ESTIMATIOK OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
1227
The Estimation of Diffusion Coefficients for Ternary Systems of Strong and Weak Electrolytes
by Richard P. Wendt (Received October 14, 1964)
Coates Chemical Laboratory, Louisiana State University, Baton Rouge, Louisiana
Equations are derived for estimating phenomenological coefficients and diffusion coefficients for two classes of ternary systems: class I, containing only strong electrolytes as solutes, and class 11, containing both weak and strong electrolytes as solutes. The approxiniate equations derived relate each of the four phenonienological coefficients for a ternary system to the concentrations and limiting equivalent conductivities of the ions in the system. From estimates of the phenomenological coefficients and from either known or estimated values for the chemical potential derivatives, the four diffusion coefficients for the system can be calculated. Estimated values for the coefficients are compared with observed values in the literature for four systems: H20-KaCl-KC1, H20-LiCl-KCl, H20-LiCl-NaC1, and H20-NazS04-H2S04. Agreement between estimated and observed values is good for the first three systems but poor for the last system. From these comparisons the concentration range of validity of approximations made during the derivatives is inferred.
Introduction Diffusion coefficients are difficult to measure for niulticomponent systems. Since 1955, when Dunlop and Gosting’ at the University of Wisconsin first measured diffusion coefficients for a three-component system, the diffusion process has been studied for only several nonelectrolyte and electrolyte ternary systerm2 Especially because of the many multiconiponent electrolyte systenis of industrial and biological interest, we feel it is important to present a general method for estimating phenomenological coefficients, ( L J 0 , and diffusion coefficients, (Dzj)o, for such systems froni readily obtainable conductivity data. The coefficients to be estimated appear in the following two sets of equations, either of which, according to nonequilibrium thermodynamic^,^, may be used to describe the isothermal diffusion process in a system containing n neutral components n- 1
( J J o=
-E
(L,Jograd
CC,(i =
1,
n - 1) (1)
(i = 1,
n - 1) (2)
3-1
n-1
(Jib = -C (D,,)o grad
cj
5’1
Throughout this paper the notation corresponds to that used by Hooyman5 and Gosting and co-workers.216
The subscript 0 refers to the solvent; the solventfixed flows of matter, (Ji)o, are defined relative to the local velocity of the solvent and have here the dimensions moles/(cm.2 sec.) ; the gradients of chemical potentials, grad p j (ergs/mole of j ) , are taken with time held constant; c, is the concentration of j , in moles/cc. ; and the solvent-fixed diffusion coefficients, are related to the solvent-fixed phenoinenological coefficients, ( L z J 0by , n-1 (Dtj)O
Here the
pkj pkj
=
(Lik)Oplci
k=l
(i, j
=
1,. . n - 1)
(3)
are chemical potential derivatives = dpk/bc,
(k,j
=
1 , ..n
=
1)
(4)
(1) P. J. Dunlop and L. J. Gosting, J . Am. Chem. Soc., 77, 5238 (1955). (2) Brief summaries of work in this field are given as part of recent papers by L. A. Woolf, J . P h y s . Chem., 67,273 (1963); J. G. Albright, i b i d , 67, 2628 (1963); and P. J. Dunlop. ibid., 68, 26 (1964). (3) S. It. de Groot and P. Mazur, “Non-Equilibrium Thermodynamics,” Interscience Publishers, Inc., New Tork, N. T.,1962. (4) D. D. Fitts, “Nonequilibrium Thermodynamics,” McGraw-Hill Book Co., Inc., New Tork, N. T., 1962. (5) G. J. Hooyman, Physica, 22, 751 (1956). (6) P. J. Dunlop and L. J. Gosting, J . Phys. Chem., 63, 86 (1959)
Volume 69, ‘Vumber+! . .
April 1965
RICHARD P. WENDT
1228
The coefficients actually measured in diffusion experiments are cell-fixed, (Dt,)c, which are very nearly equal' to volume-fixed diffusion coefficients, ( D i j ) v; the ( D J o are related to the latter coefficients by5s6 n-1
+ COVO c
(Dtj)O = (D*,)v
k-1
t;(D,,)v
( i , j = 1 , . .n = 1) ( 5 ) where Vk is the partial molar volume, in cc./mole, of component k. It can be seen from (5) that near infinite dilution ( D f j ) o= (D,,)V, but even in relatively concentrated solutions (ca. 1.0 total molarity), (D& and (D,,)v may differ by only 1-5%.6 Therefore in the absence of values for the 7, the equations derived for estimating the (Df,)o also provide estimates of the (DiJV.
