I J P
A
— an international journal
An e stim a tio n o f v e lo c ity c o rr e la tio n co efficien ts in te r n a r y e le c tro ly tes
H a im a n ti C h a k r a b a r ti a n d S h re e k a n th a Sil S a h a I n s titu te of N u c le ar P h y sics
S e c to r 1, B lo ck -A F , B id h a n n a g a r, C a lc u tta - 700 064
A b s t r a c t : T h is w ork gives an d stim a te of th e velocity c o rre la tio n coefficients for te r n a ry e le c tro ly te so lu tio n s ( th e sy ste m m ay have tr a c e r ion as an y one of th e c o m p o n e n t), o rig in a tin g fro m th e lin e a r resp o n se th e o ry u tiliz in g th e m e a su re d tr a n s p o r t coefficients in o rd e r to g et a d e e p e r in sig h t in to th e m ic ro d y n a m ic s tr u c tu r e o f th e so lu tio n . A ssu m in g O n s a g e r’s re la tio n to b e valid we o b ta in te n se ts o f v elo city c o rre la tio n coefficients for C s ion tra n s p o rtin g in a q u e o u s so lu tio n o f C sC l a n d KC1.
K e y w o r d s : V elocity c o rre la tio n coefficient, T e rn a ry e le c tro ly te , D iffusion C oefficient, R a d io a ctiv e tra c e r.
P A C S n u m b e r : 66.0, 05.60
I n t r o d u c t i o n
T h e v elo city c o rre la tio n fo rm a lism evoked from th e lin e a r resp o n se th e ory is o f g r e a t im p o r ta n c e fo r en v isag in g th e m o le c u la r m o tio n a n d th e p a rticle -p a rticle in te ra c tio n in e le c tro ly te so lu tio n s, sp ecially in c o n c e n tra te d solutions (few te n th m o la r o r h ig h e r th a n t h a t ) w h ere O n sa g e r’s th e o ry a n d its ex ten sio n s fails to give re a l p h y sic a l p ic tu re . E sse n tia lly th e w orks of McCall a n d D o u g lass [1] a n d D ouglass an d F irsch' [2] w hich show ed first how m u tu a l a n d self diffusion d a ta can b e in te r p re te d in te rm s of in te g ra l over velocity self c o rre la tio n fu n c tio n s a n d velocity cross c o rre la tio n fu n c tions. Since t h e n th e field h a s b e e n ex ercised w ith g re a t in g e n u ity , b u t m o st
© 19j)5 IACS
134 Haimanti Chakrabarti and Shreekantha Sil
o f th e s e w orks deal w ith b in a ry sy stem s a n d even i f w ith m u ltic o m p o n e n t sy stem s no b rid g e was m a d e to cover th e d ifferen ce o f t h e fra m e s for the fluxes in th e lin e a r resp o n se th e o ry a n d in th e m e a s u re m e n t o f t h e tra n s p o r t coefficients. In th is c o n n e ctio n it is w o rth y t o m e n tio n t h a t in lin ear resp o n se th e o ry th e sy ste m s tu d ie d is in e q u ilib riu m : t h e m o m e n tu m ex
change w ith th e b o u n d a rie s is zero a n d so th e b a ry c e n tr ic re fe re n c e fra m e is th e n a tu ra l choice; w h ereas, in la b o ra to ry e x p e rim e n ts t h e b o u n d a rie s of the sy stem stu d ie d ex ch an g e m o m e n tu m w ith th e s o lu tio n a n d th is d e p e n d s on th e design of th e e x p e rim e n ta l set up. It is th e p io n e e rin g w o rk o f S chbnert [3] which first d e lt w ith th e p ro b le m of c o rre la tin g th e fra m e s o f references.
H e rep laced th e b a ry c e n tric referen ce fra m e by th e so lv e n t fix ed fra m e of ref
erence w hich tak e s in to acco u n t of th e e x p e rim e n ta l b o u n d a r y c o n d itio n s.
H ow ever, th a t w ork was also r e s tric te d to th e case o f b in a r y s o lu tio n , with th e b e a u ty of in c o rp o ra tin g th e c o n c e p t o f in tr a - ( tr a c e r ) d iffu sio n in to the irrev ersib le th e rm o d y n a m ic s to reg ain th e K u b o re la tio n s . S o t h e w ork does n o t give d ire c tly an e s tim a te of th e v elo city c o rre la tio n co efficien ts for more g en eral situ a tio n o f (i) lim itin g diffusion o f t h ir d sp e cie s in a s u p p o rtin g ele c tro ly te so lu tio n , (e.g. T ra c e r A in th e s o lu tio n o f B C ) o r ( « ) a n y of the te rn a ry (m ix tu re of tw o b in ary e le c tro ly te h a v in g o n e c o m m o n c o n s titu e n t or any ele c tro ly te w hich d isso ciates in to th re e d iffe re n t io n ic sp e c ie s in so
lu tio n . In th is w ork we a re g iving an e s tim a tio n o f th e v e lo c ity co rre la tio n coefficients for any of th e a fo re m e n tio n e d t e r n a r y s y s te m s in te r m s o f the e x p e rim e n tally m e a su ra b le tra n s p o r t coefficients. T h e g e n e ra l m orphology is b ased on tw o w orks - one from S c h o n e rt [3] a n d t h e o t h e r fro m M iller [4].
