Influence of ionic association on viscosity of electrolyte solutions. II . Methods of determination of the Jones-Dole equation

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

Adam Bald*

INFLUENCE OF IUNIC ASSOCIATION ON VISCOSITY CF ELECTROLYTE SOLUTIONS

I I . METHOOS OF DETERMINATION OF THE JONES-DOLE EQUATION

S eve ra l methods of s o lv in n the forms of the Jones-Oole equatation presented e a r l i e r [ l ] have been suggested fo r incom pletely d is s o c ia te d e le c t r o ly t e s of KA, KA2, K2A, KAj and KjA types where K and A are c a tio n arid anion, r e s p e c t i­ v e ly . ihe d if f e r e n t v a ria n ts taking in to account the values of io n ic e q u ilib riu m constant have been considered. The aim of sugested methods is the c o rre c t d eterm ination of B c o e ff ic ie n t s of the Jones-Dole equation.

ELECTROLYTES OF KA TYPE

In the case of 1-1 e le c t r o ly t e s which are com letely d is s o c ia te d the Jones-Dole equation is given in the form:

t] r s 1 + AVcT + Be (1 )

or at higher co n cen tratio n s of e le c t r o ly t e 2

Tjr = 1 ♦ AVcT + Be + Dc (2)

The A and B c o e f f ic ie n t s can be determined i f the above equa­ tio n s are rearranged to the forms:

( U )

and

- *

_{= A ♦ BV}

c

T ♦ OcJ/2

(2a)

Reationship ( l a ) I s polynomlnal of f i r s t order in r e l a t i o n to

■Jc.

Using the l e a s t squares method the A and

B

c o e f f i c i e n t can be

c a l c u l a t e d . Relation (2a) i s polynomlnal of thir d order in r e l a t i o n

to

and the l e a s t squares method l e t s us to determine the a l l A,

B and 0 c o e f f i c i e n t s . They can be a ls o c a lc u l a t e d by use

other

methods proposed by

H a r t i n u s

e t a l . [2]. In my opinion

the more c e r ta in way of the determination the . a l l parameters of

equations

(1) and (2) i s based on using in the c a l c u l a t i o n s the

t h e o r e t i c a l A c o e f f i c i e n t s . That

i s because for other so l u t i o n s

than

in water

the agreement of the values of t h e o r e t i c a l and empi­

r i c a l A c o e ff ic ie n t s is not r e a lly

s u f f i c i e n t

[3 ]. Hence

A should

be obtain from

the

F a l k e n h a g e n equation

[4].

The symbols in equation (3 ) have their usual meaning. Therefore, the lim itin g io n ic io n ic c o n d u c tiv itie s are required which may be determined from c o n d u c tiv ity measurements in e le c t r o ly t e s o lu tio n s . I f the A c o e f f ic ie n t are known, equations (1 ) and (2 ) can be rearranged in the fo llo w in g way

(3)

( | z 2U ° - |zt | X ° ) 2 U j

1

♦ l z 2 l 1 1 / 2 1 2 T ^ \ J ■rçr - 1 - AVc B _{( l b )} c

and

qr - 1 - AVc*

Equation ( l b ) and (2b ) can be treated as polynim inal of zero and f i r s t order against of molar co n c e n tratio n , c, r e s p e c t iv e ly . The s o lu tio n of equation ( l b ) reduces then to c a lc u la tio n of the a rith m e tic mean and equation (2b ) can be solved the le a s t squares method. The above ways of c a lc u la tio n c o e f f ic ie n t s g ive r e a lly the b e tte r r e s u lts because of the lower number of c a lc u la te d parameters. So lvin g equation ( l b ) and (2b) the in d is p e n s a b ility of taking in to account the 0 c o e f f ic ie n t can be also estim ate. When e le c t r o ly t e is not q u ite d is s o c ia te d the e q u ilib riu m of the process: K+ + A'

I / o

KA can be described the a s s o c ia tio n constan t:

K = I . 9 ! .

a 2 2

cot y +

where cl is the degree of d is s o c ia tio n and y+ the mean a c t i v i t y c o e f f ic ie n t . The Oones-Oole equation fo r th is case, acco rd in g ly with 0 a v i e s and M a 1 p a s s [5 ], is given by:

= l ♦ A V c “ + B ^ c a + B p C ( l - a ) ( 4 a )

where B . * B + B , and B is c o e f f ic ie n t of ion p a ir.

