SUPPLEMENTARY MATERIAL to
Temperature dependence of the parameter of the SIT model for activity coefficients of 1:1
electrolytes, submitted to J. Solut. Chem..
A.V. Plyasunov*, E.S. Popova
a
Institute of Experimental Mineralogy, Russian Academy of Sciences, Chernogolovka, Moscow Region
142432, Russia, E-mail address: andrey.plyasunov@gmail.com
1). The expression for the osmotic coefficient of a solution, containing an arbitrary number
of ions and neutral solutes, for the SIT model
1. General definitions
The osmotic coefficient  of a solution is defined by the expression:

nw
ln aw ,
 mi
(1)
i
where aw is the water activity, nw is the number of moles of water, and the sum includes all
solute species. Activities of water and solutes ( ai  mi  i ) are interconnected by the GibbsDuhem equation, which at constant T and P can be written as follows:
nw d ln a w   mi  d ln( mi  i )  0 .
(2)
i
Combining Eqs. (1) and (2) one obtains
d (   mi )   dmi   mi d ln  i ,
i
i
(3)
i
or in the integral form
   mi   mi    mi d ln  i .
i
i
(4)
i
The following expression is valid for the activity coefficient of a cation c in the SIT model:
ln  c  ln 
LR
 ln 
SR,c  a
 ln 
SR,c  n

z c2 A I 0.5
1  1.5I 0.5
    (c, a)  ma    (c, n)  mn ,
a
n
(5)
where ln  LR is the long-range (electrostatic) contribution, ln  SR,ca is the term for the shortrange contributions due to interactions, involving a cation c and anions; ln  SR,cn is the term for
the short-range contributions due to interactions, involving a cation c and neutral species; A is
the limiting Debye-Hückel slope for the natural logarithm of the activity coefficient; 1.5 is a
constant;   (c, a ) is the cation-anion binary interaction coefficient;   (c, n) is the cation-neutral
species binary interaction coefficient. The corresponding equation for an anion a is obtained by
interchanging a for c.
The equation for the activity coefficient of a neutral species n in a mixture of ions and
neutral species is as follows
ln  n  ln  SR, n  c  ln  SR, n  a  ln  SR, n  n  ln  SR, n  n 
,
   (c, n)  m     (a, n)  m     (n, n)  m
c
c
n
a
a
(6)
  (n, n)  mn
n
where ln  SR, n  c , ln  SR, n  a , ln  SR, n  n , ln  SR, n  n are the terms for the short-range contributions
due to interactions between the neutral species n and cations, anions, non-identical (n) and
identical (n) neutral species, respectively.
2. The long-range (electrostatic) contribution to the osmotic coefficient
In order to obtain the expression for the electrostatic contributions to the osmotic
coefficient it is necessary to evaluate the integral
  m d (ln 
j
LR
) , where the index j covers all
j
ions, both cations and anions. The ionic strength I is defined as I  1 / 2 m j z 2j . It is useful to
j
introduce the quantity t  1  1.5I 1 / 2 , from where I 
t 1
(t  1) 2
and I 1 / 2 
. Now the integral
2
1.5
1.5
for the long-range contribution to the osmotic coefficients can be evaluated:

A z 2j I 0.5 

0.5 
1

1
.
5
I


 LR   m j    m j d (ln  LR )    m j d  
j
j
j
 I 0.5 
 I 0. 5 
(t  1) 2  t  1 
2




m
(

A
z
)
d


2
A
Id


2
A
d

 
 
 j j  j  1  1.5I 0.5 
 1  1.5I 0.5 
1.52  1.5t 


(7)
2 A 
(t  1) 2
1
 3
dt


t  2 ln t  
2
3 
1.5
t
1.5 
t
2 A
3. The short-range contributions to the osmotic coefficient
Let’s consider the case of a cation c in an aqueous mixture of cations, anions, and neutral
species. In the framework of the SIT model the following binary interactions are to be
considered: the cation-anion and the cation-neutral species ones. In order to obtain expressions
for the short-range contributions to the osmotic coefficient due to such interactions, it is
necessary to evaluate the integrals
the SIT model we have:
 m d (ln 
 m d (ln 
SR ,c a
c
SR,c  a
c
) and
 m d (ln 
SR ,c n
c
) . In the framework of


