Influence of ionic association on viscosity of electrolyte solutions. I . A new approach to the Jones-Dole equation

A C T A U N I V E R S I T A T I S L O O Z I E N S I S FOLIA CHIMICA 9, 1991

Adam Bald*

INFLUENCE OF IONIC ASSOCIATION ON VISCOSITY OF ELECTROLYTE SOLUTIONS

I . A NEW APPROACH TO THE JONES-DOLE EQUATION

The extended forms of Jones-Dole equations have been proposed taking in to account the occurence of io n ic e q u i l i ­ b r ia in e le c t r o ly t e s o lu tio n s of the type KA-, R,A and K,A where K means c a tio n and A anion r e s p e c tiv e ly .

The v is c o s it y measurements in e le c t r o ly t e s o lu tio n s are one of the methods which allowed us to the more comprehensive d e s c rip tio n of ion-ion and io n-d ip o les of so lv e n t in te r a c tio n s . This is one of the methods of study phenomen of io n ic s o lv a ta tio n . Up t i l l now s e v e ra l equations d e sc rib in g r e la tio n s h ip between v is c o s it y and co ncen tratio n of e le c t r o ly t e in s o lu tio n have been suggested. One of mostly ap p lied is the J o n e s-0 o l e equation [ l ] !

where :

T\r - is r e l a t i v e v is c o s it y of s o lu tio n ,

and - the dynamic v i s c i s i t i e s of s o lu tio n and s o lv e n t r e ­ s p e c tiv e ly ,

c - is molar co n ce n tratio n of e le c t r o ly t e .

The c o e f f ic ie n t A is connected w ith e le c t r o s t a t ic ion-ion inter­ a c tio n s and can be c a lc u la te d from F a l k e n h a g e n equation [2].

In s t it u t e of Chem istry, U n iv e r s ity of L d d i, Poland. i /2

The e x c e lle n t agreement of the em p iric a l A c o e f f ic ie n t s and th e o r e tic a l ones was confirmed in aqueous s o lu tio n s . In organic and mixed solven t the agreement is poorer e s p e c ia lly in the case where io n ic as s o c ia tio n occurs [3]. The obtained e m p iric a lly c o e f f ic ie n t 8 is a measure of the ion-solvent in te r a c tio n and is a sum of io n ic c o n trib u tio n s fo r ca tio n and anion.

In te rp re ta tio n of the values of 8 c o e f f ic ie n t may be based on the d if fe r e n t models [4-7]. E s p e c ia lly the values of 3B/9T c o e ffi- cent a ffo rd valu ab le conclusion [8-12] with regard to the e f f e c t of the e le c t r o ly t e on the the so lv en t s tru c tu re .

The Jones-Dole in form (1 ) is obeyed in the lim ite d concentra­ tio n range of e le c t r o ly t e e .g . in water to ca. 0.1 mol -dm’ 3. In organic and mixed s o lve n ts the range of co ncen tratio n should be determined experim entally fo r each e le c t r o ly t e and o c c a s io n a lly terms in v o lv in g higher powers of the molar co ncen tratio n must be added and equation (1 ) can be w ritte n in the fo llo w in g form:

The m ajo rity of the lit e r a t u r e data of v is c o s it y fo r e le c t r o ­ ly t e s o lu tio n s concern the 1-1 e le c t r o ly t e s and the in flu e n ce of io n ic a s s o c ia tio n on v is c o s ty is not u su a lly taken into account. Io n ic a s s o c ia tio n causes th at the co n cen tratio n of ions is lower ihan the e le c t r o ly t e one and 8 c o e f f ic ie n t , being a measure of e le c t r o ly t e so lven t in te r a c tio n , is not a simple sum of BK+ and Ba-.