The same general procedure is followed here in two separate derivations. The first derivation is for ternary systems containing two strong electrolyte solutes having a common ion, and the second is for a class of ternary systems containing two electrolyte solutes which may only partially dissociate (see eq. 36). For both classes of systems the derivations rigorously follow the principles of nonequilibrium thermodynamics to obtain sets of exact equations. Approximations are then made for certain quantities in the exact equations so that the four coefficients (hi,),, can be estimated from values for the limiting equivalent conductivities, 110,and the concentrations, ci, of all ions in the given system. The four diffusion coefficients (Dt,)o can then be calculated from eq. 3 if values for the chemical potential derivatives prc, are available. Methods are also discussed for estimating the k k l t so that the and to a good approximation the ( D f , ) v ,can be estimated only from values for the Ato and ct of all ions. An earlier approximate theory, derived by Gostings for ternary systems containing strong electrolytes as solutes, suggested the general form of the theory presented here, but it differs from ours in an important way: all approximations are made early in the former derivation, with the result that explicit expressions are obtained for estimating only the (Df,)ofrom values for :1 and ct of the ions. The coefficients (Lf,)ocannot be directly estimated, nor can available values for the p,, be used to improve the estimates of (Df,)o by using eq. 3. The equations derived here should therefore provide more accurate estimates of the (Dt,)o for the systems for which the pk, are available, and the several approximations used for those systems can be more carefully examined by comparing estimated and observed values for the ( L f , ) oand (Db,)". For the The Journal of Physical Chemistry
systems of class I for which no observed values of the & j are available, our theory and Costing's theory make equivalent predictions of values for the ( D J 0 . A more recent theory derived by Stockmayergis applicable only to binary weak-electrolyte systems such as HzOHzS04, but its rigorous derivation suggested several methods of achieving generality in the present multicomponent theory. The equations derived here are tested by using available thermodynamic and diffusion data for the ternary systems HzO-NaCl-KCl, H20-LiC1-KCl, HzOLiCl-SaCl, and Hz0-NazS04-HzS04. These data were only available a t relatively high solute concentrations (ca. 0.5-3.0 total molarity) where the approximations made for quantities in the final equations were not expected to hold. Nevertheless, for the three strong electrolyte systems the predicted and observed values for the ( D , j ) oare found to agree remarkably well. For the weak-electrolyte system H20-Na2SO4H2S04the agreement is relatively poor, but the order of magnitude, algebraic sign, and approximate concentration dependence of the (Ltj),,and (Dt,)o are correctly predicted.
Theoretical Outline. The two classes of systems will be separately treated but both derivations will follow the same general procedure. The first part of each derivation presents simple thermodynamic and stoichiometric relations between the chemical potentials, the stoichiometric coefficients, the concentrations, and the mass flows of all ions and neutral solutes comprising the system. Then the entropy-production expression is used to specify the independent flows and forces in the system. The condition of zero net electrical current flow is used in the third part to obtain exact relationships between the ionic and the neutral-solute phenomenological coefficients (Lf,)owhich appear in the flow equations for ions and for neutral solutes, respectively. Approximations for the ionic (&,)a are made in the next part of the derivation which enable the neutral solute (Li,)oto be estimated from values for the concentrations and limiting equivalent conductivities of all ions in the system. Finally; simple methods for estimating the neutral-solute chemical potential derivatives p k 3 are discussed. The diffusion coefficients (Df,)o can then be calculated from (7) (Dij)v = (Dij)c if the T h are independent of concentration throughout the diffusion experiment; see G. J. Hooyman, et al., Physica, 19, 1095 (1953). (8) L. J. Gosting in "Advances in Protein Chemistry," Vol. S I , Academic Press, Inc.. New York, N. T..1956,pp. 536-539. (9) W.H. Stot~krnrtyer,J . Chem. Phys., 33, 1291 (1960)
ESTIMATION OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
either known or estimated values for the neutralby using eq. 3. solute (L& and For any system of electrolytes not in the classes discussed here, the derivation would follow the general procedure outlined above, regardless of the kind or number of coniponents in the system.
1229
The principle of conservation of mass is applied to the concentrations c, and the flows (,I of,) the,, ionic species 3, 4, 5 to obtain 2
c
VtjCt
t=1
=
(j
cj
3,4,5)
(13)
( j = 3, 4, 5)
(14)
=
2
Theory for Systems of Class I Thermodynamic Relations. Consider two neutral strong electrolyte solutes AYlSCYll and B,,,C,, which completely dissociate according to the reactions
+
AplaCvli + V13ABa YISC"
+
+ V24BE4
Bp,C,,
(64
(7) Here and throughout the derivation we define VI4 = V23 = 0. By taking the gradient of both sides of (7) and letting X, = -grad pi.we obtain
= (JAO
vt,(Jt)o
Because of (13), equations similar to (14) are valid for any reference frame.l0 Entropy Production. For three-ion systems the entropy produced by isothermal diffusion is given by 5
(6b)
v26cz6
The first subscript on each stoichiometric coefficient ut, refers to the neutral solute i; here i = 1 and i = 2. designate the solutes AVIsCYI6 and BY1,CYU,respectively. The second subscript on vi? refers to the ion; j = 3, j = 4, and j = 5 designate the three ions A'', Cs6, and B", respectively. Valences of the ions (including the algebraic sign) are represented by z t . Chemical potentials for the neutral solutes are defined by
x,=
c
a=l
Tu
=
c (J,)o(Xt + z t m
Here u is the local rate of production of entropy per unit volume, T is the temperature of the isothermal system, f is the absolute value of the charge per mole of electrons,ll and E' is the local electric field. We replace each ( J J 0 ,i = 3,4, 5 in (15) by its equivalent from (14) ; after rearranging, (15) is rewritten 2
=
5
cc
a=1
2
vtjx,
(8)
(i = 1, 2)
5
+fE c c
(Jt)OVl.,Xj
(Jt)OV4,
2=13=3
3-3
(16)
where as before V14 = V 2 3 = 0. The coefficient of fE vanishes because of (ll),and because of (8) the first term reduces to 2
Tu
=
5
3=3
(15)
2=3
a=l
(J3oXi
(17)
Thus the entropy production can be expressed by either (15) or (17) and the linear laws can be written
Equation 8 is solved for X3, X 4 , and XF,,finding 3
X,
(i
a,,X,
=
= 3,4,5)
(9)
j=l
(10) Macroscopic velocities fined by
vi
of neutral components can be de-
where the coefficients a,, have the values a31 = 0
a32= 0 v2b
a41= - -
a42
VlSv24
=
a33= 1
1
V24
where u 1 is the macroscopic velocity of ion j. Flows in general are measured with respect to an arbitrary reference frame R of velocity
v13v26
a43= - (10)
VR
V16 v24
(J$)R
=
CI(&
-
(b)
UR)
and cell-fixed flows are obtained by letting B R = 0
( J , ) c = c,v,
V16
v16
The solutes 1 and 2 are neutral; thus
Thus (a) also reads
-3
5)
(c)
v,,(J,)c = ( J , ) c . Both sides of (b) are multii=l
5 ?