)   mc d     (c, a)ma   mc    (c, a) ma . In general
a
 a

case the summation should cover all cations
  m d (ln 
SR,c  a
c
)     (c, a)  mc ma . The
c
c
a
situation with the cation-neutral species interactions is similar and results in the following double
summation:   mc d (ln  SR,cn )     (c, n)  mc mn .
c
c
n
For anions their interactions with cations are already accounted for. The expression for
interactions between anions and neutral species is
  m d (ln 
a
a
SR,a  n
)     (a, n)  ma mn .
a
n
For neutral species their interactions with cations and anions are already accounted for.
For interactions between neutral species it is necessary to distinguish two cases: the interaction
between non-identical species n and n (say, between aqueous CO2 and H2S), and the interaction
between identical neutral species.
The interaction between non-identical neutral species n and n is analogous to the case of
the interactions between cations and anions (or between neutral species with cations or anions)
and results in the following term in the expression for the osmotic coefficient of a mixture -
   (n, n)  m m
n
n n n
n
.
For the case of a single neutral species n in its binary mixture with water (only n-n binary
interactions), the following contribution to the osmotic coefficient of a solution is obtained:
 m d (ln 
SR, n  n
n
1
)   mn d   (n, n)  mn     (n, n)  mn2 . In a general case of a solution with
2
various kinds of neutral species n there should be a sum over all kinds of similar neutral species,
i.e.
  m d (ln 
SR,nn
n
n

 1
)    mn d     (n, n)  mn      (n, n)  mn2 .
n
 n
 2 n
4. The final expression for the osmotic coefficient of a solution, containing an arbitrary
number of ions and neutral solutes, for the SIT model
By adding all terms in Eq. (4) one obtains the final expression for the osmotic coefficient
of a solution:


2 A t  2 ln t  t 1
 1   3

1.5
 mi
i
1

 mi
i

   (c, a )mc ma     (c, n)mc mn     (a, n)ma mn 
c
n
a
n
c a

1
  (n, n)  mn mn     (n, n)  mn2 

2 n
n n n

(8)
2). The expression for the excess Gibbs energy of a system, containing an arbitrary number
of ions and neutral solutes, for the SIT model
1. General definitions
The deduction follows the presentation given by Prausnitz et al. [4]. The expression for
Gibbs energy, G, of a system, containing 1 kg of water and a set of solutes, is given by
G  nw  w   mi i  nw  (  w  RT ln aw )   mi  ( i  RT ln ai ) ,
i
(9)
i
where  w and  i stand for the standard chemical potentials of water and a solute i,
respectively.
Taking into account the definition of the osmotic coefficient, see Eq. (1), one obtains
G  nw w  RT   mi   mi    RT  mi ln mi  RT  mi ln  i .
i
i
i
(10)
i
For the ideal solution  i  1 and   1 and the Gibbs energy of an ideal solution, G id , is defined
as
G id  nw  w  RT  mi   mi i  RT  mi ln mi .
i
i
(11)
i
The expression for the excess Gibbs energy follows from its definition:
G E G  G id

 (1   ) mi   mi ln  i .
RT
RT
i
i
(12)
2. The expression for GE for the SIT model
From Eq. (8) one obtains
(1   ) mi 
i


2 A
t  2 ln t  t 1     (c, a)mc ma     (c, n)mc mn
3
1.5
c
a
c
n
(13)
    (a, n)ma mn     (n, n)  mn mn 
a
n
n n n
1
  (n, n)  mn2

2 n
In order to complete the expression for G E in the framework of the SIT model, it is
necessary to evaluate the term
 m ln 
i
i
which are indicated explicitly below:
i
of Eq. (12) for various types of binary interactions,
 m ln    m ln    m
i
i
c
i
c
a
c
 m ln 
a
LR
a

a
 ln 
SR, c  a
a
n
 ln 

ln  a   mn ln  n   mc ln  cLR  ln  cSR,c  a  ln  cLR ,c  n 
LR , a  n
a
c
   m ln 
a
n
SR, c  n
n
 ln 
SR, a  n
n
 ln 
SR, n  n 
n
 ln 
SR, n  n
n

. (14).
n
m
It is convenient to separate the term
c
ln  cLR   ma ln  aLR  m j ln  LR
j , which can
c
a
j
be expressed as a function of ionic strength only. Remember that the index j cover only ions,
both cations and anions, that by definition t  1  1.5I 1 / 2 , and that the following identity holds
m z
j
2
j
 2 I . Now one obtains
j
 mc ln  cLR   ma ln  aLR  m j ln  LRj  
c
a
j
2 A I 1.5
1  1.5I 0.5

2 A (t  1)3
.
1.53
t
(15)
By accounting for various short-range interactions as accepted in the SIT model, one
obtains from Eq. (14):
 mi ln  i  
i
2 A (t  1) 3
 2   (c, a)  mc ma  2   (c, n)  mc mn 
1.53
t
c
a
c
n
(16)
2   (a, n)  ma mn  2   (n, n)  mn mn     (n, n)  mn2
a
n n n
n
n
By combining all terms in Eqs. (13) and (16), the following expression for the excess Gibbs
energy in the framework of the SIT model is obtained:


2A
GE
 (1   ) mi   mi ln  i   3 t 2  4t  2 ln t  3     (c, a)  mc ma 
RT
1.5
i
i
c
a
(17)
   (c, n)  m m     (a, n)  m m     (n, n)  m m
c
c
n
n
a
a
n
n
n n n
n
n

1
   (n, n)  mn2
2 n