < < 1. T h e r e f o r e , R T I A <~ < ~ A P~ f o r t h e p e r m e a n t chmponents k . Under t h e s e c o n d i t i o n s t h e Gibbs-Duhem r e l a t i o n become s
---fLTl C
w
(
n
1
Ack
k=l
rn
1
+
Acj)
(2.15)
j=n+l
Combination o f Eq. ( 2 . 1 5 ) w i t h Eq;. gives f o r the pressure difference
(2.12)
and ( 2 . 1 4 )
rn
+
(iVw
k = l c L V A~ p = R T
j=1 n+l
AC j
(2.16a)
or rn
( 1
-
j=n+l
1
= RT j=n+l A c j
c;V\)*.
(2.16b)
F o r d i l u t e s o l u t i o n s one h a s rn
and one g e t s
c
rn Ap = R T
j=n+1
Ac
j
which i s v a n ' t H o f f ' s l a w .
(2.17)
The c o n c e n t r a t i o n d i f f e r e n c e
FRIEDRICH A. SAUER
26
o f t h e p e r m e a n t components i s g i v e n by (2.18) j=n+l
ELECTROLYTE SOLUTION WITHOUT CHEMICAL REACTIONS
111.
Homogeneous e l e c t r o l y t e s o l u t i o n s must f u l f i l l t h e e l e c t r o n e u t r a l i t y condition: n
rn
1
eknk
+
k=l
1 e ~j.n j = n +1
=
o
(3.1)
where t h e e k and e . are t h e e l e c t r i c a l c h a r g e s p e r mole of t h e i o n s . T h a t ’ c o n d i t i o n h a s c o n s e q u e n c e s f o r t h e d e f i n i t i o n a n d measurement of t h e c h e m i c a l p o t e n t i a l s of t h e ions. ( G i b b s , 1961; Guggenheim, 1 9 6 7 ) . Every v a r i a t i o n of t h e mole numbers must f u l f i l l Eq. ( 3 . 1 ) a g a i n . Because of Eq. ( 3 . 1 ) rn - 1 i n d e p e n d e n t v a r i a t i o n s of t h e m o l e numbers a r e p o s s i b l e . Choosing t h e n t h i o n a s t h e key i o n , o n e g e t s en
6nn -
n-1
- 1
m
-
6 nk
ek
k= 1
1 j=n+l
e
j
6n
j
(3.2)
F o r t h e v a r i a t i o n of t h e Helmholtz f r e e e n e r g y a t cons t a n t T I v, and nW o n e g e t s n-1 6
~
1
=
k=l
( n1
vk
rn 6nk
+
1
j=n+l
v j( n ) 6 n j
(3.3)
where p i n ’ and p j n ) a r e t h e c h e m i c a l p o t e n t i a l s of t h e i o n s . They depend on t h e c h o i c e o f t h e key i o n c o r r e s p o n d i n g t o Eq. ( 3 . 2 ) . The c h e m i c a l p o t e n t i a l s of t h e i o n s so d e f i n e d are m e a s u r a b l e q u a n t i t i e s . Choosing a n o t h e r i o n 1 # n a s t h e key i o n , o n e g e t s
MEMBRANE EQUILIBRIUM
27
n
m
(3.4)
and n
rn (3.5)
Comparing E q s . ( 3 . 2 ) and ( 3 . 3 ) w i t h E q s . ( 3 . 4 ) and ( 3 . 5 1 , one g e t s t h e r e l a t i o n s between t h e c h e m i c a l pot e n t i a l s o f i o n s d e f i n e d f o r d i f f e r e n t key i o n s n and
(3.6)
(3.7)
(3.8)
(3.9)
( j = n
+
1,
...,
rn)
(3.10)
S i m i l a r c o n s i d e r a t i o n s must be a p p l i e d f o r t h e d e f i n i t i o n of t h e p a r t i a l molar volumes of i o n s . They depend on t h e c h o i c e of t h e key i o n . One g e t s t h e t h e r m o s t a t i c r e l a t i o n f o r t h e p a r t i a l molar volume v i n ) :
FRIEDRICH A. SAUER
28
(3.11) Now w e c o n s i d e r t h e v a r i a t i o n of t h e f r e e e n e r g y a r o u n d t h e e q u i l i b r i u m s t a t e and g e t f o r e a c h p h a s e and " n-1 k=l
(3.12)
n-1 k=l
for I
and w i t h t h e component n b e i n g t h e key i o n . Because t h e s y s t e m as a whole i s c l o s e d , w e have t h e c o n d i t i o n s 1
bnw
+
I1
6nw = 0;
1
6nk
I1
+
6nk = 0
(k =
1,
...,
n
-
1)
(3.14) Then t h e e q u i l i b r i u m c o n d i t i o n (1.1) h a s t h e form (3.15) From E q .
(3.15) we conclude t h a t
Apw = 0;
"k
(n) = 0
(k =
1,
...,
n
-
1)
(3.16)
These a r e t h e e q u i l i b r i u m c o n d i t i o n s f o r e l e c t r o l y t e sol u t i o n s , which a r e i n a c c o r d a n c e w i t h t h e e l e c t r o n e u t r a l i t y condition (3.1). To g e t t h e e l e c t r i c a l p o t e n t i a l d i f f e r e n c e s a t e q u i l i b r i u m , w e i n t r o d u c e p a i r s o f i d e n t i c a l , r e v e r si b l e e l e c t r o d e s i n t o t h e p h a s e s ' and " and p a s s a s m a l l amount of c h a r g e q a c r o s s t h e system. The e q u i l i b r i u m c o n d i t i o n (1.1) c h a n g e s t o 6F = 6F'
+
6F" = E l
69
(3.17)
MEMBRANE EQUILIBRIUM
29
f o r t h e permeant i o n s and t o
+
6 F = 6F'
6F" = E
j
(3.18)
6g
f o r t h e impermeant i o n s , Here E l and E j a r e t h e e l e c t r i c a l p o t e n t i a l d i f f e r e n c e s measured a t e q u i l i b r i u m w i t h
t h e h e l p o f r e v e r s i b l e e l e c t r o d e s f o r k and j , r e s p e c t i v e l y . E q u a t i o n s ( 3 . 1 7 ) and ( 3 . 1 8 ) e x p r e s s t h e f a c t t h a t t h e change o f t h e f r e e e n e r g y must be e q u a l t o t h e r e v e r s i b l e e l e c t r i c work done o n t h e system by c h a r g e t r a n s f e r . I f we t a k e a p a i r of e l e c t r o d e s r e v e r s i b l e f o r t h e l t h p e r m e a b l e i o n and t r a n s f e r a c h a r g e 6 q , we have f o r t h e v a r i a t i o n of t h e mole numbers of t h e impermeant i o n s 6n
j
= 6n
j
= 0
( j
= n + 1,
1
+
II
(k =
6nk = 0
1,
(3.19)
For t h e p e r m e a n t i o n s w e
because t h e system i s c l o s e d . have a g a i n 6nk
..., rn)
...,
(3.20)
n)
T h i s e q u a t i o n i s v a l i d a l s o f o r t h e l t h permeant i o n . Because of t h e charge t r a n s f e r , we g e t 1
6n 1 = 6 q / e l
+ 6nl ,
II
6nl = - 6 q / e l
-
-
6n 1
(3.21)
where 6 q / e l e q u a l s t h e amount o f component 1 coming from t h e e l e c t r o d e and 6nl i s t h e exchange of 1 v i a t h e membrane. Combination of Eqs. ( 3 . 2 1 ) l e a d s t o Eq. ( 3 . 2 0 ) . The v a r i a t i o n of t h e f r e e e n e r g y becomes n-1 6F'
+
6F"
=
1
A p k( n ) tin'
k
k= 1
+ ~u~
tin:
(3.22)
Because of t h e e q u i l i b r i u m c o n d i t i o n s ( 3 . 1 6 ) , t h i s e x p r e s s i o n i s z e r o and w e c o n c l u d e from Eq. ( 3 . 1 7 ) t h a t a t equilibrium E
1
= 0
( 1 = 1,
...,
n)
or A?
:e E = 0 1 1
( 1 = 1,
...,
n)
(3.24)
30
FRIEDRICHA. SAUER
where A q l i s c a l l e d t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e . T h i s means t h a t a t e q u i l i b r i u m t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e (ECPD) of a l l p e r m e a n t i o n s i s zero. The ECPD i s measured w i t h a p a i r of i d e n t i c a l r e v e r s i b l e e l e c t r o d e s . These e q u i l i b r i u m c o n d i t i o n s a r e a g a i n i n accordance w i t h t h e e l e c t r o n e u t r a l i t y c o n d i t i o n ( 3 . 1 ) . I f w e t a k e a p a i r of i d e n t i c a l revers i b l e e l e c t r o d e s f o r a n impermeant i o n , s a y j = y I t h e c o n d i t i o n (3.19) changes i n t o I
6n
Y
II
= 6q/ey;
6n
+
n-1
6F" =
(3.25)
= -6q/ey
( 3 . 3 ) we g e t f o r t h e v a r i a t i o n of t h e
and b e c a u s e of Eq. free energy 6F'
Y
1
1
A p k( n1 6 n k
+
I
Apw
6nw
+
Apy( n ) 6 q / e
k= 1
(3.26) Under e q u i l i b r i u m (3.16) t h i s becomes
and t h e ECPD f o r a n impearmeant i o n becomes
Taking a n o t h e r impermeant i o n , s a y m , as t h e key i o n , one g e t s b e c a u s e of Eq. ( 3 . 1 0 ) (3.29) P u t t i n g j = m and o b s e r v i n g t h a t A u E ) E 0 , one g e t s (3.30) Combining Eqs.
( 3 . 2 9 ) and ( 3 . 3 0 ) y i e l d s e .
Aqj
= Apjm)
+
e m An rn
(3.31)
MEMBRANEEQUILIBRIUM
31
a r e l a t i o n between ECPDs and c h e m i c a l p o t e n t i a l d i f f e r e n c e s o f i o n s . N o t e t h a t t h i s r e l a t i o n c o n t a i n s measurable q u a n t i t i e s only. I t should n o t be mistaken f o r t h e nonthermodynamic r e l a t i o n A n = Ap
+
(3.32)
e A$
where A $ i s t h e s o - c a l l e d "membrane p o t e n t i a l " and Ap i s a s i n g l e i o n chemical p o t e n t i a l d i f f e r e n c e . F i n a l l y , Eq. ( 3 . 3 1 ) i s v a l i d f o r any c h o i c e o f t h e key i o n . So w e have a l s o e
k + e
Ank = ApLn)
An
n
(3.33)
n
I n terms o f t h e ECPDs t h e e q u i l i b r i u m c o n d i t i o n s r e a d Apw = 0;
Ank = 0
( k = 1,
f o r t h e permeant components. meable i o n s are g i v e n by
...,
n)
(3.34)
The E C P D s f o r t h e imper(3.35)
i f Eqs. ( 3 . 3 4 ) are f u l f i l l e d and t h e key i o n i s a p e r meable i o n . I t s h o u l d b e mentioned t h a t t h e ECPDs f o r uncharged components i w i t h e i = 0 , b e c a u s e of Eq. ( 3 . 3 3 ) , g o o v e r So w e have i n t o t h e chemical p o t e n t i a l d i f f e r e n c e s A p i . Ani
(3.36)
= Api
= Apln)
f o r ei E 0 . Coming back t o t h e e q u i l i b r i u m c o n d i t i o n s f o r t h e p e r m e a b l e i o n s ( 3 . 1 6 1 , w e found t h a t "k
(n)
= 0
(k =
1,
...,
n
-
1)
(3.37)
i f n ( t h e k e y i o n ) i s a l s o permeant. Choosing a n impermeant i o n s a y m , a s t h e key i o n , Eq. ( 3 . 3 7 ) c h a n g e s i nt o
"k
(m
_
- ee-
m
(3.38)
FRIEDRICH A. SAUER
32
U s e h a s been made of Eq.
