1830
0. W. EDWARDS AND E. 0. HUFFMAN
The formation of metastable phases as intermediate products during heterogeneous chemical reactions is, of course, a well-known phenomenon expressed in "Ostwald's Stufenregel."g However, the present case seems to be different from any observation reported previously in the literature. Our case is one in which a phase of one structure first transforms to a metastable phase of a different structure which then in turn reverts back to the original structure, all at a constant temperature and constant 0 2 partial pressure. This is made possible because compositional variations occur in one of the structures involved in the equilibrium. The observation has a very significant bearing on criteria used for judging what the true equilibrium phase is in heterogeneous reactions. It is generally assumed that when a crystalline phase A of a certain struc(9) W. Ostwald, Z . physik. Chem., 22, 289 (1897).
Vol. 63
ture is observed experimentally to convert to a second phase B of a different crystalline structure under a given set of experimental conditions, then structure B is stable relative to structure A under these conditions. The present investigation has shown that this is not necessarily so. Compositional variations in one of the phases may cause formation of the metastable phase B although A is the truly stable crystalline structure under the prevailing conditions. The present experience has shown that very critical evaluation of observations made in phase equilibrium studies is necessary in order to avoid wrong conclusions. Acknowledgments.-The ideas presented in this paper resulted from experimental work carried out as part of a research project on phase equilibrium relations in oxide systems sponsored by the American Iron and Steel Institute.
DIFFUSION OF AQUEOUS SOLUTIONS OF PHOSPHORIC ACID AT 25' BY 0. W. EDWARDS AND E. 0. HUFFMAN Division of Chemical Development, Tennessee Valley Authority, Wilson Dam, Alabama Received March 81. 1969
Diffusion coefficients of phosphoric acid solutions at 25" were measured with a two-lens Gouy diffusiometer over the concentration range 0.036 to 16 M . The c,oefficienta decrease from 10.41 X 10-6 cm.2 sec.-l at 0.036 M to 8.29 X 10-6 at 1 M, remain nearly constant between 1 and 5 M , and decrease from 7.84 x 10-6 cm.2 sec.-l a t 5 M to 1.32 X 10-6 cm.2 sec.-l a t 16 M. The recision of the D values is f0.3%. An extrapolation leads to 7.6 X 10-6 em.* sec.-1 for Do,, the hypothetical limiting difksion coefficient for undissociated phosphoric acid a t 25'. This value, along with hydrodynamic considerations, suggests that a hydration shell surrounds the undissociated molecule. The ratio D o ~ * ~ , - / D oism1.15.
Extensive inquiry1-'8 into the physicochemical properties of phosphoric acid has paralleled a remarkable growth in the industrial importance of the acid in recent years. Study of the diffusion of aqueous solutions of phosphoric acid offers means for gaining additional insight into the nature of this complex electrolyte. Of particular interest are the relative mobilities of the phosphate anion and the neutral molecular species-a comparison of the type recently reported for the weaker electrolytes acetic acid14and citric acid. l5 (1) J. H. Christensen and R . B. Reed, Ind. Eng. Chem., 47, 1277 (1955). (2) E. P. Egan, Jr., B. B. Luff and Z. T . Wakefield, THISJOURNAL, 62, 1091 (1958). (3) K. L. Ellnore, C . M . Mason and J. H. Christensen, J. A m . Chem. Soc., 68, 2528 (1946). (4) N. N. Grenwood and A. Thompson, Proc. Chem. Soc. (London), 352 (1958). (5) E . 0. Hnffman, J. D. Fleming and A. J. Smith, Ind. Eng. Chem.. Chem. Eng. Data Series, 3 , 17 (1958). ( 6 ) 0 . W. Edwards and E. 0. Huffman, ibid., 3, 145 (1958). (7) C. M . Mason and J. B. Culvern, J . A m . Chem. Soc., 71, 2387 (1949). (8) A. J. Smith and E . 0 . Huffman, Ind. Eng. Chem., Chem. Eng. Data Series, 1, 99 (1956). (9) C . M . Mason and W. M . Bluni, J . Am. Chem. Soc., 69, 1246 (1947). (10) R . Ripen and C. Liteanu, Acad. Rep. Populare Romdne, Bul. StiinL., A l , 387 (1949). ( 1 1 ) H . Sadek, J . Indian Chem. Sac., 29, 84G (1952). (12) RI. Kerker and W. F. Espenscheid, J. A m . Chem. Soc., 80, 776 (1958). (13) K . N. Bransoombe and R. P. Bell, Disc. Faraday SOC.,NO.24, 158 (1957). (14) V. Vitagliano and P. A. Lyons, J. A m . Chem. SOC.,7 8 , 4538 (1950).
