E L S E V I E R Fluid Phase Equilibria 106 (1995) 17-25

m E

On the calculation of phase equilibria in aqueous two-phase systems containing ionic solutes

Chris toph Grol3mann, Gerd Maurer *

Lehrstuhl fiir Technische Thermodynamik, Universitiit Kaiserslautern, D-67653 Kaiserslautern, Germany

Received 26 March 1994; accepted in final form 20 October 1994

Abstract

The equilibrium conditions governing isothermal, isobaric phase equilibrium in aqueous two-phase systems containing ionic solutes are derived from basic thermodynamics assuming that each of the coexisting phases is electrically neutral. For nonionic species the result reduces to the common equilibrium condition, i.e. the equal activity criterion. For ionic species the equal activity criterion does not hold, and has to be modified. However, like the activity, the modification can be calculated from an expression for the Gibbs energy of the uncharged mixture. Usually the concept of electrical potential difference is used to express the phase equilibrium for ionic components. It is proved that this concept only provides an alternative way of expressing phase equilibria. However, the new derivation gives an explanation for this electrical potential difference and allows its calculation.

Keywords: Theory; Chemical equilibria; Equations of state

1. Introduction

Several authors have discussed aqueous two-phase systems, for example those formed by dissolving a hydrophylic polymer and a strong electrolyte in water, for the recovery of biomolecules from a fermentation broth (e.g. Albertsson, 1958, 1977, 1978, 1979, 1986; Kula, 1979, 1985; Albertsson et al., 1982; Kula et al., 1982; King et al., 1988). When the aqueous phases contain ionic species, an electrical potential difference may exist between the coexisting phases, although the mixture itself is electrically neutral. As discussed in the literature (e.g. Albertsson, 1986; Brooks et al., 1984; Haynes et al., 1991; Johannson, 1974; King et al., 1988), it is often assumed that the potential difference is caused by small differences in the electric

* Corresponding author.

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18 C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25

charge of both phases. These differences are too small to be detected by ordinary chemical analytical methods (e.g. Adamson, 1990; Guggenheim, 1967). As shown by King et al. (1988), a very small potential difference between two phases can have a large influence on the partitioning of dissolved biomolecules such as proteins, because these substances may carry an appreciable net charge, depending on the pH of the solution.

This work presents a derivation of the appropriate phase equilibrium equations.

2. Common treatment of phase equilibrium in aqueous two-phase systems

Consider an aqueous two-phase system at constant temperature T and pressure p. For any species partitioning between the phases, the equilibrium equation is

/~e ~' = p7 ~" i = 1 ..... N (1)

where ' and " designate the phases, N is the number of species and pe I is the electrochemical potential of species i:

it el = I.~i -k- z i F ~ (2)

The "normal" chemical potential, p (without superscript "el") is given by 1

#i = (3) T,p ,n j ~ i

where G is the "normal" Gibbs energy and ni is the number of moles of i; z~ is the number of elementary charges on the ionic species i (for a cation, z~ > 0; for an anion, zi < 0 ) ; F is Faraday 's constant and • is the electric potential.

It is convenient to rewrite Eq. (1) by introducing the activity a~ to describe the deviation of the chemical potential of species i f rom its reference state:

#~ = #~ef + RTlna~ (4)

It is also convenient to use the same reference potential for both phases:

~fef, = ~ef,, (5)

and to express the solute activity as the product of molality m and activity coefficient y:

a i : m i ] ) i (6)

Substituting Eqs. (4) - (6 ) and (2) into Eq. (1) gives the well-known phase equilibrium equation2:

a / ' / a / = KiT/'/Ti' = exp( - z i F A ~ / R T ) (7)

Applying Eq. (3) to an ionic species i would require a change in the electric charge of the solution. However, because only neutral components - - from which ionic species may dissociate - - are mixed, it is common practice to neglect this effect in applying Eq. (3) to ionic species (see, for example, Robinson and Stokes, 1954). 2 Eq. (7) indicates that, in general, even if the same reference state is used for all phases, the equilibrium equation is not a/ = a/'.

c Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25 19

where Aq~ = ( b " - 4 ' is the difference in electrical potential between the coexisting phases, and the distribution coefficient Ki is defined by Ki = m / ' / m / .

To indicate the influence of A~ on K,, consider a biomolecular anion with zi = - 10, which partitions at 300 K between two phases with a potential difference A(I)= 2 mV. Neglecting all other effects which may influence the partitioning of species i (i.e. assuming that 7/' -- 7g') gives K~ =2.2.

There is no general, reliable experimental method to measure A~. However, as discussed by Haynes and co-workers (1991, 1993), it is possible to calculate A(I) provided that an expression for the Gibbs energy of the solution is available.

