THEOREMS CONCERNING T H E ACTIVlTY COEFFICIENTS AND OSMOTIC COEFFICIENTS OF STRONG AND WEAK ELECTROLYTES PIERRE VAN RYSSELBERGHE Department of Chemistry, Stanford University, California
Received M a y $2, 1994 I. INTRODUCTION
From a strictly thermodynamic point of view we may study the properties of solutions of electrolytes by considering Dhe solute as undissociated, completely dissociated, or incompletely dissociated. We thus have three corresponding “descriptions” of the same solution and three sets of activity coefficients. Thermodynamics imposes definite mathematical relations between these various sets. The straightforward derivation of these relations is the purpose of the present paper. We shall make use of the thermodynamic method of Gibbs, as expounded for instance in Guggenheim’s (2) recent book or in De Donder’s “Affinity” (1). A comparison will be made with the method of Lewis and Randall (4). The results thus obtained are general. They lead to simple formulas for the particular cases of extremely low concentrations and of very weak electrolytes. 11. LIST OF SYMBOLS AND FUNDAMENTAL RELATIONS
T = absolute temperature: P = exterior pressure Description No. 1: the solute is assumed to be undissociated. py) =
chemical potential)
IVY) = mole fraction
= activity coefficient 911) = osmotic coefficient/
f‘,”
of the solvent.
1.
= chemical potential
N F ) = mole fraction of the solute. f‘,“ = activity coefficient m = molality of the solution M
=
number of moles corresponding to 1000 g. of solvent (for water M = 55.51). 403
'404
PIERRE VAN RYSSELBERGHE
Between these various quantities the following relations hold:
~(11)= p y ) ' ( T ,P ) + RT In NY)f\') P1( l ) = P1 ( l ) ' ( T ,P ) + gy'RT In N(11)
,(1.2)
+ RT In iVp)fil)
(1.3)
= p(21)'(T,P )
(1.1)
The PO'S are for a given solvent functions of temperature and pressure only. We also have:
Description No. 6: the solute is regarded as completely dissociated, each molecule breaking up into v+ positive ions and v - negative ions, the sum Y+ v- being represented by v.
+
= chemical potential = mole fraction
Pl( 2 )
N\2) j(2) 9'1") &),
of the solvent.
= activity coefficient = osmotic coefficient p?'
IVY),"2' fy', j?'
1
= chemical potentials of the positive and negative ions. = mole fractions = activity coefficients
We have : p ( : ) = p (12 ) O
( T ,P )
+ gi2)RTIn N i 2 ) = pY)'(T, P ) + RT In N $ ! ) f y ' = p ? " ( T , P ) + RT In N?)fL2)
p:"' = pi2)'(T,P )
p?'
+ RT In N',2)fY)
(2.1) (2.2) (2.3) (2.4)
Description No. 3: the solute is regarded as incompletely dissociated into single ions, the molality of the undissociated solute being (1 - a)m,
405
COEFFICIENTS OF STRONG AND WEAK ELECTROLYTES
that of the positive ion av+m,and that of the negative ion av-m. A more detailed treatment could take into account the possible presence of intermediate and complex ions.
1
Pl( 3 )
= chemical potential
N (13 ) f(3)
= mole fraction = activity coefficient
py), p(a)
= osmotic coefficient = chemical potentials
NY', N(')
fy),fp' PU (3)
N (213 ) f (u 3)
I
= mole fractions = activity coefficients
= chemical potential = mole fraction = activity coefficient
of the solvent.
of the positive and negative ions.
i
of the undissociated solute.
We have :
+ RT In N',3)f(13) ~ ( 1 3 )= p i 3 ) ' ( T , P ) + g':)RT In N':)
(3.2)
P+( 3 ) = p+( 3 ) O ( T , P )
(3.3)
py' =
(3.1)
+ RT In N Y ' f y )
+ RT In " 3 ' f i 3 ) = p*',3)O(T,P ) + RT in N ' , 3 ) f y )
p(a' =
pel',"'
P\~)O(T,P )
p(a)'(T, P )
(3.4) (3.5)
Since the PO'S are independent of composition, it is evident that they are related as follows: P(ll)o
=
(2)O
=
P(2)0
=
P(zl)O
=
P+
(2)O
P1
(310
P+
=
J;)O
(4.1)
.
