n
series of organic acids and the electrode is seen to function as an ideal indicator for several of the anions where the points fall on theoretical slopes. In two cases, however, namely those of nitrophenol and phenol, the curves are seen to deviate from ideal behavior. This could possibly be due to adduct formation in the organic phase between acid and alkylammonium salt and which could make the 01 coefficients pH-dependent (17). In the case of phenol, however, the surface effect may again account for the observed departure from the expected behavior since phenol has a high pK value. At a low pH the picrate ions are the dominant ions in the surface layers and the potential is independent of phenol concentration. At higher pH values the phenolate ions begin to have an effect and the titration curve rises more steeply than the theoretical slope. The electrode can therefore be used as a very sensitive indicator for phenol titration if this effect is utilized. Role of Solvent. As organic phase in a liquid membrane electrode selective to anions, methylene chloride has the advantage of being a hydrogen bonding (protic) solvent which therefore favors anions. It will consequently contribute to the exclusion of co-ions from the membrane, in this case the cations in solution. For discussion of the role of solvent, see Ref. (19). In order to examine the importance of the solvent, the electrode properties of methyl isobutyl ketone (MIBK) was tested. This is an equally good solvent for ion pair extraction with alkylammonium ions but its oxygens will tend to favor cations rather than anions. The experiments described with methylene chloride were therefore repeated with MIBK as a solvent. As expected, the potential selectivity as well as the potential slopes practically disappeared in this case. It is concluded from these experiments that a suitable solvent is important for the proper functioning of a liquid membrane electrode as well as a highly extractable and selective ligand.
Ag ; AgCl
+zoo \
+150
+loo
L
PK
I 1.\hydroxy
b-
benzoic a{id
+ 50
\
\
\
0 pK
- 50
\
be
>-,
Figure 4. The membrane potential vs. pH for a series of weak organic acids
est to examine the pH-dependence of the electrode potential measured in these acids. A 10-2M solution of the acid was equilibrated with a 10-1M solution of the corresponding tetrahexylammonium salt dissolved in methylene chloride. An equilibrium distribution of tetrahexylammonium ions between organic and aqueous phases was thereby accomplished. The electrode was immersed in the acid and the potential was measured as the acid was titrated by adding sodium hydroxide. Figure 4 shows the titration curves for a
Received for review December 4, 1972. Accepted February 12, 1973.
Ionic Hydration and Single Ionic Activities in Mixtures of Two Electrolytes with a Common Hydrated Cation and One Hydrated Anion R. A. Robinson and Roger G. Bates Department of Chemistry, University of Florida, Gainesville, Fla. 32607 A convention described in earlier contributions makes it possible to derive the activities of the individual ionic species in solutions of unassociated electrolytes and in mixtures of two uni-univalent electrolytes with a common unhydrated ion. With the aid of a thermodynamic treatment of mixtures of electrolytes, it has now been shown that this convention can also be applied to mixtures of two electrolytes with a common hydrated cation, one hydrated anion, and one unhydrated anion. The method is illustrated by deriving the single activity coefficients of potassium, fluoride, and chloride ions in mixtures of potassium fluoride and potassium chloride at a total ionic strength of 3 mol kg-'. 1684
In earlier papers (1, Z), an approach to the establishment of internally consistent numerical scales for the activities of the single ionic species to which ion-selective electrodes respond has been suggested. This method is based on the concept of individual ionic hydration numbers which account fully for the specific differences in ionic activity coefficients apparent in concentrated electrolyte solutions. Hydration numbers for electrolytes are derived from the Stokes-Robinson hydration model ( 3 ) ; (1) R . G Bates, 8. R . Staples, and R . A. Robinson, Anal. Chem., 42, 867 (1970). (2) R. A. Robinson, W. C. Duer, and R G. Bates, Anal. Chem., 43, 1862 (1971). 7 0 , 1870 (3) R . H. Stokes and R . A. Robinson, J. Amer. Chem. SOC., (1948).