Here we describe measurements of diffusion coefficients of phosphoric acid at 25" over the concentration range 0.036 to 16 M . Made by means of a Gouy interferometer, the measurements led to a value for the limiting diffusion coefficient of undissociated phosphoric acid. The results expand greatly upon the few integral values heretofore available. l6 Measurements Apparatus.-A two-lens Gouy diff usiometer was constructed by modifying a diffractometer that was similar in design and dimensions to Buerger's apparatus .I7 The light source, slit, filter and cell masks of the interferometer were essentially identical to those described14*18for single-lens instruments, whereas the camera and the plate masks were modified for convenience of operation. The two lenses were almost identical plano-convex air-spaced doublets, 6 in. in diameter, corrected for chromatic and spherical aberration. The components were mounted on an H-beam that was supported, through intervening cushiona of cork and of rubber, by a continuous concrete pier. The water-bath was mounted d!rectly on the beam between the lenses by means of an adjustable foot assembly that fit holes in the beam. The windows of the bath were single optical flats in holders that were adjustable about the two axes erpendicular t o the optic axis. The temperature of the batg was 25 f 0.005". The diffusion cell was a tall-form Tiselius cell whose top section extended above the bath liquid. The cell holder was of standard design18JQwith the usual cell masks and a rack(15) G . T. A. MWer and R. H. Stokes, Trans. Faraday Soc.. 53, 642 (1957). ( l G ) L. W. bholm, Finska Kemistsamfundets Medd., 30, 09 (1921). (17) M . J. Buerger, J. A p p l . Phus., 21, 909 (1950). (18) L. J. Gosting, E . M . Hanson, G . Kegeles and M. S. Morris, Rev. Sci. Instr., 20, 209 (1949). (19) L. J. Gosting, Thesis, University of Wisoonain, 1947.
Nov., 1959
DIFFUSION OF AQUEOUS SOLUTIONS OF PHOSPHORIC ACID
ing device for the boundary-sharpening siphon. The holder was supported by a separate bottom plate, adjustable about horizontal and vertical axes by means of screws. The components of the interferometer and the diffusion cell were optically aligned by the usual procedures.19 Calibration of the instrument against solutions of potassium chloride and of sucrose20*21 showed the optical lever arm (“b” distance) to be 232.25 cm.-a value reproducible within &0.05%. The “b” distance as determined by means of a wire grating22*2aagreed with the other calibration within
I
1831
I
0 - NERNST LIMITING VALUE
ZtO.l%.
Preparation of Solutions.-Solutions of phosphoric acid were prepared from the triply crystallized hemihydrate. Duplicate determinations of density at 25”, with correction for the buoyancy of air, agreed to 1 part in 10,000. Concentrations were found from tables of density.’ Procedure.-Procedures for photographing and analyzing the interference patterns have been d e s ~ r i b e d . ~ ~Dis#~’*~~ placements of the Gouy fringes were measured with a microcathetometer (Gaertner M930-342), which had been calibrated against a standard scale. The usual starting time correctionS4At was applied to each determination. Results.-Each diffusion coefficient reported in Table I is the average of values calculated from 14 to 19 patterns that Fig. 1.