DIFFUSION IN THE WATER-METHAXOL SYSTEM
March, 1963
of AEc" for 3-methylpyridine differ significantly. The reason for these discrepancies is not apparent. TABLE X THEhfOL.4L ENTXOPY OF 3-METHYLPTBIDINE I N THE IDEAL GASSTATEIN GAL. DEQ.-' 393.36 417.29 372.45 T , "K. 62.98 65.61 60.62 S,( liy. )" 23.63 21.40 25.77 AHv/T 0.17 0.08 0.11 s* - sb -2.76 -1.38 0.00 Iz In P"
--
___
So (obsd.) i 0.17d 83.71 85.34 87.18 a By interpolation in Table VI or extrapolation by use of eq. 1. * T h e entropy in the ideal gas state less that in the real gas state, calculated from eq. 5. 'Entropy of compression, Estimated accuracy uncertainty. calculated from eq. 3.
TABLE XI COMBUSTION CAIIORIMETRICESPEEIMEKT SUXMARY O F A TYPICAL WITH 3-METHYLPYItll)INEa m' (3-methylpyridine), g. 0.86940 1.99846 At, = tr - ti - Ate,,, deg. - 8028.73 E (Calor.) (-At,), cal. -8.92 G (Cont.) ( - A L , ) , ~cal. A E i g n , tal. 1.16 AEfdeo (€IS03 HNOz), cal. 16.19 3.94 A E , cor. to st. states: cal. 380 83 -?n''AEco/M (auxiliary oil), cal. -n%'"AEco/M (fuse), cal. 3.89 m'AEc" /;If (3-methylpyridine), ea!. AEc0/M (3-methylpyridine), cal. g.
-7631.64 -8778 05
689
Auxiliary data: E(Ca1or.) = 4017.46 cal. deg.-l; V(bomb) = 0.344 1.; AEc"/M(auxiliary oil) = -10984.1 cal. g.-I; AEc"/M €i(Cont.) ( t i - 25') Gf(Cont.) (fuse) = -4050 cal, g.-1. (25" - If A&,). cItems 81-85, 87-90, 93, and 94 of the computation form (I, ref. 24).
+
+
TABLE XI1 SUMMARY OF RESULTSOF COMBUSTION CALORIMETRY AT 298.15"K. AEc"/M (3-methylpyridine), csl. g.-1: - 8778.07, -8776.79, - 8778.05 -8776.84, -8779.20, -8778.47 Mean and std. dev.: -8777.90 i 0.42 Derived Results for the Liquid State" AEc", kcal. mole-' -817.43 i 0.12 -818.17 i 0.12 AHc', kcal. mole-' AHf", kcal. mole-l 14.75 i 0.14 A&"", cal. deg.-l mole-l -88.59 AFf", kcal. mole-% 41.16 1% Kf -30.17 With uncertainty interval equal to twice the final" over-all" standard deviation.
Acknowledgment.-The assistance of W. T. Berg and J. L. Lacina in some of the experimental measurements is gratefully acknowledged. The authors thank Drs. F. A. n/Iiller and W. G. Fateley of' the Mellon Institute for obtaining the far-infrared (grating) spectrum.
DIFFUSION I1J THE WATER-METHANOL SYSTEM AND THE WALDEN PRODUCT BY L. G. LONGSWORTH Rockefeller Instatute, New Yorlc, X.Y. Received September E l , 196% The recent work of Harned and associates affords convincing experimental evidence for the Nernst postulate that the mobility of a particle in a liquid is independent of the nature of the force responsible for the motion. From measurements, with the aid of Rayleigh interferometry, of the diffusion coefficients of non-electrolytes in dilute solutions in water, methanol, and their mixtures the limiting mobilities, XO, of uncharged particles thus have been obtained. The variation, with the solvent composition, of the Walden product, Xoq, where 9 is the viscosity of the solvent, is less for a non-electrolyte than for a monatomic ion of comparable mobility and decreases with increasing size of the particle. Modification of the Stokes-Einstein relation with the extension, to methanol, of Robinson and Stokes' suggestion that the tetraalkylammonium ions are unhydrated permits computation of solvation numbers for other solutes in both solvents. For a given non-electrolyte, this number is slightly less in methanol than in water, indicating that the low value of XU? for small particles in methanol is due primarily to the relatively large volume of the alcohol molecule in the solvation shell. In no case does the solvation number exceed the number of hydrogen bonding sites on the solute molecule.