1 K+ A" p

Equation (4 a ) may be rearranged to the form:

- r---- 1A— = _{c a} B . *_i Bn _{p ot} (4b)

and hence

the

Bi

and

Bp

c o e f f ic ie n t s can be obtained by the le a s t

s quares

method. The values of A c o e f f ic ie n t s can be obtained in two

ways. The

f i r s t one [6] is based on the e x tra p o la tio n tj - vs. Vc\ taking

the

in te rc e p ts at

VcT

=

0

as the requ ired A valu e s and assumption th at the obtained A values are c o rr e c t. The second one

[3] is based on the c a lc u la tio n the A values from equation (3 ) taking the obtained p re v io u sly values of io n ic c o n d u c tiv it ie s .

The la s t - mentioned one is more se n s ib le in my opinion.

The values of the degree of d is s o c ia tio n at each co ncentration used can be i t e r a t i v e l y c a lc u la te d from the fo llo w in g equations!

R - i s the distance of c l o s e s t approach of ions,

I - the ionic force of e l e c t r o l y t e

- J T

3 ca.

The mentioned above both methods methods of solution of equation

(4a) are based on indipendent determination of the

A

c o e f f i c i e n t s

and l i n e a r i s a t i o n of equation (4a) to the

e a s i l l y s ol va b le form

(4 b ). In my opinion equation (4a) can be solved using the nonlinear

l e a s t squares methods with simultaneous determination of

a l l

i t s

parameters i . e .

A, Bj

and

Bp .

From equation (4a) and d e f i n i t i o n of the t o t a l d i f f e r e n t i a l

i t

follows tha t:

(5a)

(5b)

where !

(

6

)

where:

- | p = c ( l - a ) ,

P

^ r ' ^ r .e x p . ~ * 2 r , c a l c . ’

Applying the le a s t squares method to the equation (6 ) the system of equation (7 ) is obtained:

aa£ —k \ i [“t s

* E ^ r 1 [ ^ ] i

AAE[%]i[S[]i ♦

i

* E m c i [ ^ ] 1 ûAE [ -i x ] i

[U;]i +

a b i E = E û t i r i[-a ilrl * ah v M !" ~ à _ l * pH aBn J l L anr dA anr 357

p

ai2r anr aBT arjr

« ¿ [ S ' ].

+ AB

x[

3*ll p H 3Br 3ti

3B

_i drl i aB,

(7)

where the unknown quantity are increments of AA, AB^ and ABp .

The f i r s t step in c a l c u l a t i o n must

, ( 0 )

'P

be an e val ua ti on of the

i n i t i a l values of Av“ / ,

and Bp° \ Then the Ka

values are used

to c a l c u l a t e the degree of d i s s o c i a t i o n ,

a

,

from equations

(5)

and the Ai ^,

dt^/dA,

ai

2

r /a B^ , and 3tjr /3Bp

values

for

a l l

experimental po i n t s . The next s u i t a b l e sums, occuring in equation

are obtained

system ( 7 ) , are c a lc u l a t e d and A^1^, B^1^ and B^ ^

i

p

so lv in g i t . As a r e s u l t i t allows us to evaluate more p r e c i s e l y the

a lue s o f : A(1) = A(0) ♦ AA( 1 ) ,

b

[ : ) * fl[0) +

AB^X) and

b

£ L) =

procedure i s repeated

B<°> .

Then the a l l c a l c u l a t i o n

B ; L) and B^l )

p

( 1

\

taking into account as the i n i t i a l values A

,

___

g e t t i n g in the second cy cle of c a l c u l a t i o n the next co rr e ct values

of searched c o e f f i c i e n t s .

The c a l c u l a t i o n s are fin is he d i f :

* n (k>

rfTJ r a n

(k+l )

<

d

(B)

where :

k - number of c a l c u l a t i o n cy cl e ,

n - number of measurement.

I t should point out that taking Into consideration the number of

p a r a l l e l searching parameters the above-method can be used only for

data of high p r e c i s io n .

