ANALYTICAL SCIENCES APRIL 1998, VOL. 14
1998 © The Japan Society for Analytical Chemistry
293
Computational Aspects of Extraction Equilibrium and
Distribution Potential of Electrolyte Distributed
between Two Liquid Polar Phases
Jiří RAIS
Nuclear Research Institute, plc, Řež, Czech Republic
It is proved that both "electrochemical" and "chemical" treatments of the equilibrium distribution of electrolytes between
two polar solvents are correct, or in other words the use of electrochemical or chemical potentials is interchangeable.
This is illustrated by calculation using a recursive iteration program for distribution of Sr2+ by dicarbollide anion in the
presence of polyethyleneglycol. Several particular situations are described: the effect of common ion, limiting distribution coefficient and influence of protonized particles on distribution of cations.
Keywords Extraction, electrolyte, distribution potential, polar solvent
The distribution of electrolytes between two non-miscible polar liquid phases is extensively studied both by
electrochemical and extraction methods. The methods
of expressions of equilibrium distribution, however,
differ. The origin of the difference is due to the fact
that, whereas in extraction chemistry as a rule an equality of chemical potentials of all ions in two phases is a
starting equation, in electrochemistry it is an equality of
electrochemical potentials. Although it might seem
that a priori the electrochemical treatment is more correct, since certainly charged particles are involved in
the transfer, it can be shown that both treatments are
equally correct and useful for describing an equilibrium
state. The treatment used in extraction chemistry will
be denoted as chemical in order to make a clear distinction between the two. The main purpose of this article
is to present a simpler and quicker method of the calculation of equilibrium distribution and distribution
potential than as a rule used in electrochemical works
and also to point out some special cases of distribution
which at first sight might be not well understood.
Experimental
Pure recrystallized cesium dicarbollylcobaltate was
obtained from Institute of Inorganic Chemistry, Czech
Ac. Sci. For experiments, demineralized water was
used and nitrobenzene of a.r. purity, which was further
purified before experiments by 2 times washing with 2
M HNO3 and 4 times with demineralized water. All
reported data here refer to 23±2˚C.
Results and Discussion
Equilibrium situation from electrochemical and chemical points of view
In standard electrochemical treatments, the condition
of equality of electrochemical potentials for each ion is
expressed and a distribution potential Dabj is calculated.
In a further step, the distribution of ions is calculated
from calculated Dabj and the known standard distribution potential of an ion Dabj 0i.
The method can be illustrated on the simplest case of
the distribution of single salt M + X – between two
phases. By expressing the equality of electrochemical
potentials of M + and X – between two polar liquid
phases w (aqueous) and o (organic) we obtain:
m0,w(M+)+RTln aw(M+)+Fjw=
m0,o(M+)+RTln ao(M+)+Fj o
(1)
m0,w(X–)+RTln aw(X–)–Fjw=
m0,o(X–)+RTln ao(X–)–Fj o
(2)
where m0 values are standard chemical potentials, a
activities, j respective inner potentials and F and R
have their usual meanings. Keeping in mind that for a
single electrolyte it must apply from the condition of
electroneutrality both in organic and aqueous phase that
aw(M+)=aw(X–) and ao(M+)=ao(X–) we obtain by subtracting (2) from (1) easily an equation for the distribution potential in the system Dwoj, if a definition of standard distribution potential Dwoj0 (i) described by Eq.(8)
is used (see e.g. ref.1):
294
Dwoj =(Dwoj X0 ––Dwoj M+)/2.
ANALYTICAL SCIENCES APRIL 1998, VOL. 14
(3)
Now, having calculated Dwoj, the last step is to determine the distribution coefficient of each ion in the system, i.e. a ratio of its activity in the organic phase to
that in water phase a oi/a wi. This is done by using a relation containing potential terms:
Di=aoi/awi=exp[(ziF/RT)(Dwoj –Dwoj 0i ).
(4)
The above method has been used in great number of
works originating from electrochemical laboratories.
The mode of calculations started from now already
"classical" paper of Hung2 who first treated more complicated cases than a single electrolyte distribution and
continued among others in papers of Vany¢ sek 3 ,
Kakiuchi4 and Girault.5 Let us notice that, during such
a type of calculation, only individual ionic properties
are used and no reference to properties of electrolytes
as a whole is given (in one paper Vanysek6 makes a reference to a property of an electrolyte as a whole, calling the value pertaining to Kex (M+, X–) (see below) as a
"combined partition coefficient").