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 9, AUGUST 1973
the values for single ionic species are obtained from a n additivity rule, together with the semiempirical assignment of a hydration number of zero to the chloride ion ( I ) . Although the single ionic activity scales so derived are clearly conventional, they meet the primary requirement of self-consistency and appear, as well, to be in harmony with individual ionic activities estimated from cells with liquid junction (4, 5 ) . Extensive use is being made of ion-selective electrodes in media (such as blood, sea water, and the like) containing more than one electrolyte. For the highest accuracy, the electrode responsive to a particular ion should be standardized in a medium of constitution similar to that in which the measurements are made. The determination or assignment of activities to the individual ionic species in mixtures of electrolytes is therefore of interest and practical import. In a recent contribution ( 6 ) , we have considered the extension of the hydration approach to a determination of the activities of the three ion species in mixtures of the’type MX-NX, where M and N are hydrated cations and X is an unhydrated anion such as chloride. Specifically, the thermodynamic derivation revealed the manner in which Y K + , Y N ~ + ,and ycl- vary with composition in a mixture of KCl and NaCl of constant total ionic strength. The calculations apply with equal force to mixtures of other uni-univalent chlorides, bromides, and iodides. It is the purpose of the present contribution to extend the earlier treatment to mixtures of the type MX-MY, where the cation M and the anion X are both hydrated but the anion Y is unhydrated. This is the case for the interesting systems KF-KC1, NaF-NaCl, KF-KBr, and KF-KI, which have been the subject of recent experimental investigations with ion-selective electrodes (7, 8).
THE MX-MY SYSTEM Consider a solution containing two salts, each of the 1:l charge type, salt A with cation M and anion X , salt B with cation M and anion Y. Let hM and hx be the hydration numbers of M and X, respectively, the anion Y being taken as unhydrated. The Gibbs energy of a solution containing S mol of water, Y A mol of A, and YB mol of B, where Y A + YB = 1, is
+
+
+
1
+
2 (hhT yAhX) lna,
-
=
YA
+
+
+ + YBfiN;
1 n f ~ N . ~Y B ~ ~ ~ B N B
yAlnfLN;
(5)
+
Inasmuch as N, = m,/(55.51 2m), where m = mA + mB, mA and mB being the molalities of A and B, respectively, and N,’= m,/[55.51 + ( 2 - hM - y ~ h x ) m ]it, follows that
Moreover, since fA‘ and f ~ in ’ the hydration theory are identified with a Debye-Huckel term ~ D H ,Equation 5 becomes 1
~ ~ l n f +. Y 4 B I ~ ~ BI ~ ~ D H - ? (+ ~ . M & x ) l n a w+
(7)
ln(N,’/N,) Furthermore, as In yz = In f z- In [l 1 In ~ D H $h,~ +
+ O.O18(2m)]
(8)
In uw
Equation 7 can be written in the alternative form 1 Y;\In 7.4 Y B In YB = In f m - 5 ( h , + yahx) In a w
+
- In [I + 0.018(2 - h,
- y,h,)ml
(9)
SINGLE IONIC ACTIVITY COEFFICIENTS By the Gibbs-Duhem equation