The failure of Walden's rule when applied to the alkali halide ions in water, methanol, and their mixtures has been the subject of several inve~tigations.l-~ The low values of XOV in methanol relative to those in water are usually ascribed to increased solvation. The decrease of some 20 ml. in the apparent molal volumes of the alkali halides4 on transfer from HzO to CH30H implies electrostriction of the solvent near the ions but could result from the higher compressibility of methanol without significant alteration in the solvation number.
There also is the quantitative failure of Stokes' relation, on which Walden's rule is based, when the solute particles are comparable in size with the solvent molecules, in which case the numerical factor is less than 6n. This is observed in both polar5s6and non-polar systems,7 a striking example being afforded by the observation of Grun and Walz8 that tetrabromoethane diffuses five times as rapidly in glycerol trioleate as may be computed from the molal volume of the solute with the aid of the Stokes-Einstein relation. I n self-diffuson,
(1) L. G. Longsworth a n d D. 9.MacInnes, J . Phys. Chem., 43, 239
( 5 ) S. Glasstone, K. J. Laidler, and H. Eyring, "The Theory of Rate Processes," rMcGraw-Hill Book Co., Inc. New York, N. Y., 1941. (6) L. G. Longsworth, "Electrochemistry in RioloEy and Medicine," T. Shedlovsky, Ed., John Wlley a n d Sons, Inc., New York, N. Y., 1955, chapter 12. (7) E. R. Hammond and R. 1% Stokes, Trans. Faradav Soc., 61, 1641
(1939).
(2) R. E. Jervis, D. R. Muir, J. P. Butler, and A. R. Gordon, J. Am. Chem. Soc., 76, 2855 (1953). (3) N. G. Foster and E. S. Amis, Z . phpsrlc. Chem. (Frankfurt), 3, 365 (1955); 7 , 360 (1956). (4) W. C. Vosburgh, L. C. Connell, and J. 9.V. Butler, J . Chem. Soc., 933 (1933).
(1955).
(8) F. Grun a n d D. Walz, HeEv. C h h . Acta, 44, 1883 (1961).
L. G. LOXGSWORTH
690
the particles are similar in size and Ottarg used, in ef~ his computation of the yiscosifect, a factor of 2 . 9 in ties of several pure liquids from their self-diffusion coefficients and molal volumes. Assuming the tetraalkylammonium ions to be unhydrated and of known size, Robinson and Stokes1”have suggested that these be used to evaluate the numerical factor in Stokes relation, and Nightingalell adopted this procedure in a review of ion hydration. Although primary solvation probably results from ion-dipole interaction, hydrogen bonding doubtless contributes to secondary solvation. It appeared of interest, therefore, to compare the mobilities of ions in the water-methanol systeni with this property of nonelectrolytes, where the solute-solvent friction is restricted to that resulting from dipole-dipole interaction in the form of hydrogen bonding. Diffusion measurements of non-electrolytes afford much the same iiiformation concerning solute mobility as do conductance and transference measurements on electrolytes. Moreover, the simple dependence of the diffusioii coefficient of a non-electrolyte on concentration facilitates extrapolation to infinite dilution. Experimental The experimental procedure for the diffusionmeasurements was the same as in the work with heavy waterI2 except that evaporation from the reservoirs during the diffusion period was minimized by displacing most of the air therein with loosely fitting tubes whose lower