(4.2)
(310
(4.3)
(3)O
(4.4)
P-
PU
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PIERRE VAN RYSSELBERGHE
III. FREE ENERGY OF THE SOLUTION I N THE THREE “DESCRIPTIONS”
Let us apply the well-known relation between the free energy, F , the chemical potentials, pi, and the numbers of moles, ni, of the various constituents:
F =
i
(5)
pini
The free energy of an amount of solution consisting of 1000 g. of solvent and m moles of electrolyte is given in description No. 1 by
F = Mp?) in description No. 2 by F = MpY)
+ mpy)
+ m(v,pL(,) +
v-11-( 2 )
(6.1) )
(6.2)
in description No. 3 by
F
= MpY)
+ m[a(v+p$!)+ v-p!!)) + (1 - 0 0 p ~ ) I
(6.3)
These three expressions of F for the same solution should of course be identical. Moreover, the contribution of the solvent is the same in the three cases, as well as that of the solute. Equations 6.1, 6.2, and 6.3 give then: pL(ll) = Pl( 2 ) = F (l3 ) (7.0 and p(21) = v+p!+!)
+ v-p(2)
+ v-p!?)) + (1 - a ) p y ’
= ~r(v+py)
(7.2)
IV. ACTIVITY AND OSMOTIC COEFFICIENTS OF THE SOLVENT
A . Relations between the three activity coeficients of the solvent From equations 7.1, 4.1, 1.1, 2.1, and 3.1 we deduce
N\l)fY) = ~
( 2 (’ 2 ) 1 $1
- N1( 3 ) fl( 3 )
(8)
and, thanks to equations 1.4, 2.4, and 3.6, f(1)
-=-m M
+
f(2)
vm
f ( 13 )
+M
- [ 1 + ( v - 1)alm
+M
(9)
B. Relations between the three osmotic coeflcients of the solvent Thanks to equations 1.2, 2.2, and 3.2, equation 8 also gives:
gy)in
~ (1 1 = )
g1( 2 ) in ivy) = g(;) in
~ (1 3 )
(10.1)
(10.2)
COEFFICIENTS O F STRONG AND WEAK ELECTROLYTES
+
C. Special case of extremely dilute solutions ( v - 1)orIm are then negligible compared to M .
m, vm, [ l 9 and 10.2 become
j'y =
- fl( 3 ) fl( 2 ) -
g'l" = v g y = [ l
407
Equations
(11)
+ (v - l)(Y]g(13)
(12)
V. RELATIONS BETWEEN THE SETS O F ACTIVITY COEFFICIENTS O F THE SOLUTE
A . General relations I n description No. 3 the ions are regarded as being in equilibrium with the undissociated part of the solute. We thus have the equilibrium condition : p y = v + p y + v-p- ( 3 ) (13) Equation 7.2 can then be written: p(Z1) = v+p+( 2 )
+ v-pL.2)
= v+p+( 3 )
+ v-p?
=p p
(14)
From this relation and from equations 1.3, 2.3, 2.4, 3.3, 3.4, and 3.5, modified on account of equations 4.1, 4.2, 4.3, and 4.4, we deduce:
Introducing the equilibrium constant K ( T , P ) defined by
It has been pointed out by Guggenheinl (3) that ionic activity coefficients are thermodynamically undetermined. Thermodynamically determined are only combinations of ionic activity coefficients of the form
IT fti
(18.1)
i
in which the quantities Xi are related to the algebraic valences zi of the ions by the relation: XiZi = 0 (18.2) i
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PIERRE VAN RYSSELBERGHE
In the case of a single electrolyte an expression of the form of equation 18.1 is the mean activity coefficient of the ions f* defined by
(f*>
=
(f+>”(f-) ”-
( 19)
It has thermodynamic significance because v+z+
+ v-z,
=
0
(20)
It is also convenient to define mean mole fractions N *
(N*)
=
( N + )”+(N-)
(21)
y-
and mean molalities (m*)
= (m+)’+(m-) ”- = vy’v.?.
my
(22)
These mean quantities (19, 21, and 22) may be used both in description 2 and in description 3. Equation 17 then becomes
KNil’f!jl’ = [ N y ’ f p ) ] ”= [ N( ,3 ) f*( 3 ’ 1” = K N (,3 ) fu( 3 )
(23)
or, thanks to equations 1.5, 1.6, 2.6,2.7, 3.7,3.8, and 3.9 ,
Equation 23 could be decomposed into the two symmetrical relations: ,(21’f!jlI
= NL3’fy
N y ’ f p = N g )fy)
(25.1) (25.2)
The corresponding relations deduced from equation 24 are less symmetrical: mf!jl)
m‘,3’
--
m + M - [l
- - f!“’ vm
+M
- [1
fy
+ ( v - l)a]m + M f:“’
+ ( v - l)a]m + M
(26.1) (26.2)
We have (27.1) (27.2)
COEFFICIENTS O F STRONG AND WEAK ELECTROLYTES
409
Equations 26.1 and 26.2 become:
(1 - c
fh"
-=
m +M
[l
- f!"' -
vm
u)p
+ ( v - l)a]m+ M a
f!"'