+
- ( 5 5 , 5 l / m ) d l n a ~ = d l n N A f h I + yAdlniV.Afs-
+
yB d In u Y - (10) Noting that
+
m,4/m.A‘ = mB/mR/ = m/m’ = 1 - O,0lNhM
.y.&x)m (11) one can also write the Gibbs-Duhem equation in the form
-(55.51/m’)dlna,
=
-(55.5l/m) d l n a w
+
+
(hhT yAhX)d In a, =
d In N,’ fhp’
+ y Ad In NA’fX-’+ 3’8 d 111~
y (12) -
On subtraction of Equation 12 from Equation 10, one obtains In f l l + yAIn f s - = In fM+’ yAIn fx-’ -
+ + ( h M+ y,\hX)In aIv + In (NA’IN.4) + Y AIn (NA’ / N.J
y~(G.4’+ 2RT In fANA)
+ yB(G$ 4-
+
+
G = Y A ~ A YBGB (11 This solution contains 1 mol of the M + cation, y 4 mol of the x- anion, and Y B mol of the Y- anion in (s - hM yAhX) moles of “free” water. Its Gibbs energy can, therefore, be written in the alternative form G = ( S - hw - Y.AhX)G, YAGA‘ Y B G ~ (2) so t h a t (hv .yAhd(G,.o RT In a,)
+
By combination with Equation 3, therefore, one obtains
=
(1 + y.$)ln f D H -
(hM+ y & d l n a L v
2RT In f B N B )= y,\(eAo’f 2RT In fa”.\’)
+ i l + y A ) l n ( N , ’ I N i ) (13)
Since f D H and N,’/Ni are the same for all the ions, a (3) general expression for fM+is where the primes refer to hydrated species and N is the mole fraction. In the limit of infinite dilution, aw, f ~f ,~ , In JM- = lnf,, - [F,(h,) + F , ( h ? ; : ] l n a ~ l n ( N , ’ / N , ) f ~ ’and , f ~all ’ equal unity and, therefore, (14) (h, yAh,y)G,’ yAGAn yBGBo = YAGA“ ~ $ 2 ‘ where Fl(hh1) and Fz(hx) are, as yet, unknown functions of hM and hx, respectively. Combining this expression (4) with Equation 7 , one finds that
+ y~(Gp,”’+ 2RT In
fB’NB’)
+
+
(4) (5) (6) (7) (8)
+
+
+
A. Shatkayand A. Lerrnan, Anal. Chem., 41,514 (1969). J. Bagg and G. A. Rechnitz, Anal. Chem., 45, 271 (1973). R . A. Robinson and R . G. Bates, Anal. Chem.. 45, 1666 (1973). J. V . Leyendekkers, Anal. Chem., 43, 1835 (1971). J. Bagg and G. A. Rechnitz. Anal. Chem.. 45, 1069 (1973)
In f w =
Y A
In f A
+ yB In f R +
ANALYTICAL CHEMISTRY, VOL. 45, NO. 9, AUGUST 1973
1685
In the limit of YA = 1, when the solution contains only the salt A In f M + = In f A
+
5 [ ( h +~ h, - 2F,(h,) - 2F2(h,)l In a, 1
(16)
Equation 6 of an earlier paper (2) can be written In
In
fM+ =
fq
+1
(h, - hM) l n a w
(17)
so that Equation 16 can be true only if F l ( h ~=) hM and Fz( h x ) = 0. Similarly, for In fxIn f x - = In f D H - [ F , ( h d + FAhx)l In u w + ln(N,//N , ) 1 = Y.4 In f A YB 111 f B + 3 [hM
+
+
yAhX
- 2 F d h d - 2Fdhx)l In aw (18)
and, w h e n y ~= 1
In f x -
= In
fA
+ 31 [ h M+ hx -
-
or, sinceyA = 1 y~ log yB = log yA - 0.00782hXm4 (29) For a solution containing MX only (YA = 1, YB = 0), the trace activity coefficient of B is given by log yBtr= log ya0 - 0.00782hxm@,40
(30) where -yAO and c p ~ Oare the activity coefficient and the osmotic coefficient, respectively, of A (that is, MX) in its own solution at molality m. By Harned's Rule (9)
+ 3(
fM+
= yA In f a
1
(20)
~In )aw
(21)
and
In f x -
= yA In
fA
+ YB In f B + 21 [AM - ( 2 - yA)hsl In aw (22)
or, alternatively
+ YB log yB -0.00782(yAh, - h,)m@
log yM+= y A log