ends were closed. Densities were also measured as in that work. The methanol was the middle fraction of a synthetic material and was used without further treatment since its density a t 25O, the temperature a t which all measurements were made, was that of the anhydrous alcohol, 0.78655. Although the solvent mixtures were prepared by direct weighing of both components into glass-stoppered flasks, the subsequent de-aeration in an oven a t 45”, and occasional brief exposures to the atmosphere made i t preferable to determine the composition of a solvent from a density measurement at the time the solutions were prepared. The density-composition data of Gibson13 were used for this pirpose. With only a few exceptions, the solvent composition differed insignificantly from che round values given in Table I1 and where nesessary the diffusion coefficients have been corrected to the tabulated mole fraction of methanol. Solute concentrations, m, are expressed as moles of solute per thousand grams of solvent. The solutes were of reagent grade and, in the case of the solids, were used without further purification except for vacuum desiccation. The formamide and glycerin were distilled under reduced pressure and the fractions having densities of 1.12929 and 1.25787, respectively, were used. The ethylene glycol had a density of 1.10982, whereas that of the polyethylene glycol of average molecular weight 600 (PEG 600) was 1.12212. The polyethylene glycol of average weight 3350 (PEG 3350) was a solid. In all cases where comparison with diffusion data of modern precision is possible, ;.e., aqueous urea,14 formamide,’j aoetamide,la glycolamjde,l7 dextrQse,ls sand a u c r ~ s ethe , ~ ~piesent results agree within0.1Ojo or better. If the solutes were sufficientIy soluble, diffusion measurements were made over three or more concentration intervals to permit linear extrapolation. A concentration difference was used that gave approximately 50 fringes in a 2.5 cm. channel. A solution (9) B. Ottar, “Self-diffusion and Fluidity in Liquids,” Oslo University Press, 1958; of.* Acta Chem. Scand., 9, 344 (1955). (10) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions,” 2nd Edition, Butterworths Scientific Publications, London, 1959, p. 125. (11) E. R. Nightingale, J . Phys. Chem., 68, 1381 (1959). (12) L. G. Longsworth, ibid., 64, 1914 (1960). (13) R. E. Gibson, J . Am. Chem. Soc., 67, 1551 (1935). (14) L. J. Gosting and D. F. Akeley, ibid., 74, 2058 (1952). (15) J. G. Alhright and L. J. Gosting, J . Phys. Chem., 64, 1537 (1960). (16) A. Biancheria and G. Kegeles, J . Am. Chem. Soc., 79, 5908 (1967). (17) P. J. Dunlop and L. J. Gostinn, i b i d . , 76, 5073 (1953). (18) L. G. Longsworth, ibid., 76, 5705 (1953). (19) L. J. Gosting and M. S. Morris, ibid., 71, 1998 (1949).
Vol. 67
of about the same concentration as that below the boundary in one experiment was the dilute solution above the boundary in the experiment at the next higher mean concentration. The small departures from ideal diffusion in the mixed solvents were not significantly different from those in the pure ones and no attempt has been made to correct for possible flow coupling in the threecomponent systems. Unless limited by the solubility, e.g., erythritol and dextrose, the apparent molal volumes, +, were determined at a molality of 0.5.
