MOLAL VOLUMES OF SOLUTES
619
MOLAL VOLUMES OF SOLUTES. IV OTTO REDLICH Department of Chemistry, State College of Washington, Pullman, Washington Received October 9, 1959
The relationship between the apparent molal volume of a strong electrolyte and the concentration has been represented by Masson (37) and Geffcken (8, 9) through the empirical rule
4 =
40
+ k’c1/2
(1)
where 4 is the apparent molal volume and c is the concentration in moles per liter. From the definition of the apparent molal volume
V
= nlv:
+ n24
(2)
where V is the volume of the solution, n1and nz are the number of moles of solvent and solute, respectively, and v; is the molal volume of the solvent, the relation follows between apparent molal volume and specific gravity:
we - -,1000 s - s o 4 =so c so
(3)
(we is the molal weight of the solute; s and so are the specific gravities of the solution and solvent, respectively.) A formula similar to equation 1 has been derived from the DebyeHuckel theory in the first papers of this series (41, 43, 44). According to this theory the free energy, F , of the solution of a strong electrolyte is given in the limit of low concentrations by’ F = nl$
+ nz[F; + vRT In m - K(D, T)w”*c”~]
(4)
where F; and F: are the partial molal free energies of solvent and solute, respectively, in the reference states; D is the dielectric constant of the solvent; m is the molality; K is a function of D and T given by the DebyeHuckel theory; w = 3 2 v d ; v i is the number of ions of species i formed by one molecule of the electrolyte; v = Zv, ; and z i = valence. Applying a well-known thermodynamical relation we obtain
1 Bachem (2) expressed the opinion that in the logarithmic term of the following formula c should be used rather than m. As this term folloss from the laws of the perfect dilute solution, we cannot see any basis for this opinion (cf. Lewis and Randall (36)).
620
OTTO REDLICH
where 6 is the compressibility of the solvent. The function k is given by
k
2e2
ais
(">"'(lOOOT
!)
1dD DdP 3
where R is the gas constant in cc.-atmos., t is the electric charge of a univalent ion, b is Boltzmann's constant, N is Avoeadro's constant, and the pressure is in atmospheres. Comparing equations 2 and 5 we have
4 = 40
+ kW8'2C'/2
(7)
Although similar to Masson's rule (equation l), this theoretical formula haa an entirely different meaning. While the empirical rule is claimed to represent the data over a considerable concentration range, formula 7 cannot be expected to be more than a limiting law for low concentrations. This fact was stressed as early as 1931 (43,44). But there is another important point. The empirical rule has always been used with different coefficients for different electrolytes; the theoretical equation, however, postulates a single coefficient k, common to all electrolytes and dependent only on the temperature and the nature of the solvent. Although we believed that the experimental material available in 1931 was sufficient to confirm equation 7 (44), various objections have since been raised. These objections, summarized by Stewart (49)(cf. 50), concern (a) the question as to whether the square-root law is characteristic of electrolytes alone or applies to non-electrolytes as well, (b) the experimental proof with improved data, and ( c ) the valence factor. A thorough discussion of all objections seems desirable, as formula 7 was derived from the theory of Debye and Huckel by means of thermodynamics alone. Any failure of equation 7 would indicate therefore the invalidity of the theory. NON-ELECTROLYTES
A linear relationship a t low concentrations between molal volumes (or some other properties like the logarithms of the activity coefficients, molal heat contents, and capacities) and the square root of the concentration is characteristic of strong electrolytes alone. According to extensive investigations by Hildebrand (23,24,25), Scatchard (46),Guggenheim, London, Kratky and other authors, deviations from the laws of the perfect solution are to be represented by first-power term in the case of non-electrolytes. This holds true even for strong dipoles (32). Contrary to some earlier statements, very accurate data for sucrose (42) and urea (22)give evidence for the linear relationship between the apparent molal volumes and the first power of the concentration. Some amino acids, though strong dipoles, are characterized by a linear relationship