B esides o ur a tt e m p t t th e w ork of R a in e ri a n d T im m e rm a n [5] a lso d e a ls with th e V C C ’s in te rn a ry e le c tro ly te s b u t th e e q u a tio n s t h e r e n e e d t o re w ritte n in term s of e x p e rim e n ta l q u a n titie s to be u se fu l w hich is r a t h e r co m p ic ate d .
T h e o r e t i c a l :
O u r sy stem s consists of th re e different s o lu te c o m p o n e n ts i = 1)2,3 (an y one am o n g th e s e m ay b e th e iso to p e o f e ith e r o f t h e tw o ) disso lv ed in
the sam e so lv e n t, i = 0. T h e e n tro p y p ro d u c tio n a ol th is s o lu tio n in th e iso th erm al iso b a ric c o n d itio n is given by
3
T o = £ ( 7 ') w * . ( ! )
i=0
where w d e sig n a te s th e velocity o f th e referen ce. T h e (Ji)w a n d X ; a re th e co rresp o n d in g fluxes a n d forces o f th e tth c o m p o n e n t. F o r e x a m p le in th e solvent fixed referen ce fra m e
3
Ta = ^ ( J , ) 0X j [because (J0)o = 0] (2 )
i
and in th e b a ry c e n tric referen ce fra m e
3 v
T o = (3 )
i = 0
where v is th e v elo city o f th e so lv en t. A s th e fluxes (./*)<,) o f th e so lu te c o n stitu e n ts a re re fe rre d to th e v elo city of th e so lv e n t, th e p h en o m e n o lo g ic a l tra n s p o rt e q u a tio n s of th e irre v e rsib le th e rm o d y n a m ic s w h ich d e sc rib e o u r system for a flow lin e a r alo n g th e sp a ce c o -o rd in a te x a re
(Ji)o = ' t L^ X k (t = 1 ,2 ,3 )
k = 1
Q
with X k = + zkF<t>)
or, in m a tr ix n o ta tio n ,
An estimation of velocity correlation coefficients in ternary electrolytes. 135
J
0 = L X (4)where L{k a re t h e p h e n o m e n o lo g ic a l coefficients c o n s titu tin g L , &re th e chemical p o te n tia l, zk a re th e sig n e d ch a rg e n u m b e r o f th e c o n s titu e n ts k , F is th e F a ra d a y c o n s ta n t a n d <(> is th e e le c tric a l p o te n tia l.T h e p h e n o m e n o logical coefficients Lik o b e y th e O n s a g e r’s re c ip ro c ity re la tio n s . T h e s e co
efficients c a n b e r e la te d to th e c o n v e n tio n a lly u se d t r a n s p o r t coefficient:
th e c o n d u c ta n c e A, th e tra n sfe ra n c e n u m b e rs U a n d t h e s o lv e n t fix e d th e r m o d y n a m ic diffusion coefficients a n d th e v o lu m e fix ed in te rd iffu sio n coefficients D ,*.
Now, as we consider th e case from th e view p o in t o f t h e lin e a r response th eory, th e reference velocity is th e v elo city o f th e c e n te r o f m a s s , w = v and as th e to ta l m ass flow is zero
(J0)v ~
“ 77" (5)m ° .= i
\
A gain from G ib b s-D u h em e q u a tio n we have 3
£ C , - * ; = 0 (6)
i=0
Now e lim in a tin g (J„)v a n d (A ,,) from e q u a tio n (3 ) a n d we re w rite th e en- tro p y p ro d u c tio n e q u a tio n in th e b a ry c e n tric fra m e as
T o
=
j z W . Y i. (7)
1=1 136 Haimanti Chakrabarti and Shreekantha Sil
w here Yi = is th e force c o n ju g a te t o t h e flow W ith th is th e tra n s p o r t coefficient m a tr ix in th e b a ry c e n tr ic re fe re n c e fram e com es to b e,
ft = PtL@ (w h ere ik th e le m en t o f /? is = $,•* +
K now ing th e ft m a tr ix , th e n o rm a lise d v e lo c ity c o rr e la tio n coefficient / , ; can b e c a lc u la te d u sin g th e re la tio n
/ o = < v,o ( 0 ) . u ^ ( t ) > dt (*'»>
RTSljj _ DfSij
C{ C j C{
0 , 1 , 2 , 3 ) (8)
w here, f t tJ is th e tra n s p o rt coefficient m a tr ix e le m e n t o f i t h ro w j t h co lu m n in th e b a ry c e n tric referen ce fra m e , D* is th e in tr a ( tr a c e r ) - d if f u s io n coefficient, ( h ere th e co n cep t of th e lab e lle d c o n s titu e n ts h a s b e e n in tr o d u c e d in to the form alism o f irre v ersib le th e rm o d y n a m ic s ), V is v o lu m e o f all t h e ensem ble, is th e velocity of ath p a rtic le o f th e ith sp e cie s in t h e b a r y c e n tr ic frame of reference, R is th e u n iv ersal gas c o n s ta n t, N0 is th e A v a g a d ro N u m b er.
