Diffusional titration errors in coulometry

Electroanalytical Chemistry and Interfacial Electrochemistry

255

Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

DIFFUSIONAL TITRATION ERRORS IN COULOMETRY PART I. ANALYTICAL SOLUTIONS FOR NON-STATIONARY D I F F U S I O N T H R O U G H NON-INTERACTING MEMBRANES

JOHAN CLAESSON

Department of Mathematical Physics, Lurid Institute of Technology, Box 725, S-220 07 Lund (Sweden) JAN LINDBERG

Department of Analytical Chemistry, University of Umett, S-901 87 Umed (Sweden) (Received 1st May 1972)

INTRODUCTION

This investigation deals with non-steady state diffusion for conditions relevant to coulometric titrations. The aim has been to calculate concentration distributions and the amount of sample transported across the diaphragm separating the sample and connecting chambers. The one-dimensional macroscopic diffusion inside the membrane is assumed to be governed by Fick's law in terms of an appropriate proportionality constant, K. This is the approach of Goodknight et al. 1. Fatt's investigation of sintered glass membranes 2 indicates that they have no significant amount of dead-end pores. Many separators in coulometric cells are of this type. The analytical solutions given apply to such cases. The concentration distribution in the membrane and the connecting chamber and the total sample amount inside these are calculated for both linearly and exponentially decreasing types of titration. Analytical solutions under the simplified assumption of linear concentration distribution at all times inside the membrane are also given. Many authors 3'~" have used a relation between K and electrical membrane resistance. One method of deriving this relation is presented in this paper. GENERAL CASE

Equations Figure I represents the geometry for a simplified coulometric cell. The diffusion SAMPLE COMPARTMENT

CONNECTING CHAMBER

MEMBRANE

>.v

X---I

Xffi0

Fig. 1. Geometry of the coulometric cell where diffusion occurs in the x-direction.

J. Electroanal. Chem., 40 (1972)

256

J. CLAESSON, J. LINDBERG

occurs in the x-direction. The sample compartment, x < - l, is stirred and the liquid for x > - l is quiet. The concentration in the sample compartment is a known function of time. The thickness of the membrane is I. The connecting chamber is assumed to be semi-inf'mite. The constant starting sample concentration for x > - l defines the concentration zero level. To calculate the non-stationary mean one-dimensional diffusion inside the membrane it is necessary to relate the mean concentration in the membrane C1 (x, t) to the total flux J(x, t). The basic assumption for the membrane is: OC, J (x, t) = - K "0x

(1)

K is an empirical proportionality constant which may be related to the electrical resistance across the membrane. This is discussed below. The continuity equation for a membrane without dead-end pores is: ~x (S (x, t)) = ~~ (C, (x, t) ~v)

(2)

e, is the porosity. This means that e, is also the free area per unit membrane area. The left-hand side represents the net flux into x per unit time. This equals the time derivative of the mass content per unit area at x. Inserting (1) in (2) gives the equation for diffusion inside the membrane: d2C 1 1 0C1 0x 2 - D~ 0t

l
t>0

(3)

= K/evO

The concentration outside the membrane, x > 0, is denoted C2 (x, t). Here C 2 satisfies the usual diffusion equation: ~2C 2

~0x-

10C 2 D 0t

x>0,

t>0

(4)

Initial conditions: C,(x, 0) = 0

C2(x, 0) = 0

(5)

Boundary conditions: C1 = C2

K

(~C1/Ox =

D 8C2/Ox

C, = / ( t )

for x = 0

(6)

for x = 0 for x = - I

(7) (8)

The boundary conditions (6) and (7) mean that concentration and flux are continuous at x = 0; f(t) is the known concentration in the sample compartment. The boundary condition (8) means that the concentration at x = - 1is continuous. The total amount of sample that has diffused into the membrane and the connecting compartment per unit membrane area during time t is:

M(t) =

f

'

(-K)

0

J. Electroanal. Chem.,

(x, t') ~x

• dt' x -- - 1

40 (1972)

(9)

257

NON-STATIONARY D I F F U S I O N T H R O U G H M E M B R A N E S

Calculations Laplace transforms of (3) and (4) using (5) give:

C,(x,p)=A(p)exp[x{P)']+

B(p)exp

j

[-x (P'fl ocj j

I3')

C 2 ( x , p ) = C , p ) e x p [ - x ( P ) ½]

(4')