On the other hand, in extraction studies no reference
is given as concerns the distribution potential in the
system. The systems are treated as purely chemical
systems while evaluating the respective extraction constants of different electrolytes. A conspicuous example
of this is the recent detailed review on extraction of
alkali metal cation electrolytes by Moyer and Sun7,
where the phrase "distribution potential" does not
appear even once throughout. It might be argued that in
the latter treatments an important omission is made, not
properly taking into account the electrified nature of the
involved processes.
It may be shown that both treatments are correct. If
instead of subtracting Eqs. (1) and (2) we add them, the
terms with inner potential drop and we obtain:
[m0,o(M+)–m0,w(M+)]+[m0,o(X–)–m0,w(X–)]=
–RT ln(a Mo +a Xo – /a Mw + a wX–).
(5)
The right-hand term has dimensions of a constant of
chemical reaction and the left-hand side is a sum of
Gibbs energies of transfer of cation and anion from
water to organic solvent:
DGtr(M+, w®o)+DGtr(X–, w®o)=
–RT ln Kex(M+,X–).
(6)
The dimension of the extraction constant Kex (M+,X–)
of the electrolyte is exactly the same as is used in
extraction chemistry. This is a theoretical proof of the
equivalence of using "electrochemical" or "chemical"
description in treating the transfer of ions. Without further description here, it can be easily shown that the situation is not restricted to the above case of partition of
a single electrolyte, but is a general property of the systems under consideration.
It has been proposed to divide Kex (M+,X–) according
to some "extrathermodynamic" assumption into its
ionic parts, in the form of individual extraction constants Ki, ref.8:
log Kex(M+,X–)=log Ki(M+)+log Ki(X–).
(7)
The quantities pertaining to the Gibbs energy of
transfer of ion from water to polar organic solvent are
interrelated as:
m0,o(i)–m0,w(i)=DGtr(i, w®o)=–RT ln Ki=ziFDwoj 0i. (8)
In our simple example of distribution of only M+X–
between water and polar organic solvent Eq.(8) applies
(because of condition of electroneutrality in each phase,
or [M+]o=[X–]o and [M+]a=[X–]a, where the values in
brackets are activities of respective ions in respective
phases, these may be often equaled to concentrations if
the latter are low):
DM+=DX–=[Kex(M+,X–)]1/2
(9)
Now, the last step in "chemical" treatment, after calculating the compositions of two phases, is to determine the interphase potential Dj. This is simply done
by using the following relation resulting from Eqs. (1)
and (2) (see below for definition of D iidion):
Dwoj =(RT/ziF)(ln D iidion–ln Ki).
(10)
This yields two relations:
Dwoj =(RT/2F)[ln(Kex(M+,X–))0.5–ln Ki(M+)]
(11)
and
Dwoj =(RT/2F)[ln Ki(X–)–ln(Kex(M+,X–))0.5].
(12)
By adding the two and using relation (8), we obtain
the relation used in electrochemistry, viz.(3). In fact in
the "chemical" treatment using the earlier calculated
distribution coefficient of any one ion in the system and
provided we know its individual extraction constant,
the distribution potential can be calculated.
It is important to note here that in extraction chemistry often no distinction is made between the distribution coefficient of a single species and of different
species. For example, the distribution coefficient of
Na+ can be considered as a ratio of its concentrations in
all forms in the organic phase (Na+, NaA, NaL+ etc.) to
Na+ concentration in the aqueous phase. However, in
Eq. (10) the distribution coefficient of single ion must
be used in conjunction with its Ki. Di is to be either
(Na+)o/(Na+)w or (NaL+)o/(NaL+)w. If formulating more
precisely Eq. (10), the distribution coefficient entering
the relation is a distribution coefficient of identical
ionic species and ought to be denoted as D idion.
ANALYTICAL SCIENCES APRIL 1998, VOL. 14
Complex equilibrium in the system
In certain situations one needs to calculate an equilibrium distribution in multicomponent mixtures containing a number of ions and their various associated and
complexed forms. This may be the case in biochemical
systems; in our laboratory we needed to calculate the
distributions of mixtures of ions for developing dicarbollide technology of isolation of radioactive Cs+ and
Sr2+ from nuclear waste by liquid-liquid extraction of
electrolytes using extremely hydrophobic dicarbollide
anion.9 The calculation according to "chemical" model
necessitates knowledge of the respective extraction
constants, but these are usually determined just from
obtained extraction data.