+ &I - [1 + (v - ~)culm+ M
(28.1) (28.2)
B. Special case of extremely dilute solutions When m, vm, [l (v - l)cu]mare negligible compared to M , equations
+
12, 23.1, and 23.2 become:
f p = ( 1 - cu)f?' f!"' = a f y
(30.1) (30.2)
These equations are approximate but general, since they do not involve non-thermodynamic assumptions. VI. THE LIMITING VALUES OF THE ACTIVITY COEFFICIENTS AND OSMOTIC COEFFICIENTS AT INFINITE DILUTION
The coefficients
9Y,
fY,
d3) f P , f Y ' , A?
9Y,
are all thermodynamically indeterminate when m tends towards 0. It is necessary, in order to determine them all, to make non-thermodynamic assumptions regarding: (a) the limiting values for m = 0 in one of the descriptions, and (b) the behavior of the function LY = a(m)as m approaches 0. We shall assume that the solution becomes ideal in description No. 2 when m tends towards 0. We thus have: [lim f\2)]m-o = 1
(31.1)
[lim gy']m-o = 1
(31.2)
[Iim jy)Jm-o = 1
(31.3)
Moreover we shall assume that
[limcul,=, = 1
(31.4)
410
PIERRE VAN RYSSELBERGHE
regardless of the actual form of the function a = a(m). We then deduce from equation 11 (the formulas obtained above for small molalities become more and more accurate as m tends towards 0) : limfp) = limf(12) = l i m f y ) = 1
(32)
From equation 12 Iim g y ) = v ~ i m g y = ) v
(33.1)
lim g y ) = lim 9‘1“)= 1
(33.2)
From equations 29 and 22: (34.1) or limf(z1) = 0
(34.2)
From equation 30.1:
fil
l i m j v ’ = lim
___ )
(35)
l-CY Since lim f(z‘) = 0 and lim (1 equation 30.2
-
a) = 0, f f l
is indeterminate. From
. f!“’ l i m j g ) = Iim - = 1 a
(36)
VII. PARTICULAR CASE OF VERY WEAK ELECTROLYTES
Instead of making use of the assumptions 31.1, 31.2, and 31.3 of the foregoing section, it will be more convenient, in the case of very weak electrolytes, to assume that: [lim f‘,”)jmPD
=1
(37.1)
[Iim gy)],=o = 1
(37.2)
[lim f!3)Im-0 = 1
(37.3)
[lim f
(37.4)
~ ) ] m ~= O
1
[lim CY],,^ = 1
(37.5)
If the electrolyte is so weak that f f ) and jg’ approach unity much more rapidly than a,equations 30.1 and 30.2 become: f‘,” = 1 -
f!“’ = (y
.
CY
(38.1)
(38.2)
41 1
COEFFICIENTS O F STRONG AND WEAK ELECTROLYTES
These relations, approximately valid for small molalities, are thus partly non-thermodynamic. For electrolytes obeying exactly Ostwald’s dilution law: av mv-l
1--ar
= K
(39)
we may replace our assumptions (37) by: f‘,“’= 1
(40.1)
9‘1“) = 1
(40.2)
(fy)~ =j p
(40.3)
relations holding in the same range of extreme dilutions as the dilution law (39). Equation 38.2 is a justification for referring, as is often done, to as to a “thermodynamic degree of dissociation.” The considerations developed in the present paper should clearly indicate the limitations of this concept (5) *
fy’
VIII. COMPARISON WITH THE METHOD OF LEWIS AND RANDALL
The activity a2 and the activity coefficient y2 of a solute species are defined by Lewis and Randall (reference 4, p. 255) as follows:
F
- FO
=
RT In a2 = RT In
(my,)
(41)
F and FO are molal free energies. The superscript 0 refers to a “standard state.” For given temperature and pressure Fo is independent of composition. It is moreover assumed that (lim
~ 2 ) ~ == 0
1
(42)
I n the notation of the present paper equation 41 would read: (1)
p2
=
(110’
p2
(T,
+ RT In (mr‘z‘))
(43)
From equations 1.3 and 1.5 we deduce: (1)
Pz
-P2(l)O +
Comparing equations 43 and 44 we get:
[ ~
(44)
(45)
and (46)
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PIERRE VAN RYSSELBERGHE
or, in terms of the mole fraction A’?) of the solvent, = ( 1 ) . N(1) f2 1
(47)
The activity coefficients y of Lewis and Randall coincide with the “rational” activity coefficients f only a t very low concentrations. From equation 47 we see that, even for ideal solutions, yz will be appreciably different from unity a t high concentrations. I n the same way we have in description No. 2:
and in description No. 3: y p
f!“’
z
1
+ [1+ (v - 1)aI M f!?