~
621
MOLAL VOLUMES OF SOLUTES
to the first power, while the corresponding salts obey a square-root relation (5 ). A comparison of the plots given in the two papers quoted above (42,22) can hardly leave any doubt as to the validity of the linear relationship to the concentration. However, a still more significant criterion of general applicability may be indicated. We have for apparent molal volumes (and similar properties) : Electrolytes:
finite, therefore
($z)c-o
):f
Non-electrolytes :
finite, therefore (d$)c-o
(8) =
0-0
The curves representing the apparent molal volumes of sucrose and urea are in striking accord with relation 9. Relation 8 explains, by the way, why some early and promising attempts to develop the deviations from the laws of the perfect solution in a Taylor series with respect to c (27,38) could not succeed in the case of strong electrolytes. The only condition of convergence, that all differential quotients be finite, is not satisfied. TEST OF EQUATION
7
Some questions arising in an experimental test of equation 7 were discussed in 1931 (44). From the data given by Baxter and Wallace (4),by far the most accurate then available, the values
k = 2.8 (OOC.); k = 1.7 (25OC.); k = 1.5 ( 5 0 O C . ) (10) were derived for aqueous solutions. Only data of outstanding accuracy may be used for a decisive test. Such data have been published by Geffcken, Beckmann, Kruis, and Price (11, 12, 13, 14, 34), by Gibson (15), and by Wirth (52). From the results obtained by Jones and his collaborators (28, 29, 31) the figures given for ionic strengths higher than 0.02 have been included in the following diagrams. Prang’s data (40) are not quite as precise as Wirth’s, although the same method was used.’ * A n additional confirmation of equation 7, however, may be seen in Prang’s results. That Prang’s valence factor for uni-divalent salts does not agree with Ceffcken’s figure is explained by the fact that he calculates molal volumes while Geffcken uses equivalent volumes. One must not, as Prang does, assume the polarizability for long waves to be independent of pressure in the case of a dipole liquid like water. In fact, the correct order of magnitude of Prang’s figure for k is caused by a numerical error only. The formula given by Prang for the apparent molal volumes applies actually to the partial molal volumes, the coefficients differing by the factor 2/3.
622
OTTO REDLICH
The following discussion is not claimed to comprise all pertinent material. The author believes, however, that the data used furnish an appropriate basis for a conclusive test of equation 7 and for a numerical value of k a t 25OC. and, moreover, that a discussion of any number of less accurate data would be of little value in testing equation 7. In figure 1 the quantity 9 - 6' of a few uni-univalent salts is plotted against cl'*. This diagram agrees exactly with what we would expect: namely, a common limiting slope, excellently defined by each series of
r3
0.5 FIG.1. Apparent molal volumes of uni-univalent salts KC1. . . . . . . . . . . . . . . . . . . . NaCl.. . , . . . . . . . . . . . . . . . .
KBr . . . . . . . . . . . . . . . . . . . . . N a B r . ,, , , , . . . . . . . . . . . , ,
11
.I'
Geflcken
1
Jones
X
I
A
0
...... '
' '
0
1
Wirth
..........
....., ., ..
.... ...... 9 ..... . ... . ........, .
b
measurements, and individual deviations from the limiting tangent increasing with increasing concentration. in part considerably different from the Table 1 contains the values of figures of previous authors. These values have been used in plotting figures 1 and 2. W r t h (52) has already pointed out the inconsistency of the 9' values to be derived for these salts. The value 9'(KBr) = 33.71, which we find from the figures of Jones and Bickford (28), would eliminate the dis-
MOLAL VOLUME0 OF 0OLUTE8
623
crepancy between chlorides and bromides but the discrepancy between these salts and the sulfates would still remain. The discrepancy exceeds considerably the uncertainty of the extrapolations involved. There must be a systematic error which we are unable to locate in some of the experimental data. The value k = 1.86 f 0.02 (25°C.)