3.10). O t h e r w i s e Eq. ( 3 . 3 7 ) r e m a i n s form i n v a r i a n t a s l o n g a s t h e key i o n i s a p e r meable i o n . Summarizinq t h e r e s u t s of t h e f o r e g o i n g c o n s i d e r a t i o n s , we f o u n d - t h e e q u i l i b r i u m c o n d i t i o n s Avw = 0 ;
and p h a s e s
( k = 1,
(n) = 0
and
n
...,
n
-
1)
(3.39)
obey t h e e l e c t r o n e u t r a l i t y c o n d i t i o n
'I
rn
l e k n k + k=l
1
j = n +1
(3.40)
e n = O j
j
A t e q u i l i b r i u m t h e s e c o n d i t i o n s are v a l i d r e g a r d l e s s o f
whether w e p u t e l e c t r o d e s i n t o t h e p h a s e s ' and I' o r n o t . A c t i v a t i n g one more d e g r e e of freedom by c h a r g e t r a n s f e r with t h e h e l p of r e v e r s i b l e e l e c t r o d e s [because of Eq. ( 3 . 4 0 ) t h i s i s i m p o s s i b l e w i t h o u t e l e c t r o d e s ] , w e f i n d f o r t h e ECPDs t h a t
Ank=o
(k=l,
..., n ) ;
A n j = A p j( n )
( j = n + l ,
..., rn) (3.41)
T o c a l c u l a t e p r e s s u r e and c o m p o s i t i o n o f t h e I' p h a s e , if p r e s s u r e and c o m p o s i t i o n of t h e I p h a s e and t h e c o n c e n t r a t i o n s of t h e impermeable components j i n t h e I' p h a s e are g i v e n , w e p r o c e e d a s w e d i d i n S e c t i o n 11. For d i l u t e e l e c t r o l y t e s o l u t i o n s w e make t h e Ansatz
From Eqs.
vW
( 3 . 3 9 ) and ( 3 . 4 2 ) w e g e t
AP = - ( A ' w ) P = p "
For small c o n c e n t r a t i o n d i f f e r e n c e s w e g e t
(3.44)
MEMBRANE EQUILIBRIUM
33
(3.45) Making u s e of Eqs. we f i n d
(3.39) and t h e Gibbs-Duhem e q u a t i o n ,
n-1
= c
I
(n)
‘kV c k1 Ap
k= 1
W
rn
- -RC TI w
1 j=n+l
( c j
-2GAc) n c
n
(3.46) and t h e r e f o r e from Eq.
(3.44)
F o r d i l u t e s o l u t i o n s , b e c a u s e of
(3.47)
(3.48) N e g l e c t i n g t e r m s o f t h e form
FRIEDRICH A. SAUER
34
and making u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n s , (3.48) goes over i n t o
Eq.
n Ap =
1
n
rn
1
e k ( e k - e .)c' J k
AC
k=l
j
1
(3.49)
T h i s i s t h e g e n e r a l i z a t i o n of v a n ' t H o f f ' s l a w f o r electrolyte solutions. Taking j u s t one permeable i o n ( n = 1) and one i m p e r m e a b l e F i o n (rn = 2 1 , E q . ( 3 . 4 9 ) becomes Ap = R T ( 1
-
(3.50)
e2/el)Ac2
F o r t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e of t h e j t h impermeant i o n w e g e t from E q . ( 3 . 2 8 )
(j = n
+ 1,
..., rn)
(3.51)
Combining E q s . ( 3 . 4 5 ) and o b s e r v i n g t h e e l e c t r o n e u t r a l i t y c o n d i t i o n , o n e ge t s
(3.52) I n t r o d u c t i o n of E q .
(3.52) i n t o Eq.
(3.51) g i v e s
(3.53) I n t h e s p e c i a l c a s e of E q .
(3.50) one g e t s
MEMBRANE EQUILIBRIUM
35
(3.54) I f a g a i n c '2 ~ 2
< < 1, t h e n AC
IV.
2
(3.55)
NONELECTROLYTE SOLUTION W I T H CHEMICAL REACTIONS
I n a d d i t i o n t o t h e components k and j , w e i n t r o duce components t , which c a n n o t p e r m e a t e t h e membrane b u t r e a c t a t t h e s i d e of t h e membrane f a c i n g t h e ' p h a s e according t o
where t h e v t arg! t h e s t o i c h i o m e t r i c numbers. change of t h e n t i s g i v e n by 1
6n
t
= v
t
65
( t = rn
+
1,
...,
Every (4.2)
r)
where 6 5 i s t h e change i n t h e p r o g r e s s of t h e r e a c t i o n . Because t h e t-components c a n n o t p e r m e a t e t h r o u g h t h e membrane, w e have II
An, = 0
( t = rn
+ 1,
...,
(4.3)
r)
F u r t h e r m o r e , we have components s which c a n p e r m e a t e t h e membrane o n l y i f 6 6 i s u n e q u a l t o z e r o . The s - c o m p o n e n t s do n o t undergo a c h e m i c a l change b u t a r e t r a n s f e r r e d a c r o s s t h e membrane. For a c l o s e d s y s t e m w e have 1
6ns
-
-
6n
It
S
= ts 6 5
(s = r
+
1,
...,
u)
(4.4)
FRIEDRICH A. SAUER
36
Here t h e t , a r e t h e t r a n s f e r e n c e numbers. They s h o u l d n o t be confused w i t h t h e s t o i c h i o m e t r i c numbers v t , because t h e components s a r e c o n s e r v e d ( s remains s ) . F o r t h e change of t h e f r e e e n e r g y i n b o t h p h a s e s w e get n
r
U
and
(4.5) n 6F"
U 11
=
s=r+l
k=l
U s e h a s been made of E q s .
tsuz
65
+ vw
and ( t h e system i s c l o s e d and j i s impermeant). rium w e have 6F'
+
(4.41,
6F" = 0
and t h e r e f o r e n
1
(4.31,
'Pk
6 n k1
k=l
II
6 nw I
n j
-
n
II
= O
A t equiiib-
(4.6)
+(
t s Aps
-
+
.)tic
Apw 6 n i =
0
s=r+l
(4.7)
F o r t h e c l o s e d system i t w a s assumed 6n
I
k
+
II
6nk = 0;
6n
I
W
+
II
6nw = 0
Because 6 n k , f n w r and 6 5 a r e i n d e p e n d e n t v a r i a t i o n s , w e g e t the equilibrium conditions Apk = 0
(k = 1,
..., n ) ;
Allw=
0
U
1
s=r+l
t s Aps
-
A'
= 0
(4.9)
MEMBRANE EQUILIBRIUM
37
Here (4.10)
i s t h e D e Donder a f f i n i t y of t h e c h e m i c a l r e a c t i o n ( 4 . 1 ) . To c a l c u l a t e t h e p r e s s u r e and t h e c o m p o s i t i o n of t h e 'I p h a s e f o r a g i v e n p r e s s u r e and c o m p o s i t i o n of t h e I phase, t h e e q u i l i b r i u m c o n d i t i o n s are incomplete i f t h e number o f s-components i s l a r g e r t h a n o n e . One g e t s
t h e a d d i t i o n a l e q u a t i o n s by i n t e g r a t i o n of Eqs. ( 4 . 4 ) . T h i s i n t e g r a t i o n s t a r t s w i t h an a r b i t r a r y n o n e q u i l i b r i u m s t a t e 0 a n d g o e s t o t h e r e a l e q u i l i b r i u m s t a t e 5,. Therefore 11
II
n S - n so = - tS
-
w i t h A t = 5, 50. c o n s t a n t , one g e t s c
11
s
- c
II
SO
= - -t S V I#
A5
(s = r
+
1,
...,
A f t e r d i v i s i o n by
At
(s
= r
+
1,
(4.11)
U)
v",
which i s k e p t
..., u )
(4.12)
From Eqs. ( 4 . 1 2 ) one sees t h a t t h e e q u i l i b r i y m d i s t r i b u t i o n d e p e n d s on t h e i n i t i a l c o n c e n t r a t i o n s c s o I i f t h e number o f s-components i s l a r g e r t h a n o n e . E l i m i n a t i o n o f A 5 i n Eqs. ( 4 . 1 2 ) g i v e s II
c
s
II
= c
s o + -t u
uo
(s
= r
+
1,
...,
u
-
1)
(4.13) The s e t of e q u a t i o n s ( 4 . 1 3 ) t o g e t h e r w i t h t h e e q u i l i b rium c o n d i t i o n s ( 4 . 9 ) e n a b l e one t o c a l c u l a t e t h e e q u i l i b r i u m p r e s s u r e and c o m p o s i t i o n o f t h e I' p h a s e . In g e n e r a l t h i s l e a d s t o t h e s o l u t i o n of a system of t r a n s cendental equations. I n t h e following we g i v e an approximate s o l u t i o n for t h e s p e c i a l case of a d i l u t e s o l u t i o n . The compon e n t s w i l l b e w a t e r (w), one s-component ( s = r + 1 1 , and t h e r e a c t a n t t ( t = m + 1, , r ) Because t h e r e a c t a n t s do n o t p e r m e a t e a c r o s s t h e membrane, t h e i r c o n c e n t r a t i o n s are p r e s c r i b e d . l , T ~d e t e r m i n e t h e p r e s s u r e p " and t h e c o n c e n t r a t i o n c ~ + w ~ e , have t h e equations
. ..
.
FRIEDRICH A. SAUER
38
Under t h e assumption of a d i l u t e s o l u t i o n t h i s l e a d s t o I
r
cw Vw AP = - c w ( A ~ w ) p = p = ,l R T (
If E q .
c
1
Act +
A'Cr+l
t=m+l
1
(4.15)
1
( 2 . 7 ) i s v a l i d and cWvw > > c ~ + ~ V ~ one + ~ g, e t r
1
Ap = R T
t=m+l
AC
+
t
1
Acr+l
Cr + l
A'
I
C1
A1/tr+l
r+l
(
Act
t=m+l
+
'Ic'+')
RTtr+l
(4.16)
T h i s means t h e o s m o t i c p r e s s u r e d i f f e r e n c e Ap depends On A ' . I f 1 >> c we g e t r + l'r+1'
1
(4.17) F o r more t h a n o n e s-component w e make u s e of E q s . and rewrite them i n t h e form Acs
= AcsO
+tS
(Acu
-
Acuo)
(s = r
+
1,
(4.13)
..., u )
tU
(4.18) where t h e l a s t e q u a t i o n f o r s = u i s a n i d e n t i t y . The e q u i l i b r i u m c o n d i t i o n s f o r d i l u t e s o l u t i o n s and s m a l l c o n c e n t r a t i o n d i f f e r e n c e s Acs and A c t go o v e r i n t o
MEMBRANE EQUILIBRIUM
39
(4.19) and U
U
1
t V Ap s s
+
RT
tS i-
s=r+l
s=r+l
Combination of Eqs.