( i = 1,
2
v,3z, = 0
(2
= 1,2)
(11)
plied by v 2 , and summed over i = 1,2; (a) and (13)are introduced, and the general result which can be specialized to (14) is obtained
The z , are related by z, = ai3z3 (i = 3, 4, 5)
where tjhe ai3are given by eq. 10.
(12) (11) f is given by eq. h of the Appendix
Volume 69, aVumher 4
April 1965
RICHARD P. WEXDT
1230
5
for flows of t,wo neutral solutes (eq. 1 with n = 3) or for flows of three ions 5 (Ji)o
:=
(Lt,)o(xj 3=3
-k z ~ f E ) (i = 3, 415 ) (18)
according to the linear-law postulate of nonequilibriuni thermodynamics.3 Relations hetween Phenomenological Coeficients. For the irreversible process considered here, namely diffusion with no net electrical conduction, the flows of ions are related by the condition of zero net electrical current flow 3-3
=
0
2
(19)
We apply this condition to eq. 18 and find an expression for fE 5
5
c c Zl,(Ll,JOXi fE=---k - 3 1-3 5
2
m-3
(20)
5
(Jl)O
=
y13-l
(J2)o
=
u24-1
5 - 1 1-3
J
(i = 3, 4, 5 )
(21)
The valences zt in (21) are replaced by their equivalents according t'o (12), obtaining 5
=
( J J O
C Q2,iXl (i = 3 , 4 , 5 ) 1-3
(22)
where we define 5
n,,
=
5
C
5
1'1
1-3
(L,)o
=
V13-l
(-bj)o
=
V24-l
C 1-3
a11831
( j = 1, 2)
(27a)
1, 2)
(2%)
5
1=3
acJO41 ( 2 =
Approximations for (Ltl)o. This derivation has so far been rigorously performed according to the principles of nonequilibrium thermodynamics. If the nine ionic coefficients in (18) were known at any particular composition, then the four neutral-solute coefficients ( L J 0 in eq. 1 (with n = 3) could be calculated and compared with their measured values at that composition. Unfortunately, such data are not available; therefore, to make this derivation practically useful for estimating the neutral-solute (Lf3)o, we make approximations which enable us to relate some of the ionic (L,,)o to the limiting equivalent conductivities of the ions in the system.
&l(Lmn)fins
= 3,4,5)
(23)
5
= 1-3
Xjaijaf1 (1:
=
3, 415)
(24)
The term in X 3 has been eliminated by using the important identity12 The Journal of Physical Chemiatry
. .._ ..
CC ama(Lm,)oa,a m n
After using eq. 9 for Xi (22) becomes 3'1
2
(Ltj)@13at3(Lti)0
(i,b
(Jf)C
c c X,a,,Q4,
5
m-3 n-3
2
(264
(12) The equivalent of Oij, as defined by (23) or (50), is introduced into the left-hand side of ( 2 5 ) or (51) and the sum of sums is written with a common denominator
3 - 3 E-3
(LJO -- -5
X~aljfi31
5
n-3
m n
5
The coefficients of X1 and X z in the above equations are then equated to the coefficients ( L J 0 in eq. 1 (with i = 1, 2) to obtain a set of exact relations between the phenomenological coefficients for neutral solutes and those for ions
zm(-Ln)gn
Here new subscripts k , I , m, and n have been introduced for convenience. Equation 20 could be used to calculate liquid junction potentials or diffusion potentials if the coefficients (Lkl)oand the gradients of ionic chemical potentials X Lwere all known. We use (20) to eliminate fE from eq. 18, which after rearrangement becomes
L
(25)
Thus for diffusion with no net electrical current flow the flows of ions 3, 4, 5 can be written as linear functions of only two variables, XI and X z , the chemical potential gradients of the neutral solutes. Ry using eq. 14 with13j = 3, 4, we obtain a new set of linear relations between the flows of neutral solutes and their conjugate forces
5
c4 J J O
(i = 3, 4, 5)
a,3fif, = O 2-3
In the triple sum at the right-hand side of the numerator the subscripts j, k, and 1 are replaced by I , m , and n, respectively: because all summations run over the same range both triple sums are equivalent, the numerator vanishes, and the identities (25) and (51) are proven. (13) The two eq. 14 with j = 3, 4 are chosen for convenience; any two of the three j = 3, 4 , 5 could be used.
ESTIMATIOS OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
1231
First we assume that the cross-term phenomenologi(.a1 coefficients in the ionic 00w eq. 18 are identically zero at all concentrations
(LiJo
=
0
(z #
j)
(i,j
3, 4,5 )
=
(28)
The product of concentrations can be written 5
II c p
Then the expressions for the Q i l become
=
j=3 ai1
=
(L)06il
-
x-1(LIi)oai3a13(L11)o
+
( v ~ ~ c ) ~ ~ ~ ( v v26cz)”25 ~ ~ c ~ (33) ) ~ ~ ~ ( Y ~ ~
by using (13). Thus (31) becomes
(i, I
=
3 , 4,5 )
(29)
where 5
8
ak3’(Lkk)O
=
k=3
vt3
and 6,z is the Kronecker 6, i.e., 6 i z= 1 for i = I and agJ = 0 for i # 2. The main-term coefficients ( L I J Oor (L1,)o in (29) can be approximately related to their respective ionic limiting equivalent conductivities, X1.0 or A,: by14
(LJo
=
X & / ( / z i / F 2 X €0’)
(2 =
3,.