3. Derivat ion o f equilibrium condit ions - - results

Because it is not possible to measure A(I) independently, Eq. (7) is not useful for equilibrium calculations. It is therefore desirable to obtain an equilibrium equation which does not require A(1), but instead follows as a direct result of the second law of thermodynamics and the assumption that each of the coexisting phases is electroneutral.

Fig. 1 shows a macroscopic two-phase system kept under a constant pressure, immersed in a constant-temperature bath. Assuming that the only way to transfer work to or from the system is by a volume change, and combining the first and the second laws of thermodynamics, we have

T d S - p d V > > . d U + dUel (8)

where S, V and U represent the entropy, volume and internal energy in a zero-electrical potential, while U ej represents the contribution to the total internal energy from an electrical potential.

p -- COn.el,.

water , po lymer , ions

I phase boundary

water , po lymer , ions

T = c o n s t .

- s y s t e m boundary

Fig. 1. Scheme of the heterogeneous thermodynamic system consisting of two homogeneous phases at constant temperature and pressure.

20 C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25

In a heterogeneous system consisting of two homogeneous phases ' and ", the following conservation laws hold:

dS = dS' + dS"

d V = d V ' + d V "

d U = dU' + dU"

d U el = d U el' + d U el" (9)

Substituting Eq. (9) into Eq. (8) gives

d U' - T 'dS ' + p 'd V' + d U ev + d U" - T"dS" + p"d V" + d U e~" <<. 0 ( 1 O)

The Gibbs energy G is defined in the usual way:

G = U + p V - TS (11)

Because temperature and pressure are constant, Eqs. (10) and (11) give

dG' + dG" + d U el' + d U ep' ~< 0 (12)

Defining

G el = G + U el (13)

yields the expected result

d(G e~' + G e~") <<. 0 (14)

At equilibrium (G"ev" + G ''el''') reaches a minimum. It is now convenient to replace U e~ by

U el= f ~ ( ~ n i z i ) (15)

Electrical neutrality for each of the coexisting phases requires that

~(n/z i ) = 0 (16a)

and

~(ni"zi) = 0 (16b)

Substitution of Eqs. (15), (16a) and (16b) into Eq. (14) gives

d(G ev + G el') = d(G' + G") ~ 0 (14a)

Eq. (14a) presents a key result: the condition of equilibrium can be written without taking into account any electrical contributions. Thus it can be formulated in the customary way as follows.

At constant temperature and pressure, equilibrium is reached when

(G' + G") is a minimum (17)

We relate the Gibbs energy G' to the chemical potentials # / by

G' = E ( n i ' ]2i') (17a)

C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25 21

Similarly

G" = ~(n/'/~/') (17b)

where the sums are over all species. For the entire system, the total number of moles of any species i is given by n~:

t ! t! n~=ni +n~ =constant (18)

For each phase, we use the isothermal, isobaric Gibbs-Duhem equation:

Y,(n/du/) = 0 (19a)

~(n/'d/xi") = 0 (19b)

Combining the Gibbs-Duhem equation (Eqs. 19a and 19b), and the mass balance (Eq. 18) and equilibrium conditions at constant temperature and pressure (Eqs. 17, 17a and 17b), gives

N

( / t / - /~ / ' ) d n / = 0 (20) i = 1

The fluid mixture contains N* neutral and ( N - N*) ionic species. Species designated by i ) N *

represent ionic species. Let us arbitrarily select one ionic species k and express its change in mole numbers in phase ' by rewriting Eq. (16a):

N

E ( ' ) Z k n k t ~ _ Zi l , l i

i = 1 , i ~ k

or in differential form: N

d n k ' = - ~ ( z i d n i ' ) / z k (16c) i = l , i C k

The material balance (Eq. 18) requires that

dna" = - d n k ' (18a)

We combine Eqs. (16c) and (18a) with Eq. (20): N

~] [(#/' - Z~/Zk#k') -- (#i" -- z~/zktx~")]dn/ = 0 (20a) i = 1 , i ~ - k

Recalling that for i = 1 ..... N, but i 4 = k, all dn/ are independent variables, we arrive at the equilibrium condition:

# / - # / ' = ( z i / zk ) (#~ ' - #~") i = 1,...,N but i =/= k (21)

Using the same reference state for species i in both phases, Eq. (21) becomes

ln (a i" /a i ' ) = ( z i / z~ ) ln (ak" /a~ ' ) i = 1 .... , N but i ~ k (21a)

Eqs. (21) and (21a) indicate that, contrary to phase equilibrium relations for nonionic systems, the condition of phase equilibrium in ionic systems is not written for all N species, but only for ( N - 1) species; the relation for the arbitrarily chosen ionic species k is omitted, because it is a