(11)
has been derived from the data represented in figure 1. The molal volumes of ammonium nitrate are less suitable for determining the value of k , since ammonium salts behave somewhat irregularly (cf. 47). However, even in this case the limiting slope has been confirmed (12, 14). Batuecaa (3) did not extend his determinations of the densities of sodium and potassium chloride at 0°C. to very low concentrations. According to his figures a provisional value k = 2.4 for 0°C. may be assumed. TABLE 1 Molal volumes at infinite dilution
THEORETICAL VALUE OF THE COEFFICIENT
Using Falckenberg's (7) value for dD/dP we had calculated (43), according to equation 6, k = 1.8 f 0.6 (16°C.) (12) This value w e e s with equation 11. Gucker (18)' however, pointed out that more recent determinations of the dielectric constant of water at high pressures by Kyropoulos (35) should be used, resulting in the value
k = 2.53 (20OC.) (13) The discrepancy between this figure and all others cannot be satisfactorily explained. From Falckenberg's determination at 200 atmospheres the value (l/D)dD/dP = 46 X lo-' atmos.? is obtain@. The results of Kyropoulos, who gives smoothed values in intervals of 500 atmospheres up to 3000 atmospheres, can be represented by D = 80.79(1 59.2 X 104P - 3.28 X 10-DP' - 1.61 X 10-"p8 0.42 X lO-"P') (14) so that (l/D)dD/dP = 59.2 X lo4 a t atmospheric pressure.
+
+
624
OTTO REDLICH
The assumption that the measurements of both authors are correct gives an inflexion point between 200 and 500 atmospheres,-a result perhaps not implausible in the case of a liquid as abnormal as water. However, using a somewhat indirect but fairly probable method we can invalidate this assumption. The contraction caused by the electric charges of the ions in the limit of infinite dilution, the “electrostriction,” has been found by Drude and Nernst (6) to be proportional to (1/D2)dD/dP.* Kritschewsky (33) has shown that the electrostriction may be made responsible for practically the entire variation of the limiting values, Vi , of the molal volumes with increasing pressure. One may therefore expect Ti , a t different pressures, to be a linear function of (1/D2)dD/dP; this has been confirmed by Kritschewsky in the cases of sodium chloride and potassium sulfate on the basis of Kyropoulos’ result^.^ The whole set of Kyropoulos’ data must therefore be considered inconsistent with the other figures. Only new experimental work is likely to bring about a final decision. POLYVALENT ELECTROLYTES
The values of 9 - 9’for strontium chloride, sodium sulfate, and potassium sulfate are represented in figure 2. The dashed straight line indicates the slope resulting from equations 7 and 11 with tu = 3. The limiting slope of the strontium chloride curve agrees precisely with the predicted value. The experimental material for the two sulfates is somewhat scanty a t high dilutions, so that no definite conclusion can be drawn. The molal volumes of lanthanum chloride, computed from data of Jones and Bickford (28), can be represented by a straight line in a cl” plot with the slope €i3” X 0.79 between c = 0.025 and c = 1. It is possible, however, that the molal volume curve turns below c = 0.025, as Jones and Bickford found that deviations of the conductivity persisted to still lower concentrations, The available density data do not give any hint of this turn. NON-AQUEOUS SOLUTIONS
The discussion of non-aqueous solutions is restricted by the fact that adequate data for both dD/dP and densities are available only in the case a The electrostriction can be deduced from the common formula of the electric energy of a charged sphere in the same way as formula 5, which represents, indeed, nothing else but the dependence of the electrostriction on concentration (cf. Gross (17)). 4 Values of VI obtained by means of the c’/z relationship should be used instead of the values given by Adams (1). In addition, we find minor differences from Kritschewsky’s values of (l/DZ)dD/dP when using formula 14. But the general conclusion is not altered by these modifications.