s
+ 2)
~1.10 + RT
5
(i
In
( Y ~ C ~ ) ~ E J = j
1, 2 )
c1
+
v14
In
c2
+
Vi6
To obtain estimates of the
In
(V15c1
p r k we
+
mc2)]
(31 a)
assuine”
(34) b In ( y k ) i / d c j = 0 ( i ,j = 1, 2) Then the derivatives pik:of p i in (31a) can be estimated from
(30)
Here X1.0 has the units cm.? ohm-’ equiv.-l, s is the number of ions in the system, and F is the Faraday constant with the units coulombs/equivalent. Approximations (28) and (30) amount to the assuinption that the mobilities of the ion are the same at appreciable concentrations as they are at infinite dilution; the approxiniations are exact near infinite dilution. Values of (L,,)o for the neutral solutes can, however, be estimated at any values for all ci by using eq. 30, 29, and 27, and values in the literature for X,O Estzmatzon of p i k . Values for the chemical potential , ’ ~ example, for the derivatives p i k are a ~ a i l a b l e , ~ for system H20-KaC1-KC1 of class I ; these data, and the estimates of (Li,) obtained from (27) above, could be used to calculate approximate values for the four diffusion coefficients ( D i J Ofor that system according to (3). has discussed methods for estimating the p r b in lieu of appropriate experimental data. The siniple method which we use here is to write expressions for each chemical potential derivative and then assume that the mean ionic activity coefficients are independent of concentration. For systems of class I the chemical potentials p i for neutral solutes, defined by eq. 7, may be written pt =
In
(31)
=3
where y3 is the activity coefficient of ion j and p t is the chemical potential at the standard state for neutral solute i. We define mean ionic activity coefficients for the neutral solutes
(i,k where
=
1, 2)
(35)
is the Kronecker 6 .
Theory for Systems of Class I1 This class of systems differs from class I because of the presence of four ions (instead of three for class I) and one equilibrium reaction among three of the ions (instead of none for class I). Thermodynamic Relations. A system in this class contains one neutral undissociated solvent (designated by the subscript 0 ) and two neutral solutes (designated by the subscripts 1 and 2) which dissociate according to the reactions
+ V16Cz6
Av,,Cv,, + v14AZ4 Bw2ICm
-
D 2 3 A
wB“
V?6Cz6
+ v36Czfi
~35B”
(36%) (36b) (36c)
See eq. k of the Appendix. P. J. Dunlop, J . Phys. Chem., 63, 612, 2089 (1959) D. G. Miller, ibid.,63, 570 (1959). For activity coefficients y, defined as functions of molality, m , , it can be shown that the assumptions (14) (15) (16) (17)
imply ideality for the solvent as well as the solutes, but eq. a does not imply eq. 34; -instead, eq. a implies small nonzero fixed vaiues, given Z 0.02, for the derivatives d In ( y * ) J b c , at the solby ~,/(lOOOcoVo) k ute concentrations considered here (ca. 10 - 3 mole/cc.). Therefore at very large solute concentrations, or for nonaqueous systems, care should be taken in making assumptions concerning activity coefficients. See ref. 16 and J. G. Kirkwood and I. Oppenheim, “Chemical Thermodynamics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1961, p. 171, for relevant thermodynamic derivations.
Volume 69, Number 4
April 1965
1232
RICHARD P. WEXIIT
It must be assumed that the equilibrium reaction (36c) is very fast so that local equilibrium exists throughout the diffusion p r o c e ~ s . ~ The chenlical potentials of the neutral solutes 1 and 2 and the incompletely dissociated ion 3 are defined by
Equation 45 is valid not only for the solvent-fixed but for any reference frame.'O Entropy Production. For the systems of class I1 the expression for the entropy production is 6
6
=
~i
C
Tu vijpj
3=4
(i = 1, 2, 3)
(37)
Here we have set vI6 = P24 = v34 = 0. It follows from (37) that the cheniical potential gradients, - X , , of ions and solutes are related by = 1,2,3)
3
= v-'
C Z ~ , X(, j = 3 , . . .6)
is1
(39)
Here v (without subscripts) is the determinant of the matrix of coefficients vi, on the right-hand side of eq. 36
0
v16
v = '0
v26
V26
0
v36
V36
VI4
~
v3fXt
(Jj)O
1
2=4
(40)
Tu
= v
vatxi)
f
a=4
=
c (JJOXi
(47)
211
3 =3
(L,j)o(xj
+ zjfE)
(i = 3 , . . . . .6) (48)
(i = 1 , 2 , 3 ) ( j = 4, 5, 6)
a33
-
Therefore, two independent flows ( J , ) oand forces X , of neutral solutes exist for this class of systems, and either theneutral-solute flow eq. 1, with i = 1, 2, or the ionic flow equations (JJo =
a,,, = (-l)i+j+lvij = a32 =
6 (J3)0(x3
The coefficients of and f E vanish because of (38) and (42), respectively; the teriii which remains is further simplified by using (38), obtaining
6
and the a,, are given by
a31
+
(38)
These linear equations may be solved for X 3 . . . . . X a , obtaining
X j
(46)
2=3
6
2
= )=1
(2
c (Ji)O(X, + z f f m
Three of the ion flows ( J l ) o i, = 4,5 , 6 are eliminated by using eq. 45; after rearrangement,, (46) becomes
6
x , = j=4 v,,xj
=
0
(41)
may be used to describe the isothermal diffusion process for this class of systems. Relations between Phenomenological Coefficients. The 6
condition of zero net electrical current flow,
The v" are the complementary minors of v. Conservation of charge for the dissociation reactions (36) implies
2-3
z,(JJ0=
0, is used to eliminate j E from eq. 48, which then reads
6
c
v+,
=
0
=
23
3=4
(i
= 1, 2)
(42%)
where 6
6
C
v 3 j ~ j
6
j 1 5
Solving for
z3,.
