OSMOTIC AND ACTIVITY COEFFICIENTS: THE RATIONAL TYPE VERSUS THE PRACTICAL TYPE* PIERRE VAN RYSSELBERGHE and GILBERT -1. HUNT ---University of Oregon, Eugene, Oregon
INTRODUCTION
The almost exclusive use, in the literature on the physical chemistry of solutions, of the practical osmotic coefficients (designated in this paper by q) and of the practical activity coefficients of solutes (designated by The chemical potential or partial molar free energy of 7) in place of the rational osmotic coefficients (desig- the solvent is given by the expression (8, 4, 5, 8): nated by g) and of the rational activity coefficients PI = PI" gRT In NI (6) (designated by f) seems to indicate a general belief that mathematical complications prohibit the use of in which fix0 depends only on pressure and temperature, the latter. This, however, is not the case. The only R designates the molar gas constant, T the absolute step which may, a t first inspection, foster such a belief temperature, and in is the symbol for natural logarithms. is the integration of the Gibhs-Duhem equation con- Formula (6) can be regarded as the definition of the necting the g's and f's which does indeed appear some- rational osmotic coefficient g. In terms of the practical what more difficult than the corresponding integration osmotic coefficient q we have (4, 8) : ' with the q's and 7's. The purpose of the present paper is to present the detailed integration of this equation for the rational coefficients in the case of strong electrolytes. No textbook of thermodynamics or physical In terms of x we have chemistry, to our knowledge, offers this treatment in a pl = P,' - QRTIn (1 + z) (8) complete manner (9, 6). Yet such a treatment has and pedagogical significance since, as should be well known, PI = PI' - pRTz (9) the practical coefficients q and 7 are not exact measures of the departures from ideality (5, 7,8). We thus have, between g and q, the relation
+
DEFINITION OF RATIONAL AND PRACTICAL OSMOTIC COEFFICIENTS AND ACTIVITY COEFFICIENTS
Let us consider the case of a solution containing one electrolyte dissociating into v+ positive ions of valence z+ and v- negative ions of valence z-, with u+ v- = u. The molality of the electrolyte is m, the molecular weight of the solvent is MI. The mole fractious are N1 for the solvent and N , for the electrolyte. They are connected with m and MI as follows (8, 19):
+
N, = N* =
1000/M, 1000/M, urn
+
Vm 1000/M1
+
Introducing the notation 2 =
M 2
1000 vrn
-
+ 2)
Q ln (1
(1)
(10)
= fl
and it is only when x is very small'that a and q. aDproach each &her on account the well-kn& approximation
of
A
ln(1 + z ) S z
(11)
They are identical and both equal to unity a t infinite dilution. For the electrolyte (component 2 of the twocomponent system, solvent-neutral electrolyte) we have (4, 5, 8, 13). m = up,
= v[P*'
+ RT In (N,f,)l
(12)
in terms of the ratiohal activity coefficient f,, or
(2)
~n = vwa = Y [ P - " +
RTln(mr,)l
in terms of the practical coefficient y ,. show that
(3)
f,
formulas (1) and (2) become
=
r,(1
+ 2)
(13)
One can easily (14)
and that .
*This paper wes presented st the second Pacific Northwest Regional Meeting of the American Chemicd Society a t Pullman, Weshington, May 3, 1947.
a,"
87
= p , " l + R T l n - 1000 "MI
(15)
88
JOURNAL OF CHEMICAL EDUCATION
Formula (14) shows t h d j, and r * are identical only at infinite dilution, when they are both equal to unity. If we write (13) in the form 112
w, = ~[r,'
=
+ R T l n (zr,)l
(16)
the MAd', of (12) and (16) are equal to each other. Summarizing and decomposing the chemical potentials w and p2 into an ideal portion and a non-ideal one we have
- RT2 "ideal"
p, = p,*
+ (1 - p)RTz
(18)
"non-ideal"
whiie (27) can be rewritten
The obviously simpler appearance of (29) in contrast with (28) is one of the reasons for the general use of the practical coefficients,. INTEGRATION OF THE GIBBS-DUHEM EQUATION
When m moles of solute are gradually added to 1000 grams of pure solvent the osmotic coefficients gradually change from 1 to g or p and the activity coefficients gradually change from 1 to f, or y , in accordance with the integrated form of the Gibbs-Duhem equation. For the rational coefficients we have from (28),
r," 1+z + + ~ ~f ~ ,]l n(19) ~~nz ideal non-ideal
rz
=
vr,
=
vir,"+RTlnz+ "ideal"
RTIny,] "nan-ideal"
(20)
The, quotation marks recall that the separation into ideal and non-ideal terms in the case of the practical coefficients is not exact.
For the practical coefficients we have from (29),
which simplifies immediately to
THE GIBBS-DUHEM EQUATION
The Gibbs-Duhem equation, for conditions of constant pressure and temperature, is (4, 6, 8, 12):
Completion of the integration requires the experimental or theoretical knowledge of t h e dependence of pons. Integration of (30) is only slightly more laborious than that of (31) and has never been presented before. The lirst integral on the right-hand side of (30) can be integrated by parts as follows: Let
It should be identically satisfied for an ideal solution.
