836
D. R. MAY B S D I. 11. KOLTHOFF
The authors are indebted to Professor A. J. Allmand of King's College, London, for the use of laboratory facilities during the war-time evacuation of University College. REFERENCES (1) BERG: Trans. Faraday Soc. 35, 445 (1939). (2) BERGAND MESDELSOHN: Proc. Roy. Soc. (London) 168, 168 (1938). (3) COHNAKD GOODEVE: Trans. Faraday Soc. 36, 433 (1940). (4) COHNA N D HEDVAL:J. Phys. Chem. 47, 603 (1943). (5) GOODEVE AKD KITCHEXER: Trans. Faraday Soc. 34, 570 (1938). (6) HILSCHAXD POHL:Trans. Faraday Soc. 34, 883 (1938). (7) PADOA: Atti accad. Lincei, 1909-1916. (8) PRESTOS:J. Optical Soc. Am. 31, 15 (1930). (9) SHEPPARD, WIGHT.IIAS, AND QUIRK:J. Phys. Chem. 38, 817 (1934). (10) STOBBE:Ber. 40, 3372 (1907). (11) TAYLOR:J. Optical Soc. Am. 24, 192 (1934).
STUDIES OS THE -4GISG O F PRECIPIT.'i'I'ES A S D COPRECIPI'Ta\TIOS. XI,
THESOLUBILITY
O F I,E.iD
CHHO\f.%TE .\-i .%
FLITTIOXO F
THE
PARTICLE SIZE' D. R. K i Y 2 AXD I. 11. KOLTHOFF School of Chemistry, U n i v e w i t y of Minnesota, Minneapolis 14, M i n n e s o t a
Received October 23, 2947 ISTRODUCTIOS
Willard Gibbs (10) in 1878 was the first t o relate the particle size of a solid to its solubility. Ostwald (20) somewhat later derived an expression which was improved by Freundlich (8) and which is generally known asIthe OstwaldFreundlich equation. -According to this equation the soluhility of a solid is B function of its particle size:
where R is the gas constant, 1' the absolute temperature, S and S , the solubility of large solid particles and of particles having small radii T , and M, u , and d are the molecular weight, surface tension, and density of the solid. The Ostn-ald-Freimdlich equation doe5 not take into acoount the possible ionic From a thesis submitted by D. It. May to the Graduate School of the University of Minnesota in partial fulfillment of the requirements for the degrw of Doctor of Philosophy July, 1944. * Present address .inierican ('yanainid Companr, Stamford. Cnnnrct irut.
SOLUBILITY AS FUXCTION O F PARTICLE SIZE
837
dissociation of the solid in solution. T o correct this, Dundon (6) in 1923 introduced van’t Hoff’s number i.
On the basis of thermodynamics a slightly different expression is derived. I n a heterogeneous system such as that of a slightly solubIe solid suspended in water, the region between the solid and liquid phases has a different composition from either the solid or the liquid phase. This region, having a small but finite thickness of the order of several molecular diameters, is described as the surface phase. Thus three phases must be considered: the solid phase a, the liquid phase p, and the surface phase s. Consider a system made up of a solid substance in equilibrium with its saturated solution. If the dissolved substance is a strong electrolyte, then the diwolution of the solid may be represented by the equation
-&q2.
+ vqA” +
(3) . . . represent the ionizing atoms or radicals of the substance and where vl, “.v . . represent the number of atoms or radicals involved. a\t equilibrium *
= VIA’
* *
zv,p2 = 0
(4)
where p i is the chemical potential of the ithconstituent. .it equilibrium the chemical potentials of a given constituent in the solid, surface, and solution phases are all equal. Thus pa = v1pf
+ vqpz + B
(5) where p* is the rhemical potential assigned to the surface phase, and p f , p$ . . . are the chemical potentials of the ionizing atoms or radicals in the solution phase. The right-hand side of equation 3 is evaluated from the equation for a constituent ion in solution p? = p{, RYIna, (6) * * *
+
where p$ is the chemical potential of the ithconstituent ion, pci is a constant depending upon the substance, the temperature, and the pressure, R is the gas constant, T is the absolute temperature, and a, is the activity of the i t h conrtittient . Substituting rquation 6 into equation 5 : 0 ps = PO
+ R2’ In ala2
+
.