and
(23)
log YX- = Y A log ?A
+ YB log Y B 0.00782 [ h -~(2 - y ~ ) h ~ ] m 4(24)
where p is the osmotic coefficient defined by -In a, = 0.018 vmp. Furthermore, as
In f M+ + yAIn f x -
+ yBIn
fy-
= 2yA In
f A
+
= (1 /
log
7.4
-Q A Y B ~ = log -yAo - aAm
= log
log
h -~hx) In aLr
+ YB In f B + 2 ( ~ a h x- h
(32)
or
+ 2yB In
f H
(25) then, by Equations 21, 22, and 25, bearing in mind that y A yR = 1, one obtains
+
(35)
YAO
(36)
so t h a t aA= ( l / m ) l o g ( y A 0 / y B o-) 0.00782hx~B0 (37)
from which it follows that F&M) = 0 and Fc(hx) = h x , so that one can write In
log yBtr = log yBo- a B m
m) log (yB0/7:) 0.00782hx4A0 (33) For a solution containing B (that is, MY) only, where Y A = 0 and yB = 1, Equation 29 gives log yBo= log yAtr- 0.00782hxm&,0 (34) But, by Harned's Rule (YB
Equation 7 of the earlier paper ( 2 ) can be written In f x - = In fa
(31)
?Bo
Hence,
2F,(hhf)- 2F4(hx)]In a, (19)
1
- ffBy.4m
log YB = 1% or
APPLICATION TO THE SYSTEM KF(A)-KCl(B) It has been shown (2) that, for potassium fluoride, hK = h~ = h = 1.9. At m = 3 mol kg-l, ~ A O= 0.705, = 0.569, p . 0 = 1.048, and p B O = 0.937 (10). Substituting these values in Equations 33 and 37, it is found that aA = 0.0171 and (YB = -0.0154. It can be shown (11) that the excess free energy of mixing of solutions of two 1:1 electrolytes is given by the equation
AGE = -2.3026yAyBRTm2(a.4+ cyB) (38) For the KF-KC1 system at m = 3 mol kg-l, with Y A = Y B = 0.5, AGE = -5 cal kg-1. This excess free energy of mixing is known (12-14) to be a measure of interaction between the two like charged ions in the mixture. It is seen to be small for F--Cl- interaction, as has been found (15) also for C1--Br- interaction, where A G E = 5 cal kg-1 for the NaC1-NaBr system at m = 3 mol kg-I and 3 cal kg-I for the KC1-KBr system at this molality (15). By means of emf measurements, Owen and Cooke (16) have found (YA = 0.016 and a g = -0.017 for the HC1 (A)-HBr (B) system so that AGE must be very small. In contrast, interaction between cations is more pronounced; for example, AGE = -40 cal kg-I for K+-Na+ interaction (15) at m = 3 mol kg-1. If h M = hx = h, Equations 23,24, and 27 become
log YK'
=
log YF- =
YA
log Y A
+ yB log YB
-
0.O0782(yA- l)hm@ (39)
HARNED RULE COEFFICIENTS Addition of Equations 23 and 27 gives
+
log y>f+ log y y - = 210g yg = 2ya logy,
+
2yB log yB- 0.00782(2yAhx)m4 (28) 1686
ANALYTICAL CHEMISTRY, VOL. 45, NO. 9 , AUGUST 1973
H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed., Reinhold, New York, N. Y., 1958,p 603. R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," 2nd revised ed, Butterworths, London, 1970,pp 476,484,and 494. H. S. Harned, J. Phys. Chern., 63, 1299 (1959). G. Scatchard, J. Amer. Chern. SOC.,83, 2636 (1961). E. A. Guggenheim, Trans. FaradaySoc., 62, 3446 (1966). P. J. Reilly and R. H. Wood, J. Phys. Chem., 73, 4292 (1969). A. K. Covington, T. H. Lilley, and R. A. Robinson, J. Phys. Chem., 72, 2759 (1968). B. B. Owen and T. F. Cooke, J. Arner. Chem. SOC., 59, 2277
(1937).
