STUDIES OF CELLS WITH LIQUID-LIQUID JUNCTIONS. PART 11. THERMODYKARIIC S I G N I F I C d S C E X E D RELATIONSHIP TO ACTIVITY COEFFICIENTS B Y E. A. GCGGEXHEIM
In a previous paper' the author introduced a thermodynamic function called the "electrochemical potential" of an ion, the difference between its values in two phases being definable as the work of transferring reversibly at constant temperature and constant volume one gram ion from the one phase t o the other. The electrochemical potential of an ion of type i in the phases I, 11, . . will be denoted by Fi1, jii", . . , . If phases I and I1 are both ideal solutions of the ion i at concentrations Ci' and Ci" in the same solvent, we have
.
-11
-
-1
Ci"
+
R T log, ---Zi~ E(4'' - $I) (1) Ci where R is the gas constant, T the absolute temperature, - E the charge of a gram electron, Z ~ Ethe charge of one gram-ion of i and $I1 - $I the electric potential difference between the two phases. The electrochemical potential being defined as above, equation ( I ) may be regarded as the physical definition of the electric potential difference between the two phases, since the common electrostatic definition involves the conception of a hypothetical fluid '(electricity" which has no physical existence. If however the solvents in the two phases are different, one might write pi
-11 pi
-
-1 pi
pi
=
CiII - R T log, -
Ci'
+ R T log, li +
(4" - $I) ,
Z ~ E
(2)
where li would be the "partition coefficient" of the ion i. Finally, if the two solvents are the same but the solutions are not ideal, one might write
where f:'/f: would be the "ratio of the activity coefficients" of the ion i. But just as equation ( I ) defines $I1 - $I, so equations (2) and (3) do no more than define RT log, li z i e ($" - $I) and
+
c
I1
and the resolution of each of these expressions into two terms is arbitrary and without physical significance. This principle was expressed by the authorz 1
Guggenheim: J. Phys. Chem., 33, 842 (1929). J. Phys. Chem., 33, 842 (1929).
* Guggenheim:
1759
CELLS WITH LIQUID-LIQUID J U S C T I O S S
in the form that the electric potential difference between two points is a conception without any physical significance unless the two points are in the same media; if the media are solutions, the “same medium” connotes not merely the same solvent but also ideal solutions. For the case of non-ideal solutions in a given solvent the principle had already been stated by Taylor‘ in the form that the activity coefficient of an ion is “not an experimental quantity but a mathematical device.” h similar remark might be equally truly applied to the “partition coefficient” of an ion for the case of two ideal solutions in different solvents. If instead of considering the transference of one ionic species i, we consider the simultaneous transference of several species, say X i gram ions of each species i, such that the net transference of electric charge is zero, that is such that 2
X i zi
=
0,
(4)
1
then the terms containing transference 2
Xi
(#I1
- #I) vanish and we get for the work of
Ci“ Ci
(11’- 11) = Z RT X i log, 7+ Z RT X i log, 1
Z Xi f l i ” I
ZiI) =
Z RT 1
Xi
log,
Ci”
li
i
(5)
+ Z R T X i log, fi’ fi”
, I Ci
i
corresponding respectively to ( 2 ) and (3). These formulae define the products II(li)xi and I I ( f i l r , ’ f i l ) A i , so that these have a real physical significance, 1
1
though the individual l i and f i “ ’ f i ’ have not. As a special case the “mean partition coefficients” and “mean activity coefficients” of any salt are thus defined. So are similarly the “partition coefficient ratios” and the “activity coefficient ratios’’ for two ions of the same electric type. A third important example is that particular combination of activity coefficients which occurs in the expression for the E.1I.F. of a cell with liquid-liquid junction. This case will now be discussed in detail.