. . . . . z 6 , we obtain
(i, b where the aj3are given by (41). Conservation of mass requires
=
3 , . .6) (50)
(i = 3,. . .6)
(51)
The identitylz 6 ~23Qij 1-3
=
0
has been used to obtain (49), which relates the flows of ions to chemical potential gradients of the two neutral solutes. The first two equations in t'he set (45), which relate The Journal of Physical Chemistry
ESTIMATION OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
flows of ions to flows of neutral solutes, are rearranged to read
(J2)O
=
(52a)
V14-1(J4)0
(Jl)O
+
Y26-1V35(J3)0
(52b)
v26-1(J6)0
Expressions (49) for (J3)0, ( J 4 ) 0 , and (Js)oenable the flows and (J-JOto be written as linear functions of X1and X 2 2
=
(Jl)O
v-1V14-1
2
( J ~ =o
V-~VZS-~ j-1
XjatjQ41
j-11-3
6
Xj[c
+
alj(v%Q3t
1=3
6 a,3(v%Q32
V-~V~A-'
2=3
ptk =
RT
P,O
%t)]
[
5 VtGVkC
(vi,/ct)
611,
(53b)
+
f ---__ v16c1
j-4
v26c2
1
( i , k = 1, 2) pf
(59)
Estimates of ptk may also be obtained by defining the as functions of all ion concentrations c , 6
+ QsJ
pt =
+ RT In II (y,cj)vI1
I/'
3
(i
=3
=
1, 2)
(60)
The activity coefficients for the ions, y,, are not measurable quantities but nevertheless we can make the
(k
( j = 1,2) (54b)
=
1, 2)
Then the expressions for estimating the pit are
(L,,)o = 0 ( i # 1) (i,j
=
3 , . . , . ,6)
(55)
Then the coefficients a,,, defined by eq. 50, become = 6tz(Lti)o
I1
(534
Approximations for ( L J 0 . As before, to obtain practical and useful expressions for predicting the four coefficients from the exact eq. 54, we make the a p proxiniation that all cross-term ionic phenomenological coefficients are zero
Qtz
where (grt){is now called the practical mean-ionic activity coefficient. By using eq. 44 (with c3 = 0) to introduce c1 and c2 into (57)) and again making the assumption (34) equations are obtained for estimating the chemical potential derivatives p t b in systems of class
6
The coefficients (LJO of XI and Xz in (1) with i = 1 , 2 are equated to the corresponding coefficients in the flow eq. 53, obtaining the desired relations between neutral-solute and ionic phenomenological coefficients
(Lzj)o =
1233
- s-'(Li~)o~iraza(Lii)o (i, 1 = 3,. . .6)
6
pn
=
RT
where
bc,
(i, k
=
1, 2)
(62)
act
If the ionic concentrations c, can be wriaen as functions of the neutral solute concentrations c l , c2 then the derivatives bc,/dck in (62) can be evaluated; the desired functions for c1 and c2 may be derived by using eq. 44, the equilibrium-quotient expression for reaction 36c Q
(56)
vt3
-j,3c,
(63)
=
and the condition of electrical neutrality 6 ZfCt
=
0
2-3
The main-term coefficients (Ltt)o and (Llt)0 in (56) are est'imated from values for the limiting equivalent conductances according to eq. 30, which applies to all ions in any system. Estimation of p l r . For systems of this class, which contain a partially dissociated species, D (denoted by the subscript 3)) the chemical potentials may be defined by ignoring the existence of D 6
pt = p/'
+ RT In II 3-4
As before, we define
(y,cj)vij
(i = 1, 2)
(57)
Thus values of Q for the ternary system must be available to estimate the ion concentrations et for use with eq. 30 and to calculate the derivatives bc,/bc,. The accuracy of values estimated for the from eq. 54 and 3 depend on the possibly crude approximations 30, 55 and 59, 62 for the ionic ( ~ 5 ~ ~ ) ~ and the neutral-solute p t k , respectively, which must be used in lieu of experimental data. In general, the estimates for p f k obtained using (59) and (62) will differ from each other for a given system in class I1 because one procedure neglects ion 3 and the other includes that ion. For a given system of class I , approximations similar to (59) and (62) would lead to equivalent estimates of the p t , because no ions are neglected by either procedure. Of course for any Volume 69, Number 4
April 1966
RICHARD 1'. WENDT
1234
system the exact or measured values of the p i a are independent of the explicit expressions used for the p r . Tests of the Equations