Hence in te r a c tio n u nd isso ciated p a rt of e le c t r o ly t e for example an ion p a ir with solven t should be also taken in to co n sid e ra tio n . Therefore i f io n ic a s s o c ia tio n , connected with the process:

= 1 ♦ AVcT + Be Dc^

₍

₎

K+ ♦ A' Ä KA

occurs equation (1 ) takes the fo llo w in g form:

rjr = 1 + AVoTc' + B j a c + Bq(1 - oî )c

where :

a - the degree of d is s o c ia tio n , K _o - the a s s o c ia tio n constan t,

8j - B c o e f f ic ie n t fo r io n s: fl^ * a« * * 0A~!

B0 - 6 c o e f f ic ie n t fo r u n d isso ciated form of e le c t r o ly t e .

The degree of d is s o c ia tio n , can be obtained from a s s o c ia tio n

d e s c rip tio n of v i s c i s i t y in e le c t r o ly t e s o lu tio n s .

In the case of e le c t r o ly t e s KA2 type two p o s sib le e q u ilib r ia may be e x is t :

where:

Ka - a s s o c ia tio n co nstan t, Krf - d is s o c ia tio n co nstan t.

For each of them i . e . (4 ) and ( 5 ) , r e s p e c t iv e ly , the a s s o c ia ­ tio n co nstan ts, K_, may be w ritte n :_a

constant using the r e la t io n s :

(3 a ) -Aqh-VSc (3b) 1 Rfi0HVo.c where: y4 - mean molar a c t i v i t y c o e f f ic ie n t ;

ADH’ ®DH " c o e if ic ie n t3 of the Debye-Hückel equation; R - d ista n ce parameter of io ns.

D a v i e s and M a 1 p a s s [13] used equation (3 ) fo r

(4 )

K (5 )

c KA2yKA2 a l CKA+ CA ' yKA+ yA~

K 3 °KA+ KA _____ (7 )

*2 c - c . y - y . U )

\ K 2 A K 2 A

•»

Taking y^- » y ^ * 3 Y~ sod y«^ “ 1 ar*d also d escrib in g concentrations in fo llow in g way: 2

c ka2 * C ( l

'« 0

CA* I£ C I1 ^ + °* 2^

c k +2 = c c i i ^ j . ( 8 )

equation (6 ) and (7 ) can be w ritte n in forms: 1 - Olj K_{a 1 J l} / 1 w 2 \ 2 coi^d - « 2)y^

1 - a 2

Very freq u en tly the e q u ilib r ia (4 ) and (5 ) s a t is f y the condi-tio n

0 » Kfll « Ka2 (11)

which means, th at in the s o lu tio n undissociated form, KA2, does not p r a c t ic a lly e x is t , th at is oi ^ = 1. Thus only the eq u ilib riu m (5 ) describes e le c t r o ly t e s o lu tio n and the a s s o c ia tio n constant, Ka2* may be w ritte n in forms:

Ka2 = ca2( l + « 2 ^ + 2 i1 2 ^

In th i3 s p e c ia l case the Jones-Dole equation should be w ritte n in form [1 4 ]:

nr - 1 + AVcSÇ ♦ Bk*2 ck+2 ♦ Bka* c KA+ * Ba - c

a-or

Tj r * I + A + B ^ + 2 c o2 + 0 « A 4 " a 2^ * B A " * “ 2^

which may bo transform in to form:

rjr * 1 + AVcotj' * (B + 2Ba ~) cc<2 (B ^ + BA~) c ( l - oi^) O 3)

K K A

Denoting: B^+2 + 2BA- » B2, B + ♦ B _ > B j, a = o ij the f o l l o ­ wing r e la tio n s h ip is obtained; KA

*lr * 1 ♦ f^JccT + B2ca ♦ B ^ c (l - cx ) (14)

Such form of Jones-Oole equation was used by Q u i n t a n a e t a l. [14] in order to obtain the B^ and B2 c o e f f ic ie n t fo r Na2S04 in w ater-etanol m ixtures. They used the s a lt of K2A type and in th at case: B2 * BA-2 * 2BK* and ®1 * ®KA~ + BK+ where: K+ = Na+ and A '2 = SO-2.