625
MOLAL VOLUMES O F SOLUTES
-
e
0
Q b)
-5
0
-
//
4 b
-
0
///
///
8
O 0
@eoo
do
? ///&0O0 e,’,’
0
/
p-p - e’&
/IO
/
/
I
I
I
05
Geflcken
I
I
j
Gibson
~ &SO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Na&04... . . . . . . . . . . . . . ..................
I
Jones
b
........
I
LO
Wirth
-
1 7
::::::::I:::::::: 43 0
X
i
A x
KI
MI AaCl Li Cl 0.5 FIG. 3. Apparent molal volumes of uni-univalent salts in methyl alcoholic solution
slope, based on Falckenberg’s (7) measurements and indicated by a dashed line, is very satisfactory. The results of Jones and Fornwalt (30) on
~
6%
OTTO REDLICH
ammonium chloride and potassium iodide also agree fairly well with the theoretical equation. Stark and Gilbert (48)did not extend their determinations to very low concentrations. APPLICATIONS
The interpolation of specific gravities of dilute solutions of strong electrolytes and particularly the extrapolation should be carried out according to the theoretical formula 7. A relation between s and c may be obtained by eliminating 4 from equations 3 and 7. We omitted to state this relation in our first papers, because we wished equation 7 to be considered as a limiting law only. A corresponding formula, given by Root (45), has been used by several authors with individual coefficients consistent with Masson's rule rather than with the theoretical equation. Nevertheless it has proved useful for interpolation in many instances. A correct representation of the apparent molal volumes is given by the series
We obtain from equations 3 and 16 lOoO(s -
$0)
=
+ + ycS + . . .
ac - sokwa~zca/z j3cz
(17)
where
a = w z - 4 s0
0
The value s'k = 1.85 should be used for aqueous solutions at 25°C. The number of individual constants a, j3, y . . required in equation 17 depends on the nature of the electrolyte, the range, and the precision desired. In many cases the term Bc2 will be su5cient. The following method is likely to be the most convenient for determining the coe5cients a, 8, y of equation 17.' The quantity
is computed from experimental data and equation 19. This quantity can in almost any case easily be plotted against c with the precision required. If the experimental points may be connected by a straight line, the constants a and 6 are immediately given. In the case of a distinct curvature, 6
A similar method has been devised already by Gucker (20);cf. also Gibson (16).
MOLAL VOLUMES OF SOLUTES
627
three points of a smoothed curve are used for calculating CY, 8, and y. The specific gravities should be recalculated from the resulting formula for all experimental points, and minor adjustments of the coefficients should be carried out if systematic deviations occurred. The quotient (s - so)/c loses little accuracy in an extrapolation to zero concentration according to equation 20. Up to a certain concentration the differences s - so are found, therefore, with higher precision by extrapolation than by direct measurement. Numerous series of measurements have been extended far below that limit. The limit is determined by the probable error of the numerical value of k;for aqueous solutions at 25'C. the limits are as follows: wc = 0.03 if the accuracy of s equals lo-' = 0.1 if the accuracy of s equals lo-' = 0.6 if the accuracy of s equals lo-'
There is no use in extending measurements much below these limits (say to one-tenth of the limit concentration). On the other hand, measurements should be extended a t least to these limits to furnish the best possible basis for determining the numerical value of k as well as individual deviations from the limiting law. Many authors, early and recent, have attempted to investigate molecular changes occurring in solutions by discussing volume changes. This aim has often been frustrated by the interfering effect of the electrical interaction between ions. Geffcken and Price (14) succeeded in splitting the two effects in the case of the second dissociation of sulfuric acid and of the hydrolysis of sodium carbonate. A similar application was made by Hoather and Goodeve (26) in a discussion of the molal volumes of sulfuric and sulfurous acids. Gucker (18, 19) and Geffcken (10) derived a square-root formula of the apparent molal compressibility. The experimental data then available were hardly sufficient. I n the meantime this formula has been confirmed by measurements carried out with the aid of ultrasonic waves (2, 39). A similar formula has been deduced and discussed by Gucker for the apparent molal expansibility. SUWRY
The daerence between Masson's empirical rule and the square-root relation between molal volumes and concentration derived from the theory of Debye and Huckel is stressed. The latter is verified by means of the available data for aqueous and non-aqueous solutions. The value k = 1.86 f 0.02 is derived for aqueous solutions at 25°C.