Calculations. Phenomenological coefficients, chemical potential derivatives, and diffusion coefficients are estimated for three systems in class I : H20(0)NaCl(l)-KC1(2), HzO(0)-LiCl(l)-KC1(2), and HzO(O)-LiCl(l)--NaCl(2) , where the numbers in parentheses designate the respective neutral components in all subscript notations. Corresponding to the dissociation reactions (6) for solutes 1 and 2, the stoichionietric coefficients vi, and zt for all three systems have the values VI3 = V16 = V24 = V26 = 1, V14 = V23 = 0,23 = z4 = +1, and z6 = -1. To estimate the main-term ionic phenomenological coefficients (L,,)o from eq. 30, the limiting equivalent conductivities were taken to heL8XNa+O = 50.10, X g + O = 73.50, XL,+O = 38.68, and icl-O = 76.35 cm.2ohm-' equiv.-l. Values used for the quantities F', R , and T were 96,493 coulombs equiv.-', 8.3144 x lo7 ergs mole-' deg.-', and 298.15"K., respectively. Concentrations of ions were calculated a t each composition point el, e2 according to (13), and the four coefficients (Lij)ofor flows of solutes 1 and 2 were estimated by using eq. 29 and 27. For the system HzO-NaC1-KC1 two methods were used to estimate the four diffusion coefficients (D,j)oaccording to (3). In method A, values used for the chemical potential derivatives p i r were estimated from eq. 35; in method B, observed value^^^^^ for pi, were used. Because no appropriat~e thermodynamic data are available, only method A was used to est'iniat,ediffusion coefficients -for the systems HzO-LiC1-KC1 and HzOLiC1-NaCl. Only for the system Hz0-XazSO4-H2SO4of class I1 are there sufficient therniodynaniic and diffusion data to compare estimated and observed values for diffusion coefficients and phenomenological coefficients. We designate the solutes and ions in this system as follows: Na2S04= 1,H2S04= 2, HS04- = 3, Kaf = 4, H f = 5 , and = 6. The reactions corresponding to (36) are
Kerker's value for Aso was used,lg and values for the other X,O were taken from ref. 18. Concentrations of the ions, c3,. . .ea, are given in Table I ; they were calculated a t c1 = c2 = 0.5 X lop3 and c1 = e2 = 1.0 X lop3niole/cc. by using eq. 44, 63, 64, and a derived equation for c6 cfi
= '/z(c2 -
ci - Q)
+
'/2[(~2
-
ci
-
Q)2
+ 8Q~z]'/'
(65) Values for Q were obtained from Baes' estimates20 of the concentration dependence of Q for this ternary system. The four phenomenological coefficients were estimated by using eq. 30, 56, and 54 with values for the ai3 calculated from (41). The p,k for the system Hz0-NazS04-HzS04 were estimated in two ways, according to eq. 59 (est. 1) and eq. 62 (est. 2 ) . Equation 65 for c6 and appropriate equations from (44) for c3, c4, and c6 were differentiated with respect to c1 and c2 while holding Q constant to obtain expressions for the derivatives bc,/dck in (62). The apparently more accurate values obtained by est. 2 were used in method A2, and the observed values2I for the p t k were used in method B, to calculate the (D,j)o from eq.
".9 Tab,e
Ion Concentrations in the System
Ia,b:
H~~-N~~so~-H,s~~ C,X 1 0 3 (NazSO,)
c2
x
103
Q x 10'
(H~so~)
cJ
x
103
x
10:
C6
x
IOJ
(Nac)
(H+)
1,ooo 2.000
0 ,448 0.904
o,
o,
o , 363
o , 552
1.0
1.0
0.745
1.096
a
cr
(HSOI-1
Concentrations ct in moles/cc. estimates in ref, 20,
* Values for
cex I O J
0 ,448 0.904
Q obtained by
Results Estimated and observed values for the neutral-solute phenomenological coefficients for the systems HzONaC1-KC1 and HzO-NaS04-HzS04 are given in Table 11. The only assumptions used to estimate the four (Ltj)o for each system concern the ionic phenomenological coefficients and are given by eq. 28 and 30. At total c2 of less than for the solutes concentrations el NaCl and KC1, these assumptions seem to be valid; at
+
Thus the stoichiometric coefficients v i j and zi have the values v14 = v26 = 2, v16 = v26 = v36 = v36 = 1, v16 = V24 = v34 = 0,2 3 = -1, 24 = z6 = +1, and z6 = -2. = 51.2, X4O = 50.10, Values used for the A2 were As0 = 349.8, and ken = 80.02 C111.2 Ohm-' equiV.-'. The Journal of Phyaical Chemiati-g
(18) R. A. Robinson and It. H. Stokes, "Electrolyte Solutions," Academic Press, Inc., New Tork. N. T.,1959, p . 463. (19) XI. Kerker, J . Am. Chem. Soc., 79,3664 (1957). (20) c. F. Baes, Jr., ibid,, 79, 5611 (1957). (21) R. P. Wendt, ,I. Phlj.9 Chem., 66, 1279 (1962).
ESTIMATION OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
~~~~~~~~
~
~~
1235
~
~
~
Table 11" : Estimated and Observed Values for Phenomenological Coefficients and Diffusion Coefficients c1
Systemb
(Lll)O X 10lg
cz
X 101 X 103
Method
(L1z)o X 1019
(Ldo X
1OIQ
(Lzdo X
1019
(Dido
(Dda
(Dzdo
(Dido
x
X 106
105
105
Method
10:
x
x
H20(0)-NaCl(l)-KC1(2)
0 . 2 5 0 . 2 5 Est.' 1 , 1 0 1 -0.358 Obsd.d 1.057 -0.320
-0.358 -0.294
1.448 AQ 1.409 Bh Obsd.i
1.461 0.014 1.428 -0.046 1.380 -0.011
0.186 1.977 0.075 1.865 0.150 1.836
H20(0 )-KaCl( 1)-KCl( 2)
0.50 0.50 Est." 2.203 -0,716 Obsd.d 2.071 -0.611
-0,716 -0,585
2.897 A' 1.340 Bh Obsd.(
1.461 0.014 1.478 - 0 , 0 1 9 1.403 0.026
0.186 1.977 0 . 0 9 5 1.889 0.173 1.859
H20(0)-NaCl(l)-KC1(2)
1 . 5 0 1 . 5 0 Est.' 6 . 6 0 8 -2.147 Obsd.d 5.318 -1.510
-2.147 -1.630
8.691 As 7.677 Bh Obsd.{
1.461 1.788 1.464
0.014 0.174 0.199
0.186 1.977 0 , 3 4 5 2.118 0.387 1 . 9 0 1
H20(0)-LiCl( 1)-KC1(2)
0 . 2 5 0 . 2 0 Est."