C
A c S = A'
(4.20)
s
(4.18) and (4.20) leads to
U
-
1
(4.21) C
s=r+l
S
The contribution of the s-components to the osmotic pressure difference becomes then
-
2 -1
s=r+1
R*(
U
A' + -R T
c
Cs
ts J
U
c
s = r + l v=r+l U
[
1
s=r+l v = r + l tvf
-
>)Ac CS
1
s=r+l
Here we assume that U
2)
U
tstvV" <<
1
(4.22)
so
FRIEDRICHA. SAUER
40
V.
ELECTROLYTE SOLUTION W I T H CHEMICAL W A C T I O N S
Assuming t h a t t h e r e i s a t l e a s t one i o n of t h e t y p e k p r e s e n t which can permeate t h e membrane i n d e p e n d e n t l y of t h e r e a c t i o n , w e t a k e t h i s i o n as t h e key i o n . From t h e f o r e g o i n g p a r a g r a p h s w e f i n d t h e e q u i librium conditions
"k
(n) = 0
(k =
1,
..., n -
1 ) ; Apw = 0
s=r+l
where n i s t h e key i o n . Acs = Acso
+
tS
Together w i t h
(Ac
U
U
-
Acuo)
(s = r
+ 1,
..., u ) (5.21
Eqs. ( 5 . 1 ) allow o n e t o c a l c u l a t e t h e e q u i l i b r i u m d i s t r i b u t i o n o f t h e components and t h e p r e s s u r e i f t h e I phase i s g i v e n . For t h e ECPDs w e g e t
(s = r
+
1,
...,
u)
(5.3)
T h e r e f o r e , w e c a n w r i t e t h e g e n e r a l i z e d e q u i l i b r i u m cond i t i o n s i n t h e form
U
1
ts A n s
-
A'
= 0
(5.41
s=r+l
T h i s h a s been d e r i v e d u n d e r t h e c o n d i t i o n t h a t t h e r e e x i s t s a n i o n which can permeate i n d e p e n d e n t l y of t h e chemical r e a c t i o n . I f t h i s i s n o t t h e case, w e have t o d i s t i n g u i s h two s i t u a t i o n s .
MEMBRANE EQUILIBRIUM
41
L e t us assume t h e o n l y i o n s p r e s e n t a r e t h o s e o f W e take t h e ion w i t h t h e index u as t h e the s-type. key i o n . For t h e v a r i a t i o n o f t h e f r e e e n e r g y one g e t s u-1
1
bF =
(u)
ts
' -
A'
6ns
65
+
Apw
I
6nw
(5.5)
If t h e system a s a whole i s c l o s e d (no e l e c t r o d e s )
have
I
tins = t
65
S
(s =
r
+ 1,
...,
u
-
1)
,
we
(5.6)
and b e c a u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n u-1
I
=
6n
c
-
eS
e ts
(5.7)
6E
s=r+l
A s on t h e o t h e r s i d e u i s o f t h e s - t y p e ,
w e have
65
6nU = t
(5.8)
ni
To g e t v a r i a t i o n s of t h e i n t h e closed system w i t h o u t v i o l a t i n g the e l e c t r o n e u t r a l i t y condition, U
1
e t
s s
= O
(5.9)
s=r+l
m u s t be f u l f i l l e d . l I f Eq. ( 5 . 9 ) i s n o t f u l f i l l e d , v a r i a t i o n s of t h e n s a r e p o s s i b l e o n l y by means o f e l e c t r o d e s . Assuming t h e v a l i d i t y o f E q . ( 5 . 9 ) , one g e t s the equilibrium conditions u-1
1
A v w = 0;
t s A p s( u )
-
A'
= 0
(5.10)
s=r+l
Together w i t h the material c o n d i t i o n s Ac
S
= Ac
so
tS + tU
(AcU
-
Ac
uo
)
(s =
r
+ 1,
...,
u
-2)
(5.11)
FRIEDRICH A. SAUER
42
t h e e q u i l i b r i u m d i s t r i b u t i o n i s determined. Under t h e c o n d i t i o n ( 5 . 9 ) it i s i m p o s s i b l e t o p a s s e l e c t r i c a l c h a r g e a c r o s s t h e membrane w i t h t h e h e l p of r e v e r s i b l e electrodes. L e t u s assume w e have a p a i r o f e l e c t r o d e s r e v e r s i b l e t o t h e i o n u . Then w e have (5.12) 1
6nS
--
From Eqs.
ts 65
s
#
(5.13)
u
( 5 . 1 2 ) and ( 5 . 1 3 ) w e g e t
U
U
1
es 6 n i = 6q
+
s=r+l
1
t s e s 65
(5.14)
s=r+l
Because of Eq. t i o n we g e t
( 5 . 9 ) and t h e e l e c t r o n e u t r a l i t y c o n d i (5.15)
6q = 0
T h i s means t h a t it i s i m p o s s i b l e t o p a s s e l e c t r i c a l c h a r g e a c r o s s t h e membrane. T h e r e i s no e l e c t r i c cont a c t between b o t h p h a s e s and no ECPD i s d e f i n e d . On the o t h e r side i f w e release the condition (5.91, w e have from Eq. ( 5 . 1 4 ) (5.16) I f 6 q = 0 ( c l o s e d s y s t e m ) . I f c:=r+l t e # 0, we f i n d 6 5 = 0 b e c a u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n . T h i s means no v a r i a t i o n of t h e mole numbers are poss i b l e i n t h e c l o s e d system. For an open system ( w i t h e l e c t r o d e s ) one g e t s from Eq. (5.14)
c U
6q =
Because of
-
s=r+l
tses 65
(5.17)
MEMBRANE EQUILIBRIUM
43
6q e
(5.18)
6F =
U
t h e e q u i l i b r i u m c o n d i t i o n becomes
(5.19) and t h e r e f o r e
(5.20) Making u s e of E q .
( 3 . 3 3 1 , one g e t s
(5.21) E q u a t i o n s ( 5 . 2 0 ) c a n be u s e d t o c a l c u l a t e t h e p r e s s u r e d i f f e r e n c e Ap and t h e ECPD A n u , which b u i l d up a t e q u i l i b r i u m . Because C;=r+l e t # 0 and no o t h e r p e r m e a t i n g i o n s a r e p r e s e n t , t h e d i s t r i b u t i o n o f t h e comp o n e n t s w i l l n o t change. I n t h e s p e c i a l case t h a t Acs - AcsO = 0 w e g e t
(5.22)
and Ans
= Anu es/eu
( s = r + 1,
.. .,
u
-
1)
(5.23)
I n t h e g e n e r a l c a s e , when a l l t h e d i f f e r e n t k i n d s o f components ( k , j , t , s ) a r e p r e s e n t , o n e g e t s t h e e q u i librium conditions
FRIEDRICH A. SAUER
44
Apw
= 0;
"k
..., n
( k = 1,
(n) = 0
-
1)
U
(5.24) s=r+l
where n i s t h e k e y i o n . T o g e t h e r w i t h Eq. ( 4 . 1 8 ) I Eqs. ( 5 . 2 4 ) d e t e r m i n e t h e e q u i l i b r i u m d i s t r i b u t i o n of p r e s s u r e and components. F o r t h e ECPDs o n e g e t s ( k = 1,
Aqk= 0
... , n ) ;
(t=m
+
1,
Aqs
- A y s( n )
..., r ) ; ( j = n
+
A
(s=r ~
1,
~
+ 1,
= (n A) V
..., rn)
..., u )
~
(5.25)
With t h e h e l p of Eqs. ( 5 . 2 5 ) t h e " c h e m i c a l " e q u i l i b r i u m c o n d i t i o n c o u l d be w r i t t e n i n t h e form U
(5.26)
I n t h e f o l l o w i n g w e compare t h i s r e s u l t of e q u i l i b r i u m thermodynamics w i t h t h e s t o p - f l o w s i t u a t i o n o f nonequil i b r i u m thermodynamics. For s i m p l i c i t y w e assume t h a t o n l y s- and t-components are p r e s e n t . Then we g e t f o r t h e s-components U II
Js =
w=r+l
'sh
Anh
+ Ls 5
A'
(s = r
+
1,
...,
u)
(5.27) and f o r t h e r e a c t i o n r a t e
J
5
U
J
5
=
1
h=r+l
L S h Anh
+
L
55
A'
(5.28)
For t h e t i m e b e i n g w e release t h e c o n d i t i o n o f c o m p l e t e coupling. Then t h e c o n d i t i o n s of u n c o n s t r a i n e d thermodynamic ' e q u i l i b r i u m are .
MEMBRANE EQUILIBRIUM
45
II
J s = 0;
J
5
= 0
(s
= r
+
...,
1,
u)
(5.29)
From t h a t w e c o n c l u d e i f t h e Onsager m a t r i x i s nons i n g u l a r (independent flows) t h a t = 0;
A'
Anh = 0
a t equilibrium.