628
W l O REDLICH
The molal volumes of non-electrolytes in dilute solutions depend linearly on the first power of the concentration. The theoretical formula should be used as a limiting law only and with a common coefficient varying only with the temperature and the solvent. A formula for interpolating and extrapolating molal volumes is indicated. Below a certain limiting concentration densities are determined by means of extrapolation better than by direct measurement. REFERENCES
(1) ADAMS,L. H.: J. Am. Chem. Soc. 59, 3769 (1931);64, 2229 (1932). (2) BACHEM,CH.: Z. Physik 101, 541 (1936). (3) BATUECAS, T.:Z. physik. Chem. A182, 167 (1938). (4) BAXTER,G. P., AND WALLACE, C. C.: J. Am. Chem. Soc. 38, 70 (1916). (5) COHN,E.J., et a l . : J . Am. Chem. SOC.67, 637 (1935);68,415 (1936). (6) DRUDE,P., AND NERNST,W.: Z. physik. Chem. 16, 79 (1894). (7) FALCKENBERQ, G.: Ann. Physik [4]61, 145 (1920). (8) GEFFCKEN,W.: Naturwissenschaften 19, 321 (1931). (9) GEFFCKEN,W.: Z. physik. Chem. A166, 1 (1931). (10) GEFFCKEN, W.: Z. physik. Chem. A167, 240 (1933). (11) GEFFCKEN,W., BECKMANN, CH., AND KRUIS,A.: Z. physik. Chem. B20, 398 (1933). (12) GEFFCKEN,W., A N D KRUIS, A.: Z. physik. Chem. B23, 175 (1933). (13) GEFFCKEN,W., KRUIS,A,, AND SOLANA, L.: Z. physik. Chem. BS6, 317 (1937). (14) GEFFCKEN, W., A N D PRICE,D.: Z. physik. Chem. B26, 81 (1933). (15) GIBSON,R. E.: J. Phys. Chem. 31, 496 (1927). (16) GIBSON,R. E.:J. Phys. Chem. 38, 319 (1934). (17) GROSS,PH.: Z.Elektrochem. 37, 711 (1931). (18) GUCKER,F. T.,JR.: Chem. Rev. 13, 111 (1933). (19) GUCKER,F. T.,JR.: J. Am. Chem. Soc. 66, 2709 (1933). (20) GUCKER,F. T., JR.: J. Phys. Chem. 38, 307 (1934). (21) GUCKER,F. T.,JR.: J . Am. Chern. SOC.66, 1017 (1934). (22) GUCKER,F. T.,JR.,GAGE,F. W., A N D MOSER,CH. E.: J. Am. Chem. Soc. 80, 2583 (1938). (23) HILDEBRAND, J. H.: J. Am. Chem. SOC.67, 66 (1929). (24)HILDEBRAND, J. H. : Solubility of Non-electrolytes. American Chemical Society Monograph Eo. 57. Reinhold Publishing Corporation, New York (1936). (25) HILDEBRAND, J. H., AND WOOD,S. E.: J . Chem. Phys. 1, 817 (1933). (26) HOATHER,R. C., A N D GOODEVE,C. F.: Trans. Faraday SOC. SO, 630 (1934). (27) JAHN, H.: Z.physik. Chem. 37, 490 (1901); 41, 257 (1902). (28) JONES, G., AND BICKFORD, CH. F.: J. Am. Chem. SOC.66,602 (1934). (29) JONES, G., A N D CHRISTIAN,S. M.: J . Am. Chem. SOC.69, 484 (1937). (30)JONES, G., AND FORNWALT, H. J.: J. Am. Chem. SOC.67, 2041 (1935). (31) JONES, G., A N D RAY,W. A , : J. Am. Chem. SOC.69, 187 (1937). (32) KIRKWOOD, J.: J. Chem. Phys. 2, 351 (1934). (33) KRITSCHEWSKY, I. R.: Acta Physicochim. U.R.S.S. 8, 181 (1938). (34) KRUIS, A.: Z. physik. Chem. BS4, 1 (1936). (35) KYROPOULOB, S.: Z.Physik 40, 507 (1926). (36) LEWIS, G. N., AND RANDALL, M.: Thermodynamics and the Free Energy of Chemical Substances, Chap. XIX, XX. McGraw-Hill Book Company, Inc., New York (1923).