-Diffusion coefficients for phosphoric acid a t 25’. were obtained within a 2- to 2.5-hr. period. The precision of With exclusion of the At corrections a t the two highest conD is about ZtO.3%. The deflection constant Ct did not drift significantly when the average concentration exceeded centrations, At averaged 22 sec. and ranged from 7 to 44 sec. 0.15 M. A drift of 0.730 2.0% in Ct at concentrations be- Although At amounted to 75 and 144 sec. a t the two highest low 0.14 M suggested that these boundaries were non-Gaus- concentrations, the precision of D a t these points was sian, probably in the sense that they were skew. The con- Zt0.2%. The two high values of At may have reflected atcentration dependence of D 111 the concentration range 0.036 tack on the lubricant between the flanges. A plot of D against t/C is shown in Fig.1. to 0.14 M (Table I) was about 10 times that a t concentrations above 9 M , and was many times that a t intermediate Discussion concentrations. Although generally small a t low concentrations. Ac doubtless was high enough a t concentrations For a partially ionized electrolyte, Fick’s first (2) below 0.14 to give skew boundaries because of the high law leads to the equation concentration dependence of D. D = lOOORT TABLE I GOUYDATAFOR PHOSPHORIC ACID SOLUTIONSAT 25’0 where a,, the activity of the undissociated species in
(”) c d,l;nacu
D X lo6 ( A n l l c ) x lo4 C Ac jm 0.0719 32.46 0.0360 10.41 98.49 .0724 32.46 10.43 97.81 .0362 .0972 43.16 10.20 96.87 .0486 .1434 61.53 9.912 93.61 .0714 .1434 61.57 9.948 93.67 .0717 .1408 56.63 .1422 9.121 87.71 .1426 55.62 8.908 86.G2 .2143 .1443 55.91 8.693 84.52 .3586 .1976 76.08 8.530 84.00 .4992 1.0846 .2115 79.40 8.297 81.90 1.9112 .1669 59.23 8.141 77.42 .1823 64.34 2.643 8.03, 77.00 ,2155 73.39 7.978 74.30 3.940 ,1680 57.31 4.872 7.848 74.60 5.650 .2125 68.68 7.698 70.51 .2299 72.98 6.439 7.454 68.30 .1732 53.33 6.870 67.17 7.772 .2742 79.24 6.268 63.04 8.940 10.238 .1876 52.51 5.302 61.06 .2732 67.72 3.048 54.08 13.257 .2767 59.61 15.992 1.322 47.00 Symbols: 5 = average concentration, moles/l.; Ac = difference in concentration across the boundary; j, = total number of fringes in the Gou pattern; D = diffusion coefficient a t E in cm.2 sec.-l; A n L c = (jm/Ac)(A/a), where a = length along optic axis of diffusion cell (2.503 cm.) and X = wave length (5460.7 A.).
(20) L. J. Gosting and M. 8. Morris, J. Am. Chem. Soc., 71, 1998
(1949). (21) L. J. Gosting, ibid., 7 2 , 4418 (1950). (22) E. Leibhardt, J. O p t . SOC.Am., 43, 1221 (1953). (23) F. A. Jenkins and H. E. White, “Fundamentals of Optics,” 2nd Ed., MoGraw-Hill Book C o . , Inc., New York, N . Y.. 1950, p. 323. (24) L. G. Longsworth, J . A m . Chem. Soc., 69, 2510 (1947).
solution, is equal to the solute activity, u~~~~~ c_ is the solute Concentration in moles per liter, and M is the mobility. Activity data for phosphoric acid,a together with the diffusion and other data, make possible an evaluation of the mobility term. An important additional consideration, however, is how to account correctly for the effect of changing viscosity of the solution on the mobility. A correction for viscosity has been applied26as in the equation
):(
D = lOOORT
* dlna
(d)(2)
where qo denotes the viscosity of the solvent and q that of the solution, and ( M / c ) * is the mobility term a t infinite dilution. The factor qo/q is of limited a p p l i ~ a b i l i t y ~it~ ~ often ~ ~ ; tends toward o v e r c o r r e ~ t i o n . ~ ~Stokes - ~ ~ and S t o k e ~ ~pro~t~~ posed a new correction factor (3)
where Aso is the limiting equivalent conductance of an electrolyte in sucrose solution of specified viscosity and Aw0is the limiting equivalent conductance of the same electrolyte in water. The factor R generally provides less correction than does qO/q and (25) P. Van Rysselberghe, THIS JOVRNAL, 39, 403 (1935). (26) A. R. Gordon, J. Chem. Phys., 6 , 522 (1937). (27) A. R.Gordon, J. Am. Chem. Soc., 73,4840 (1950). (28) P.A. Lyons and C. L. Sandquist. ibid., 76, 3896 (1953). JOURNAL, 62, 497 (1958). (29) J. M.Stokes and R . H. Stokes, THIS (30) W.H.Green, J. Chem. SOC.,93, 2049 (1908). (31) C. L. Sandquist and P. A. Lyons, J . Am. Chem. SOC.,76, 4641 (1954). (32) J. M. Stokes and R. H. Stokes, THIBJOURNAL, 60, 217 (1956).