Assuming otj » 1 the whole co n cen tratio n of e le c t r o ly t e can be d iv id e fo rm ally in to two p a rts which d is s o c ia te according to equa­ tio n s:

p a rt 1 KA2^ --->• KA+ + A” as sym m etrical e le c t r o ly t e

p a rt 2 KA^2 ) --->• K*2 * 2A" as asym m etrical e le c t r o ly t e (15) The co n cen tratio n c ( l - c<2) corresponds the p art 1 and co<2 the p a rt 2 r e s p e c t iv e ly . Pro p o rtion between both c o n trib u tio n s dependent on co n cen tratio n of e le c t r o ly t e i . e . at the lower concen­ t r a t io n the q u a n tita tiv e c o n trib u tio n of the p a rt 2 in c re a s e s . At c = 0 th is c o n trib u tio n becomes to t a l i . e . e le c t r o ly t e is com plete­ ly d is s o c ia te d a c c o rd in g .to the scheme:

where Is the number of types of the ions formed from molecule and

T - *?*2 ^

* 1

The other symbols have th e ir usual meaning.

As i t seen the value of A depend on the type of e le c t r o ly t e and the lim m iting io n ic m o b ility .

In the case i f a l l e q u ilib r ia described equations (4 ) and (5 ) take place 01 ^ < 1 then in s o lu tio n a l l chemical forms e x is t and

th e ir co ncentrations are described equations ( B ) .

Hence e le c t r o ly t e may pe trea ted as the composed of three forms: K A ^ - u ndissociated one,

K A ^ - d is s o c ia te d one according to the scheme KAj —^KA* ♦ A ',

K A ^ - d is s o c ia te d one according to the scheme K A j—vK*2 + 2A". The co n cen tratio n s! c ( l - o i j ) , co^Cl - and c a ^ c o rre s ­ pond these forms KA^0 ) , KA^X) and Ka£2 ) , r e s p e c tiv e ly .

In th is most general case the Jones-Dole equation fo r e le c t r o ­ ly t e of KAj type may be taken on form equation (17)

Tlr = 1 + 'Wca

♦ oijLBjCU - a 2) + B ^ o ^ ] + c ( l - c^JB g

(17)

where!

80 1 BKA2* B1 = 8KA+ + 8A'* 02 s BK+2 * 2Ba- In the case of e le c t r o ly t e of K2A type i t w i l l be: B0 = BKA7’ B l = b kA” + BK+> B2 = BA” 2 4 2Bk + r e s p e c tiv e ly .

0x2 ( l z 2 | - j z j

* i Z | i . i z 2r 1/2 ■i U i Z 2l

Assuming Ka ^ « K&2 and cij » 1 the form KA2 is p r a c t ic a lly absent in the s o lu tio n equation (17) takes on form (18 ) id e n t ic a l w ith equation (14)

rjr = I ♦ AVcolj ♦ B i c ( l " a 2^ * B^ca2 (18)

As besides Kfl2 has in co n sid e ra b le value i . e . oi2 “ 1 then equation (18) can be w ritte n in form:

i^r = I ♦ AjVc" + Be

id e n t ic a l with one fo r com pletely d is s o c ia te d e le c t r o ly t e s of KA2 type.

For unsym etrical e le c t r o ly t e s of KAj or K} A type the fo llo w in g p o ssib le e q u ilib r ia should be taken in to account:

For unsymmetrical e le c t r o ly t e s of KAj or K-jA type the f o l l o ­ wing

Kal

KA* + A' KA, (19a)