Results The experimental results in water and methanol are summarized in Table I, whereas those in the mixed solvents are given in Table 11, together with the pertinent values from Table I for completeness. I n Table I the limiting value of the diffusion coefficient, D , is given together with the slope, ADlArn, of the line through the experimental points. The figure in parentheses to the right of each slope in Table I is 100 times the highest mean moIaIity a t which diffusion measurements were made. The slopes for aqueous urea,14 formamide,ls and glycolamidel’ are taken from the more complete work of Gosting and associates, whereas the values for HzO in mater are those of HDO,lZ the isotope effect found by Wang, Robinson, and Edelman20 being neglected. The L‘~~lvatioii” numbers, n, in the last two columns of Table I are considered in the discussion. There the Stokes radius, rs, which is inversely proportional to the Walden product, frequently will be used for convenience in comparison with other estimates of particle size. The relationslo are ys
=
IcT/GayD
=
~F~/6irN7Xo
where x is the valence of the ion whose limiting equivalent conductance is Xo, F is the Faraday equivalent and the other quantities have their usual significance. For interconversioii of a volume and a radius the relation V = 47rNrva/3has been used, V being the molal volume of the pure component. Discussion I n Fig. 1 the logarithm of the Stokes radius in methanol is plotted as ordinate against the same function of this radius in water as abscissa. The values for non-electrolytes were computed from the data of Table I, whereas those for ions were taken from the compilations of Robinson and Stokes1”and Evers and Knox.21 On this plot any solute on the left of the diagonal line through the origin has a larger Stokes radius, ie., a smaller Walden product, in methanol than in water. It will be noted that the alkali halide ions are farther to the left of this line than the non-electrolytes, only the higher tetradkylammonium ions having larger Stokes radii in water than in methanol. The electrical screening effect of the large alkyl groups and their inability to form hydrogen bonds justifies the assumption that these ions are unhydrated. Froni estimates of the radii, r h , of the tetraalkylammonium ions, and a plot of against their Stokes radii, rs, in water, Robinson, and StokeslO have computed hydration numbers, n, for the monatomic cations having comparable rs values with the aid of the relation n = ( V , - V)/Vo
Here V h = 47rAVrh8/3, V is the molar volume of the solute and Va that of the solvent, 18 cc./mole for water (20) J. H. Wang, C. V. Robinson, and I. S. Edelman, z h d , 76,466 (1953). (21) E. C. Evers and A. G. Knox, zbzd., 75, 1739 (1951).
DIFFUSION IN
March, 1963
THE
691
WATER-METHANOL SYSTEM
TABLE I DIFFUSION COEFFICIENTS AND APPARENTMOLAL VOLUWES IN WATERAXD METHAKOL AT 25' ----105~, cm.2/s---
r_-
- 106 AD/Am-
7 4 , ml./mole-
-nCHsOH
2.2 1.8 2.3 2.0 2.4 2.8 2.4 2.2 2.9 2.4 3.5 4.4 6.7"
1.0
1.597 1.658 0.9225 .7303 ,6339 .2756 .355 .465 37.6 44.2 0.542
0.8 -
0.6-
109 rs (CH30H!
cdbK+ Os4-
OBr'
EG
5
ethylene glycol
(3-
Gly
i
glycolomide
MU = methyl urea
M,U= dimethyl ureo
%*
P = propionomide
0.2
or 40 cc./mole for methanol. This procedure effectively modifiea the Stokes factor for small solutes. Over the range in r, covered by the alkylammonium ions in water, Le., 2.0-5.2 A., the relation is ?'h
k 0.04 = 0.546~8$- 2.408 A.
(1)
and this has been used in lieu of a graph for estimating the solvation numbers in column 8 of Table I. In methanol, column 9 of Table I, the relation for the reference ions is
rt,
f
0.07 = 0.75r,
+ 2.10 A.
