252
mRBSRI '' S. HARNED.
[CONTRIBUTION FROM THE JOHN HARRISON LABORATORY OF CHEMISTRY OF THE UNIVERSITY OF PENNSYLVANIA.]
ACTIVITY COEFFICIENTS AND COLLIGATIVE PROPERTIES OF ELECTROLYTES. BYHER~ERT S. HARNED. Received November 9, 1921.
As a result of considerable study in recent years on the thermodynamic properties of dilute and concentrated solutions of electrolytes, the author has arrived a t the following means for testing the accuracy and consistency of available activity data, for calculating other thermodynamic data therefrom, and for organizing a convenient system by means of which such properties as activity coefficients, osmotic coefficients, vapor pressures and osmotic pressures may be accurately calculated. In some instances, the method suggested by Lewis and Linhart' for computing activity coefficients from freezing-point data is employed but, in the main, the present method differs from that employed by the above authors and by Lewis and Randall.2 Further, the results obtained, especially for solutions of potassium and sodium chlorides, differ considerably from those of Lewis and Randall. Since these differences exist, and there will be shown to be much evidence favoring the calculations here presented, i t was thought t o be highly desirable to state this view of the matter. 1. Derivations of Equations. I n a recent communication3 i t was shown that the individual ion-activity coefficients (activity of the ion divided by the molal concentration) of the hydrogen, chlorine, potassium, sodium and lithium ions can be calculated by means of the empirical formula log
F a = ac
- flcm
(1)
where F , is the activity coefficient of the ion, c is the concentration of the electrolyte in mols per 1000 g. of water, and a , 0,and IN are constants which differ for the different ions. The mean activity coefficient2 of any electrolyte, F o r , may also be expressed by an equation of similar form to (1) Thus log F~'=(Y'c--P'c~'
(2)
where c is the molal concentration, and a ' , p', and w' are constants. uni-univalent electrolytes
where a1,PI, ml refer to one ion, and as,8 2 , defined by the equation
refer t o the other. Fa' is
~ 2 2
2
Lewis and Linhart, THISJOURNAL, 41, 1951 (1919). Lewis and Randall, ibid., 43, 1112 (1921).
8
Harned, ibid., 42, 1080 (1920).
1
For
ACTIVITY COEFFICIENTS AND COLLIGATIVE PROPERTIES. a*
Fa‘= c(
y+
253
-y
+ x v
In this equation v + is the mol number of cations present, v- is the mol number of anions present, and Y is equal t o v + + v - (e. g., for BaC12, v + = l , v - = 2 ; for Laz(SOS3,v f = 2 , v - = 3 ) and a* is the vth root of the product of the activities of all the ions. When the anion and cation have identical activity coefficients, Gquations 1 and 2 are identical. In very dilute solutions, the term a’c becomes negligible and Equation 2 approaches the form of the empirical equation of Lewis and Linhart,’ namely log Fn’= -p’t“’.
(3)
In the present paper, Equation 2 will be used principally in testing the consistency of and in coordinating available activity data, and will be shown to be valid for all electrolytes studied throughout a concentration range from 0.0001 M to 3 M . :By combining Duhem’s equation in its exact form and Equation 1 or 2, it becomes possible t o calculate within narrow limits the vapor pressures and osmotic pressures of aqueous solutions of all electrolytes whose activity coefficients have been accurately measured. In addition, the osmotic coc:fficients, or, for uni-univalent electrolytes, i- 1, where i is van’t Hoff’s i, may be computed. This purely thermodynamic quantity has usually been called “the degree of dissociation.” Owing to the fact that very little reliable data are t o be found on these colligative properties a t 25”, i t was thought highly desirable to construct a convenient and unified method for their calculation. (a). Vapor Pressures from the Square Root of the Activity Products of Uni-univalent Electrolytes.-Duhem’s equation may be used in the follohh-ing form Nid Fi = - Nzd Fz (4) where N1, F1 are the mol fraction and free energy of the electrolyte, and N2, F2 are the mol fraction and free energy of the water.4 F1 is related t o the ion activities by Fl = RT In alaz 4-const. whence d Fl= RTd In alal wliere al and a2 are the activities of the ions of the electrolyte tion 2 log
Fa’ =lOgJFnFo=
a‘c-/3’cm’
or In ~ ~ = 2 . 3 0 2 6 a 1 c - 2 . 3 0 2 6 / 3 ’ c ”+In ’ c
or .
~ In ala2=2 (2.3026afc-2.3026/3‘c“ ‘Lewis, Proc. A m . Acad. Arts Sci., 43, 259 (1907).