#
J~
0;
+
(h = r
1,
...,
u)
(5.30)
I n t h e stop-flow s i t u a t i o n
J~ = 0
(s =
r
+
1,
...,
U )
(5.31)
we g e t U
(5.32) o r i n vector notation (5.33) where Lo h a s t h e e l e m e n t s L s h and t h e v e c t o r L5 h a s t h e components ~ c = ; L, ~ assuming Onsager r e c i p r o c i t y . Here t h e s u p e r s c r i p t "sf" d e n o t e s s t o p flow. F u r t h e r Introduction more, w e assume t h a t Lo i s n o n s i n g u l a r . of t h e v e c t o r t of t h e t r a n s f e r e n c e numbers d e f i n e d by
... , n ) (5.34) leads t o
(5.35) The r e a c t i o n r a t e a t s t o p f l o w i s g i v e n by
FRIEDRICHA. SAUER
46
(5.36)
55
s=r+l
Because t h e t o t a l Onsager m a t r i x is p o s i t i v e d e f i n i t e , w e have (5.37) T h i s means t h a t t h e f a c t o r i n f r o n t of A ' i n Eq. ( 5 . 3 5 ) is positive. From t h e p o s i t i v e v a l u e o f t h e d e t e r m i n a n t of t h e t o t a l Onsager m a t r i x o n e can show t h a t t h i s f a c t o r i s less t h a n u n i t y . The p r o o f g o e s a s f o l l o w s . W e have t h e r e l a t i o n ( d e t Ltot)
=
5
)
(5.38)
[ H i n t : One g e t s t h i s r e l a t i o n by e x p a n s i o n o f ( d e t & , t o t ) a b o u t t h e e l e m e n t s o f t h e l a s t row.] Because ( d e t Ltot)
0
and ( d e t L ) are p o s i t i v e , w e h a v e
T 0-laL 0 < (L5*L
5)'L5s
< 1
(5.39)
Taking t h e a b s o l u t e v a l u e of Eq. ( 5 . 3 5 ) and making u s e of i n e q u a l i t y ( 5 . 3 9 ) , one g e t s t h e i n e q u a l i t y (5.40) From r e l a t i o n s ( 5 . 3 5 ) and (5.39) one c o n c l u d e s t h a t A ' and E T * A n S f have t h e same s i g n . T h e r e f o r e , w e c a n w r i t e two p o s s i h e i n e q u a l i t i e s : (5.41) or
MEMBRANE EQUILIBRIUM
47
(5.42) s=r+l
d e p e n d i n g on t h e s i g n of A ' . The i n e q u a l i t i e s (5.41) and (5.42) a r e d i r e c t c o n s e q u e n c e s of t h e s e c o n d law o f thermodynamics. Comparing t h i s r e s u l t w i t h Eq. (5.36) I w e come t o t h e c o n c l u s i o n t h a t €or an i n c o m p l e t e c o u p l e d s y s t e m ( a l l f l o w s i n c l u d i n g J~ a r e i n d e p e n d e n t ) a t s t o p flow i n general
#
Jsf
5
(5.43)
0
For a c o m p l e t e l y c o u p l e d s y s t e m i n n o n e q u i l i b r i u m t h e c o n d i t i o n (4.4) g o e s o v e r i n t o II
J~ = -t
J
s 5
(s = r
+
1,
. . . I
u)
(5.44)
i n e a c h s t a t e of t h e s y s t e m and n o t o n l y i n l e v e l flow. T h i s means t h e f l o w s a r e d e p e n d e n t now and t h e t o t a l Onsager m a t r i x becomes s i n g u l a r . Comparing Eqs. (5.44) w i t h Eqs. (5.27) and (5.281, one g e t s 0
(5.45)
L s h = Ls5Lh(/L5E
and
J5 = L5
5
(4'
-
s=r+l
is
Ans)
which f u l f i l l s c o n d i t i o n (5.44). F o r a c o m p l e t e l y c o u p l e d s y s t e m o n e h a s one i n d e p e n d e n t f l o w and o n e i n d e p e n d e n t d r i v i n g f o r c e o n l y . The f l o w m i g h t be t h e J 5 and t h e d r i v i n g f o r c e i s t h e n
A'
is
s=r+l
Aqs
(5.47)
FRIEDRICHA. SAUER
40
The s t o p - f l o w s i t u a t i o n c o i n c i d e s w i t h t h e c o n s t r a i n e d thermodynamic e q u i l i b r i u m . A t s t o p f l o w w e have
and t h e c o n d i t i o n f o r t h e d r i v i n g f o r c e (5.47) becomes U
A'
-
1
(5.49)
is
s=r+l
t h e same c o n d i t i o n a s w a s d e r i v e d ,at Eq. ( 5 . 4 ) . Then, and o n l y t h e n , t h e i n e q u a l i t i e s ( 5 . 4 ) and ( 5 . 4 2 ) become equalities. I f one u s e s Eq. (5.49) € o r t h e e v a l u a t i o n o f t h e a f f i n i t y A ' o f t h e d r i v i n g chemical r e a c t i o n , one must be s u r e t h a t t h e system is c o m p l e t e l y c o u p l e d . T h i s means Eqs. (5.44) must b e f u l f i l l e d i n e a c h s t a t e o f t h e system. O t h e r w i s e , Eq. (5.35) must be used. Here t h e knowledge o f t h e Onsager m a t r i x i s needed. R e w r i t i n g Eq. (5.351, one g e t s
L
55
c s=r+l
(5.50) 0-1
tsthLsh h=r+t
I f o n l y some of t h e f l o w s are c o m p l e t e l y c o u p l e d , t h e stop-flow s t a t e a g a i n c o i n c i d e s w i t h t h e c o n s t r a i n t e q u i l i b r i u m . L e t u s assume w e have s-components (5.51)
...,
and k-components ( k = 1, n ) which are n o t completel y c o u p l e d . Then t h e s t o p - f l o w c o n d i t i o n s II
Js =
0
(s = r
+
1,
...,
u);
II
Jk = 0
( k = 1,
..., n ) (5.52)
because of Eqs.
(5.51) imply
MEMBRANE EQUILIBRIUM
J
5
49
= o
(5.53)
T h i s means c o n s t r a i n e d e q u i l i b r i u m w i t h t h e e q u i l i b r i u m conditions Ank = 0
( k = 1,
...,
U
n);
A'
-
1
is
Ans
= 0
s=r+l
(5.54) T h e r e w i l l be no b u i l d u p of E C P D s f o r t h e k-components. T h i s a l s o happens i n t h e c a s e of n o n v a n i s h i n g Lk It s h o u l d be mentioned t h a t s u c h a s y s t e m , n e v e r t h e i e s s , shows a l e v e l f l o w
.
I n a system f o r which p a r t s o f i t a r e c o m p l e t e l y c o u p l e d , t h e s t o p - f l o w e x p e r i m e n t g i v e s no i n d i c a t i o n f o r t h e a c t i v e t r a n s p o r t of t h e incompletely coupled components. C o n s i d e r a t i o n s b a s e d on the i n e q u a l i t i e s ( 5 . 3 7 ) a l l o w e d Kedem and Caplan ( 1 9 6 5 ) t o d e r i v e t h e i r r e s u l t s on thermodynamic e f f i c i e n c y of membrane s y s t e m s w i t h active transport.
VI.
SUMMARY
U s u a l l y components which p e r m e a t e a c r o s s a memb r a n e do n o t c o n t r i b u t e t o t h e o s m o t i c p r e s s u r e d i f f e r ence a t theryodynamic e q u i l i b r i u m i n d i l u t e s o l u t i o n s . As l o n g as ckvk i s s m a l l compared t o u n i t y , t h e i r cont r i b u t i o n c a n be n e g l e c t e d . T h a t i s n o t t r u e f o r components which are completel y c o u p l e d t o a chemical r e a c t i o n . These components c a n b u i l d up l a r g e r d i f f e r e n c e s o f t h e i r c h e m i c a l potent i a l s , which t h e n c o n t r i b u t e t o t h e o s m o t i c p r e s s u r e d i f f e r e n c e . For t h i s osmochemical e q u i l i b r i u m o n e g e t s Eq. ( 4 . 1 6 ) :
50
FRIEDRICH A. SAUER
T h i s means t h a t t h e a f f i n i t y A' of t h e chemical r e a c t i o n g i v e s an a d d i t i o n a l c o n t r i b u t i o n t o t h e osmotic p r e s The s u r e depending on t h e t r a n s f e r e n c e number t,+l. s i g n of t h e t e r m w i t h A' depends o n t h e s i g n of t h e t r a n s f e r e n c e number and t h e s i g n of A' i t s e l f . T h i s res u l t can e a s i l y be e x t e n d e d f o r membranes where t h e r e a c t i n g components p e r m e a t e t h e membrane and o t h e r i m permeant components j a r e p r e s e n t . W e g e t them t h e osm o t i c p r e s s u r e d i f f e r e n c e between d i l u t e s o l u t i o n s
By a d j u s t m e n t of t h e A c j i t i s p o s s i b l e t o v a r y t h e Ap. I f t h e r e i s more t h a n one component p r e s e n t which i s c o m p l e t e l y c o u p l e d , w e g e t memory e f f e c t s i n t h e osmot i c pressure. T h i s means t h e f i n a l osmotic p r e s s u r e d i f f e r e n c e depends on t h e i n i t i a l c o n c e n t r a t i o n d i f f e r of t h e c o m p l e t e l y coupled components. F o r ences A c t h a t comB8re Eq. (4.22). The w i d e l y used r e l a t i o n o f Eq. (5.49):
which r e l a t e s t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e s i n a s t o p - f l o w s t a t e w i t h t h e a f f i n i t y A ' o f t h e metab o l i c r e a c t i o n , i s r i g o r o u s l y v a l i d i f t h e components are c o m p l e t e l y c o u p l e d t o t h e chemical r e a c t i o n . Then t h e s t o p - f l o w s t a t e becomes a c o n s t r a i n e d thermodynamic e q u i l i b r i u m . For i n c o m p l e t e l y c o u p l e d systems Eq. (5.49) must be r e p l a c e d by t h e i n e q u a l i t y (5.40) :
which i s a consequence o f t h e second l a w o f thermodynamics.
MEMBRANE EQUlLlBRlUM
51
REFERENCES Gibbs, J. W. (1961). "The S c i e n t i f i c Papers," pp. 83-85, 331-349, 4 06-4 12. Guggenheim, E. A, (1967). "Thermodynamics," Chap. 8. North Holland, Amsterdam. Kedem, O., and Caplan, S. R. (1965). T r a n s . F a r a d a y SOC. 6 1 , 1897.
The Membrane Equilibrium with Chemical Reactions FRIEDRICH A. SAUER Max-Planck-Imtitut filr Biophysik Frankfurt, Federal Republic of Germany
I.
INTRODUCTION
Many b i o l o g i c a l membranes show a t i g h t c o u p l i n g between f l o w and chemical r e a c t i o n s , which o c c u r i n o r a t t h e membrane. I f t h i s coupling i s complete, t h e s t o p f l o w o r s t a t i c head s t a t e of t h e membrane s y s t e m In becomes a c o n s t r a i n e d thermodynamic e q u i l i b r i u m . t h i s a r t i c l e w e w i l l a n a l y z e t h i s e q u i l i b r i u m and comp a r e it w i t h t h e r e s u l t s of n o n e q u i l i b r i u m thermodynamics. W e d i s c u s s t h e thermodynamic e q u i l i b r i u m between two p h a s e s I and 'I s e p a r a t e d by a membrane which i s r i g i d and c a n n o t move. Phases and have t h e volumes V' and v " , r e s p e c t i v e l y ( F i g . 1 ) . Without loss of g e n e r a l i t y , w e assume t h a t t h e system ( b o t h p h a s e s and t h e membrane) have t h e common t e m p e r a t u r e T. Both p h a s e s c o n t a i n t h e component water ( d e n o t e d by s u b s c r i p t w t h r o u g h o u t ) , t h e s o l v e n t f o r which t h e membrane i s permeable. There are s o l u t e conponents of t h e f o l l o w i n g kinds: components k ( k = 1, n ) , f o r which t h e
...,
21
Copyright 0 1983 by Academic Press, Inc. All nghts of reproduction in any form reserved. ISBN 0-12-1533190
22
FRIEDRICHA. SAUER
V I
V
T
T
I1
F i g . 1 . A membrane i n contact with two homogeneous phases and Selective reversible electrodes are used t o measure electrochemical potential d i f f e r e n c e s .
'
".
...,
membrane i s permeable; components j ( j = j + 1, m), f o r which t h e membrane is impermeable; and components t ( t = m + 1, r ) , f o r which t h e membrane i s i m permeable. They react a t t h e s i d e of t h e membrane f a c u ) , for i n g t h e t phase. Components s (s = r + 1, which t h e membrane i s permeable o n l y i n complete c o u p l i n g w i t h t h e chemical r e a c t i o n . According t o Gibbs ( 1 9 6 1 , pp. 83-85) o n e g e t s t h e e q u i l i b r i u m c o n d i t i o n s from t h e v a r i a t i o n s of t h e Helmholtz f r e e e n e r g y F . More p r e c i s e l y , t h e f i r s t - o r d e r v a r i a t i o n o f F = F ' + F " around e q u i l i b r i u m must be zero, when v', v " , and T a r e k e p t c o n s t a n t and t h e system a s a whole i s c l o s e € o r m a t e r i a l exchange. T h i s means t h a t
...,
6F = 6F'
+
6F"
...,
= 0
(1.1)
o b s e r v i n g t h e above-mentioned c o n s t r a i n t s . Because t h e d e f i n e d system i s t o o complex, w e d i s c u s s it s t e p by s t e p , a t f i r s t l e a v i n g o u t many o f t h e components i n t r o duced above and d i s c u s s i n g i n d e t a i l t h e d i f f e r e n c e s between e l e c t r o l y t e s and n o n e l e c t r o l y t e s . A t t h e end w e come back t o t h e g e n e r a l e q u i l i b r i u m c a s e and d i s c u s s p o s s i b l e consequences.
MEMBRANE EQUILIBRIUM
11.