A bar denotes a Laplace transform in t. The Laplace transforms of eqns. (6), (7) and (8) determine A, B and C. A (p) + B (p) = C (p)

(6')

F [A (p)- B (p) ] = - C (p)

(7')

A (p) exp [ - l (p/DCF] + B (p) exp [l (p/D¢) ½] = f(p)

(8')

F = K/D~ ~ This gives: co

a' {exp [-(l(2v+ 1)+x)(p/D~) ½]

(x, p) = v=0

o- exp [ - ( l (2v + 1)- x)(p/De) ½]}f(p)

-

2F

~ a'exp{

C z (x, p) = 1 +----fir=o

-

[l(2v+l)

¢~

+X

](D)'}

(3")

f(P)

M(p,=FD½p-½ { l + 2 . 2 l a ~ e x p [-2vl (-~)½]}f(p)

(4")

,

(9')

G = (l-F)/(1 +F) In deriving (3") and (4"), the formula for the sum of an infinite geometric series was used : 1 = ~ a ~ exp ~- 21v 1 -o- exp [-21(p/D~) ~] v=o The following cases are considered:

(10)

f(t) = Co ;

f(t) = Cot/t1 ; f(t) = Co exp(-flt)

(11)

i.e. f(p) = PC°'

f(p) = ~-1 Co 71' respectively f(p) = P+ Cofl

(12)

In the solutions below, Co, t 1and fl > 0 are arbitrary constants. The following functions corresponding to the Laplace transforms in eqns. (Y'), (4") and (9') inserting (12) are needed to complete the solutions.

(4t)n/2 i" erfc (~-~) ~ J. Electroanal. Chem., 40 (1972)

exp(--~P p1+,/2 ½)

n = 0 , 1,2,3

(13)

J. CLAESSON,J. LINDBERG

258 exp ( - a p ½) p+/~

(14)

fl-T"Im [Wj(t, a, fl)] ~ exp(-~p~) ¢(v+

(15)

Re [Wj(t, ~, fl)]~

,

1

i° erfc = erfc, i.e., the complementary error function. The functions i" erfc are repeated integrals of erfc. Tables of these functions are found in ref. 6. Wi is complex : N(t,

= exp

t"

( a - i(fit)½)

erfc

"

(16)

The complex function exp (~2/4t)W~(t, ~, fl) is tabulated in ref. 6. Expressions (14) and (15) are deduced from the table of Laplace transforms in ref. 5, while (13) is given there explicitly.

Solutions (1) f(t)= C o

(

/l(2v+l)+x\

C,co(X,t) = v=o ~" a"/erfc ~. ~

C2(x,t) Co

2F ~

.

(l(2v+l)-x'~(

{l(2v+l)/~+x]

- I + F ,=o ~ a* eric k

(4-~f

(18)

)

M(t) = FD~ { 2 ( t ) ½ + 4t½ ~ a~ierfc ( l Co

(17)

) - a erfc k (4DCt).~ / j

)}

(19)

v=l

(2) f(t)= Cot/t 1 C1 (x, ¢) Co

•.., a v i2 erfc t~ v=o

C2(x't--~)=CoI+F2F4t~.t, ~=o a' iz erfc \ _

_

- ai 2 erfc

(4Vet) ~ ]

(4Dt)½

_

Mco(t)= FD½t,(4t)} (~ ---~1 6n +2 ~~

(20)

k (4DCt)½

/

(21)

a ' i 3erfc { ,_Iv ,, ~~

(22)

\(D~t:/)

V=I

(3) f(t) = C Oexp ( - fit)

Cl(X, t) = ~.,..,a~Re[VV~ (t l(2v+l)+x fl) _ a W i ( l(2v+l)-x Co

~=o

'

(D~)½

'

(D~)½

' (23)

-C 2-(x, t) Co

2F ~ a ~ Re IWJ (t l(2v+l)/~+x fl) ] = 1 +~--F v=O

Co = --~-" Im J. Electroanal.Chem.,40 (1972)

'

D½

(t, O, fl)+ 2 ~ trvWj

'

(24) (25)

NON-STATIONARYDIFFUSIONTHROUGH MEMBRANES

259

SIMPLIFIED CASE

Equations and solutions Suppose : [ K ~9C1

~a ~'f J-,

(x', t)dx' I for

-l
(26)