I Solution
sible to cation lager
J. Phgs. Chem., 31, 1478 (1927).
Electrode reversible t o cation A
reversible. If all the solutions prcscnt including thc bridgc golution and those in the transition layers were ideal, the electro-motive force E of the cell could be regarded as the sum of the electrode potential and the diffusion potential. We should in fact then have
where ti denotes the transport number of the ion of type i and both integrals extend over both transition laycrs in the direct,ions I to 111 and 111 to 11. When thc solutions are not ideal we might write instead of ( 7 )
This is formally correct but involves individual ionic activity coefficients, that is quantities with no physical significance. These must therefore occur combined in such a manner as to be expressible in terms of mean activity coefficients of salts and activity coefficient ratios of either two cations or two anions. That this is the case is easily shown by observing t,hat
z tR
+
+ x-t x =
1p
so that the fourth term on the right hand side can be transformed as follows:
RT
-log, e
fA1’
-i= fA
“fd €
log,f*
Substituting in (8) we get
E
= ICE1
+ ED +
ES,
where the “ideal electrode potentials” EEIare given by
the “ideal diffusion potentials” ED by
and the “salts effects potential”
by
If as bridge solution is used a concentrated solution of a salt whose two ions have nearly the same mobility tho value of J?D may sonietirnes be mado smal[
CELLS K I T H LIQUID-LIQUID JUNCTIONS
1761
and perhaps even negligible, but however this may be the salt effects as measured by Es are given by (13) and to split up $ s into electrode terms and diffusion potential terms as appear in (8) is quite arbitrary and without any physical significance. I Bjerrum and L-nmack* give a conventional definition of individual activity coefficients by arbitrarily assigning the value zero to the sum of the last two terms in (S), which is equivalent to the definition
RT Es = -log,
f
2-
e
fAI
for the particular cells in which the bridge solution is 3.5 K KCl and the ~ Other authors in the same junctions are of the “continuous m i ~ t u r e ”type. field have either had no clear idea of what they meant by f a or else have implicitly used a similar definition. There is no logical objection to this convention provided it is remembered that the ratio of the ionic activity coefficients in solutions I1 and I then depends not only on the thermodynamic properties of these solutions themselves, but also on those of the intervening solutions and on the various transport numbers in the two transition layers. As pointed out by Rjerrum and Unmack this definition of f.4 sstisfies the condition that at infinite dilution, where all the f R X tend to unity, fa also tends to unity. I t is however important to realise that at great but not infinite dilution, where all uni-univalent salts have the same activity coefficient different from unity, fa does not become equal to this mcan activity coefficient. In fact setting fRS = fAS =
f*
(15)
in (13) and (11)we get
and so even when
(I5
) holds, in general i
unless by chance Z tR
3.
B t, = In view of this result it is not justifiable for extrapolation to infinite dilution to assume for these conventional “ionic activity coefficients” formulae known t o hold for mean activity coefficients. This is possibly the explanation why Bjerrum and Unmack when they assume that f H f follows a law of the form
-
=
log,
fHf
= -
0.3
fi
pr,
r being the ionic strength, obtain such surprisingly large /3 values that they themselves draw attention to the fact. ’Taylor: J. Phys. Chem., 31, 1 4 j 8 (1927). * Bjerrum and Unmack: Kgl. Danske Vid. Selsk. M a t . Fys. sled., ( 9 ) I (1929) aGuggenheirn. J. 4 m . Chem. SOC.,52, (1930).
1762
E. A. G C G C E S H E I M
Some measurements being published elsewhere by Miss Unmack and the author* confirm the inequality ( I ; ) and to some extent the equality (16) in so far as it is possible to compute accurately the right hand side of (16). Thus retaining the above mentioned conventional definition of ionic activity coefficients it was found experimentally that these differed by ;.Cc in . I 9 HCI and . I LiC1, although the mean activity coefficients differ by less than 0 . 3 5 . The effect measured is due not to any differences between the thermodynamic properties of the solutions, but simply to the great difference in the mobilities of H+and Li-. lloreovcr this effect remains considerable even at much higher dilutionp. Thus in . O I S solution the coresponding difference was found experinientally to be j.?, in satisfactory agreement with a computed 6 . 5 , The effects at .ISand . O I X concentrations are in opposite directions. The relegation of the conception of ionic activities to the insignificant position of a mathematical device may possibly at first seem repugnant’ or a t least strange to those accustomed to regard this conception with greater respect. In particular those engaged in ‘ ‘ P R measurements” are liable sometimes to attach great importance to the “hydrogen ion activity” as opposed to the “hydrogen ion concentration.” Such an attitude is however due to a misconception even more fundamental than the failure t o realise that the former quantity is physically undefined. I t is due in fact t o an incomplete realisation of why hydrogen ion determinations are important. If the hydrogen ion content of an indicator solution or any other solution is important, it is presumably because the colour of the former or some equally important property of the latter depends on the state of some acid-base equilibrium, which in turn is controlled by hydrogen ion. The case of reaction rates is no exception, if we take account, of the equilibrium between the reacting molecules or ions and the reacting coniples.’ An acid-base equilibrium is defined according to BronstedJ by the scheme
A $B
+ H+,
where A represents an acid and B the conjugate base. Both acid and base may have any charge positive, zero or negative subject only to the condition that the charge of A esceeds that of B algebraically by one positive unit. Let us as usual denote concentrations by C and activity coefficients by f . Then the colour of the indicator solution, or analogous interesting property of some other solution, is determined by the ratio Ca,’Cg and this is given according to the law of mass-action by:
* rnmark
and Guggenheirn: Kgl. Danske. Vid, Selsk .\fat. Fys. Med , 1930. For references see Guggenheirn: J . Phys. Chem.. 33, 842 irgzg). *BrOnsted: Rec. Tray. chim., 42, 718 l1923); J. Phys. Chem., 30, 777 :Igz61; Chern. Rev., 5 , V S o . 3 (1928).