(7)
Since the activity product, /C, is represented by
K
= a;lai2
equation 7 may be n-ritten::/h: p8 =
+RYlnK
The chemical potential of the surface phase, ps, in terms of surface tension is not so readily evaluated. Thermodynamically an expression may be derived
838
D. R. MAP AND I . &I. KOLTHOFF
for the chemical potential based upon the surface work function. However, for the complete evaluation of F~ the treatment must be extended to include the energy supplied by the charges known to exist in the solid-liquid interface. At constant temperature and composition of the surface phase when only the work function is considered, dF8
=
adA8
(91
where dFa is an infinitesimal change in the free energy assigned to the surface phase, B is the surface tension, and d-4s is an inSnitesima1 change in the surface area. The chemical potential of the surface phase a t a given temperature and pressure is equal to a constant, &, characteristic of the constituent in the surface phase plus the partial derivative of d P with respect to the number of moles, n, keeping the other constituents constant.
The termaAa/an can be evaluated if the substance is considered t o be divided into spherical particles of radius r. The surface area of a single particle is then:
A8
= 4w-Z
(11)
Differentiating with respect t o n : dAa dr. - = 8nr dn dn The volume of the solid substance is equal to the number n of moles of the suhstance times the molar volume. Thus
where N is the molecular weight and d is the density of the substance. ferentiating equation 13
and substituting into equation 12
equat'ion 10 becomes :
In view of equation 1(i,equation 8 becomes 2uM p ~ + ?~d - , u ~ + R T l n K 8
Dif-
SOLUBILITY AS FUXCTIOh- O F PARTICLE SIZE
839
Consider two systems which differ in that the particles have different radii, rl and r2,and corresponding activity products, K , and Kz. It follows from equation 17 that IC, 2uAf 1 KTln- = K1 d (TI-;) Equation 18 may be simplified by letting r1 be the radius of particles of macro size. Therefore, K1 would be the normal activity product K . Let K 2 and r2 he the activity product and radius of a h e l y divided substance and represented by Ti,. and r. Thus 1 ’1.1 will be very small as compared to 1 r2 and equation 18 may hc w i t t e n as:
Equation 19, derived above, does not take into consideration the surface energy derived from the electrical charge on the surfacc of the particles. W. C. 31. Lewis (17) and Knapp (14) suggested that electrical charges would decrease the solitbility, and Einapp extended the Ostu-ald-Freundlich equation to include an expression for the electrical charge>. His general development can be adapted t o the derivation above and an expression involving the activity products obtained (in place of solubilities as in linapp’s derivation). &lccordingto Helmholtz’s theory there is an electrical “double layer’’ at the solid-liquid interface. The surface of the particle is charged by an excess of either positive or negative ions and is surrounded by the oppositely charged ions which lie at a fixed distance from the surface. If each particle is regarded as a rigid double-layer condenser, its electrical energy, vhich is also the free energy due to the electrical forces, is given by the expression
li’, = .