23
NONELECTROLYTE SOLUTION WITHOUT CHEMICAL REACTION
The s y s t e m u n d e r d i s c u s s i o n c o n t a i n s w a t e r ( w ) a s t h e s o l v e n t , t h e p e r m e a n t component ( k ) , and t h e impermeant component ( j ) . A l l components a r e assumed t o b e nonelectrolytes. The v a r i a t i o n s o f F ' and F " a r e g i v e n by
where t h e p are c h e m i c a l p o t e n t i a l s and t h e n a r e t h e mole numbers of t h e components. I t i s assumed t h a t 6 T = 0;
6V'
(2.2)
= 6V" = 0
and f o r t h e components ( j ) w e have 6n
I
j
= 6n
II
j
= 0
(2.3)
Because t h e s y s t e m as a whole i s c l o s e d , one h a s t h e conditions 6n
I
W
+
(1
6 n w = 0;
6n
1
k
+
11
6nk = 0
(2.4)
I n t r o d u c i n g Eqs. ( 2 . 1 ) and ( 2 . 4 ) i n t o Eq. ( l . l ) ,o n e g e t s t h e e q u i l i b r i u m c o n d i t i o n i n t h e form
If, as w i t h A b e i n g t h e d i f f e r e n c e between I and ' I . w a s assumed, t h e components ( k ) and water a r e i n d e p e n d e n t components, one c o n c l u d e s from Eq. ( 2 . 5 ) t h a t Apk = 0
( k = 1,
...,
n);
Auw = 0
(2.6)
FRIEDRICHA. SAUER
24
The e q u i l i b r i u m c o n d i t i o n s ( 2 . 6 ) e n a b l e u s t o c a l c u l a t e p r e s s u r e and composition o f t h e 'I p h a s e , i f p r e s s u r e and c o m p o s i t i o n o f t h e I phase and t h e concen phase t r a t i o n s of t h e impermeant components ( j ) i n t h e are known. T h i s u s u a l l y l e a d s t o t h e s i t u a t i o n t h a t o n e h a s t o s o l v e a system of t r a n s c e n d e n t a l e q u a t i o n s . In case t h e s o l u t i o n i s d i l u t e , one h a s 'k
-
'kO ( T , p)
+
I n ck
RT
(2.7)
where p i s t h e p r e s s u r e and C k t h e c o n c e n t r a t i o n . Then o n e c a n f i n d a n approximate s o l u t i o n of Eqs. ( 2 . 6 ) . R e w r i t i n g Eqs. ( 2 . 6 1 , one g e t s 1
I
Pk(P'
I
Cj)
CkI
-
1
Vk(P",
I
Ck,
c
.I
J
and
= Pw(P",
II
Ck,
II
c
.I J
-
PW(PflI c
I
I
k'
c.) J
(2.9)
E q u a t i o n s ( 2 . 8 ) and ( 2 . 9 ) are of t h e form
-
(AIJ)c=c'
(2.10)
-(AP)P=pll
I n t r o d u c i n g Eq. ( 2 . 7 ) i n t o Eq. ( 2 . 8 ) and n e g l e c t i n g t h e p r e s s u r e dependence of t h e p a r t i a l molar volumes vk gives
vk
I
II
AP = - R T l n ( c k / c k )
I n a s i m i l a r way t h e l e f t - h a n d s i d e of Eq. be e x p r e s s e d and one g e t s
vw
AP =
-
(AVw)p=pll
(2.11)
( 2 . 9 ) can (2.12)
where vW i s t h e p a r t i a l molar volume o f t h e w a t e r . F o r small c o n c e n t r a t i o n d i f f e r e n c e s t h e e q u a t i o n s c a n be s i m p l i f i e d f u r t h e r . I f (2.13)
MEMBRANE EQUILIBRIUM
Eq.
25
( 2 . 1 1 ) becomes 1
Vk
AP = -RT
A ck/ c k
...,
( k = 1,
n)
(2.14)
F o r d i l u t e s o l u t i o n s w e have v k c l < < 1. T h e r e f o r e , R T I A <~ < ~ A P~ f o r t h e p e r m e a n t chmponents k . Under t h e s e c o n d i t i o n s t h e Gibbs-Duhem r e l a t i o n become s
---fLTl C
w
(
n
1
Ack
k=l
rn
1
+
Acj)
(2.15)
j=n+l
Combination o f Eq. ( 2 . 1 5 ) w i t h Eq;. gives f o r the pressure difference
(2.12)
and ( 2 . 1 4 )
rn
+
(iVw
k = l c L V A~ p = R T
j=1 n+l
AC j
(2.16a)
or rn
( 1
-
j=n+l
1
= RT j=n+l A c j
c;V\)*.
(2.16b)
F o r d i l u t e s o l u t i o n s one h a s rn
and one g e t s
c
rn Ap = R T
j=n+1
Ac
j
which i s v a n ' t H o f f ' s l a w .
(2.17)
The c o n c e n t r a t i o n d i f f e r e n c e
FRIEDRICH A. SAUER
26
o f t h e p e r m e a n t components i s g i v e n by (2.18) j=n+l
ELECTROLYTE SOLUTION WITHOUT CHEMICAL REACTIONS
111.
Homogeneous e l e c t r o l y t e s o l u t i o n s must f u l f i l l t h e e l e c t r o n e u t r a l i t y condition: n
rn
1
eknk
+
k=l
1 e ~j.n j = n +1
=
o
(3.1)
where t h e e k and e . are t h e e l e c t r i c a l c h a r g e s p e r mole of t h e i o n s . T h a t ’ c o n d i t i o n h a s c o n s e q u e n c e s f o r t h e d e f i n i t i o n a n d measurement of t h e c h e m i c a l p o t e n t i a l s of t h e ions. ( G i b b s , 1961; Guggenheim, 1 9 6 7 ) . Every v a r i a t i o n of t h e mole numbers must f u l f i l l Eq. ( 3 . 1 ) a g a i n . Because of Eq. ( 3 . 1 ) rn - 1 i n d e p e n d e n t v a r i a t i o n s of t h e m o l e numbers a r e p o s s i b l e . Choosing t h e n t h i o n a s t h e key i o n , o n e g e t s en
6nn -
n-1
- 1
m
-
6 nk
ek
k= 1
1 j=n+l
e
j
6n
j
(3.2)
F o r t h e v a r i a t i o n of t h e Helmholtz f r e e e n e r g y a t cons t a n t T I v, and nW o n e g e t s n-1 6
~
1
=
k=l
( n1
vk
rn 6nk
+
1
j=n+l
v j( n ) 6 n j
(3.3)
where p i n ’ and p j n ) a r e t h e c h e m i c a l p o t e n t i a l s of t h e i o n s . They depend on t h e c h o i c e o f t h e key i o n c o r r e s p o n d i n g t o Eq. ( 3 . 2 ) . The c h e m i c a l p o t e n t i a l s of t h e i o n s so d e f i n e d are m e a s u r a b l e q u a n t i t i e s . Choosing a n o t h e r i o n 1 # n a s t h e key i o n , o n e g e t s
MEMBRANE EQUILIBRIUM
27
n
m
(3.4)
and n
rn (3.5)
Comparing E q s . ( 3 . 2 ) and ( 3 . 3 ) w i t h E q s . ( 3 . 4 ) and ( 3 . 5 1 , one g e t s t h e r e l a t i o n s between t h e c h e m i c a l pot e n t i a l s o f i o n s d e f i n e d f o r d i f f e r e n t key i o n s n and
(3.6)
(3.7)
(3.8)
(3.9)
( j = n
+
1,
...,
rn)
(3.10)
S i m i l a r c o n s i d e r a t i o n s must be a p p l i e d f o r t h e d e f i n i t i o n of t h e p a r t i a l molar volumes of i o n s . They depend on t h e c h o i c e of t h e key i o n . One g e t s t h e t h e r m o s t a t i c r e l a t i o n f o r t h e p a r t i a l molar volume v i n ) :
FRIEDRICH A. SAUER
28
(3.11) Now w e c o n s i d e r t h e v a r i a t i o n of t h e f r e e e n e r g y a r o u n d t h e e q u i l i b r i u m s t a t e and g e t f o r e a c h p h a s e and " n-1 k=l
(3.12)
n-1 k=l
for I
and w i t h t h e component n b e i n g t h e key i o n . Because t h e s y s t e m as a whole i s c l o s e d , w e have t h e c o n d i t i o n s 1
bnw
+
I1
6nw = 0;
1
6nk
I1
+
6nk = 0
(k =
1,
...,
n
-
1)
(3.14) Then t h e e q u i l i b r i u m c o n d i t i o n (1.1) h a s t h e form (3.15) From E q .
(3.15) we conclude t h a t
Apw = 0;
"k
(n) = 0
(k =
1,
...,
n
-
1)
(3.16)
These a r e t h e e q u i l i b r i u m c o n d i t i o n s f o r e l e c t r o l y t e sol u t i o n s , which a r e i n a c c o r d a n c e w i t h t h e e l e c t r o n e u t r a l i t y condition (3.1). To g e t t h e e l e c t r i c a l p o t e n t i a l d i f f e r e n c e s a t e q u i l i b r i u m , w e i n t r o d u c e p a i r s o f i d e n t i c a l , r e v e r si b l e e l e c t r o d e s i n t o t h e p h a s e s ' and " and p a s s a s m a l l amount of c h a r g e q a c r o s s t h e system. The e q u i l i b r i u m c o n d i t i o n (1.1) c h a n g e s t o 6F = 6F'
+
6F" = E l
69
(3.17)
MEMBRANE EQUILIBRIUM
29
f o r t h e permeant i o n s and t o
+
6 F = 6F'
6F" = E
j
(3.18)
6g
f o r t h e impermeant i o n s , Here E l and E j a r e t h e e l e c t r i c a l p o t e n t i a l d i f f e r e n c e s measured a t e q u i l i b r i u m w i t h
t h e h e l p o f r e v e r s i b l e e l e c t r o d e s f o r k and j , r e s p e c t i v e l y . E q u a t i o n s ( 3 . 1 7 ) and ( 3 . 1 8 ) e x p r e s s t h e f a c t t h a t t h e change o f t h e f r e e e n e r g y must be e q u a l t o t h e r e v e r s i b l e e l e c t r i c work done o n t h e system by c h a r g e t r a n s f e r . I f we t a k e a p a i r of e l e c t r o d e s r e v e r s i b l e f o r t h e l t h p e r m e a b l e i o n and t r a n s f e r a c h a r g e 6 q , we have f o r t h e v a r i a t i o n of t h e mole numbers of t h e impermeant i o n s 6n
j
= 6n
j
= 0
( j
= n + 1,
1
+
II
(k =
6nk = 0
1,
(3.19)
For t h e p e r m e a n t i o n s w e
because t h e system i s c l o s e d . have a g a i n 6nk
..., rn)
...,
(3.20)
n)
T h i s e q u a t i o n i s v a l i d a l s o f o r t h e l t h permeant i o n . Because of t h e charge t r a n s f e r , we g e t 1
6n 1 = 6 q / e l
+ 6nl ,
II
6nl = - 6 q / e l
-
-
6n 1
(3.21)
where 6 q / e l e q u a l s t h e amount o f component 1 coming from t h e e l e c t r o d e and 6nl i s t h e exchange of 1 v i a t h e membrane. Combination of Eqs. ( 3 . 2 1 ) l e a d s t o Eq. ( 3 . 2 0 ) . The v a r i a t i o n of t h e f r e e e n e r g y becomes n-1 6F'
+
6F"
=
1
A p k( n ) tin'
k
k= 1
+ ~u~
tin:
(3.22)
Because of t h e e q u i l i b r i u m c o n d i t i o n s ( 3 . 1 6 ) , t h i s e x p r e s s i o n i s z e r o and w e c o n c l u d e from Eq. ( 3 . 1 7 ) t h a t a t equilibrium E
1
= 0
( 1 = 1,
...,
n)
or A?