(2)
692
L. G. LOXGSWORTFI
Vol. 67
xm Fig. 2.-The
Stokes radii, normalized to unity in water, of representative electrolytes and non-electrolytes as a function of the mole fraction of the alcohol in water-methanol mixtures
and the interval is 1.7 < r, < 4.2. I n this solvent carbon tetrachloride, with rs = 1.745, r, = ?h = 3.37 and a scattering radius of 3.2 is., affords an additional reference particle and was used in obtaining eq. 2. I n both solvents the tetramethylammonium ion exhibits the largest deviations from eq. 1 and 2 and accounts for much of the indicated uncertainty in r h . This uncertainty is reduced appreciably if rv for (CHI)$ replaces r h for (CH3)4N+. I n most cases the value of n for a solute in methanol is somewhat less than in water, from which it is clear that the low value of the Wdden product in alcohol is due to the volume of this molecule in the solvation shell. I n no instance does n exceed the number of hydrogen bonding sites on a solute molecule if both proton donor and acceptor sites are counted. However, the correlation of n with the volume coiitraction, columns 6 and 7 of Table I, is poor. Over the size range of the reference ions the Stokes factor in water is nearer 6n for a given solute than in methanol. Moreover, eq. l indicates that 67r becomes the appropriate factor a t q, = 7, = 5.3 in water, whereas in methanol, eq. 2 must be extrapolated to ? h = r, = 8.4. For small polar solutes in a polar solvent the Stokes factor is clearly not a simple function of the size of the solute particle relative to that of the solvent. The solvation numbers of Table I are less than those of monatomic ions of comparable size. In the case of the alkali halide ions the modified Stokes relation leads to a fairly consistent set of numbers. I n water, n decreases monotonely with increasing crystal radius from 7.0 for Li+ to 3.3 for Cs+'O and from 3.2 for C1- to 2.6 for I-, whereas in methanol the decrease is from 7.5 to 3.6 for the cations and from 4.5 to 3.1 for the anions. The higher values for the anions in methanol
than in water may not be significant since eq. 1 has been used somewhat outside of its range of validity for these ions in water but there is also the possibility that the methyl group of the alcohol has enhanced the affinity of the proton for anions. Two-Component Solvents.-The variation of the Walden product in the mixed solvents cannot be coiisidered independently of the non-ideal behavior of the viscosity. As the mole fraction of methanol, x,, is increased the viscosity a t first increases from the value for water a t 25", 0.00895 poise, to a maximum of 0.0160 at x, II 0.27 and then decreases to 0.00542 for methanol. The positive departure of the viscosity from additivity is accompanied by a contraction in volume on mixing the components that reaches a maximum value of 1 cc. per mole of mixture a t x, = 0.48. This negative departure of the solvent volume from additivity is accompanied, in turn, by positive deviations of the volumes of the two solutes that were studied, urea and thiourea, Table 11. I n Fig. 2 the Stokes radii, normalized to unity in water, for a large and a small non-electrolyte from Table I1 are compared, as a function of x,, with those for representative electrolytes, Le., hydrochloric acidJZ2 tetraethylammonium bromide3 and picrateJZ3and man~ transference ganese (11) benzene d i ~ u l f o n a t e . ~Since data are not available for computation of limiting ion conductances in the mixed solvents the values of ra for the electrolytes are a harmonic mean of the anion and cation. The minimum change in the Walden product is that for the large non-electrolyte and amounts to 7%, (22)
T.Shedlovsky and R. L. Ihy, J. Phys.
(23) F. Accascina,
Chem., 60, 151 (1950).
A. D'Aprano, a n d R. M. Fuoas, J . Am. Chem. Soc., 81,
1058 (1959). (24) C. J. IIallada and G. Atkinson, abid., 83, 3759 (1901).