TERNARY SYSTEMS
629
(37) MASSON,D.0 . : Phil. Mag. [71 8, 218 (1929). (38) NERNST,W.: Z.physik. Chem. 38, 487 (1901). (39) PASSYNSKI, A.: Acta Physicochim. U.R.S.S. 8, 385 (1938). (40) PRANQ, W.: Ann. Physik [5]31, 681 (1938). (41) REDLICH,0.:Naturwissenschaften 19, 251 (1931). (42) REDLICH,O.,A N D KLINGER,H.: Sitsber. Akad. Wiss. Wien IIb, 143, 489 (1934) or Monatsh. 66, 137 (1934). Data from PLATO, F., DOMKE,J., AND HASTINGS, H. : Wiss. Abhandl. Xormaleichungskommission (J. Springer, Berlin (1900)). (43) REDLICH,O.,AND ROSENFELD, P.: Z. physik. Chem. A166, 65 (1931). (44) REDLICH,O.,AND ROBENFELD, P.: Z. Elektrochem. 37, 705 (1931). (45) ROOT,W. C.: J. Am. Chem. SOC. 66, 850 (1933). (46) SCATCHARD, G.: Chem. Rev. 8, 321 (1931);Trans. Faraday SOC.33, 161 (1937). (47) SCATCHARD, G., AND PRENTISS, 5. S.: J. Am. Chem. SOC.64,2696 (1932). (48) STARK,J. B., AND GILBERT,E. C.: J. Am. Chem. SOC.69, 1818 (1937). (49) STEWART,G. S.: Trans. Faraday SOC.33, 238 (1937). (50) STEWART, G. S.: J. Chem. Phys. 7, 381 (1939). (51) VOSBURGH, W. C., CONNELL, L. C., AND BUTLER,J. A. V.: J. Chem. SOC.1933, 933. (52) WIRTH, H. E.:J. Am. Chem. SOC.69, 2549 (1937).
T H E TERNARY SYSTEMS ETHYLENE GLYCOL-POTASSIUM CARBONATE-WATER AND DIOXANE-POTASSIUM CARBONATE-WATER KENNETH A. KOBE
AND
JOSEPH P. STOXG, JR.
Department o j Chemical Engineering, University of Washington, Seattle, Washington Received October 3, 1939
The study of ternary systems of water-miscible organic liquids, salts, and water was initiated by Frankforter and Frary (2), who studied methanol, ethanol, and 1-propanol and various salts and reported that potassium carbonate was the only salt that would salt out methanol, while many salts salt out ethanol. Frankforter and Cohen (1) studied acetone and several salts, and Frankforter and Temple (3) studied some of the higher alcohols. Further work has been done since 1930 by Ginnings and Chen (4),who used 2-propanol and various salts a t 25OC.; by Ginnings and Robbins (7) , who used tertiary butyl alcohol (2-methyl-2-propanol) ; by Ginnings, Herring, and Webb (6), who used 1-butanol; and by Ginnings and Dies ( 5 ), who used allyl alcohol. Ginnings and coworkers developed mathematical relationships for the equilibrium curve. This work was started to study the salting-out effect on the dihydric