+ In c )
(5)
By Equa-
254
HERBERT S. HARNED.
Differentiation of Equation 6 gives d l n al@=2(2.3026a’dc
+ d lnc-2.3026.i3’m’~”‘-~dc).
(7)
From (4) and ( 5 ) is obtained Nld1n ala¶= - N2d In a3 where a3 is the activity of the water. Substitution of (7) in (8) gives
(8)
+ d In ~-2.3026.i3’m’c”‘-~dc) = - d
2”(2.3026a’dc Na
In as.
(9)
The activity of the water molecule in these solutions is related to the vapor pressure by the equation Pc
0
f dlnao= f d l n p C
(10)
4
on the assumption that water vapor a t these low pressures (about 20 mm.) obeys the ideal gas law. In the above equation, po is the vapor pressure of pure water and p is the vapor pressure of the solution. Consequently, the integration of the left hand member of Equation 9 will be between the limits c and 0, and the right hand member between the limits p and po. Further
--
55.5 Nl=-’ c Nz 55.5+c’ 55.5+c
and Hence js(2.3026a’dc
+ d In c-2.3026
/ 3 ’ m ’ ~ ” -dc) ~ =-
j.
d In p .
(11)
Po
4
Integration gives 4.6052P’m’
p 55.5 m’+1 (b). The General Equation for Any Electrolyte.-In write Duhem’s equation N l d l n a i a ? .. . . a , = - N 2 d l n a ,
where a, is the activity of the water molecules.
this case, we (13)
According to Equation 2
ACTIVITY COEFFICIENTS AND COLLIGATIVE PROPERTIES.
IXff erentiation of (14) gives d In(alaz. . . .a,) =~(2.3026a'dc-2.30268'm'c"-~dc
+ d In c.
255
(15)
Substituting in (13), we obtain as before G
J2
55.5
L2.3026a'dc
+ dIn~-2.3026B'm'G"'-~dc]=-
* which becomes on integration
[.+
In--=*O
p
.
55.5
Be
2.3026 va'c - 2.3026 vp'm' cmJ] m' 1
+
(16)
(c) Vapor Pressures of Uni-univalent Electrolytes from Individual Ion Activities.-Lewis4 has deduced the equation [N1dlnal+Nzdlnaz+ dN
. . . . . I p2T
(17)
= 0.
.VI, Nz.. . . . . are the mol fractions of the molecular and ionic species of a mixture; al, a 2 .. . . . ., their activities. This, as was shown by Lewis, is Duhem's equation modified so as to be thermodynamically exact. In the present case, we write N1 d In al
+
rV2
d In a2
+ N3 d In as=O
(18)
where A'I, al, and h-2,a2refer to the mol fractions and activities of the cation and anion, respectively; and N 3 , a3,the mol fractions and activities of the water molecule. From Equation 1
a
log = LYC - Be" G
or In a=2.3026ac-2.3026Pc'"
+
111
c
where a is the activity of the ion. Differentiation gives d In a=2.3026
CY
dc
+ d In c-2.3026pmcm-'dc.
(19)
Hence
+ +
d In al=2.3026aldc d In c-2.3026~1m1cml-'dc d In az=2.3026azdc d In c-2.3026Bzm2cml-'dc
}
(19a)
where subscripts 1 and 2 refer to the cation and anion, respectively. Further N-N--.
C
2-55.5+2c'
l-
n;3 ---.
55.5 55.5+2c
Rearranging (18) Ni [ d In a1 + d In az] = - d In a3. N3
Su'bstitute in 18a the right hand members of Equation 19a for d In al and d In a2, the values obtained above for N I and NB,and collect coefficients. The following is obtained,
256
HERBERT S. HARNED.
C
+
5 ~ 5 [ 2 . 3 0 2 6 ( ~ l cuz)dc
+ 2d lnc - 2.3026/31nticml-'dc - 2.3026,3zm2c"r-1dc]= -d
In u Q .