The left-hand side of eqn. (26) is the absolute value of the diffusion flux at x = - I. The right-hand side is the absolute value of the time derivative of the mass in section - l to x. Thus, the rate of mass change for any portion of the membrane is neglected compared with the influx at x = - I. This assumption is reasonable if the membrane is thin and the concentration changes at the boundaries are slow. Integration of the diffusion eqn. (2) between - l and x using (26) gives:

-~x

ax

~=

_,

-,-~ (Cl (x', t)e,)dx' ~O

(27)

This assumption gives that the flux in the membrane is independent of x. Then from eqn. (1) the flux across the membrane will be given by:

J = - (K/l)(C2 (0, t)-f(t))

(1')

This gives essentially simpler solutions. The validity of this assumption may be checked by comparison of the numerical result with the corresponding general case. This gives the following equations: a 2 C 2 1 aC2 - ax 2 = -D- - atx>0, t>0 (28) C2(x, 0) = 0

gC2 = K (C2(0, t)-f(t)) D ~ - x x=o

M(t)

=

(29)

fro (-O) 3C2ax (x' t') tx=o I dt'

Proceeding as before gives:

h f '+h P'exp[:x(P)~] C (x, p) = p-r

(30,

M(p) - hD~ f(P) p~ p½+h

(31)

where h -- K/lD ~. (1) f(t)= Co. This case is found in ref. 7. (2) f(t) = Co t/t1.

C2(x,t)Co

=

h2-~-1{erfc ( ~ ) x ~ - exp ( ;

+ ht J. Electroanal.Chem.,40 (1972)

- 2ht ½ i erfc

+ th2~/ x + 4h 2 ti 2 erfc

(32)

260

J. CLAESSON, J. LINDBERG

Co = ~

exp(th2)erfc(ht½)- I +

2ht½ - h 2 t + 4ha t~ n½

3re* I

(33)

(3) f(t) = Co exp ( - fit).

C2(x,t) Co

-

h

h2 + fl

{

h Re

[Wj (t x fl)] ,~ , +

+ fl½lm[Wj (t, D~x--r,fl)l-hexp(~i+th2) x (34)

x erfc((4~t)~ + ht½)l Coo =

exp(th2) erfc(ht½) + h

+ ~ Im [-Wj(t, O,/~)]-exp(-flt)

(35)

In deriving cases 2 and 3 C2 (x, p) and M(p) have been separated into suitable partial fractions in order to get recognizible Laplace transforms.

Stirred connecting chamber The case when the connecting chamber is stirred may also be of interest. As before the concentration in the sample compartment is given by f(t). The volume of the connecting chamber is V~1 and the concentration is C2 (t). Then :

Vc1 ~C2 (t) = AK f ( t ) - C 2(t) dt

(36)

l

c2(0) = 0

A is the cross-sectional area of the membrane. The solution of (36) is :

C2(t) = e -'t

f' ~e,"f(t')dt'

(37)

o

~,= AK/IV~, SUPERPOS!TION

Through superposition of the derived solutions for cases 1, 2 and 3 new analytical solutions may be derived for any function f(t), which is piecewise linearly and/or exponentially decreasing. Two cases are considered that are of interest in this work.

Case A f(t) is falling linearly from Co for t = 0 to zero for t=t 1. For t >tl,f(t)=O. This gives :

f(t) = C o [1 - ( t / t l ) + H ( t - t l ) ( t - t l ) / t l ] J. Electroanal. Chem., 40 (1972)

t >10

(38)

NON-STATIONARY DIFFUSION THROUGH MEMBRANES

261

H(t) equals one for positive arguments and zero for negative arguments. The third term represents the linear increase, starting from zero at t = tl, necessary to balance the decrease of the first two terms. Superposition gives the solution directly in terms of the earlier solutions:

Ci(x, 0 = Ca(x, t)-C? (x, t)+ H ( t - t l ) C2(x, t - t 1 )

(39)

i=1,2

M (t) = M 1(t)- M 2 (t) + H ( t - tl) M z ( t - t,)

(40)

The upper index refers to the cases 1, 2 and 3.

Case B f(t) is falling linearly from Co for t = 0 to C,, for t = t~, thereafter exponentially to zero. That is:

f(t) = Co + (Ct,- Co)((~tO + H ( t - t,){C, exp [ - fl ( t - tl)] - C,, - ( C .

