THEPARTIAL M O L A L VOLUME OF IONS IN VARIOUS SOLVENTS superimpose the curves into one if the range of variables covered in the experimental studies is more limited. Therefore it appears that the proposed rate equations are in qualitative agreement with the available experimental results. 4. Comparison with Flow. We hope that the present formulism also helps to clarify the conceptual difference between spreading and flow. Spreading occurs at the interface only whereas flow assumes the liquid layer adjacent to the substrate to be stagnant. Hence, the surface properties are dominant in spreading and become irrelevant in flow. The mechanics and the physical variables which control the two transport processes are also different. This can be seen when eq 12 is compared with Poiseuille's law for the flow of Newtonian liquid through cylindrical tubes dV - nR4P dt 871
2417
where dV/dt is the rate of flow in terms of volume per unit time, R is the radius, 1 is the length of the tube, and P is the applied pressure. Therefore the driving force is supplied externally for viscous flow whereas the spreading force is intrinsic; the dimensions of the confining walls are always present in the flow equation whereas only the inherent volume of the droplet is pertinent to spreading. Both the spreading and flow rates are inversely proportional to the viscosity of the liquid. Temperature exerts an enormous effect on the rates through viscosity. I n addition, the surface properties of the liquidsubstrate pair are temperature dependent.
Acknowledgment. I am indebted to Dr. H. Schonhorn of the Bell Telephone Laboratories who alerted me to his publication. Mrs. Phyllis Gilmore of the Du Pont Engineering Department programmed the evaluation of the definite integrals.
The Partial Molal Volume of Ions in Various Solvents1 by Frank J. Millero Contribution No. 1043 from the Institute of Marine Sciences, University of Miami, Miami, Florida 83149 (Received January 16, 1989)
The partial molal volumes of ions in the solvents methanol, water, and N-methylpropionamide (NMP) have been examined by using the Frank and Wen model for ion-solvent interactions. The partial molal volume of an ion is assumed to b_e due to three major components, Vo(int), the intrinsic partial molal volu_me (equal to the crystal volume), Vo(disord), the disordered partial molal volume (void space effects), and Vo(elect), the electrostriction partial molal volume. The results of fitting the Vo(ions) in these solvents to various semiempirical equations indicate that Vo(disord) is related to the structure of the solvent. This disordered effect may be visualized as the void space caused by the solvated ion rather than improper packing in the electrostricted region. The po(elec_t)of the ions in the various solvents is proportional to l/r (r = crystal radius) and in the predicted order (Vo(elect) of methanol > water > NMP).
Introduction Using the Frank and Wen2 model for ion-solvent interactions, the partial molal volume of an ion at infinite dilution can be attributed to the following componentsa& VO(ion) = P(int)
+ P(e1ect) + 'C70(disord)
+ Po(caged)
Po(caged)is the caged partial molal volume (due to the formation of caged or structured water around ions with hydrocarbon tails). &nce the various ComPonents of eq 1 may be small when compared to the absolute size of the ion (i.e,, the crystal volume), it is difficult to determine the importance of the individual components of P(ion) or
(1)
where ro(int) is the intrinsic partial molal volume (the crystal volume), P(e1ect) is the electrostriction partial molal volume (the decrease in volume due to ion-solvent interactions), Vo(disord) is the disordered partial molal Volume (normally attributed to void space effects), and
(1) Paper presented at Southeastern Regional Meeting of the American Chemical Society, Tallahassee, Fla., Dec 1968. (2) H. 5. Frank and W. Y . Wen, Discussions Faraday SOC., 24, 133 (1957). (3! (a) F. J. Millero, J . Phys. Chem., 72, 4589 (1968); (b) F. J. Millero and w. Drost-Hansen, ibid., 72, 1758 (1968). Volume 73, Number 7 July 1969
FRANK J. MILLERO
2418 separate overlapping effects. I n recent paper^,^-^ we have tried to solve this problem by examining the effect of temperature on the Vo(ions) or the partial molal expansibilities of ions, Eo(ions). Other methods that can be used to elucidate the various components of Vo(ion) are to examine the effect of pressure on the VO(ion), the partial molal compressibility,6 or to examine the Vo(ions) in various solvents. I n this paper we will examine the Vo(ions) in the solvents water, methanol, N-methylpropionamide (NMP), and seawater by a simple model for ion-solvent interactions. The results will hopefully lead to a better understanding of the major components of the Po(ion) in these solvents.
Table I: The Partial Molal Volume of Ions in Various Solvents at 25' (ml/mol) Ion
H+ Li + Na + K+ Rb +
cs
+
Fc1BrINOa-
Crystals
... 1.2 2.2 5.9 8.2 12.2 6.3 14.9 18.7 25.4 21.1
NMP~
Seawatere
3.4
-3.7
6.0 11.1
-4.4 5.9
... ... ... ...
24.7 29.8
...
33.6
...
... ... ...
23.3 30.3 41.4 34.8
Waterd
-4.5 -5.2 -6.7 4.5 9.6 16.9 3.4 22.3 29.2 40.8 33.5
Methanole
-14.6 -14.0 -12.6 -2.6
...