-0,260
1.184
1.195 1.141
0.013 0.012
0 , 2 5 1 1.841 0 . 2 2 1 1.822
1.246 1.105
0.138 0.107
0.173 1 . 4 2 1 0 . 2 0 3 1.356
0 , 8 6 8 -0,260
AQ Obsd.'
H20(O)-LiCl(l)-??aC1(2)
0 . 2 5 0 . 2 0 Est.'
0.877
-0.193
-0.193
0.853
AQ Obsd.'
Hz0(OtNapSO4(1)-H2SO4(2) 0 . 5 0 0 . 5 0 Est.' Obsd.'
H20(0)-Na&3O4(l)-H~S04(2) 1 . 0 0 1 . 0 0 Est." Obsd.'
1.167 -0.456 0.9t50 -0.262
-0,456 -0.212
3.795 A2k 2.164 B' Obsd."
1.591 -0,498 1.371 -0.781 1.096 -0.496
-0.757 -1.175 -0.617
3.795 4.900 2.785
2.337 1.564
-0.918 -0.529
7.612 A2k 3.174 B' Obsd."
1.588 -0,497 1.385 -0.849 0 . 9 2 3 -0.401
-0,748 -0.706 -0.380
3.802 6.520 2.723
-0.918 -0.419
Units: c i , moles/cc.; ( L t j ) ~moles2/(erg , em. sec.); ( D t j ) ~cm.l/sec., , corresponding to flows having units of moles/(cm.Z see.). Numbers in parentheses designate components in subscript notation. Estimated by using eq. 30, 29, and 27. Calculated according to eq. Id-lg, ref. 6, by using values for ( D i j ) and ~ in ref. 23 and ref. 6, 15, respectively. e Estimated by using eq. 30, 56, Observed values given in ref. 21. Calculated from eq. 3 by using estimates of p i k in Table I11 and estimates of ( L , j ) oin and 54. this table. * Calculated from eq. 3 by using obsd. values for pik in Table I11 and estimates of (LV)oin this table. Observed values Calculated from eq. 5 by using observed values for ( D t j )and ~ V k in ref. 16. * Calculated from eq. 3 by using est. 2 given in ref. 23. values for pik in Table I11 and estimates of (LiI)oin this table. Calculated from eq. 3 by using observed values for p i k in Table I11 Calculated from eq. 5 by using observed values for (Dij)o and Vt in ref. 21. ~ this table. and estimates of ( L i j ) in
'
'
'
'
Table 111": Estimated and Observed Values for Chemical Potential Derivatives Systemb
el X 10'
c2
X 102
Method
1111
X 10-14
BIZ
X 10-14
w21
x
10-1'
#*%
x
10-1'
HzO(0)-KaC1( l)-KCl( 2)
0.25
0.25
Est.' Obsd.d
1.4874 1.4283
0.4958 0,4096
0.4958 0,4047
1.4874 1,3884
HzO(0)-NaCl( l)-KCl( 2)
0.50
0.50
Est.c Obsd.d
0,7437 0,7411
0.2479 0.2211
0.2479 0.2157
0.7437 0.7068
H20(0)-NaCl( l)-KCl( 2)
1.50
1.50
Est,." Obsd.e
0.2479 0.3082
0.0826 0.1146
0.0826 0.1159
-0.2479 0.2720
HzO(O)-NazSOa( 1)-HzS04(2)
0.50
0.50
Est. 1' Est. 2Q Obsd.h
1.2395 1.3483 1.1056
0,2479 -0.0373 -0,1729
0.2479 -0.b373 - 0.1767
1.240 0.9955 1.2701
HzO(0)-Na&Oa( 1)-&So4(2)
1.00
1 .oo
Est. 1' Est. 2Q Obsd.h
0,6197 0.6730 0.5838
0.1239 -0.0171 -0.0279
0,1239 -0.0171 -0.0223
0.6197 0.4974 0.8531
H20(0)-LiC1( l)-KC1(2)
0.25
0.20
Est.c
1.5425
0.5509
0.5509
1.7904
H20(0)-LiCl( l)-NaCl( 2)
0.25
0.20
Est.'
1.5425
0,5509
0,5509
1.7904
Units: c t , moles/cc.; ptk, cc. ergs/mole2. Numbers in parentheses designate components in subscript notation. Estimated Observed values in ref. 6. E Observed values in ref. 15. Estimated by using eq. 59. Estimated by using eq. 62 by using eq. 35. and values for Q and ct given in Table I. * Observed values in ref. 21.