I was reported e a r l i e r [l] that in e l e c t r o l y t e s ol ut io n of KA2

type the following e q u i l i b r i a can be expectet:

ELECTROLYTES OF KAj AND KjA TYPES

KA* + A" * = * KA2 and K*2 ♦ A'

* = *

KA+

for which the equilibrium constants are expressed:

K

_{( 10)}

K

₍

11

)

where

In y± =

( 12)

1 ♦

rboh

VT

*ao h^

In equation (12) and (13) the AQH and BDH c o e f f i c i e n t s are the

Debye-Hiickel ones.'The parameter of the c l o s e s t approach of

ions,

R, according to J u s t i c e ' s s ug ges tio n, can be

assumed to the BJer-

rum’ s d ist an ce q * | z j Z 2 | e 2/2kT

[7, 8],

The J o n e s-D o l e equation takes the general form [l ]

tjc = 1 ♦ Ayfcdja^

*

o»^[B^c(l -

0

*

2

) + BjCoij] ♦ c ( l - o(j)Bq (14)

where

and

c ij

are the degrees of d i s s o c i a t i o n s r e l a t e d to the

and Ka2 equilibrium co n s ta n ts , r e s p e c t i v e l y . The Kal and

Ka2

values may be found from conductivity measurement

of e l e c t r o l y t e

s o l u t i o n s .

The degrees of d i s s o c i a t i o n

oi^

and

01

2 for each e l e c t r o l y t e

concentration used are c a lc u l a t e d from Kal and Ka2

so l v i n g the

equation system (1 0 ) - ( 1 3 ) using the

i t e r a t i v e method of succ es ive

approximations. In t h i s aim subroutine c a l l e d ALFA 12 can be used.

Equation (14) contains as many as 4

unknown parameters.

The

simultaneously c a l c u l a t i o n of them would be, in my opinion* theore­

t i c a l l y p o s s i b l e using the nonlinear

l e a s t squares method

i f the

whole procedure was converged. The obtained r e s u l t s would be burden

with s i g n i f i c a n t e r r o r s . I t i s more reasonably

to determine

the

l im it in g ionic c o n d u c t i v i t i e s from conductivity

measurements

in

e l e c t r o l y t e s o l u t i o n s and using them to c a l c u l a t e the A c o e f f i c i e n t

from equation ( 3 ) .

The other parameters B^, B2 and B

q

can be c a l c u l a t e d

in two

ways :

1° Using equation (14) in the form of t o t a l d i f f e r e n t i a l

An r s A 0 i + â B ^ A B 2 +

30

^ * Bo ( 1 5 >

where a n r / 9flo = c ( x " “ i * » 3nr / 3Bi = coli (1 "

a 2^ > dllc / d B 2

3

and from the a p p l ic a ti o n of the l e a s t squares method the following

equation system can be writ te n:’

anr

4,

o

E [ ^ ] 1 f e ] , *

* “ j l [ 2 j ] l [ 2 j ] i

=I > v i

ABoE[^]i ^ i + ABlE[S[]i[§-"]i * AB2E[^]i[^]i

•£aV i i B o ï t e ] i a0i dn r + AB.

* “ E[

5

î]‘

! 3e dB-' W . i 3Tît dB_{2 J} (16)

Finding increments of ABq, AB^ and AB2 the more p re c is e ly values of the B c o e ff ic ie n t s can be c a lc u la te d and the whole proce­ dure is repeated as mentioned above analogous method for incomple­ t e ly d is s o c ia te d 1-1 e le c t r o ly t e s .

2 ° Equation (14) may be lin e a r iz e d to the form:

Tjr - 1 - c ( l

-cetjO<2

b2

+ B i

I

-(17)

The l e f t hand

assuming the value of Bq c o e f f ic ie n t .

sid e of equation (17) can be c a lc u la te d Then B^ and B2 c o e f f ic ie n t s can be obtained from the le a s t squares method. The c a lc u la tio n must ;,e repeated fo r the Bq values changing w ith the d e f in it e step t i l l to fin d the minimum of the standard d e v ia tio n of the fu n ctio n

e s ta b lis h in g in th is way the optim al values of the BQ, and B2 c o e f f ic ie n t s .

I t may be noticed th at in both methods the experim ental data of high p re c is io n should be used. L u c k ily , as a ru le , Ka « Ka2 and a d d itio n a lly “ 0. In th is case otj = 1 and equation (14) has a form:

r/r = 1 ♦ Ary6oi2 B jC d - a 2) B2cw2

(

18

)

This equation can be solved with help of the both e a r l i e r pro­ posed methods and only two param eters, B ( and B2 , to be determined. Both the methods w i l l be s ig n if ic a n t y s im p lifie d from the fo llo w in g reaso ns:

1 The determ ination of the c*2 value reduces to the i t e r a t i v e s o lu tio n of the system

1 - oi ‘ a2 _{co.2( l + oi2) y ^ 2} and OH I f , y --- — K 1 + RBq h-vT where I = c ( l + 2c*2).