:e E = 0 1 1
( 1 = 1,
...,
n)
(3.24)
30
FRIEDRICHA. SAUER
where A q l i s c a l l e d t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e . T h i s means t h a t a t e q u i l i b r i u m t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e (ECPD) of a l l p e r m e a n t i o n s i s zero. The ECPD i s measured w i t h a p a i r of i d e n t i c a l r e v e r s i b l e e l e c t r o d e s . These e q u i l i b r i u m c o n d i t i o n s a r e a g a i n i n accordance w i t h t h e e l e c t r o n e u t r a l i t y c o n d i t i o n ( 3 . 1 ) . I f w e t a k e a p a i r of i d e n t i c a l revers i b l e e l e c t r o d e s f o r a n impermeant i o n , s a y j = y I t h e c o n d i t i o n (3.19) changes i n t o I
6n
Y
II
= 6q/ey;
6n
+
n-1
6F" =
(3.25)
= -6q/ey
( 3 . 3 ) we g e t f o r t h e v a r i a t i o n of t h e
and b e c a u s e of Eq. free energy 6F'
Y
1
1
A p k( n1 6 n k
+
I
Apw
6nw
+
Apy( n ) 6 q / e
k= 1
(3.26) Under e q u i l i b r i u m (3.16) t h i s becomes
and t h e ECPD f o r a n impearmeant i o n becomes
Taking a n o t h e r impermeant i o n , s a y m , as t h e key i o n , one g e t s b e c a u s e of Eq. ( 3 . 1 0 ) (3.29) P u t t i n g j = m and o b s e r v i n g t h a t A u E ) E 0 , one g e t s (3.30) Combining Eqs.
( 3 . 2 9 ) and ( 3 . 3 0 ) y i e l d s e .
Aqj
= Apjm)
+
e m An rn
(3.31)
MEMBRANEEQUILIBRIUM
31
a r e l a t i o n between ECPDs and c h e m i c a l p o t e n t i a l d i f f e r e n c e s o f i o n s . N o t e t h a t t h i s r e l a t i o n c o n t a i n s measurable q u a n t i t i e s only. I t should n o t be mistaken f o r t h e nonthermodynamic r e l a t i o n A n = Ap
+
(3.32)
e A$
where A $ i s t h e s o - c a l l e d "membrane p o t e n t i a l " and Ap i s a s i n g l e i o n chemical p o t e n t i a l d i f f e r e n c e . F i n a l l y , Eq. ( 3 . 3 1 ) i s v a l i d f o r any c h o i c e o f t h e key i o n . So w e have a l s o e
k + e
Ank = ApLn)
An
n
(3.33)
n
I n terms o f t h e ECPDs t h e e q u i l i b r i u m c o n d i t i o n s r e a d Apw = 0;
Ank = 0
( k = 1,
f o r t h e permeant components. meable i o n s are g i v e n by
...,
n)
(3.34)
The E C P D s f o r t h e imper(3.35)
i f Eqs. ( 3 . 3 4 ) are f u l f i l l e d and t h e key i o n i s a p e r meable i o n . I t s h o u l d b e mentioned t h a t t h e ECPDs f o r uncharged components i w i t h e i = 0 , b e c a u s e of Eq. ( 3 . 3 3 ) , g o o v e r So w e have i n t o t h e chemical p o t e n t i a l d i f f e r e n c e s A p i . Ani
(3.36)
= Api
= Apln)
f o r ei E 0 . Coming back t o t h e e q u i l i b r i u m c o n d i t i o n s f o r t h e p e r m e a b l e i o n s ( 3 . 1 6 1 , w e found t h a t "k
(n)
= 0
(k =
1,
...,
n
-
1)
(3.37)
i f n ( t h e k e y i o n ) i s a l s o permeant. Choosing a n impermeant i o n s a y m , a s t h e key i o n , Eq. ( 3 . 3 7 ) c h a n g e s i nt o
"k
(m
_
- ee-
m
(3.38)
FRIEDRICH A. SAUER
32
U s e h a s been made of Eq.
3.10). O t h e r w i s e Eq. ( 3 . 3 7 ) r e m a i n s form i n v a r i a n t a s l o n g a s t h e key i o n i s a p e r meable i o n . Summarizinq t h e r e s u t s of t h e f o r e g o i n g c o n s i d e r a t i o n s , we f o u n d - t h e e q u i l i b r i u m c o n d i t i o n s Avw = 0 ;
and p h a s e s
( k = 1,
(n) = 0
and
n
...,
n
-
1)
(3.39)
obey t h e e l e c t r o n e u t r a l i t y c o n d i t i o n
'I
rn
l e k n k + k=l
1
j = n +1
(3.40)
e n = O j
j
A t e q u i l i b r i u m t h e s e c o n d i t i o n s are v a l i d r e g a r d l e s s o f
whether w e p u t e l e c t r o d e s i n t o t h e p h a s e s ' and I' o r n o t . A c t i v a t i n g one more d e g r e e of freedom by c h a r g e t r a n s f e r with t h e h e l p of r e v e r s i b l e e l e c t r o d e s [because of Eq. ( 3 . 4 0 ) t h i s i s i m p o s s i b l e w i t h o u t e l e c t r o d e s ] , w e f i n d f o r t h e ECPDs t h a t
Ank=o
(k=l,
..., n ) ;
A n j = A p j( n )
( j = n + l ,
..., rn) (3.41)
T o c a l c u l a t e p r e s s u r e and c o m p o s i t i o n o f t h e I' p h a s e , if p r e s s u r e and c o m p o s i t i o n of t h e I p h a s e and t h e c o n c e n t r a t i o n s of t h e impermeable components j i n t h e I' p h a s e are g i v e n , w e p r o c e e d a s w e d i d i n S e c t i o n 11. For d i l u t e e l e c t r o l y t e s o l u t i o n s w e make t h e Ansatz
From Eqs.
vW
( 3 . 3 9 ) and ( 3 . 4 2 ) w e g e t
AP = - ( A ' w ) P = p "
For small c o n c e n t r a t i o n d i f f e r e n c e s w e g e t
(3.44)
MEMBRANE EQUILIBRIUM
33
(3.45) Making u s e of Eqs. we f i n d
(3.39) and t h e Gibbs-Duhem e q u a t i o n ,
n-1
= c
I
(n)
‘kV c k1 Ap
k= 1
W
rn
- -RC TI w
1 j=n+l
( c j
-2GAc) n c
n
(3.46) and t h e r e f o r e from Eq.
(3.44)
F o r d i l u t e s o l u t i o n s , b e c a u s e of
(3.47)
(3.48) N e g l e c t i n g t e r m s o f t h e form
FRIEDRICH A. SAUER
34
and making u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n s , (3.48) goes over i n t o
Eq.
n Ap =
1
n
rn
1
e k ( e k - e .)c' J k
AC
k=l
j
1
(3.49)
T h i s i s t h e g e n e r a l i z a t i o n of v a n ' t H o f f ' s l a w f o r electrolyte solutions. Taking j u s t one permeable i o n ( n = 1) and one i m p e r m e a b l e F i o n (rn = 2 1 , E q . ( 3 . 4 9 ) becomes Ap = R T ( 1
-
(3.50)
e2/el)Ac2
F o r t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e of t h e j t h impermeant i o n w e g e t from E q . ( 3 . 2 8 )
(j = n
+ 1,
..., rn)
(3.51)
Combining E q s . ( 3 . 4 5 ) and o b s e r v i n g t h e e l e c t r o n e u t r a l i t y c o n d i t i o n , o n e ge t s
(3.52) I n t r o d u c t i o n of E q .
(3.52) i n t o Eq.
(3.51) g i v e s
(3.53) I n t h e s p e c i a l c a s e of E q .
(3.50) one g e t s
MEMBRANE EQUILIBRIUM
35
(3.54) I f a g a i n c '2 ~ 2
< < 1, t h e n AC
IV.
2
(3.55)
NONELECTROLYTE SOLUTION W I T H CHEMICAL REACTIONS
I n a d d i t i o n t o t h e components k and j , w e i n t r o duce components t , which c a n n o t p e r m e a t e t h e membrane b u t r e a c t a t t h e s i d e of t h e membrane f a c i n g t h e ' p h a s e according t o
where t h e v t arg! t h e s t o i c h i o m e t r i c numbers. change of t h e n t i s g i v e n by 1
6n
t
= v
t
65
( t = rn
+
1,
...,
Every (4.2)
r)
where 6 5 i s t h e change i n t h e p r o g r e s s of t h e r e a c t i o n . Because t h e t-components c a n n o t p e r m e a t e t h r o u g h t h e membrane, w e have II
An, = 0
( t = rn
+ 1,
...,
(4.3)
r)
F u r t h e r m o r e , we have components s which c a n p e r m e a t e t h e membrane o n l y i f 6 6 i s u n e q u a l t o z e r o . The s - c o m p o n e n t s do n o t undergo a c h e m i c a l change b u t a r e t r a n s f e r r e d a c r o s s t h e membrane. For a c l o s e d s y s t e m w e have 1
6ns
-
-
6n
It
S
= ts 6 5
(s = r
+
1,
...,
u)
(4.4)
FRIEDRICH A. SAUER
36
Here t h e t , a r e t h e t r a n s f e r e n c e numbers. They s h o u l d n o t be confused w i t h t h e s t o i c h i o m e t r i c numbers v t , because t h e components s a r e c o n s e r v e d ( s remains s ) . F o r t h e change of t h e f r e e e n e r g y i n b o t h p h a s e s w e get n
r
U
and
(4.5) n 6F"
U 11
=
s=r+l
k=l
U s e h a s been made of E q s .