0. W. EDWARDS AND E. 0. HUFFMAN
1832
Vol. 63
TABLE I1 DIFFUSIONCOEFFICIENTS AND SUPPLEMENTARY DATAFOR PHOSPHORIC ACIDSOLUTIONS AT 25"" Density, d In au Dobsd x los, ~
moi&.
g./ml.
0.1 .3 .5 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 a Density from HC1, ref. 29.
Fig. 2.-Effect
d In c
R
90/1
cm.*sec.-'
/(D) X
lokg
F(D) X
1019
3.731 3.745 1.0026 1.0431 0.9734 9.392 0,970 1,0130 1.1074 3.392 ,9273 .938 8.734 3.431 3.343 1.0234 1.1389 8.514 3.420 .8818 ,902 3.472 3.176 1.0490 1.2393 .7776 .850 8.295 3,096 1.0744 1.3654 8.195 3.530 .6859 .782 2.959 1.0996 1.5219 8.126 3.561 .6049 .728 3.652 2.898 1.7001 .5237 ,660 8.061 1.1246 8.026 3.660 2.740 1.9059 ,4641 .620 1.1495 3.710 2.144 7.956 1,1987 2.4240 .3569 ,540 3.770 2.246 3.0465 .2752 .462 7,836 1.2472 1.977 7.597 3.692 3.8808 .2139 .399 1.2953 ref. 1; d In a u l d In c calculated from activity data in ref. 3; $/q from ref. 6; R derived from data for
of viscosity corrections on mobility term.
may more reliably represent the effect of viscosity on mobility.33 Factor R varies somewhat with the electrolyte (loyodifference between HC1 and tetramethylammonium iodide in 20y0 sucrose) but is not a highly specific function of the e l e c t r ~ l y t e .The ~~ hydrogen ion is considerably less affected by increase in viscosity than are other i o n ~ and , ~ ~we have assumed that the values of R for hydrochloric acid a t a given value for qo/q are applicable to phosphoric acid solution of the same viscosity. Values for the limiting mobility ratio, represented as a function of diffusion coefficient, were calculated from the equations
I n addition to the viscosity correction, a complete equation for predicting diffusion coefficients of a partially ionized electrolyte must reflect the influence of dielectric constant, hydration and degree of dissociation, a,in relatibn to concentration. A treatment has been advanced which incorporates corrections for a! and for hydratioqa4but the data available on phosphoric acid are not adequate for a test. The hypothetical diffusion coefficient of the undissociated phosphoric acid a t infinite dilution, Dm0, may be estimated from somewhat less information. Phosphoric acid is too strong an electrolyte to permit confident use of the method that Vitagliano and Lyons14 used for acetic acid, however, because the method requires data a t concentrations where a S 0. Furthermore, the degree of dissociation of phosphoric acid, as calculated from either conductance or e.m.f., passes through a minimum at about 1 M-a phenomenon presumably attributable to association or hydration reactions of uncertain detail, although several complex species have been s~ggested.'~-'~Thus, any use of a at concentrations above 1M may lead to unreliable results. Muller and Stokes16 evaluated Dm" for the undissociated citric acid molecule from data in a range of concentration where a! varied from 0.06 to 0.09. They plotted a function of the observed and the Nernst limiting diffusion coefficients against concentration and extrapolated to zero concentration. The function may be written
where Di0 is the Nernst limiting diffusion coefficient and 1/R replaces the quantity q/qo. At zero concentration The data required and results of the calculations lim D,' = D,o C-CO are shown in Table 11. The importance of viscosity The approximations and limitations inherent in in the mobility of phosphoric acid is shown in Fig. 2, where mobility ratios from equations 4 and 5 are equation 6 were discussed.16 As a! approaches compared with the values derived from the experi- unity, the calculated value for Dm' will approach inmental diffusion coefficients when no viscosity fac- finity. Even with this relationship, data should tor was used. Since q o / q is more highly concen- be used for concentrations corresponding to low tration-dependent than R, it has a much greater values of a,and the extrapolation should be made effect on the mobility term, as has been found for from the linear portion of the curve corresponding other material^.^^-^^ (34) B. F. Wishaw and R. H. Stokes, J. Am. Chem. Soc., 76, 2065 Dabsd
F(D) =
1OOORT
(%)
(5)
R
(33) J. C. M . Li and P. Chang, J . Chem. Phys., 28, 518 (1955).