Hence 2.3026(al
r
i
+ cU2)dc+2dZn c -2.3026p1m~c~l-' dc -2.3026&mzcml-'dc]= - d In p fiD
Integration of this last equation gives -
+ a&
2.3026(a1
-2.3026P1m~c,l- 2 . 3 0 2 6 8 ~ m ~ ml
2
+1
mz
+1
(d). Osmotic Pressures and Osmotic Coefficients.-Let the expressions within the brackets of the right hand members of Equations 12, 16, and 20 equal [i]. Then
The osmotic pressure can be calculated from the vapor pressure (neglecting the small term due to the compressibility of the solvent) by the formula RT #Q RT r=-ln-=-14 vo p vo 55.5 where r is the osmotic pressilre, V Othe molal volume of the water a t temperature T and R is the gas constant. On inspection, i t is seen that [i] is van't Hoff's [i],and that fo,b
($2) is equal 1 21-
or the classic degree of dissociation. x
to the osmotic coefficient
At 2.5" Equation 22 reduces to
(atms.) =24.42 c [i],
(23)
on substituting 298.1 for T , 0.08207 for R, and 0.018053 for Vo. 2. The Activity Coefficients of Potassium Chloride Solutions. In order to determine the parameters of Equation 2 and thus the activity coefficients of potassium chloride solutions, a method has been adopted which combines the method suggested by Lewis and Linhart' for computing activity coefficients from freezing-point data with the method employed by the author3 for computing the same a t higher concentrations from electromotive-force measurements. Lewis and Linhart found that, in dilute solutions of all electrolytes, the following empirical relationship holds, namely,
Bi' ( :) is as before the number of different ions produced in the solution log v i -
=all log c +log
(24)
where y by the dissociating substance, X the molal freezing-point lowering of water (1.858' C.), 0 the freezing-point lowering, G the concentration, and aI' and PI' are constants (al', PI' are the constants a1, pi in Lewis and Linhart's 5
Bjerrum, 2 . anorg. Chem., 109, 275 (1920).
ACTIYITY COEFFICIENTS AND COLLIGATIVs PROPBRTD%.
papler). By plotting lop( v k - : )
257
against log c, straight lines are obtained
from which the values of al’ and pl’ can be evaluated.
By combining
this equation with the thermodynamic equation
which relates the activity coefficient with the freezing-point lowering in dilute solutions, and integrating between the proper limits, they obtained log F’,=
Since for a given electrolyte equivalent t o (3) where
all,
3l’(LUl’+l)C~I‘. - !2.3026 V X ai’
&’, v, and X are constants, this equation is
Thiis, by plotting freezing-point data in accordance with Equation 24, the parameters P’ and WL’ of Equation 2 may be evaluated. Employing this method with solutions of potassium chloride, Lewis and Linhart obtained al’=0.535; P I ’ = 1.223 which according to Equations 27 give the values /3’ = 0.401 ; nz’ = 0.535. The author obtained p’ = 0.318 and m‘= 0.3!37, calculated from electromotive-force data and based on the value F’a(o.l) =0.754 a t 0.1 M concentration of Noyes and MacInnes;6 Lewis and
no
?>
30
Fig. 1.
Randall2 obtained F’a(o,,)= 0.798. The large differences between the val.ues of these constants as determined by Lewis and Linhart and by the author make a recalculation imperative. Instead of basing the calculation entirely upon the freezing-point data of Adams,7i t was thought preferable to include also the data of Fliigels and of Jahn.g In Fig. 1, the values of a Noyes and MacInnes, THISJOURNAL,42,239 (1920). Adams, ibid., 37,494 (1915). Fliigel, 2.physik. Chem., 79, 585 (1912). Jahn, ibid., 59, 35 (1907).
.258 log
HERBERT S. HARNED. (vh-a)
are plotted against log c.
I t can be seen that these points
group along a straight line between log c = i.00 and log c=2.00, and lie near a straight line a t lower concentrations. At concentrations abow 0.1 111, Equation 24 is no longer valid. The dotted line represents thc straight corresponding t o values of CY^' and 01’ obtained by Lewis and Linhart. Their plot passes through some points obtained from Adams’ data, and does not represent the line drawn through the points of all the data. The representative curve gives al‘ = m‘ = 0.394 and pl’ = 0.692 from which, by (27), p’= 0.286. Employing these values in Equation 3 F’a(o,l)= 0.766. But Equation 3, is not applicable up to concentrations as high as 0.1 M ,and consequently, in order to obtain a more exact value for F’a(o,,), Equation 2 is employed, assigning to CY’ the value 0.070, and to p’ and m‘ the values obtained above. I n the previous article, CY’ was assigned the value 0.080. The corrected value for F’a(,,l) is 0.779. Using this value, the activity coefficients have been computed a t higher concentrations from the electromotiveforce data of Noyes and MacInnes6 and Harned’o and compiled in Table I. TABLEI ACTIVITY COEFFICIENTS FROM ELECTROMOTIVE FORCE DATA F,’ (obs.)
C
Noyes and MacInnes
0.03 0 .0;i 0.10 0.20 0.30 0.50 0 .70 0.75
1 .oo 1 .