An algorithm for calculation of the equilibria in mixtures uses a recursive iteration on the distribution ratio
of anions in the system (since in extraction studies usually one hydrophobic anion is used in its bare form as a
reagent anion, whereas a number of inorganic cations
either bare or complexed are extracted). In an example
in Fig. 1, the experimental and theoretical curves are
given for partition of Sr2+ in the presence of synergist
Slovafol 909 ( p-nonylphenol nonaethyleneoxide, P)
and dicarbollide anion (A–). The supposed species in
both phases are Sr2+, SrP2+, H+, HP+, L, A–, however,
the calculations could be broadened to simultaneous
presence of further ions like Cs+, Rb+, Na+, Ba2+.10 The
constants used for calculation are given in the legend of
Fig. 1. Although DSr passes through a maximum, the
ionic distribution coefficients are monotonous (see calculated DAidion) and so is the distribution potential, calculated on the basis of individual extraction constants of
anions, supposed to be approximately equal to the
value log K(A–)=8.8 for a given mixture of solvents.
Some particular cases of electrolyte distribution
It has been shown by Diamond11 that in partition of
electrolytes the distribution coefficient of each ion is
influenced by the distribution of other ions. Hence,
some concepts used in extraction of non electrolytes
may not apply here. Several cases which may be
sources of possible pitfalls in interpreting the data will
be given below.
Common ion effect
Such an effect was already treated by Diamond.11 An
electrolyte M+A– (M+ is inorganic cation as Na+ and A–
is a hydrophobic anion like dipicrylaminate–) is extracted in the presence of an excess of inorganic salt M+X–
at aqueous phase and M+A– is fully dissociated in polar
organic phase. At such conditions if we plot the dependence of log DA vs. log(equilibrium concentration of
A–) in the aqueous phase, a straight line of a slope –1/2
results.
According to classical log–log plot analysis, A– associates in the aqueous phase forming dimer A22 –. This is
not true, since applies, as easily derived from respective
constants and condition of electroneutrality in the
organic phase:12
295
Fig. 1 Distribution of microamounts of strontium from 3 M
HNO 3 into 60% vol. nitrobenzene+40% CCl 4 containing
0.065 M H+[(C2B9H8Cl3)2Co]– depending on the concentration
of p-nonylphenol nonaethylene oxide (Slovafol 909).
,
experimental points, full line (DSr) calculated with constants:
K(H +,A –)=39.8, K(HP +,A –)=3.02´10 5, K(Sr 2+,2A –)=5012,
K(SrP2+,2A–)=7.08´1011 and dP=[P]o/[P]a=7.5. All data were
obtained at 23±2˚C.
DA–=([A–]o+[MA]o)/[A–]a
=(Kex(M+,A–)[M+]a/[A–]a)1/2+Kex(M+,A–)[M+]a/K2.
(13)
Here it is possible to substitute organic ionic activities
for concentrations. Then using concentrations also for
M+ in the aqueous phase means that reported constants
are conditional, valid just for the respective ionic
strength of the aqueous phase. The constant K 2 is
defined as K2=[M+]o[A–]o/[MA]o i.e. dissociation constant of MA in the organic phase. According to Eq.
(13) the distribution coefficient of the anion decreases
with the square root of its equilibrium concentration in
the aqueous phase, if full dissociation occurs in the
organic phase. This type of behavior has been
observed by us and others in many instances. 12–14
Characteristic results are given in Fig. 2.
Limiting distribution coefficient
It could be inferred that (in comparison with extraction of non-charged molecules) if extracting an ion present in microamounts in the presence of macroamount
of non-extractable and non-complexing salt, the characteristics of the ion extractibility can be directly
obtained. This is not true since again the competitive
action of the "inert" electrolyte must be considered.
296
ANALYTICAL SCIENCES APRIL 1998, VOL. 14
Fig. 3 Extraction of Cs+[(C2B9H11)2Co]– from demineralized
water to nitrobenzene and nitrobenzene–CCl 4 mixtures.
Nitrobenzene content (vol.%) in the mixture was as follows
(number of curve): 100(1), 60(2), 50(3), 40(4), 30(5), 20(6).