tsuz
65
+ vw
and ( t h e system i s c l o s e d and j i s impermeant). rium w e have 6F'
+
(4.41,
6F" = 0
and t h e r e f o r e n
1
(4.31,
'Pk
6 n k1
k=l
II
6 nw I
n j
-
n
II
= O
A t equiiib-
(4.6)
+(
t s Aps
-
+
.)tic
Apw 6 n i =
0
s=r+l
(4.7)
F o r t h e c l o s e d system i t w a s assumed 6n
I
k
+
II
6nk = 0;
6n
I
W
+
II
6nw = 0
Because 6 n k , f n w r and 6 5 a r e i n d e p e n d e n t v a r i a t i o n s , w e g e t the equilibrium conditions Apk = 0
(k = 1,
..., n ) ;
Allw=
0
U
1
s=r+l
t s Aps
-
A'
= 0
(4.9)
MEMBRANE EQUILIBRIUM
37
Here (4.10)
i s t h e D e Donder a f f i n i t y of t h e c h e m i c a l r e a c t i o n ( 4 . 1 ) . To c a l c u l a t e t h e p r e s s u r e and t h e c o m p o s i t i o n of t h e 'I p h a s e f o r a g i v e n p r e s s u r e and c o m p o s i t i o n of t h e I phase, t h e e q u i l i b r i u m c o n d i t i o n s are incomplete i f t h e number o f s-components i s l a r g e r t h a n o n e . One g e t s
t h e a d d i t i o n a l e q u a t i o n s by i n t e g r a t i o n of Eqs. ( 4 . 4 ) . T h i s i n t e g r a t i o n s t a r t s w i t h an a r b i t r a r y n o n e q u i l i b r i u m s t a t e 0 a n d g o e s t o t h e r e a l e q u i l i b r i u m s t a t e 5,. Therefore 11
II
n S - n so = - tS
-
w i t h A t = 5, 50. c o n s t a n t , one g e t s c
11
s
- c
II
SO
= - -t S V I#
A5
(s = r
+
1,
...,
A f t e r d i v i s i o n by
At
(s
= r
+
1,
(4.11)
U)
v",
which i s k e p t
..., u )
(4.12)
From Eqs. ( 4 . 1 2 ) one sees t h a t t h e e q u i l i b r i y m d i s t r i b u t i o n d e p e n d s on t h e i n i t i a l c o n c e n t r a t i o n s c s o I i f t h e number o f s-components i s l a r g e r t h a n o n e . E l i m i n a t i o n o f A 5 i n Eqs. ( 4 . 1 2 ) g i v e s II
c
s
II
= c
s o + -t u
uo
(s
= r
+
1,
...,
u
-
1)
(4.13) The s e t of e q u a t i o n s ( 4 . 1 3 ) t o g e t h e r w i t h t h e e q u i l i b rium c o n d i t i o n s ( 4 . 9 ) e n a b l e one t o c a l c u l a t e t h e e q u i l i b r i u m p r e s s u r e and c o m p o s i t i o n o f t h e I' p h a s e . In g e n e r a l t h i s l e a d s t o t h e s o l u t i o n of a system of t r a n s cendental equations. I n t h e following we g i v e an approximate s o l u t i o n for t h e s p e c i a l case of a d i l u t e s o l u t i o n . The compon e n t s w i l l b e w a t e r (w), one s-component ( s = r + 1 1 , and t h e r e a c t a n t t ( t = m + 1, , r ) Because t h e r e a c t a n t s do n o t p e r m e a t e a c r o s s t h e membrane, t h e i r c o n c e n t r a t i o n s are p r e s c r i b e d . l , T ~d e t e r m i n e t h e p r e s s u r e p " and t h e c o n c e n t r a t i o n c ~ + w ~ e , have t h e equations
. ..
.
FRIEDRICH A. SAUER
38
Under t h e assumption of a d i l u t e s o l u t i o n t h i s l e a d s t o I
r
cw Vw AP = - c w ( A ~ w ) p = p = ,l R T (
If E q .
c
1
Act +
A'Cr+l
t=m+l
1
(4.15)
1
( 2 . 7 ) i s v a l i d and cWvw > > c ~ + ~ V ~ one + ~ g, e t r
1
Ap = R T
t=m+l
AC
+
t
1
Acr+l
Cr + l
A'
I
C1
A1/tr+l
r+l
(
Act
t=m+l
+
'Ic'+')
RTtr+l
(4.16)
T h i s means t h e o s m o t i c p r e s s u r e d i f f e r e n c e Ap depends On A ' . I f 1 >> c we g e t r + l'r+1'
1
(4.17) F o r more t h a n o n e s-component w e make u s e of E q s . and rewrite them i n t h e form Acs
= AcsO
+tS
(Acu
-
Acuo)
(s = r
+
1,
(4.13)
..., u )
tU
(4.18) where t h e l a s t e q u a t i o n f o r s = u i s a n i d e n t i t y . The e q u i l i b r i u m c o n d i t i o n s f o r d i l u t e s o l u t i o n s and s m a l l c o n c e n t r a t i o n d i f f e r e n c e s Acs and A c t go o v e r i n t o
MEMBRANE EQUILIBRIUM
39
(4.19) and U
U
1
t V Ap s s
+
RT
tS i-
s=r+l
s=r+l
Combination of Eqs.
C
A c S = A'
(4.20)
s
(4.18) and (4.20) leads to
U
-
1
(4.21) C
s=r+l
S
The contribution of the s-components to the osmotic pressure difference becomes then
-
2 -1
s=r+1
R*(
U
A' + -R T
c
Cs
ts J
U
c
s = r + l v=r+l U
[
1
s=r+l v = r + l tvf
-
>)Ac CS
1
s=r+l
Here we assume that U
2)
U
tstvV" <<
1
(4.22)
so
FRIEDRICHA. SAUER
40
V.
ELECTROLYTE SOLUTION W I T H CHEMICAL W A C T I O N S
Assuming t h a t t h e r e i s a t l e a s t one i o n of t h e t y p e k p r e s e n t which can permeate t h e membrane i n d e p e n d e n t l y of t h e r e a c t i o n , w e t a k e t h i s i o n as t h e key i o n . From t h e f o r e g o i n g p a r a g r a p h s w e f i n d t h e e q u i librium conditions
"k
(n) = 0
(k =
1,
..., n -
1 ) ; Apw = 0
s=r+l
where n i s t h e key i o n . Acs = Acso
+
tS
Together w i t h
(Ac
U
U
-
Acuo)
(s = r
+ 1,
..., u ) (5.21
Eqs. ( 5 . 1 ) allow o n e t o c a l c u l a t e t h e e q u i l i b r i u m d i s t r i b u t i o n o f t h e components and t h e p r e s s u r e i f t h e I phase i s g i v e n . For t h e ECPDs w e g e t
(s = r
+
1,
...,
u)
(5.3)
T h e r e f o r e , w e c a n w r i t e t h e g e n e r a l i z e d e q u i l i b r i u m cond i t i o n s i n t h e form
U
1
ts A n s
-
A'
= 0
(5.41
s=r+l
T h i s h a s been d e r i v e d u n d e r t h e c o n d i t i o n t h a t t h e r e e x i s t s a n i o n which can permeate i n d e p e n d e n t l y of t h e chemical r e a c t i o n . I f t h i s i s n o t t h e case, w e have t o d i s t i n g u i s h two s i t u a t i o n s .
MEMBRANE EQUILIBRIUM
41
L e t us assume t h e o n l y i o n s p r e s e n t a r e t h o s e o f W e take t h e ion w i t h t h e index u as t h e the s-type. key i o n . For t h e v a r i a t i o n o f t h e f r e e e n e r g y one g e t s u-1
1
bF =
(u)
ts
' -
A'
6ns
65
+
Apw
I
6nw
(5.5)
If t h e system a s a whole i s c l o s e d (no e l e c t r o d e s )
have
I
tins = t
65
S
(s =
r
+ 1,
...,
u
-
1)
,
we
(5.6)
and b e c a u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n u-1
I
=
6n
c
-
eS
e ts
(5.7)
6E
s=r+l
A s on t h e o t h e r s i d e u i s o f t h e s - t y p e ,
w e have
65
6nU = t
(5.8)
ni
To g e t v a r i a t i o n s of t h e i n t h e closed system w i t h o u t v i o l a t i n g the e l e c t r o n e u t r a l i t y condition, U
1
e t
s s
= O
(5.9)
s=r+l
m u s t be f u l f i l l e d . l I f Eq. ( 5 . 9 ) i s n o t f u l f i l l e d , v a r i a t i o n s of t h e n s a r e p o s s i b l e o n l y by means o f e l e c t r o d e s . Assuming t h e v a l i d i t y o f E q . ( 5 . 9 ) , one g e t s the equilibrium conditions u-1
1
A v w = 0;
t s A p s( u )
-
A'
= 0
(5.10)
s=r+l
Together w i t h the material c o n d i t i o n s Ac
S
= Ac
so
tS + tU
(AcU
-
Ac
uo
)
(s =
r
+ 1,
...,
u
-2)
(5.11)
FRIEDRICH A. SAUER
42
t h e e q u i l i b r i u m d i s t r i b u t i o n i s determined. Under t h e c o n d i t i o n ( 5 . 9 ) it i s i m p o s s i b l e t o p a s s e l e c t r i c a l c h a r g e a c r o s s t h e membrane w i t h t h e h e l p of r e v e r s i b l e electrodes. L e t u s assume w e have a p a i r o f e l e c t r o d e s r e v e r s i b l e t o t h e i o n u . Then w e have (5.12) 1
6nS
--
From Eqs.
ts 65
s
#
(5.13)
u
( 5 . 1 2 ) and ( 5 . 1 3 ) w e g e t
U
U
1
es 6 n i = 6q
+
s=r+l
1
t s e s 65
(5.14)
s=r+l
Because of Eq. t i o n we g e t
( 5 . 9 ) and t h e e l e c t r o n e u t r a l i t y c o n d i (5.15)
6q = 0
T h i s means t h a t it i s i m p o s s i b l e t o p a s s e l e c t r i c a l c h a r g e a c r o s s t h e membrane. T h e r e i s no e l e c t r i c cont a c t between b o t h p h a s e s and no ECPD i s d e f i n e d . On the o t h e r side i f w e release the condition (5.91, w e have from Eq. ( 5 . 1 4 ) (5.16) I f 6 q = 0 ( c l o s e d s y s t e m ) . I f c:=r+l t e # 0, we f i n d 6 5 = 0 b e c a u s e of t h e e l e c t r o n e u t r a l i t y c o n d i t i o n . T h i s means no v a r i a t i o n of t h e mole numbers are poss i b l e i n t h e c l o s e d system. For an open system ( w i t h e l e c t r o d e s ) one g e t s from Eq. (5.14)
c U
6q =
Because of
-
s=r+l
tses 65
(5.17)
MEMBRANE EQUILIBRIUM
43
6q e
(5.18)
6F =
U
t h e e q u i l i b r i u m c o n d i t i o n becomes
(5.19) and t h e r e f o r e
(5.20) Making u s e of E q .
( 3 . 3 3 1 , one g e t s
(5.21) E q u a t i o n s ( 5 . 2 0 ) c a n be u s e d t o c a l c u l a t e t h e p r e s s u r e d i f f e r e n c e Ap and t h e ECPD A n u , which b u i l d up a t e q u i l i b r i u m . Because C;=r+l e t # 0 and no o t h e r p e r m e a t i n g i o n s a r e p r e s e n t , t h e d i s t r i b u t i o n o f t h e comp o n e n t s w i l l n o t change. I n t h e s p e c i a l case t h a t Acs - AcsO = 0 w e g e t
(5.22)
and Ans
= Anu es/eu
( s = r + 1,
.. .,
u
-
1)
(5.23)
I n t h e g e n e r a l c a s e , when a l l t h e d i f f e r e n t k i n d s o f components ( k , j , t , s ) a r e p r e s e n t , o n e g e t s t h e e q u i librium conditions
FRIEDRICH A. SAUER
44
Apw
= 0;
"k
..., n
( k = 1,
(n) = 0
-
1)
U
(5.24) s=r+l
where n i s t h e k e y i o n . T o g e t h e r w i t h Eq. ( 4 . 1 8 ) I Eqs. ( 5 . 2 4 ) d e t e r m i n e t h e e q u i l i b r i u m d i s t r i b u t i o n of p r e s s u r e and components. F o r t h e ECPDs o n e g e t s ( k = 1,
Aqk= 0
... , n ) ;
(t=m
+
1,
Aqs
- A y s( n )
..., r ) ; ( j = n
+
A
(s=r ~
1,
~
+ 1,
= (n A) V
..., rn)
..., u )
~
(5.25)
With t h e h e l p of Eqs. ( 5 . 2 5 ) t h e " c h e m i c a l " e q u i l i b r i u m c o n d i t i o n c o u l d be w r i t t e n i n t h e form U
(5.26)
I n t h e f o l l o w i n g w e compare t h i s r e s u l t of e q u i l i b r i u m thermodynamics w i t h t h e s t o p - f l o w s i t u a t i o n o f nonequil i b r i u m thermodynamics. For s i m p l i c i t y w e assume t h a t o n l y s- and t-components are p r e s e n t . Then we g e t f o r t h e s-components U II
Js =
w=r+l
'sh
Anh
+ Ls 5
A'
(s = r
+
1,
...,
u)
(5.27) and f o r t h e r e a c t i o n r a t e
J
5
U
J
5
=
1
h=r+l
L S h Anh
+
L
55
A'
(5.28)
For t h e t i m e b e i n g w e release t h e c o n d i t i o n o f c o m p l e t e coupling. Then t h e c o n d i t i o n s of u n c o n s t r a i n e d thermodynamic ' e q u i l i b r i u m are .