...
...
9.3 16.6 24.3
...
a Calculated from the equation Vo(ion) in the crystal equals Partial Molar Volume of Ions in Various Solvents 2.52r3, using Pauling crystal radii quotedoin ref 3a except for at I n h i t e Dilution Li+, where the Goldschmidt radius, 0.78 A, was used. * From The additivity of the partial molal volumes of salts, ref 9. ' From ref 14. From ref 13, except for VO(F-)calculated from VO(NaF) given in ref 15. e From ref 10. Po(salts), at infinite dilution in the solvents water,' seawater,* NMP,9 and methanol,1° has been shown; however, the division of Vo(sa1ts) into their ionic components, Vo(ions), cannot normally be made by where a and b are constants obtained by a least-square direct experimental methods. Zana and Yeager" best fit of VO(ions)in the solvent, S and water, W. reported the first estimates of the Vo(ions) in water Figure 1 shows a plot of P(ions) in the solvents S based on a direct experimental method using an plotted us. the Po(ions) in water, determined by ultrasonic technique. However, until Zana and YeaMukerjee's method (Table I). The correspondence ger's experimental method" is used in other solvents, equation is obeyed (within experimental error) showing we are forced to use nonthermodynamic methods of that the two methods of dividing Vo(salts) are essendividing lio(salts). Mukerjee12 has summarized the tially the same. We obtain constants a = 0.814, various estimates of V0(H+)in water and finds Vo(H+) 0.764, 0.976 and b = -9.0, 8.0, 1.5 ml mol-l, respec= -4.5 ml/mol at 25". This value of VO(H+) agrees tively, for the solvents methanol, NMP, and seawater. very well with Mukerjee's earlier estimatela made by The most successful semiempirical c o r r e l a t i ~ nof~ ~ ~ ~ ~ assuming the BO of the monovalent monoatomic cations lio(ions) in water have been made using an equation of and anions were a smooth function of the crystal radii the form cubed. Since Mukerjee's method of dividing Vo(ions) Vo(ion) = Ar3 - B Z 2 / r (3) gives reasonable values in water and also is simple to P(disord) use, we have used his method to examine the Vo(ions) where the first term is equal to Vo(int) and the second term is equal to Vo(elect), A and B in various solvent^.^^^^^^^ His method has the added advantage of giving one relationship for the Po of (4) F.J. Millero, W. Drost-Hansen, and L. Korson, J. Phys. Chem., cations and anions, thus providing a uniform comparison between various solvents. Table I gives the P- 72, 2251 (1968). (5) F. J. Millero and W. Drost-Hansen, J. Chem. Eng. Data, 13, 330 (ions) in the solvents water, seawater, NMP, and (1968). methanol at 25" determined by Mukerjee's method (6) F.J. Millero and F. Lepple, paper in preparation. and K. J. Laidler, Can. J. Chem., 34, 1209 (1956). using Po(sa1ts) taken from the l i t e r a t ~ r e . ~ ~ ~ (7) ~ JA. ~ M. - ~Couture ~ (8) I. W.Duedall and P. K. Weyl, LimnoE. Oceangr., 12, 52 (1967). Recently, Ellis'B has used Criss and Cobble's corre(9) F. J. Millero, J. Phys. Chem., 72, 3209 (1968). spondence method" to estimate the Vo(ions) at various (10) F.J. Millero, unpublished results. temperatures based on Po(ions) a t 25". Criss and (11) R. Zana and E. Yeager, J . Phys. Chem., 70, 954 (1966); 71, coworkers18 have also used the correspondence method 621 (1967). to determine the partial molal entropy of ions in various (12) P. Mukerjee, ibid., 70, 2708 (1966). solvents from their values in water. Since VO(ions) in (13) P.Mukerjee, ibid., 65, 744 (1961). water a t 25" determined by various methods are in (14) F. J. Millero, Limnol. Oceangr., 14, 376 (1969). agreement,12the correspondence method may be a use(16) F. J. Millero, J. Phya. Chem., 71, 4567 (1967). (16) A. J. Ellis, J. Chem. SOC.,A , 1138 (1968). ful method of dividing the Bo(ions) in various solvents. (17) C. M. Criss and J. W. Cobble, J . Arne?. Chem. SOC.,86, 5385 Using the correspondence method, one obtains the (1964). equation (18) C. M. Criss, R. P. Held, and E. Luksha, J . Phya. Chem., 72,
+
V0(ion)' = aVo(ion)w The Journal of Physical Chemistry
+b
(2)
2970 (1968). (19) L. G.Hepler, ibid., 61, 1426 (1957).