'
~~
~
Volume 69, ,Vumber
4
A p r i l 1968
RICHARD P. WENDT
1236
least the good agreement between all estimated and ob~ e r v e d (Ltj)o ~ ~ l ~a t~ the relatively low concentrations implies no inconsistency of the assumptions for this system. A simple proportionality exists between estimates of corresponding (L,J0a t each composition because all ions are univalent and because c1 = c2. This proportionality also exists inversely for the derivatives p t n in Table I11 a t the three compositions. I n eq. 3 for calculating the (Dtj)othe proportionality constants will cancel and the estimates of all (D,J0 will be independent of c1 or c2 for systems where c1 = c2, if the pfb are estimated according to (35). Diffusion data at low concentrations of the systems Hz0-Na2S04-H2S04are not available and the limits of validity of assumptions (55) and (30) for this system cannot be inferred from values in Table 11. Furthermore, the estimates of Q for this system are of unknown accuracy at these high concentrations. Estimates of for this system are, however, of correct algebraic sign and, except for ( L 2 2 ) 0 , agree with observed21 (L,,)oto within 0.20 X lo-’$ a t the lower composition. Because (L2& depends strongly on (Ls6)0(the mainterm coefficient for the flow of H+) we suggest that the approximation (30) relating to Abo is very inaccurate a t total concentrations c1 c2 greater than approximately 1.0 X For all systems none of the assumptions (28), (55), or (30) violates the Onsager relations24 (Lfj)o = (Ljt)o, i # j, for the ionic phenomenological coefficients. Therefore, as expected, the estimates for the neutralsolute phenomenological coefficients are seen to satisfy the Onsager relation for neutral solutes, (L12)o = ( 1 5 ~ ~ )Previous ~. analyses6*l 5 9 2 l of experimental errors have explained the small differences between observed values for (Al& and (L21)O. Estimates of the p i n in Table I11 are in good agreement with observed values a t all compositions considered here for the system H20-NaC1-KC1. For the weak-electrolyte systems, eq. 62 (designated est. 2 in Table 111) gives considerably better estimates of p12 and pzl than eq. 59 (designated est. l ) , but est. 1 does a somewhat better job for the derivatives p11 and p22. We chose est. 2 values to calculate the (Dfj)o for the system H20-Na~S04-H2S04because all diffusion coefficients were found to be somewhat more accurately estimated with those values than with values from est. 1. In Table 11, values for (D,,)o estimated in two ways for the systems H20-?iaCl-KC1 and H20-NazS04H2S04 are compared with observed value^,^^^^^ 50 that the effectsof assumptions used in the estimates can be semratelv considered. For the system H20-NaC1-KC1, the differences between (DLj)ocalculated by method A, which uses
+
The Journal of Physieal Chemistry
estimates of ptn, and values calculated by method B, which uses observed values of p i n , are not appreciable except at the highest concentration, c1 = c2 = 1.50 X loW3mole/cc. At that concentration the agreement and (Dzl)oand values between observed values for (D12)0 estimated by method B is deceptively good; the estimated (Lfjj0in Table I1 are not in good agreement a t the highest concentration of this system, and calculations of (Df,)ousing the poor estimates can only give fortuitously good results. The good agreement between estimated and observed26 values for for the systems H20-LiC1-KC1 and H20-LiCl-KaCl may or may not be fortuitous. KOthermodynamic activity data are available for those systems, and only if we assume the estimates of p f k for these relatively dilute systems are as accurate as estimates of prrc for the system H20-NaC1-KC1 can we then infer the approximate validity of assumptions (28) and (30) for the ionic phenomenological coefficients. Both methods used to estimate for the weakelectrolyte system, H20-Na2S04-H2S04,give results which are generally in rather poor agreement with experiment. However, for the systems of this class no other theory is availablen to estimate both the (L,,)o and (Dfl)o. The present theory enables us to regard as fortuitous the few apparently accurate estimates of (D,,)o made by method A2 (which uses est. 2 values of the ptn from Table 111) because the estimated and observed values for ( L f l ) oare not in good agreement. At the concentrations of the strong-electrolyte systems of class I considered here we conclude that this theory provides good estimates of the (L,,)o and (D,,)o and small second-order corrections to the (Dfj)o as estimated by the first-order approximate theory of Gosting and O’Donnell. For the one system of class (22) Values for (&)a for the system HzO-NaCl-KC1 that were recalculated by Fujita and Gosting (ref. 23) were used with observed values of )btk (ref. 6 and 15) to calculate observed values for the ( L i j ) according ~ to eq. Id-lg of ref. 6. (23) H. Fujita and L. J . Gosting, J . Phys. Chem., 64, 1256 (1960). (24) L. Onsager, Phys. Res., 37, 405 (1931); 38, 2265 (1931). (25) Observed values for the ( D i j ) ~for HzO-N~ZSO,-HZSO~ were calculated according to eq. 5 by using values of (Dij)v and 8, given in ref. 21. (26) For the systems Hz0-LiCl-KC1 and Hz0-LiC1-NaCl, observed values for ( D i j ) were ~ calculated by using eq. 5 and Miller’s values (ref. 16) for (Di;)v and v k (originally measured by Dunlop and Gosting, ref. 1 ) . P. J. Dunlop, in J . Phya. Chem., 68, 3062 (1964), also presents recalculated values of ( D i j ) ~for these systems. His values do not differ sinnificantlv from Miller’s. (27) A first-order approximate theory for the system HzO-NarSOr HzSOa, similar in form to that derived by Gostings for systems of class I, was presented in the author’s Ph.D. thesis (R. P. Wendt, Universitv of Wisconsin. 1961, Librarv of Connress Card No. Mic 61-680). -Values of (D;,)o calculated -from t h e present theory according to method A2, Table 11, are found to equal those Calculated according to the previous theory.
ESTIMATION OF DIFFUSION COEFFICIENTS FOR TERNARY SYSTEMS
I1 examined, the theory provides poor estimates of (A,,),,and (Dz,)oat the relatively high concentrations where data are available which can be compared with the estimates. At lower concentrations of all systems the assumptions concerning the ionic (A,,),,become more accurate, the derivatives p z n should have larger values, and the second-order corrections of the present theory should become more significant. Acknowledgment. This research was supported in part by a grant-in-aid from the University Council on Research of Louisiana State University.
Appendix Expressirms Relating (L,,)o fo A t . Limiting equivalent conductivities, A,: may be defined without using the formal theory of nonequilibrium thermodynamics (I,O)c = z