MEMBRANE EQUILIBRIUM
45
II
J s = 0;
J
5
= 0
(s
= r
+
...,
1,
u)
(5.29)
From t h a t w e c o n c l u d e i f t h e Onsager m a t r i x i s nons i n g u l a r (independent flows) t h a t = 0;
A'
Anh = 0
a t equilibrium.
#
J~
0;
+
(h = r
1,
...,
u)
(5.30)
I n t h e stop-flow s i t u a t i o n
J~ = 0
(s =
r
+
1,
...,
U )
(5.31)
we g e t U
(5.32) o r i n vector notation (5.33) where Lo h a s t h e e l e m e n t s L s h and t h e v e c t o r L5 h a s t h e components ~ c = ; L, ~ assuming Onsager r e c i p r o c i t y . Here t h e s u p e r s c r i p t "sf" d e n o t e s s t o p flow. F u r t h e r Introduction more, w e assume t h a t Lo i s n o n s i n g u l a r . of t h e v e c t o r t of t h e t r a n s f e r e n c e numbers d e f i n e d by
... , n ) (5.34) leads t o
(5.35) The r e a c t i o n r a t e a t s t o p f l o w i s g i v e n by
FRIEDRICHA. SAUER
46
(5.36)
55
s=r+l
Because t h e t o t a l Onsager m a t r i x is p o s i t i v e d e f i n i t e , w e have (5.37) T h i s means t h a t t h e f a c t o r i n f r o n t of A ' i n Eq. ( 5 . 3 5 ) is positive. From t h e p o s i t i v e v a l u e o f t h e d e t e r m i n a n t of t h e t o t a l Onsager m a t r i x o n e can show t h a t t h i s f a c t o r i s less t h a n u n i t y . The p r o o f g o e s a s f o l l o w s . W e have t h e r e l a t i o n ( d e t Ltot)
=
5
)
(5.38)
[ H i n t : One g e t s t h i s r e l a t i o n by e x p a n s i o n o f ( d e t & , t o t ) a b o u t t h e e l e m e n t s o f t h e l a s t row.] Because ( d e t Ltot)
0
and ( d e t L ) are p o s i t i v e , w e h a v e
T 0-laL 0 < (L5*L
5)'L5s
< 1
(5.39)
Taking t h e a b s o l u t e v a l u e of Eq. ( 5 . 3 5 ) and making u s e of i n e q u a l i t y ( 5 . 3 9 ) , one g e t s t h e i n e q u a l i t y (5.40) From r e l a t i o n s ( 5 . 3 5 ) and (5.39) one c o n c l u d e s t h a t A ' and E T * A n S f have t h e same s i g n . T h e r e f o r e , w e c a n w r i t e two p o s s i h e i n e q u a l i t i e s : (5.41) or
MEMBRANE EQUILIBRIUM
47
(5.42) s=r+l
d e p e n d i n g on t h e s i g n of A ' . The i n e q u a l i t i e s (5.41) and (5.42) a r e d i r e c t c o n s e q u e n c e s of t h e s e c o n d law o f thermodynamics. Comparing t h i s r e s u l t w i t h Eq. (5.36) I w e come t o t h e c o n c l u s i o n t h a t €or an i n c o m p l e t e c o u p l e d s y s t e m ( a l l f l o w s i n c l u d i n g J~ a r e i n d e p e n d e n t ) a t s t o p flow i n general
#
Jsf
5
(5.43)
0
For a c o m p l e t e l y c o u p l e d s y s t e m i n n o n e q u i l i b r i u m t h e c o n d i t i o n (4.4) g o e s o v e r i n t o II
J~ = -t
J
s 5
(s = r
+
1,
. . . I
u)
(5.44)
i n e a c h s t a t e of t h e s y s t e m and n o t o n l y i n l e v e l flow. T h i s means t h e f l o w s a r e d e p e n d e n t now and t h e t o t a l Onsager m a t r i x becomes s i n g u l a r . Comparing Eqs. (5.44) w i t h Eqs. (5.27) and (5.281, one g e t s 0
(5.45)
L s h = Ls5Lh(/L5E
and
J5 = L5
5
(4'
-
s=r+l
is
Ans)
which f u l f i l l s c o n d i t i o n (5.44). F o r a c o m p l e t e l y c o u p l e d s y s t e m o n e h a s one i n d e p e n d e n t f l o w and o n e i n d e p e n d e n t d r i v i n g f o r c e o n l y . The f l o w m i g h t be t h e J 5 and t h e d r i v i n g f o r c e i s t h e n
A'
is
s=r+l
Aqs
(5.47)
FRIEDRICHA. SAUER
40
The s t o p - f l o w s i t u a t i o n c o i n c i d e s w i t h t h e c o n s t r a i n e d thermodynamic e q u i l i b r i u m . A t s t o p f l o w w e have
and t h e c o n d i t i o n f o r t h e d r i v i n g f o r c e (5.47) becomes U
A'
-
1
(5.49)
is
s=r+l
t h e same c o n d i t i o n a s w a s d e r i v e d ,at Eq. ( 5 . 4 ) . Then, and o n l y t h e n , t h e i n e q u a l i t i e s ( 5 . 4 ) and ( 5 . 4 2 ) become equalities. I f one u s e s Eq. (5.49) € o r t h e e v a l u a t i o n o f t h e a f f i n i t y A ' o f t h e d r i v i n g chemical r e a c t i o n , one must be s u r e t h a t t h e system is c o m p l e t e l y c o u p l e d . T h i s means Eqs. (5.44) must b e f u l f i l l e d i n e a c h s t a t e o f t h e system. O t h e r w i s e , Eq. (5.35) must be used. Here t h e knowledge o f t h e Onsager m a t r i x i s needed. R e w r i t i n g Eq. (5.351, one g e t s
L
55
c s=r+l
(5.50) 0-1
tsthLsh h=r+t
I f o n l y some of t h e f l o w s are c o m p l e t e l y c o u p l e d , t h e stop-flow s t a t e a g a i n c o i n c i d e s w i t h t h e c o n s t r a i n t e q u i l i b r i u m . L e t u s assume w e have s-components (5.51)
...,
and k-components ( k = 1, n ) which are n o t completel y c o u p l e d . Then t h e s t o p - f l o w c o n d i t i o n s II
Js =
0
(s = r
+
1,
...,
u);
II
Jk = 0
( k = 1,
..., n ) (5.52)
because of Eqs.
(5.51) imply
MEMBRANE EQUILIBRIUM
J
5
49
= o
(5.53)
T h i s means c o n s t r a i n e d e q u i l i b r i u m w i t h t h e e q u i l i b r i u m conditions Ank = 0
( k = 1,
...,
U
n);
A'
-
1
is
Ans
= 0
s=r+l
(5.54) T h e r e w i l l be no b u i l d u p of E C P D s f o r t h e k-components. T h i s a l s o happens i n t h e c a s e of n o n v a n i s h i n g Lk It s h o u l d be mentioned t h a t s u c h a s y s t e m , n e v e r t h e i e s s , shows a l e v e l f l o w
.
I n a system f o r which p a r t s o f i t a r e c o m p l e t e l y c o u p l e d , t h e s t o p - f l o w e x p e r i m e n t g i v e s no i n d i c a t i o n f o r t h e a c t i v e t r a n s p o r t of t h e incompletely coupled components. C o n s i d e r a t i o n s b a s e d on the i n e q u a l i t i e s ( 5 . 3 7 ) a l l o w e d Kedem and Caplan ( 1 9 6 5 ) t o d e r i v e t h e i r r e s u l t s on thermodynamic e f f i c i e n c y of membrane s y s t e m s w i t h active transport.
VI.
SUMMARY
U s u a l l y components which p e r m e a t e a c r o s s a memb r a n e do n o t c o n t r i b u t e t o t h e o s m o t i c p r e s s u r e d i f f e r ence a t theryodynamic e q u i l i b r i u m i n d i l u t e s o l u t i o n s . As l o n g as ckvk i s s m a l l compared t o u n i t y , t h e i r cont r i b u t i o n c a n be n e g l e c t e d . T h a t i s n o t t r u e f o r components which are completel y c o u p l e d t o a chemical r e a c t i o n . These components c a n b u i l d up l a r g e r d i f f e r e n c e s o f t h e i r c h e m i c a l potent i a l s , which t h e n c o n t r i b u t e t o t h e o s m o t i c p r e s s u r e d i f f e r e n c e . For t h i s osmochemical e q u i l i b r i u m o n e g e t s Eq. ( 4 . 1 6 ) :
50
FRIEDRICH A. SAUER
T h i s means t h a t t h e a f f i n i t y A' of t h e chemical r e a c t i o n g i v e s an a d d i t i o n a l c o n t r i b u t i o n t o t h e osmotic p r e s The s u r e depending on t h e t r a n s f e r e n c e number t,+l. s i g n of t h e t e r m w i t h A' depends o n t h e s i g n of t h e t r a n s f e r e n c e number and t h e s i g n of A' i t s e l f . T h i s res u l t can e a s i l y be e x t e n d e d f o r membranes where t h e r e a c t i n g components p e r m e a t e t h e membrane and o t h e r i m permeant components j a r e p r e s e n t . W e g e t them t h e osm o t i c p r e s s u r e d i f f e r e n c e between d i l u t e s o l u t i o n s
By a d j u s t m e n t of t h e A c j i t i s p o s s i b l e t o v a r y t h e Ap. I f t h e r e i s more t h a n one component p r e s e n t which i s c o m p l e t e l y c o u p l e d , w e g e t memory e f f e c t s i n t h e osmot i c pressure. T h i s means t h e f i n a l osmotic p r e s s u r e d i f f e r e n c e depends on t h e i n i t i a l c o n c e n t r a t i o n d i f f e r of t h e c o m p l e t e l y coupled components. F o r ences A c t h a t comB8re Eq. (4.22). The w i d e l y used r e l a t i o n o f Eq. (5.49):
which r e l a t e s t h e e l e c t r o c h e m i c a l p o t e n t i a l d i f f e r e n c e s i n a s t o p - f l o w s t a t e w i t h t h e a f f i n i t y A ' o f t h e metab o l i c r e a c t i o n , i s r i g o r o u s l y v a l i d i f t h e components are c o m p l e t e l y c o u p l e d t o t h e chemical r e a c t i o n . Then t h e s t o p - f l o w s t a t e becomes a c o n s t r a i n e d thermodynamic e q u i l i b r i u m . For i n c o m p l e t e l y c o u p l e d systems Eq. (5.49) must be r e p l a c e d by t h e i n e q u a l i t y (5.40) :
which i s a consequence o f t h e second l a w o f thermodynamics.
MEMBRANE EQUlLlBRlUM
51
REFERENCES Gibbs, J. W. (1961). "The S c i e n t i f i c Papers," pp. 83-85, 331-349, 4 06-4 12. Guggenheim, E. A, (1967). "Thermodynamics," Chap. 8. North Holland, Amsterdam. Kedem, O., and Caplan, S. R. (1965). T r a n s . F a r a d a y SOC. 6 1 , 1897.