Table I. Single Ion Activity Coefficients in Mixtures of KF (A) and KCI (B) at a Constant Total Molality of 3 mol kg- l,t = 25'C YB
-1ogYKF = -log Y K t
1.o
0
0.2
0.4
0.5
0.6
0.1518
0.1622
0.1 721
0.1 776
0.1827
0.1928
0.2033
0.1985 0.2452
0.2078 0.2535
0.21 71 0.2614
0.221 7 0.2659
0.2264 0.2700
0.2357 0.2786
0.2450 0.2867
0.8
= -log Y F -
- IogYKCl - IOgYCland
,-
log *tc
y Alog
iyBlog yB
- 0.00782(y4 + l ) h W
-0.15
(40)
The osmotic coefficient cp of the mixed salt solution is calculated by means of the equation (17, 18) f#J
=
44'
+ (2.3026/ 2)YBm[YB(aA + aB) - 2 a A 1 (41)
-0.20
Y -0.25
using the values of aB and CY* given above. Equations 31 and 35 then give log yA and log YB, and finally Equations KCI 39 and 40 yield the single ion activity coefficients, YK t , YF-, andycl-. The values of these single ion activity coefficients at rn 0 0.5 1.0 = 3 mol kg-1 are given in Table I and are plotted as a KF KCI function of ye in Figure 1. Because LYA and CYBare almost YE equal in magnitude but opposite in sign, the term y ~ ( c t ~ Figure 1. Activity coefficients of potassium fluoride, potassium a B ) in Equation 41 contributes very little to curvature chloride, potassium ion, fluoride ion, and chloride ion in a mixin a plot of cp against YB. As a consequence, the three plots ture of KF (A) and KCI (B) at a constant total molality of 3 mol in Figure 1are almost linear. kg-I
+
DISCUSSION It is appropriate to examine further the hydration convention on which these individual ionic activity coefficients are based. By its very nature, a convention is not subject to proof; it is adopted, usually by general agreement of those concerned, in order to permit the assignment of useful numbers to quantities that cannot be derived exactly from theory and which must therefore remain partly arbitrary. A convention should, however, be reasonable in the light of theory and should be consistent with all of the known parameters of the systems concerned. The hydration convention as outlined above and in earlier papers treats the hydration number as a constant characteristic of each individual ionic species. In many concentrated solutions, the quantity of solvent is insufficient to permit all of the ions to combine with h molecules of water. Stokes and Robinson ( 3 ) found, for example, that their equation reproduced accurately the mean activity coefficient of hydrochloric acid up to a molality of about 2 mol kg-1 at which concentration 16 out of the 55.51 available water molecules are required for the hydration process. At higher concentrations, their equation predicts activity coefficients higher than those observed. This indicates that the hydration number decreases as the concentration increases and, for nonelectrolytes such as sucrose, the osmotic coefficient can be accounted for (19) up to satura-
tion by assuming a set of hydration stages in equilibrium with one another. Furthermore, an examination of the hydration numbers derived for simple electrolytes ( 3 ) shows that the h values for ionic species are not strictly additive. The reason for this is that the hydration model is made to account for all departures from ideality apart from those due to the operation of Debye-Huckel forces, It is probably true that the hydration effect accounts for the major part of these departures from ideality but it cannot be denied that there are other effects, such as mutual polarizability of the ions, which should be allowed for in a complete theory of electrolytes. In our view, however, the major success of the hydration convention is in attributing specific differences in the activity coefficients of electrolytes a t high ionic strengths primarily to differences in cation hydration. Although manifestly not beyond criticism, the hydration convention provides a reasonable formula for separating the mean activity coefficients of concentrated electrolyte solutions into their individual ionic components. It may be expected to be even more successful a t low ionic strengths. Moreover, this approach offers a means of deriving consistent standard scales for the activities of many of the individual ionic species for which ion-selective electrodes are now commercially available.
(17) R . A . Robinson and R . H. Stokes, "Electroiyte Solutions," 2nd rev. ed., Butterworths, London, 1970, p 440. (18) H. S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions,"3rd ed., Reinhold, New York, N. Y., 1958, p 602. (19) R. H.Stokes and R. A. Robinson, J. Phys. Chem.. 70, 2126 (1966).
Received for review January 8, 1973. Accepted March 30, 1973. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work, and to the National Science Foundation under Grant GP-14538.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 9, AUGUST 1973
1687