22 C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25

tautology. Only for nonionic species do Eqs. (21) and (21a) reduce to the common equilibrium condition:

pi ' = # / ' i = 1 ..... N * (21b)

o r

ai' = a i " i = 1 .... , N * (21c)

The important results appear in two parts: (i) for nonionic species, the equation of equilibrium is Eq. (21b) or its equivalent, Eq. (21c); (ii) for ionic species, the equation of equilibrium is Eq. (21) or its equivalent, Eq. (21a). We can now show the equivalence of Eqs. (7) and (21a). The ( N - N * - 1 ) equilibrium

conditions for ionic species may be rearranged by introducing the concept of an electrical potential difference between both phases A~ = ~ " - - ~ ' . We define A~ such that

IZk ' (T ,p ,n / ) -- p k " ( T , p , n j ' ) = A ~ F z k (22)

o r

l n (ak ' / ak" ) = A ~ O F z k / R T (22a)

Eqs. (22) and (22a) define a potential difference, only providing a different way of writing the right-hand sides of Eqs. (21) and (21a).

To indicate further how the A~ concept provides only an alternative way of expressing the criteria for phase equilibrium, we can rearrange the N - N * - 1 equilibrium Eq. (21) for ionic species to give the same number of equations for neutral salts. We consider a salt of type M~.mXv× , where Ym and Vx are stoichiometric coefficients:

PMX t = ]~MX" (21 d)

where

PMX = VmPM + VxpX (23)

In forming these neutral electrolytes, the only restriction is that each ionic species partitioning between both phases must appear in at least one neutral salt.

We consider a two-phase system that contains ionic species which partition between both phases at constant temperature and pressure. For such a system, phase equilibrium can be expressed by the common equilibrium condition for nonionic species i:

#g' =/ti" i = 1 ..... N* (21b)

and N - N* -- 1 equilibrium conditions for n e u t r a l electrolytes MvmXvx:

#MX' = ~MX" (21 d)

E x a m p l e

To illustrate, consider an aqueous two-phase system at temperature T and pressure p containing ntw moles of water, np moles of phase-forming polymer, n] moles of fully dissociated electrolyte MvrnXvx and n~ moles of fully dissociated electrolyte Cv~Ava. Eq. (21a) holds for water,

C. Groflmann, G. Maurer/Fluid Phase Equilibr& 106 (1995) 17 25 23

polymer and three out of the four ionic species present. Arbitrarily k is chosen to represent X. The five resulting equations are

ln(ai"/ai') = (Zi/Zk)ln(ak"/ak') i = w,p,M,C,A

In addition there are six mass balances:

water: ntw = nw' + nw" t t tt polymer: Hp = n p + r/p

ionic species X: v×n] = nx' + nx"

ionic species M: Vmrt] =rtM' -k- rtM"

ionic species C: vont2 = nc' + nc"

ionic species A: van[= rlA' + rlA"

and finally there is the condition of electroneutrality phase 1:

Zxnx' + zMnM' +zcnc ' + ZAnA' : 0

Because the mixture was formed by electrically neutral water, polymer and electrically neutral salts, the electroneutrality of phase " is automatically fulfilled. In this example we have 12 independent equations: five equations for phase equilibrium, six equations for material balances and one equation of charge conservation (electroneutrality). These 12 independent equations can be solved provided an appropriate expression is available for the Gibbs energy of the mixture which is required to calculate activities. These 12 independent equations give the 12 unknown mole numbers (for all six species - - water, polymer, cations M and C, and anions X and A - - in both phases). It is not necessary to know the electrical potential difference AO. However, AO can be calculated from the expression for the Gibbs energy through Eq. (22) or (22a).

The procedure derived here is generally valid as long as all coexisting phases are electrically neutral.

Acknowledgement

The authors thank the Deutsche Forschungsgesellschaft, Bonn-Bad Godesberg and the Volkswagenstiftung, Hannover for financial support, and J.M. Prausnitz (University of Califor- nia, Berkeley) for helpful discussions.

List of symbols

A a

C F G

anion activity cation Faraday constant Gibbs energy

24 C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25

i K/ k M

m

MX N ?/

N*

P R

S T U V X z

integer number distribution coefficient for species i integer number cation molality salt consisting of cations M and anions X number of species mole number number of neutral species pressure universal gas constant entropy absolute temperature internal energy volume anion charge number

Greek letters

])i A #i tt~ 1

(I)

activity coefficient of species i difference chemical potential of species i electrochemical potential of species i stoichiometric coefficient electrical potential

Subscripts

A , a

C,c i k M,m MX

P W

X,x

anion cation component component cation M salt consisting of cations M and anions X polymer water anion X

Superscripts

el ' and " ref t

electrical designating phases reference state total

C. Groflmann, G. Maurer/Fluid Phase Equilibria 106 (1995) 17-25 25

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