March, 1963
DIFFUSION IN
THE
whereas the variation in solvent viscosity is nearly 300%. No exceptions have been noted to the observation, illustrated in Fig. 2, that on the addition of methanol to water the initial effect is a decrease in the Stokes radius. This is in accord with the results of Stokes25on the limiting mobilities of ions in aqueous solutions whose viscosity has been raised by the addition of non-electrolytes of relatively large molal volume. Invariably, the depression of the ion mobility is less than the increase in the bulk viscosity but exceeds that to be expected if the added molecule is considered as an obstruction that increases the length of the path taken by the solute particle but not the viscosity of the medium in which this particle moves. This model is a stage in the transition from a fluid continuum to a porous medium and thus makes available the literature on tortuosity for the interpretation of experimental results. The model is applicable to ions and nonelectrolytes and provides, as Stokes has noted, an explanation for the modified Walden relation, Xoqa = constant, where a is less than unity. In the water-methano1 system, however, the Stokes radius of small particles also decreases when water is added to methanol. Here the decrease could be attributed to replacement of methanol in the solvation shell by the smaller water molecules. F U O Shas S~~ interpreted the variation of the Walden product for an ion in a two-component solvent, only one of which is polar, as a braking effect resulting from the orientation of the solvent dipoles ahead of, and their subsequent relaxation behind, the advancing ion. The effect increases with decreasing dielectric constant. If this braking effect also operates in the case of a moving dipole it would contribute to the observed change in the Stokes radius for the methanol-rich mixtures, since the dielectric constant in this system decreases monotonely with x,. I n this system, however, both solvent components are polar and the non-ideality of their mixtures implies that other effects also are operating. The cathodic transfer of water by hydrochloric acid and the alkali chlorides in the presence of a reference substance implies not only that the cation solvation exceeds that of the anion but that the solvation shell is rich in water. The assumption is tacit here that the anion is not preferentially solvated by the reference material. With the aid of an e.m.f. method, Feakins27 has obtained Washburn numbers for these electrolytes, with methanol a t z, = 0.06 as the reference, that are comparable with the values derived from electrophoretic (25) R. H. Stokes, in “The Structure of Electrolytic Solutions.” W. J. Hamer, Ed., John Wiley and Sons, Inc., New York, N. Y., Chapter 20, pp. 298-307.
(26) R. M.Fuoss, Proc. Nut. Acad. Sci. U.S., 46,807 (1959). (27) D. Feakins, J . Chem. SOC.,6308 (1961).
WATER-METHANOL SYSTEM
693
methods using other reference materials. As x, --* 1 a t what solvent composition, then, is the water in the solvation shell replaced by methanol? The curves of Fig. 2 suggest that the water content of this shell exceeds xw over the entire range of composition. In the case of the hydronium ion of HC1, the water shell essential for its abnormal conductance remains intact up to xm % 0.15, whereas only a trace of water destroys the methanol shell that is required for the Grotthus mechanism a t xm == 1. From a comparison of MnBDS and (Et)4NPic in Fig. 2, it is clear that the monatomic ions are chiefly responsible for the variation in the Walden product. In pure methanol r,/r,(H20) = 3.04 for MnBDS and 90% of the increase in the ratio occurs above x, = 0.7, these high concentrations being required to replace the tightly bound water around the compact divalent manganese ion with the larger methanol molecules. Although the Walden product for a small non-electrolyte varies less with xmthan that for a monatomic ion of the same mobility, the parallel behavior of the curves in Fig. 2 implies that the water content of the non-electrolyte solvation also exceeds xw. Insofar as the solvation numbers represent a preferential interaction of the solute with one component of a mixed solvent, it should be possible to relate them eventually to the cross-term diffusion coefficients in these threecomponent systems. There remains the possibility that the solvation numbers afford a measure of hydrogen bonding. The clearest evidence for such bonding is afforded, however, by work in progress on N,N-dimethylacetamide and the parent amide in methanol. Although the volume of the dimethyl compound is 82% greater than that of acetamide it diffuses 10% faster. In methanol the solvation numbers are 0.8 and 2.3, respectively. Elimination, by N-methylation, of the only two proton donor sites on acetamide has thus reduced the solvation number by 1.5 and methanol is too weakly acidic to take advantage of the proton acceptor sites on the amide oxygen. In water the effect is somewhat less, dimethylation reducing the numbers from 3.0 to 1.9. The weakly acidic character of methanol is also apparent in the case of a related solute, acetone, that is not a proton donor. The solvation numbers for this solute are 1.6 and 0.0 in water and methanol, respectively. Other factors that contribute to the strength of the hydrogen bonds in these systems have been reviewed recently by Huggins.28 Acknowledgment.-The author is indebted to D. A. MacInnes for a review of this manuscript and to Emilia Jurevicius for the analysis of the Rayleigh fringe patterns. (28) M.
L. Huggins, American Scknfbt, 60, 485 (1962).