2 ° In f i r s t method, above e a r l i e r , only the system of equations remains for so lu tio n :

ratjr ' ABl l [ l 5 7 j i æ_l-ii a b2 E 3-rjr AB_{l ^ a e j i}

r f — 1

3TIr_3B 2-> ao2-1 a J k 3B~ t - E ï k l 3B2j i . ^ r ,1

3 ° In the second method, equation (18) can be rearranged to the form:

1 - ct n r- i - A v ^

where 8^ and B2 c o e ff ic ie n t s are determined by the le a s t suares method.

I f also the K ^ value is very sm all then a 2 “ 1 and equation (1 8 ) changes i t s form to :

*»

t j f « 1 ♦ A V ? ♦ B jC ( 1 9 )

c h a r a c t e r is t ic fo r com pletly d is s o c ia te d e le c t r o ly t e and which can be solved by one from the e a r l i e r described methods.

ELECTROLYTE OF KAj AND KjA TYPES

As i t was reported e a r l i e r [ l ] in s o lu tio n ' of e le c t r o ly t e of KAj type the fo llo w in g e q u ilib r ia can be expected:

Kai Kj2 K j j

KAg ♦ A" * = * KAj, KA+2 ♦ A" 9 = * KA^, K+3 + A* *= *• KA+2

ch a ra c te riz e d by the co nstan ts:

I - a i K s o 2 (20) ° « l ( l - a 2) ( l + a 2 + o«2a 3) c y i

d , d - ot, )

K . — , --- i---i--- - (21) o i j ^ d - a 3) ( l ♦ a 2 * « 2a , ) c y 2 KA t f i d i d - G U iy * *« * K3j = f 1^ --- (22) o i ^ a j d + a 2 + a 2oi3) c y ± y +J K where; Aq u - v r In

¥±

--- — (23) i ♦ rbdhV T 4a o h v t In y

,2 -

--- ^ ---- (24) K A 1 ♦ R 8 d h V T

. in y , -

---K 1 + RB0HV Î

(25 )

and I = 201^ 2 (1 + 3 a j)c .

In the case of s o lu tio n of e le c t r o ly t e of KA^ type the Jones- -Oole equation has the form [ l ] :

I f the values of K a l , K ^ and Kaj are known, the d is s o c ia tio n degrees oi^, c*2 and ot^ fo r each c o n c e n tra tio n , c , can be i t e r a ­ t i v e l y estim ated. The subroutine ALFA 123, w ith co n sid erab ly more advanced algorithm than mentioned above subroutine ALFA 12, may be ap p lied fo r i t .

With the aim of reducing the number of parameters ( f i v e ) the values of io n ic c o n d u c tiv it ie s should be in d ip e n d en tly obtained ( I . e . by measurement of e l e c t r i c a l c o n d u c tiv ity of e le c t r o ly t e so­ lu tio n s ) and the A c o e f f ic ie n t s c a lc u la te d from equation (3 ) .

S im ila ry as in the case of e le c t r o ly t e of KA2 type the two methods of determ ination of the Bq, B^, b2 and B-j c o e f f ic ie n t s can be used.

1° From the r a la t io n s h ip :

and by applying the n o n lin ear le a s t squares method the equation system is constru cted , from which s u c c e s iv e ly decreasing increments of ABq, A B j, AB2 and AB3 values are determined and f i n a l l y the proper values of the B c o e f f ic ie n t s are obtained.

2 ° L in e a riz in g equation (26) we haves

rjf * 1 + A-^o^otjcÇ + °*i

1 " a 2^c * 02a 2^1 " a 3^c + 03OI2OI3C^ +

♦ ( 1 - o l j )cBq (26)

y *

= B, + B

- 1 - A-y/cdjOtjOtj - B q C ( 1 - o ^ ) * B jC O I^ 1 - c*2 ) co^o<2oij

The l e f t - hand sid e of the above aquation is c a lc u la te d assu­ ming the Bq and B j c o e f f ic ie n t s . The fl2 and values are determ in­ ed by the le a s t squares method. The c a lc u la tio n must be repeated fo r Bq and B^ changed w ith fix ed step down to the minimum of the standard d e v ia tio n , This procedure optim izes the values of B0 , B p B2 and B j c o e f f ic ie n t s . In my opinion the f i r s t method is in th is case much more e f f e c t iv e fo r s o lv in g equation (2 6 ). Accuracy of the experim ental must be extremely high and the number of data r e l a t i v e l y la rg e .