(37
Experimental
T h e 134Cs (tr a c e r ) diffusion coefficient for th e CsCl an d K C l a q u e o u s solution h av e b e e n m e a s u re d in o u r la b o r a to r y over th e c o n c e n tra tio n ran g e (J.009 to 10.OM [6] a n d 0.001 to 4.0M [7] resp ectiv ely . T h e se diffusion d a t a coupled w ith th e o th e r tr a n s p o r t d a ta e x istin g in th e lite r a tu r e a re u tiis e d to calculate th e v elo city coefficients for each sy ste m u p to c o n c e n tra tio n 4.0M to the g e t profile of th e c o n c e n tra tio n d e p e n d e n c e of th e V C C s. T h e
Concentration / Mol. dm"3
F ig .l C o n c e n tr a tio n d e p e n d e n c e o f th e ve- locity c o r r e la tio n c o e ffic ie n ts w h en 1MC s tra n sp o rtin g in a q u e o u s s o lu tio n o f C sC l.
diffusion coefficients are m e a su re d in o u r la b o r a to r y by a ra d io a c tiv e tra c e r te c h n iq u e u til
isin g th e slid in g cell m ec h a n ism . T h e d e ta ils a re given in an e a rlie r w ork [6]. T h e c o n c en tr a tio n d e p e n d e n c e of th e te n se ts of c a lc u la te d V C C for each sys
te m h ave b e e n show n in fig .l a n d fig.2. T h e cu rv es w ere e x te n d e d to m e e t th e diffusion coefficient axis. Now we e n d e a v e r to in te r p r e t th e tr e n d e x h ib ite d in th e c o n c e n tra tio n d e p e n d e n c e of V C C on a m icrodynam ical level. T h e velocity c o rre la tio n coefficients for th e ions having sam e ty p e of c h a rg e ( / u , /2 2, /3 3, / i2) go th ro u g h a m in im a as co n centration o f th e e le c tro ly te in creases. A t h ig h er c o n c e n tra tio n s w h ere a re the fluidity is sm a ll all m o tio n a l p ro cesses a re d a m p e d o u t ra p id ly an d so integrals over th e tim e c o rre la tio n fu n c tio n s will b e sm a lle r. V elocity cor
relation coefficients for io n s h a v in g o p p o s ite ty p e s of ch arg es /1 3, /2 3, have a marked a s sy m e try in th e b e h a v io u r. T h e m o st in te re s tin g is th e c o n c en tration d e p e n d e n c e of th e w a te r-w a te r coefficient. I t h a s th e c o m p le te ly
138 Haimanti Chakrabarti and Shreekantha Sil
o p p o site profile in th e two system s. T h e e x p e rim e n ta l d a t a is in acco rd an c w ith th e d isto rte d hydrogen bond m odel o f liq u id w a te r.
O ne inference can be draw n from an evalua
tio n o f th e p e rtie n t ex
p erim en tal d a ta is th a t th e electrical cu rren t flu ctu atio n s in high con
cen tratio n s of th e elec
tro ly te solutions con
sidered h ere are such th a t, if an in sta te n e o u s flu ctu atio n is for in stan ce p ro d u ced by th e m otion of th e positive charges th en th is flu c tu ation does n o t decay by a reversal of th is pos
itive cu rren t to g eth e r w ith th e superposition of a flow of negetive charge in th e direction
F ig .2 C o n c e n t r a t i o n d e p e n d e n c e o f t h e v e lo c ity c o r r e la tio n c o e ffic ie n ts w h e n 134Cs t r a n s p o r t i n g in a q u e o u s s o l u t i o n o f KC1.
of original positive cu rren t flu c tu a tio n , r a th e r , th e p o sitiv e c u rre n t flows alm ost com pletely back.
A d etailed p ictu re of th e m icroscpic in te ra c tio n s b e tw e en p a irs of ions can be b u ilt up from th e consideration of th e s e v elo city c o rre la tio n coeffi
cients. It can be shown also how self diffusion a n d c o n d u c tiv ity behaviour of electrolyte solution would be affected by th e se ionic in te r a c tio n s . However we shall defer this m ore d etailed analysis u n til a la t e r p a p e r.
Acknowledgement
ne of th e a u th o r (H C ) likes to th a n k C .S .I.R . for fin a n c ia l a s sista n c e during th e period of th is work.
R eferences
[1] D. W . M cCall and D. C. Douglass, J. Phys. Chem . 71 987 (1967).
[2] D. C. Douglass and H. L. Frisch, J.P hys. Chem . 73 3039 (1969).
[3] H. Schdnert, J. Phys. Chem . 88 3359 (1984).
[4] D. G. Miller, J . Phys. Chem . 71 616 (1967).
[5] F. 0 . R aineri and E. 0 . T im m erm ann, JCS Faraday Trans. 80 1057 (1990).
[6] H. C h ak rab arti, Appl. R adiat. Isot. 45 171 (1994).
[7] H. C h ak rab arti and S.Sil, to be com m unicated (1994).