*^a2 KA + 2 + A- «==> KAl (19b) Kd2

,

Ka3

+2

and the a s s o c ia tio n constants can be w ritte n fo r then, re s p e c tive ly :

cKA3 yKAj

al _{c KAt CA- yKAÎ yA-} (20a)

va2

cKA; yKA2

CKA+ yKA*

« . J • c . « » ">

K A K ° A

I f ^o<2 and oij. denote the degree o f'd is s o c ia t io n c o rre s ­

ponding to e q u ilib r ia (1 9 a ), (19b) and (1 9 c ), r e s p e c t iv e ly , the concen tratio n of a l l p o ssib le form of ions and u nd issociated e le c t r o ly t e can be obtained from equations (21)

cKAj = (1 - o l p c (21a)

cKA3 = o i j ( l - q i2)c (21b)

CKA*2 * “ l ' V 1 ‘ ' ■ (21c)

» oijC ♦ ajOijC ♦ oijoijajc (2 ld )

ck+3 = o t ^ a j c ( 21e)

When the determined as above co ncen tratio n s are s u b s titu te d in to equations (2 0 a ), (2 0 b ), (20c) then fo r

yKA3 * 1 and *I<a; 3 in ­ equations (20) take forms:

1 - a . K = --- (22a) o ijd - a 2) ( l + a 2 + a 2a 3)c y 2 0 1 ,(1 - a 2) Ka2 = -*---—— --- — (22b) d jC tjd - a 3) ( l ♦ a 2 + a 2°'3^cyKA+2 o ijC jd - ct3)y KA + 2 a3 7 1 I--- ~ (22c) ° ' l a 2a 3^1 * a 2 * c*20l3^c y - yK*3

I f a l l e q u ilib r ia (19 ) occur and a l l p o ssib le chemical forms of e le c t r o ly t e e x is t th e ir co n cen tratio n s are expressed by equation (2 1) , e le c t r o ly t e may be considered as composed of four p arts

- u nd isociated one,

- d is s o c ia te d one according to the scheme: KAj—*KA2 ♦ A ,

KAj2^ - d is s o c ia te d one according to the scheme: KAj—*KA+2 + 2A ,

K A j^ - d is s o c ia te d one according to the scheme: KAj—►K*3 + 3A .

The co ncen tratio n s c ( l - ot^), c a ^ (l - c*2) , ca^otjd - <*3) and

c a lc»2a 3 correspond the forms K A j ° \ K A j1^, KA^2^ and K A ^ , r e s ­ p e c t iv e ly .

In the above case fo r e le c t r o ly t e of KAj type the Jones-Dole equation should be presented in form (23a)

Tjr * 1 + A-yfco^o^o^ + 011 1 - * 02ot2^1 " a ^ c + B3°!2oi3c^ *

* (1 - j)c B q (23a) where: B o 1 b k a3 -bi 1 BKA2+ + V > B2 * BKA+2 + 2 0A-. B3 s BK + 3 + 3BA

-in the case of e le c t r o ly t e KjA type i t w i l l be:

8 0 =

BKjA* B1

V * BK2A->

BKA-2'

3BK+ + BA-3

r e s p e c t iv e ly .

The Jones-Qole equation fo r KA} e le c t r o ly t e shown in form (23a) is mostly general equation taking in to c o n sid e ra tio n the p o s s i b i l i ­ ty of occurance of e le c t r o ly t e in d if f e r e n t chem ical forms, which

concen tratio n depend on the values of a s s o c ia tio n constants K a21 Ka3 and the molar concen tratio n of e le c t r o ly t e .

U su a lly the e q u ilib riu m constants obey the in e q u a lity Kgl-< Ka2<Ka3’ Then oij * 1 and equation (23a) should be modified (K^j « 0 ):

* 1 + A - ^ / c* B j ( l - orfjic * " o1} ) ♦ (23b)

When moreover K32 < Ka3 and c<2 “ 1 equation (23b) is given by

t|r = 1 ♦ AVcoi-j + B 2c ( l - o tj) + B^coij (2 3 c ) I f also Kaj reaches not larg e values and oi^ » 1 then equation (23a) can be w ritte n in a form:

Tjr = 1 + AV? + BjC (23d)

which is c h a r a c t e r is t ic fo r com pletely d isso c iate d e le c t r o ly t e s of KAj or KjA type.