G e n e ra lly , the e q u ilib riu m constants f u l f i l l e d the in e q u a lity K a3 > K a2 > Kal " °* i , e ' 011 = 1 and aquation (26) s im p lif ie s to the form:

Tjr = 1 + Ay'ca^a^' + B ^ l - oi2)c + a 2[B 2( l - a 3)c + BjC ijc] (28)

fo r which the values O j, a 2 are e a s ily o b tainable and the above mentioned two method of s o lu tio n can be ap plied (th e n onlinear le a s t squares method and lin e a r is a t io n of equation (2 B ) to the s im p lifie d form of equation ( 2 6 )).

72- - 1 - AJc,oi9a ,' - B , c ( l - a „ ) c l - o / ,

y --- ^ ; c ..*---— ■ b3 * B2 — ^ (29)

F req u en tly, also the r e la t io n Kj3 > K a2 * 0 and oj2 * 1 takes p la ce . In th is case only the parameters B2 and B j remain unknown and the equation (26) changes i t s form:

rit - 1 ♦ A^ca + B2( l - o fj)c * B ^ c

which can be e a s ily rearranged to the form e a s ily s o lv ab le by using the simple le a s t squares method.

In the case of com pletely d isso c ia te d e le c t r o ly t e s of KAj type (oij = oi2 = oi3 1 1) equation (27) s im p lifie s to the form:

which can be also solved using e a r l i e r mentioned methods a p p lic a b le to the com pletely d is s o c ia te d 1-1 e le c t r o ly t e s .

As a r e c a p itu la tio n , one should claim th at u n t i l now the pro­ posed new methods of s o lv in g the Jones-Dole equation fo r the incom­ p le t e ly d is s o c ia te d e le c t r o ly t e s ( e s p e c ia lly non-sym etric ones) were no reported in the l it e r a t u r e regarded to v is c o s it y of e le c ­ t r o ly t e s o lu tio n .

I hope th at these methods w i l l be u se fu l in determ ination of the c o e f f ic ie n t s A and B of the extended Jones-Dole equation. I t is e s p e c ia lly s ig n if ic a n t , because the values of A and B c o e f f ic ie n t s r e f l e c t ion-lon and ion-aolvent in t e r a c tio n , r e s p e c t iv e ly . They are d ir e c t ly connected with s p e c ifie d io n s, which s u f f i c i e n t l y f a c i l i ­ ta te s in te r p r e ta tio n of obtained data.

REFERENCES

[1] A. B a l d , Acta Univ. Lodz., F o lia Chim ica, 9, 33 (1991). [ 2 ] N . M a r t i n i u s , C . O . S i n c l a i r , C . A . V i n ­

c e n t , E lectro ch im . Acta, 22., H B3 (1977).

[3] J . C r u d d e n , G .M . D e l a n e y , 0. F e a k l n s , P. 3. O ' R e i l l y , W . E . W a g h o r n e , K . G . L a w ­ r e n c e , 3. Chem. S o c ., Faraday Trans. I , 82, 2195, 2207 (1985).

[4] T. E r d e y-G r u z, Transport Phenomena in Aqueous S o lu ­ tio n s , Budapest 1974, p. 12B.

[5] C. W. D a v i e s , V. E. M a 1 p a s s , Trans. Faraday S o c ., 72, 2075 (1964).

[6j C. Q u i n t a n a , M. L. L 1 0 r e n t e, H. S a n c h e s , A. V i v o , J . Chem. Soc. Faraday Trans. I , 82., 3307 (1986). [7] J . C. J u s t i c e , J . Chim. P h y s ., 65, 353 (1968).

Adam Bald

WPŁYW ASOCJACJI JONOWEJ NA LEPKOŚĆ ROZTWORÓW ELEKTROLITÓW I I . METODY WYZNACZANIA WSPÓŁCZYNNIKÓW RÓWNANIA JONESA-DOLE’ A

Zaproponowano szereg metod rozwiązywania równań Jonesa-D ole’ a dla roztworów e le k tr o litó w n ie c a łk o w ic ie ¿dysocjowanych typu KA, KA„) l^A, KA,, i KtjA (K - k a tio n , A - anion) f l j . Rozważano różne w arianty uwzględniające w artości s ta ły c h równowag jonowych. Propo­ nowane metody mają na celu głównie poprawne wyznaczenie w artości współczynników B równania Jonesa-D ole’ a i powiązania tych w artości ze ś c iś le określonymi jonami.