Hence
Referring to (17) and (19) we find v = g
The integral becomes
It is interesting to note that, in spite of the inexact nature of the separation into ideal and non-ideal terms in (18) and (20), the Gibbs-Duhem formula for the "ideal" portions of the chemical potentials is neverthe less satisfied: -dz+zdlnz
(25)
= 0
The Gibbs-Duhem equation requires that, for the nonideal portions of the chemical potentials, we must have d[(l
- g) In (1 + z ) ] + zd In f ,
=
0
(26)
for the rational coefficients and d[(l
- q)z] + zd In y,
for the practical ones. Formula (26) can be rewritten
=0
(27)
We have In (Iz+ 2)
:1
=
In (1 z
+ z)
- lim In (1 2-0
+ z)
Z
(37)
The limit on the right-hand side is 1in accordance with (11). The lirst integral on the right-hand side of (36) cancels with a portion of the second integral on the right-hand side of (30), a particularly interesting feature of this treatment. We thus have from (30), (36) and (37)
W R U A R Y , 1948
+
The last integral on the right-hand side of (38) can be , In (1 y2)/ya against y. In the very dilute range, where graphic extrapolation is necessary, the Debyeintegrated by parts as follows: Hiickel limiting law for Q or g provides the safest, and u = -1 d u = - dz x 1 + ~ (39) now universally accepted, guide. (
r---+
+
+ z)
dz ( 1 -Z ) - m--In ( 1 = In -o 4 1 z) z 2=o z
xa
-
APPLICATION OF THE DEBYE-HUCKEL LIMITING LAW FOR THE OSMOTIC COEFFICIENT
In very dilute solutions of strong electrolytes in water a t 25 degrees we have (14):
+
r'B o
)
dz
(41) l - g S l - * =
Combinine: with (38) , , In j, = - In ( 1
+XI
z
0.3908 ( z + z _ ) ' / a X 18.016)/1000
d ( u
( 1 - g) -
J
(1 - 8 ) In ( 1
2%
+
Z)
dZ (42) . .
The integration of the Gibbs-Duhem equation for r e tional coefficients has now been carried out to the same point as equation (32) for the case of the practical coefficients. Completion of the integration requires experimental or theoretical information on the dependence of g on x. APPLICATION OF THE RANDALL-WHITE METHOD
Formula (32) and its analog (42) require a graphical or an analytical integration. These integrals, however, do not converge if l - Q and (1 - g) In (1 2)
+
3:
are plotted against x. The remarkable Ftandall-White method consists of using the square root of x as the independent variable and the identity (4,8, 9, 10) d In z = 2d In z'/r
(43)
(48)
The graphical methods for the two cases of rational and practical coefficients are identical in the very dilute . range. At higher concentrations the difference between (46) and (47) has to be taken into account. We have carried out the complete determination of the rational activity coefficientsf, from the isopiestic data of Robinson and Sinclair (11) for potassium chloride by the method expounded in the present paper. We have satisfied ourselves that this method is no more difficult nor more time-consumine than the more familiar one involving the practical coefficients. Workers in the field may go on preferring the direct determination of Q'S and y's, sinee it is always possible to transform them into g's and f's afterwards. On the other hand, since the rational coefficients are the only true measures of the departures from ideality, it is important to know that a complete calculation of these coefficients from experimental data is possible without any reference to the practical coefficients. LITERATURE CITED ( 1 ) DEBYE,P.,
AND
E. H~~cKEL, Phy~ik.Zeits., 24, 185, 305
11 401)
( 2 ) DE DONDER, TA., AND P. VANRYSSELBERGHE, "Thermo-
+
The ratios (1 - Q)/X and (1 - g) In (1 x)/x tend to i n f i ~ t yat infinite dilution, while the ratios (1 - Q)/x'l' and (1 - g) In (1 x)/x'/' have definite limits provided by the Debye-Hiickel theory (1). Let
+
z'/. = y
(45)
Formula (32) becomes:
and formula (42) becomes:
In the graphical determination of y, from p one uses plots of (1- 9)/y against y. In the graphical determination off, from g one would use plots of (1 - g)
dynamic Theory of Affinity, a Book of Principles," Stanford University Press, 1936, pp. 88, 114, eta. ( 3 ) GLASSTONE, S., "Textbook of Physical Chemistry," 2d ed., Van Nostrand. New York. 1946. o. 960. ~ ~ - . ( 4 ) GLASSTONE, S., "~hermod~namics for Chemists," Van Now strand, New York, 1947, pp. 389-90. ( 5 ) GUGGENHEIM, E. A,, "Modern Thermodynamics by the Methods of Willard Gibbs," Methuen. London. 1933. p. 118, etc. ( 6 ) HARNED, H . S., AND B. B. OWEN,"The Physical Chemistry of Electrolytic Solutions," Reinhold Publishing Co., New York, 1943, pp. 12-13. (7) LEWIS,G. N., AND M. RANDALL, "Thermodynamicsand the Free Energy of Cbernicd Substances," McGraw-Hill, New York, 1923. ( 8 ) MACDOUGALL, F . H., "Thermodynamics and Chemistry," . 3d ed., Wiley & Sons, New York, 1939, pp. 266-8. ( 9 ) RANDALL, M., J. Am. Chem.Soc., 48, 2512 (1926). M., AND A. M. WHITE,ibid., 48, 2514 (1926). (10) RANDALL, (11) ROBINSON, R. A., AND D. A. SINCLNR,ibid., 65, 1249
. ~ , . ~ ~ ~
.
(1943). ~, (12) VANRYBSELBERGHE, P., J. P h p . Chem., 38, 1161 (1934). (13) VANRYSSELBERGHE, P., ibid., 39, 403, 415 (1935). (14) VANRYSSELBERGHE, P.. J. Am. Cham. Sac., 65,1249 (1943). ~