Fig. 2 Distribution of sodium dipicrylaminate between water
and nitrobenzene. Aqueous phase: 3´10 –3 M NaOH.
Experimental curve recalculated from constants given in
ref.14 (log K(Na +,DPA –)=1.00, pK 2=1.32). Distribution
potential based on value of log K(DPA–)=6.9, see ref.8.
The notion of "limiting distribution ratio" was introduced by us for behavior under such conditions.15
It can be easily shown that the behavior of microcomponent (M+) reflects the behavior of "inert" electrolyte
present in excess. The macro equilibrium is given by
distribution of the latter, let us say N+X–. There are two
extraction constants available: K ex (M + ,X – ) and
Kex(N+,X–), but the macro distribution is given only by
the second one, i.e. D X–=[K ex(N +,X –)] 1/2. Now, M +
"copies" the macrodistribution and since it applies
DM+=Kex(M+,X–)/DX–, it results:
DM+,c(M)®0=Kex(M+,X–)/[Kex(N+,X–)]1/2
=K MN[(Kex(N+,X–)]1/2
(14)
where K MN=K ex (M + ,X – )/K ex (N + ,X – ) is extraction
exchange constant for exchange of M+ for N+. Hence,
for extraction of microamounts of Cs+ from 0.02 M
sodium picrate into nitrobenzene, the measured value
of DCs was 3.5, as compared with calculated from Eq.
(14): DCs=4.5. It is important to notice that this value is
higher than the value measured for distribution of pure
cesium picrate in macroamounts, namely DCs=0.11.
Since Cs+ in microamounts does not influence the overall equilibrium, the effect cannot be discerned by electrochemical methods.
Influence of water protons or protonized solvent molecules on distribution
The effect was recognized only recently. We
observed in our laboratory for a long time that distribution of some cesium dicarbollides into nitrobenzene
increases with their concentration. However, for H+
(C2B9H11)2Co–, the distribution ratio is independent of
concentration9 as is expected, since it is just a case of
distribution of single electrolyte and it applies at full
dissociation in the organic phase D M + =D A –
=[K ex(M +,A –)] 1/2. Hence, an increase of D Cs during
extraction of Cs+dicarbollide– with its increasing concentration should mean that the compound is to certain
degree associated in the organic phase. This would be
physically unbearable, since there ought to be lesser
interaction of Cs+ with anion than that of H+, and in all
our previous experiments we have found complete or
nearly complete dissociation in pure nitrobenzene.
The answer lies in competition of aqueous proton (or
protonized nitrobenzene) for the extraction of cesium.
Intuitively this should be neglected considering the
concentration of proton of dissociated water (c H
=10–7 M) is several orders lower than concentration of
cesium used in experiments (10 – 4 – 10–2 M) and extraction exchange constant KHCs is of the order 103.
However, if the distribution coefficient of cesium is
high, the competition of proton takes place since at any
condition DCs/DH=const. Theoretical DCs calculated on
supposing H+, Cs+, B– and OH– particles in the both
phases, initial pH of the aqueous phase equal to 6 (the
curve refers to 100% of nitrobenzene) are denoted as
"theoretical". Value of log K(Cs+,B–) is 6.18, hence log
DCs without taking into account influence of H+ water
ions ought to be equal to 3.09. From Fig. 3 it can be
seen that the calculation used does not explain well the
ANALYTICAL SCIENCES APRIL 1998, VOL. 14
experimental line and a better model is perhaps needed.
Nevertheless, the trend of decrease of D Cs with its
decreasing concentration is clearly apparent. The
presently described effect is a practical answer to a
strange claim that "ionic contributions to Dj do not
vanish, in general, as the ionic concentration tends to
zero" (ref. 16; see Eq. (3), where no concentration term
does really appear). In fact, it seems more plausible to
consider that at high dilution the distribution potential
will be ultimately given by the distribution of dissociated ions of the involved solvents. For water systems,
this would be a shift from a system controlled by salt
behavior to a system controlled by the distribution of
H+ and OH– ions upon high dilution of salt.
Author wishes to thank Prof. S. Kihara for helpful comments
on interdisciplinary extraction/electrochemistry area. Financial
support of Grant Agency of Czech Republic No. 109/97/0368 is
appreciated.
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(Received August 30, 1997)
(Accepted January 24, 1998)