Summarizing the suggested approaches to the Jones-Dole equation include in flu e n ce of ion-ion and ion-d ipole of so lven t in te r a c tio n fo r a l l ions and ion p a irs e x is tin g in s o lu tio n . The complete forms of these equations fo r e le c t r o ly t e of KA2 type i . e . equation (17) and e s p e c ia lly fo r KA^ one equation (23a) are e v id e n tly com plicated. However the lim itin g e q u iva le n t c o n d u c tiv itie s of ions and the a s s o c ia tio n co nstan ts, determined from the co n d u ctivity measurements, enable to c a lc u la te the values of A c o e f fic ie n ts from the Falken- hagen equation and the degrees of d is s o c ia tio n of e le c t r o ly t e in an i t e r a t i v e procedure. The values of B c o e f fic ie n ts corresponding to s p e c ifie d ions and ion p a irs can be obtained from s e rie s of rj and c data by use of s u ita b le methods.

%

The review of the proper methods to enable to s o lu tio n suggest­ ed abo forms of the Jones-Dole equation was made in paper [15] .

REFERENCES

[1] 6. J o n e s , M. D o l e , J . Amer. Chem. S o c ., 51,, 2950 (1929).

[2] T. E r d e y-G r u z, Transport Phenomena in Aqueous S o lu ­ tio n s , Budapest 1974, p . 128.

[3] J . C r u d d e n, G. M. D e l a n e y , D. F e a k 1 n s, P. J . O’ R e 1 1 1 y, W. E. W a g h o r n e , K. G. L a w ­ r e n c e , J . Chem. Soc. Faraday Trans. I , ¿ 2 , 2195, 2207 (1985).

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]

[5] R. S. S t o k e s , R. M i l l s , V is c o s ity of E le c t r o ly t e s and R elated P ro p e rtie s . Pergamon Press I n c . , New York 1961. [6] B. S. K r u m g a 1 t z, J . Phys. Chem., 82. 763 (1979). [7] D. F e a k 1 n s, D. J . F r e e m a n t 1 e, G. K. L a w ­

r e n c e , J . Chem. Soc. Faraday Trans. I , _70, 795 (1974). [8] M. K a m i n s k y , D isc. Faraday S o c ., 24,, 171 (1951). [9] D. J . P. 0 u t , J . M. L 0 s , J . S o lu tio n Chem., 9, 1 (1980). [10] M. K a m i n s k y , Z. N a t u r f., 12a, 424 (1957). [11] E. R. N i g t i n g a 1 e, J . Phys. Chem., £3, 138, 742 (1959). [12] I . M. T s a n g a r i s , M. R. B r u c e , Arch. Biochem. and B io p h y s., 112, 269 (1965). [13] C. N. D a v i e s , V. E. M a 1 p a s s , Trans. Faraday S o c ., 72, 2075 (1964). [14] C. Q u i n t a n a , M. L. L 1 o r e n t e, M. S a n ­ c h e s , A. V i v o , J . Chem. Soc. Faraday Trans. I , 82, 3307 (1986).

Adam Bald

WPŁYW ASOCJACJI JONOWEJ NA LEPKOŚĆ ROZTWORÓW ELEKTROLITÓW I . NOWE POSTACIE RÓWNANIA JONESA-DOLE* A

Zaproponowano rozszerzono w ersje równania Jone3a-Dol8’ a, które uwzględniają równowagi jonowe występujące w roztworach e le k tr o litó w n ie c a łk o w ic ie zdysocjowanych typu KA2, K? A, KA, i K,A, gdzie: K - k a tio n , A - anion. Współczynniki A i B proponowanych równań związa­ ne z oddziaływaniami typu jon-jon i jon-rozpuszczalnik przypisano ś c iś le określonym jonom i parom jonowym.