THEPARTIAL MOLAL VOLUMEOF IONS IN VARIOUS SOLVENTS
2419
401
20 t;O”)*
0
-20
Figure 1. The partial molal volume of an ion in solvent, S, QO(ion)s, plotted vs. the partial molal volume of an ion in water, P(ion)w.
are constants, 2 is the charge on the ion, and r is the crystal radius of the ion. Recently, various workers20*21r22 Figure 2. Semiempirical constants, A , A’, and a for the have attempted to separate the intrinsic size of an ion ffo(int) + ffo(discord) term of eq 3, 4, and 5 (respectively) and void space effects (ie., VO(int) and Vo(disord)) by plotted os. D,the dielectric constant of the solvents. using the semiempirical equations
+ A’r2 - B’Z2/r - B”Z2/r P(ion) = 2.52(~+
Vo(ion) = 2.52r3
(4) (5)
by assuming the void space is proportional to the surface of the ion (eq 4)20or the void space could be represented by a spherical shell around the ion (eq 5).21 The constants for these equations have been evaluated for the Po(ions) in methanol, H20, seawater, and NMP a t 25’ graphically and by a least-square best fit method (with the aid of an IBM computer). The constants are tabulated in Table I1 (along with the standard deviations) and plotted vs. D and 1/D where D is the dielectric constant of the solvent (Figures 2 and
3) * The comparison of the constants A , A’, and a (or the Vo(int) and Vo(disord) for ions) is shown in Table I1 and in Figure 2 plotted vs. D, the dielectric constant of the solvents (D(Me0H) = 31.5,23D(HzO) = 78.5,23 D(NMP) = 1769). The constants for seawater are not given in Table I1 (or shown in Figure 2 and 3) because they may be meaningless due to ion-pair formation.’* The constants A in methanol, water, and NMP are larger than the theoretical value calculated by assuming = that the ion is a perfect hard sphere (4/3 nN X 2.52). Thus, Vo(int) appears to increase by 32% in methanol, 77% in water, and 59% in NNIP. The magnitude and order of the constants A, A’, and a show no simple correlation to common physical properties such as dielectric constant (Figure 2). The constants are also not in the order expected (NRSP < HzO
B(Hz0) > B(NR4P)) based on the assumption that P(e1ect) is proportional to 1/D or p, the compressibility of the solvents. From the simple continuum model one can calculate the electrostriction of an ion using the Drude-NernstZ4 equation ~
-NZ2e2 d In D -BZ2 - =2Dr [ d P ] r
(6)
This equation serves only as an approximation and is valid only for large, isolated ions. Using the most recent value for d In D/dP and D for water26 and methanol,26 values of B = 4.2 in H2O and 24 in MeOH are obtained. A similar calculation for NMP is not possible since d In D l d P is not known. The agreement between the calculated values of B in methanol and water with the correlation values is fair, considering the approximate nature of the Drude-Nernst equation. Po(e1ect) can be calculated by more elaborate methods by considering dielectric saturation effects;27 however, the calculations are normally complex and cannot be The Jozirnat of Phusical Chemistry
A
rt 1 . 6 ml/rnol f0.B
10.4
- B’Z2/ra
B‘, ml mol-’
A
Std dev
16 12 10 11 2*2
rtl.5 10.8 f0.4
+ a)a - B”Z2/rb B”, ml mol-1
Std dev
I
Figure 3. Semiempirical constants, B, B’, and B” for the Po(e1ect) term of eq 3, 4, and 5 (respectively) plotted vs. 1/D, the recipocal of the dielectric constant of the solvents.
P(e1ect) =
a,
Std dev
16 12 8rtl 3 f 2
Eq 4, vo(ion) = 2.52~4 A V Solvent
A
ml mol-’
0.20 =I=0.05 0.46 10.05 0.34 i 0.02
16 f 2 11 1
11.3
2*2
10.4
*
a Constants determined by graphic methods. determined by least-square “best” fit method.
f0.9
Constants
made for the solvents, methanol and NRIP, due to the lack of data. If one assumes that frO(elect)is proportional to the compressibility of the solvent^,^^^^^ one would expect Po(elect) in methanol to be about 2.7 times as great as in water, which is in fair agreement with the B constants. (The compressibility of NMP has not been measured to the best of our knowledge.) Since the Po’sof polyvalent salts are not available in the solvents methanol and NMP, little can be said about the effect of charge on the electrostriction or the B constants. As more data become available for the solvent systems mentioned in this paper (and other solvent systems) for other monovalent and polyvalent ions as a function of temperature, the correlation equations can be improved and more can be learned about ion-solvent interactions in various solvents.
Acknowledgment. The author wishes to acknowledge the support of the Office of Naval Research, Grant No. NONR 4008 (02), for this study. (24) P. Drude and W. Nernst, Z. Phys. Chem., 15, 79 (1894). (25) B. B. Owen, R. C. Miller, R. C. Milner, and H. L. Cogan, J . Phys. Chem., 65, 2065 (1961). (26) J. Padova and I. Abrahamer, ibid., 71, 2112 (1967). (27) J. E. Desnoyers, R. E. Verrall, and B. E. Conway, J . Chem. Phys., 43, 243 (1965). (28) R. E. Gibson, J . Amer. Chem. SOC.,59, 1521 (1937). (29) G. S. Kell and E. Whalley, Phil. Trans. Roy. SOC. (London), 258, 1094 (1965).