I763
CELLS WITH LIQUID-LIQUID JUXCTIOXS
where K is the thermodynamic equilibrium constant of the acid-base equilibfH+.fB
rium. Thus the determining factor is CH+. -,this combination of activity fA
coefficients being of course of the type which is physically definite. In the important case that the acid-base equilibrium is of the electric type SH4f
+ ?;H3 + H+
(18)takes in dilute solution the approximate form
while in the equally common case of an acid-base equilibrium of the electric type CH,.CO?H i2 CH3,CO*-+H+
(18)takes in dilute solution the approximate form
where f, denotes the mean activity coefficient of a uni-univalent salt. Thus in the former case the determining factor is CH+and in the latter CH+.f*’. In less simple cases it may take other forms, but under no possible circumstances can it ever become CH+.fH+. Thus, without assuming that the “hydrogen ion activity” CH+.fH+is physically undefined, we have proved that, even if it were defined, it could never have the importance of either CH+or CH+.f +2. It must not be supposed that we mean in any way to disparage the many valuable results obtained by means of cells with liquid junctions of the type in general use for “ p measurements.” ~ But these should be used only as a last resort and wherever possible discarded in favour of cells without liquid junctions. Thus cells of the type
[
H2
1
Solution
HgCl
1
Hg,
if the concentration of C1- in the solution is known, may be used to measure CH+.f**, while cells of the type H2 I Solution I Na(Hg), if the concentration of S a ” in the solution is known, measure approximately CH+. But a cell such as HZ
1
Solution
I
3.5
N KCl
I
HgCl
I
Hg
may best be described as measuring CH+.f?where f, is a much more complicated function than f as it depends on the transport numbers of all the ions present a t all parts of the cell. The only possible advantage of the last type of cell over the two first is a practical one and this is the point that cannot be emphasised too strongly.
1764
E. A. GUGGESHEIM
There remains the question whether one should continue to make use of ionic activity coefficients as a mathematical device and, if so, how they should be conventionally defined. The chief advantage of an ionic activity coefficient is that the usual formulae for ideal solutions can be transformed to thermodynamically exact formulae by replacing each concentration by the product of the concentration and the corresponding activity coefficient, In the formulae so obtained the ionic activity coefficients will always occur combined in such a way that they can be transformed so as to contain only mean activity coefficients of salts (or activity coefficient ratios of pairs of ions with the same charge). We had an example of this in our discussion of cells with liquid junctions. I t is therefore quite unnecessary for this purpose to have in mind any particular conventional set of ionic activity coefficients. Another possible advantage of ionic activity coefficients is for tabulatjon purposes. Thus in any solution, containing more than five kinds of ions, of which at least two are cations and at least two anions, there will be more salts than ions present. In such a case instead of tabulating the activity coefficients of every salt present it may then be more convenient to tabulate a conventional set of ionic activity coefficients. Khilst any convention is permissible it is most satisfactory to choose one that is so far as possible rational. X convention suggested by Rronsted' has been discussed and criticized by the present author.? The definition suggested for the electric potential difference between two phases is such as practicalIy to preclude the possibility of its ever being determined. This effectually prevents Bronsted's system of ionic activity coefficients from being put into practice. Ionic activity coefficients have sonietimrs hcen computed according to the assumption3 that the ions K+ and C1- always have the same activity coefficient. As soon as one realisrs that this is a convention with no physical significance, the claim of Ii- and C1- to be singled out rather than any other pair of ions vanishes. This convention is therefore not t o be recommended owing to its lack of symmetry. Moreover it has a very real practical disadvantage. Thus, for example, to dctermine the values of say fAg+ and f N O j - in a solution of A g ~ it 0 ~would be necessary to add a trace of KCl and then determine fRx for no less than three of the four salts present. Bjerrum and Unmack'a convention, which is by contrast preeminently practical, has the disadvantage that even at great dilutions the inequality (17) holds. With this convention one would generally know very little about the behaviour of ionic activity coefficients in very dilute solutions, unless they were so extremely dilute, that all activity coefficients were unity. We shall now define a new system of conventional ionic activity coefficients which seem to the author to have simpler and more symmetrical properties than thow of any other system. Let Ii now denote any cat,ion with valency Brcinsted: Z. physik Chem., 143,301 (1929,. Guggenheim: J. Phys. Chem., 34, I j40 (1930). MacInnes: J. Am. Chem. SOC., 41, 1086 f r g i g ) ; Scatchard: 47, 696 (192j); Harned: J. .4m. Chem. Soc., 42, 1814 (1920); J. Phys. Chem., 30,433 (1926).
S dn>- aiiion with valt;nc*y-vx, so that -VX= XI.Let, i ~ ,ilimo~r s nctiT1ty cacHicit,nt of the salt whose ions :ire K and S. 1 , ~