(1954).
..
DIFFUSION OF AQUEOUS SOLUTIONS OF PHOSPHORIC ACID
Nov., 1959
1833
TABLE I11 QUANTITIES~ REQUIRED FOR ESTIMATING Omo Eauiv. C,
moles/l
Density, g./ml.
s/so
1.0081 117.71 0.33 9.946 1.047 8.16 1.0191 0.0717 1.00097 1.0163 95.64 .27 9.121 1.062 7.55 1.0382 .1422 1.00479 1.0246 83.06 .24 8.903 1.077 7.45 1.00854 1.0555 .2143 1.0945 1.0452 66.51 .20 8.693 1.108 7.41 .3586 1.01603 1.0661 64.30 * 20 8.530 1.138 7.23 1.1323 .4992 1.02302 a Density from ref. 1. 7/10 from ref. 6; R derived from data for HCl, ref. 29; equivalent conductance from ref. 8; limiting equivalent conductance (A0 = 382.74) from ref. 7; LY calculated from conductance data, ref. 8.
the diffusing particle, from intrinsic viscosity according to the method of Scheraga and Mandelkern.36 Equation 7 leads to 7.2 X sec.-1 for Dm0,and equation 8 leads to 10.8 X cm.2 see.-'. Other modifications of hydrodynamic modelsa3J7lead to even higher, and less likely, values for D m o . The relatively good agreement between the values calculated from equations 6 and 7 suggests that the limiting diffusion coefficient of molecules is approximately 7.6 X 10+ cm.2 sec.-l and lends strength to the assumption of no slip a t the surface of the diffusing phosphoric acid molecule-a condition indicating that the diffusing molecules are surrounded by an associated layer of solvent.36 The value for D m o is less than the limiting diffusion coefwhere R is in joules deg.-' mole-' and F is in ab- ficient calculated for the anion H2P04-, 8.79 X cm.2see.-', from the relation solute coulomb equivalentd1. Extrapolation of the linear portion of a plot of Dmfagainst concentration leads to the value 7.6 X 10-6 sec.-l a t infinite dilution, which is taken where Xo was taken as 33.0 (ref. 7). The ratio as an approximate value for Dm0. Values for Dm0based on classical hydrodynamical D o H 2 P 9 - / D m o is 1.15. Finding ratios for limiting considerations are of interest, even though the phos- diffusion coefficients of anions to molecules ranging phoric acid molecule may not fully meet the require- from 1.23 to 1.33, Stokes16J5 suggested that this ments of theory in regard to size and shape.35 In relationship is accounted for by the charged particle spite of its inherent limitations, the Stokes-Ein- breaking the water structure and thus giving a lubricating,effect for the ions. On the other hand, stein equation Vitagliano and Lyons14 found for the much weaker electrolyte, acetic acid, a value of Dm0that is slightly (7) greater than that of DOCH~COOcalculated from often leads to values for D in surprisingly good equation 9. Acknowledgment.-K. L. Elmore, Chief of the agreement with values determined by more direct methods.35 Equation 7 assumes no slip a t the sur- Research Branch, TVA, and H. S. Harned, Conface of the diffusing particle. With complete slip sultant to TVA, contributed guidance and encouragement during the study. P. A. Lyons, Cona t the surface the equation becomesa5 sultant to TVA, gave helpful advice about the dekT D=(8) sign and testing of the Gouy apparatus. 4~qr
to the higher concentrations. The weaker the electrolyte, the more reliable will be the value derived for D m o . Calculation of D m ' for several concentrations of phosphoric acid in the concentration range 0.07 to 0.5 M yielded the results shown in Table 111. The value of a a t each concentration was calculated from conductance data by the method outlined by Wishaw and Stokes,34except that the factor R replaced the ratio volt. Note that the values for (Y are relatively high-0.2 to 0.3. The Nernst limiting diffusion coefficient was found to be 16.05 X 10-6 set.-' by use of the usual formula
A value of 3.41 8. was calculated for r, the radius of (35) R . H . Stokes, Australian J . Sci., 19, P35 (1957).
(36) H. A. Scheraga and L. Mandelkern, J . A n . Chem. Soc., T 6 , 179 (1953). (37) B. Ottar Acta Chem. Scand., 9, 344 (1955).