y y =
1
m
+ 11 + (v - 1b.1-M
m
(49.1)
(49.2)
Replacing the various f’s by the various y’s in equation 24 we obtain, thanks to equations 46, 48, 49.1, and 49.2,
a remarkably symmetrical relation. In terms of activities we then have
or
As shown before (equation IS), K is the true “dissociation constant” of the electrolyte. It is hence not allowed by thermodynamics to assume arbitrarily, as do Lewis and Randall a t the starting point of their treatment of strong electrolytes (reference 4,p. 326)’ that
COEFFICIENTS O F STRONG AND WEAK ELECTROLYTES
413
Fortunately this relation is never used, the significant relations being : ( a y ) ) ~(a?)) + Y= (ap))Y
(54)
(@)
(55)
and p+
@))
Y-
=
( y y ) ) Y
which are in no way expressions of the law of mass action. Equation 52 is the mathematical formulation of a LLcomparison”between descriptions 1 and 2. As long as no use is made of the third description, assumption 53, although unnecessary, is perhaps permissible but, on the other hand, when all the descriptions are used (cf. Randall and Allen (5)) equations 24, 50, or 52 must be obeyed and one should not write separately:
IX. R ~ S U M I ~ .LIST OF THEOREMS
A certain number of the results obtained in the present paper may be considered as theorems. They are :
A . General theorems valid for all types of electrolytes and all concentrations 1. The activity coefficients of the solvent corresponding to the three descriptions of the solution (no dissociation, complete dissociation, incomplete dissociation) obey the relation:
2. The osmotic coefficients obey the relation:
3. The activity coefficient of the solute in the first description (no dissociation) and that of the undissociated solute in the third description (incomplete dissociation) are connected by the relation : (1) (1 - CY) f i 3 ) - f2 [l + (v - 1)a] m + M m +M 4. The average activity coefficients of the ions in the second and third descriptions are connected by the relation (see equations 19, 22, and 23) :
f:“’ vm
+M
= Ly
f!“’
[l + (v - l h ] m
+M
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PIERRE VAN RYSSELBERGHE
B. Theorems valid f o r very low concentrations Theorem 1:
fI" = f'l"' = p
I
Theorem 2 : 1
=
vg'l") = [1 + (lJ - 1)al s?'
Theorem 3 : f'," = (1 - a)f?) Theorem 4 :
fy = a f y C. Theorems valid for very weak electrolytes at very low concentrations
fy'
When a is very small and is practically equal to unity, the solution being apparently ideal, theorems 3 and 4 of the foregoing series become:
X. SUMMARY
Three thermodynamic descriptions of solutions of electrolytes corresponding respectively to an undissociated, a completely dissociated, and an incompletely dissociated solute have been studied. Between the three sets of activity and osmotic coefficients certain thermodynamic relations are shown to hold. These are obtained in general form, and some interesting simple formulas are deduced for the particular case of very low concentrations and that of very weak electrolytes. I n concluding we wish to express our gratitude to Dr. J. W. McBain and Dr. F. 0. Koenig, of this laboratory, for their valuable suggestions and comments. REFEREXCES (1) DE DONDER, TH.: L'AffinitB. Gauthier-Villars, Paris (1927). (2) GUGGENHEIM, E. A . : Modern Thermodynamics by the Methods of Willard Gibbs. Methuen, London (1933); E. P. Dutton and Co., New York (1933). (3) GUGGENHEIM, E. A . : J. Phys. Chem 33,842 (1929); 34,1758 (1930). Reference 2, pp. 134-9.
N.,AND RANDALL,M.: Thermodynamics and the Free Energy of Chemical Substances. McGraw-Hill Book Co., New York (1923). (5) Concerning the activity coefficients of weak electrolytes, see RANDALL, M., AND ALLENC . : J. Am. Chem. SOC.62,1814 (1930). (4) LEWIS, G.
