OSMOTIC AXD ACTIVITY COEFFICIESTS BY R. C. CANTELO
Debye and Huckel' in their brilliant paper on the theory of strong electrolytes, on the assumption that interionic forces exist between the ions obtain for the potential energy U,,of the ions in the solution, the expression:
where and the other symbols have the following significance : Ce = potential energy of the ions, or the electrical energy of the ,solution in ergs; N, is the total number of ion-molecules of the ith kind of valence z, in the solution; e is the charge on a univalent positive ion viz: 4.774 X I O ~ O e.s.u.; D is the dielectric constant of the medium a t the absolute temperature T ; k is the gas constant for I molecule viz: k = R 'N, where N is Avogadro's number (6.061 X 1 o Z 3 ) ;and n, is the number of ions of the ith kind per cubic centimeter of solution. Kow the Electrical Free Energy of the solution represents the potential free energy of the S O I U ~ Z O ~ ,due to the potential energy U, of the ions. I t may be defined by the equation:
This gives for F,,
Khere X, is a function of the ionic diameter a,. If now in place of a, and X, we introduce an average value of a and, therefore, of 9, we obtain:
Debye and Huckel introduce also a function u defined by the equation:
This function is so related to 9 that 2X+u=
3 + Ka
I
Debye and Huckel: Physik. Z., 24, 185 (1923)
(7)
Tow t,he simplest waj- of tenting the validity of the theory is to derive expressions for the activity coefficient or activat,ion, and osmotic eoefficirnt of an electrolyte, and then to compare the values calculated froin tliesc erpressions with the values obtained experinlentally from freezing point or electromot,ive force measurements. Debye and Hiickel derived equations for f arid @, the activation and osmotic coefficient, and these equations have been tested experimentally by many investigators. I n the present paper, the writer gives a simple slternative derivation of these equations.
Let us consider a symmetrical binary electrolyte forming a total of n ions. Then the electrical free energy of the solution is
Then
Hence,
Activity Coefficients If F0 be the free energy of the ions when they arp at the infinitely small concentration C,, and F the corresponding free energy when the ions are at the finite concentration C',, we have the relation, E'
-
Fo = IWT In a1 'as
(0)
where a. and al are the activit,ies and X is t,he number of ion-inols in thc solution. T,-sing n, the number of ion-molecules, ~ v ehave
F
=
F,
+ n kT In c l , ~ c o+- n k T I n fl 'f,
(10)
where f,, and f l are the. activationr. F is evidently the frec encrgy of the ions in the solution. Since C, is infinitely small, the last term becomes n k T I n f. If the ions were unchargrd we should hare for F,
I.' =
17,
+ n k T In el 'e3
f 11)
Hence the last term of (10) must repreqent the partial electrical free energy of the ions in the qolution. Hence from f8), we can \\rite
- lnf
or
E?Z?K
=
+ z D k T ( i + tia)
where f is the activation of a single ion, S o w Equation fornied into:
(2)
can be trans-
n-here i.1 is the ionic strength’ of the electrolyte. Khen numerical values are substituted in i 15) we obtain for z j”C, 0,3283
K
>.:
IOq
;v ’
(16J
Vsing this value of K , substituting numerical values and changing to tkcadic logarithms transforms Equation (i 4) to - logf
=
fI
+
o.;o;
z:
0.328
x
io%\.’-;-
for the activity copfficient of a single ion at z j”c‘. S o w what we are interested in is the mean actiration f of an electrolyte. B,?is defined by: The mean activation of an rlectrolytc of the type
*
where v
= VI
f
+ v?
*
= (fl”:f,”)I 7 . 1
(IS:
111,
z:
,\gain for an e!ectro!yte of t’he type -Iv.E,> vlzi
E’ron1 ( I S )
In f
1
1
=
vl
i.’
ln f ,
+
V:
=
v In f?
Lewis arid Randall: J. .im. C‘hern. Soc.. 4 3 , 1141 i i 9 2 1 !
v,z2. Hence
When numerical values are substituted, Equation (19)becomes: -
*
- logf
d /L
0.jOj Z~ZZ
= I
+ 0.328 X
108a
47
i2 0 )
If we expand the denominator, this gives -logf=k=o.joj
ZIZ?
\/i [ I - 0 .33 X 1 o ~ a . \ / ~ + 0 . 1 0 8 9 X 1 0 ' ~ a ~ p...I +
(21)
When the solution is dilute, the termshigher thanthe first beconienegligible since a is of the order of IO-~CIII.Hence for a dilute solution Equation (20) becomes equivalent to that of Debye and Hiickel for the limiting case of c -+0 . i.e.' - log f = 0 . j o j ZIZ? 4 ; (22)
*
Osmotic Coefficient The partial niold free energy of the solvent, in a perfect solution or in an infinitely dilute solution is defined by the equation:-
FI - F2 = R T l n
(I
- s)
(23)
where F2 is the free .energy of the pure solvent and s is the mol-fraction of the solute. For an infinitely dilute solution, this reduces to -
FZ-F?= - H T x
(2.4)
Equation (24) defines the behavior of the solvent for the ideal state of an infinitely dilute solution. For an actual state other than an ideal one we shall correct for the deviation in the behavior of the component from its behavior in the state of zero concentration of solute, by means of an osmotic coefficient +; and we shall define by
+
where A refers to the depression of the freezing-point of the solvent. Then for a n y solution, Equation ( 2 3 ) can be written: -
F2 - Fr
= -
IiTxd
!26)
and for several solutes,
F,
-
F? = - R T
(11
+ + + . . . ,)+ SI
~g
lVe shall now define a quantity, F?,eby the equation,
Fz,e =
-
RT
@ s - (-
R T X)
-
i.e. F2,, = - I1 T s (+ -
or for a single molecule
-
VI,,
=
1;e 1
- k T s (+ -
I)
I)
t'KSll2' -
3n
See Bronsted and LahIer: J. .im. Chem. Soc., 46, 561 (1924).
(27)
OSXOTIC
.%SD ACTIVITY C O E F F I C I E N T S
while the partial free electrical energy of the ions is n
S
= I -
$Ka
FI,,
+ Qxa2 + . . . . . ,
whence we get
.1comparison of Equations ;(a) and (29a) shows that a part of the free electrical energy of the solution must reside in the solvent. This electrical free energy of the riiediuiri is given, evidently, by Equation (28). S o w F,,,F1,, and F?,eare related according to the familar therniodynamic equation : F, = n FI,, no F?,e (30)
+
whpre n
=
number of ion-molecules, and no = number of solvent molecules.
Upon substituting the respective values from Equations
(s),
(29) and
(28) we obtain,
and for moderate concentrations we can write,
Equation (3 I ) holds for a symnretiieal binary electrolyte. From Equations ( 1 4 ) and (31) we obtain the relation 3 - Inf i I -0 u(l + h a ) ~
(32)
632
K. C. C A S T E L O
Hence for any electrolyte, we have the relation ut1
1 - 9 =
+ ha1 l n f =
Then from Equation (19)we obtain for the generai equation for
Upon substituting numerical values we have finally, for I -
4
=
G.373
0'
I
-
4,
c'.
Z,Z?Ud/;
and for the lirniting case of t.streiiic1y dilute solutions, for which u have I - 4 0 . 3 ; ; Z;Z?\'&
= I,
we
I -
Depnitmeilt of C'heimeiri), rniwrsitg of C'ii~ciii:~dz, Cii!Ci7Z71Uti,Oi!iO
13.5)