- C o ) ( t - t~)/t~

}

(41)

This gives for example:

M (t) = M 1 (t) + M 2 (t)(C,, - Co)/Co + tt ( t - t~) x x {MS(t'tl)~o-M'(t-tl)C./Co-M2(t-tO(C,,-Co)/Co}

(42)

RELATION BETWEEN THE PROPORTIONALITY CONSTANT K AND THE ELECTRICAL MEMBRANE RESISTANCE

The diffusion inside the membrane is three-dimensional. It is assumed that in the pore-liquid there is undisturbed diffusion. Then Fick's law gives :

0CI~

( 0C1 0C1 J = - O grad Cl = - D k,~x ' 0y ' ~z~]

(43)

Here C 1 is the three-dimensional concentration distribution in the pores of the membrane. J is the diffusion flux. At the walls in the pores the flow must be parallel to the walls. This means that at the pore walls0C1 = 0 0/1 wall

(44)

0/0n is the derivative in the normal direction of the walls. In a stationary case the diffusion equation is: 0 2 C1

0 2 C1

Ox---g-+ ~

0 2 C1

+ Oz2 - 0

(45)

Denote by C'1 (x, y, z) the stationary solution, when the concentration at x = - 1 is unity and zero at x = 0. C'~ ( - l, y, z) = 1

(46)

C] (0, y, z) = 0

(47)

J. Electroanal. Chem., 40 (1972)

262

J. CLAESSON, J. LINDBERG

Thus C~ (x, y, z) is the solution of (44)-07). Suppose, on the other hand, that there is a unit potential across the membrane. In the pore-liquid the electric current Jet is related to the potential V:

Jet=-xgradV=-x

~x-x' 0y' Oz

is the conductivity of the soiution. At the walls in the membrane the current must be parallel to the walls. This gives: I

0V]

= 0

(49)

The potential V satisfies: 0z V 02 V 02 V 0x ~- + ~ + ~ = 0

(50)

~-n watl

The potential at x = - l is unity and zero at x = 0: V ( - l , y, z ) = 1

(51) (52)

V(0, y, z) = 0

V(x, y, z) is the solution of (49)-(52). Thus the electrical problem and the diffusion problem are identical, i.e. :

V(x, y, z) = C~ (x, y, z)

(53)

A unit area of the membrane is regarded. The electrical membrane resistance of this area is denoted Rm, i.e., R m I = 1. I is the electric current in the x-direction of this area. I =

ff

Jet, x(x, y, z)dydz =

ff

( - x) -~x dydz =

ov

= -~ x ff i - D ) ~oc; dydz

(54)

Here (48) and (53) are used. The last integral represents the macroscopic flux in the x-direction. Let C~(x) denote the one-dimensional solution corresponding to C'1 (x, y, z). Then assumption (1) gives: 0C7 x 0-1 0x - D ( - K ) " l

x I=~(-K)

(55)

Here is assumed that C'~ix) falls linearly across the membrane. Thus : K = Ol/xRm

(56)

eo is defined as the solution resistance divided by the membrane resistance. e¢ = l/teRm

(57)

Solution layer and membrane have the same length, I. Equations (56) and (57) give : = K/O

v =

The assumptions leading to this relation were eqns. (1), (43) and (48). d. Electroanal. Chem., 40 (1972)

(58)

NON-STATIONARY DIFFUSION THROUGH MEMBRANES

263

SUMMARY

Analytical solutions are given for non-steady state diffusioh in a membrane and a semi-infmite connecting chamber. The one-dimensional macroscopic diffusion inside the membrane is assumed to be governed by Fick's law using an empirical proportionality constant. The solutions apply to membranes without dead-end pores. The concentration distributions in the membrane and connecting chamber and the total sample amount inside these are calculated for linearly and exponentially decreasing types of titration. Analytical solutions under the simplified assumption of linear concentration distribution inside the membrane are also given. REFERENCES 1 2 3 4 5

R. C. Goodknight, W. A. Klikoff, Jr. and I. Fatt, J. Phys. Chem., 64 (1960) 1162. I. Fatt, J. Phys. Chem., 63 (1959) 751. L. J. Klinkenberg, Bull. Geol. Soc. Amer., 62 (1951) 559. R. K. Schofield and C. Dakshinamurti, Discuss. Faraday. Soc., 3 (1948) 56. H~ S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 2nd ed., 1959, chap. 12. 6 M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965, chap. 7. 7 J. Crank, The Mathematics of Diffusion, Oxford University Press, London, 1956, chap. 3.

J. Electroanal. Chem., 40 (1972)