q26 2Dr(r 6 )
+
n-here q is the electrical charge on each layer, 6 is the distance betn.een layers’
D is the dielectric constant of the medium, and r is the radius of the particle’ The distance between layers i* small compared to the radius r. Thus equation 20 may he written as:
If the free energy due t o electrical forces aq n-ell as the free energy due to the surface work function is included in the total free energy, in place of equation 10, t h r rhemical potential of the surface pha,ie is expressed as: (22)
DifYerentiating equation 21 ivith reTpec‘t to n:
arc - _-- - q26 _ an
dl. D r 3 dn
840
D. R. MAY AND I. 31. KOLTHOFF
In view of equation 14
Combining equations 8, 15, 22, and 24
and
or, where rz is much smaller than rl,
Equation 2’7 is essentially thc equation developed by h a p p (14). It difYers from I h a p p ’ s equation only in that the x t i v i t y products K , and K are used in place of the solubilities S, and S. On a purely theoretical IM , the solubility would increase esponentially with decreasing particle bize on the basis of the Ostwald-Freundlich equation. But the I h a p p equation predicts that the solubility increase with decreasing particle size would be lessened hy the electrical charge and that for very small particles the solubility would approach zero. The relation hetn-een particle size :~nd solubility is undoubtedly influenced by factors other than the surface ~ o r function k and the simple electrical picture developed ahove. Recently, Harl)~iry(11) has suggested the substitution of U’ in equation 2 a5 a “catch-all” for all corrections. The theoretical soundness of this substitution of a variable u is questioned, hut empirically it has some advantages over the original equation. A number of measurements of the relative solubilities of large and small particles has been made. The method used by investigators for the determination of the differences in solubility was generally that of conductimetric measurements. pure crystalline substance ground to a fine powder showed a higher conductance than the coarse unground pon.der. Ostwald (20) and Hulett (12) in 1901 were the first to make measurements of this kind. According to Hulett’s findings, admittedly semiquantitative, the solubility of gypsum (CaS04.2H20)could be increased 20 per cent and of barium sulfate 80 per cent by grinding to particle size diameters of 0.1 micron. Mercuric oxide when ground also exhibited greater solubility. Later, Dundon and Mack (6) measured thc increase in solubility diie to particle size for a number of substances. Calculations of the surface tension w r e made from measurements of particle size and solubility, using equation I , -1 qualitative correlation was found between the calculated surface tension and the hardness. Hulett’s and Dundon’s results agreed closely in the case of barium sulfate, but not in that of gypsum. -4 plausible esplanntion Tvas given by Dundon. In
841
SOLL-BILITY .4S FVNCTIOK O F PARTICLE SIZE
the latter's experiments precautions were taken t o prevent the dehydration of gypsum by grinding. The dehydrating effect of grinding had been known and was shown by Dundon to reduce the n-ater content from 20.93 per cent (theoretical value) to 12 per cent or less, depending upon other conditions. Since it ivas well linon-n (18) that anhydrous calcium sulfate is more soluble than the dihydrate, the presence of partially dehydrated gypwm was probably the reason for Hulett 's higher solubility. The results of Hnlett and Dundon n-ere more critically examined by Balarex ( I ) . He also found a higher conductance u-ith iinely ground barium sulfate than with a coarse product. The conductance of ground p o d e r , high at first. rapidly decreased t o vahiei approximating that of a ( 3 0 ' 1 1 ' ~ iingi-ound powder. The same result9 w r e observed 11hen a drop of barium chloride iolution TI-:~S added to a solution saturated with cwme barium iiilfate: the conductance, high immediately after adding the bariiim chloride, decareased rapidly to a value very nearly equal to the conductance before the addition. The implication of this obvrvation i q that grinding might expobe impiii-ities in the coar-e powder (such as barium chloride) which would give rise to a higher conductance. -1fter standing a while the barium sulfate 11-odd ahsorl) moqt of thc baiinm chloride. Thus the higher conductance of the ground product could he raiised by inipuritie.. . Coheii and I3lelikingh 13) have pointed out another effwt which interferes in the conductimetric method for determining soliibilitiei,~-that of conductance diie to Snely dividecl, quipended material. Thih n-ah dcnionst rated by grinding salicylic acid with gold spheres. The conductance of such a salirylic acid solution is greater than the c3ondiictance of :i solution saturated n-ith coarse d i c y l i c acid. After proper filtering. hoivever. a cmdiictancr cwreapontling to normal solubility \ra. found. pointed out hy ('oheii and Rlelikingh, this greater condiictance may lie eaii~edby two factor.;: the condiictanre of charged particles (14) and the wndiictance of ions in the double layer ivhich, ncwrding t o Rutgers and Overbeek (21), conduct current 3.: if they \\-ere free in solution. 1-sing tl chloride-free, pure hariiim siilfate prepared by precipitation from concentrated sulfuric arid. ('ohm and Mekkingh found )lo rzrirlciicc o f iriflitcncc of particle size .rcpon soluhilify. Their reqiilts >ire' based iipon :L critical examination of their conductimetrir measurements and upon polarographic determination of the barium-ion (soncentration. *I slightly higher condiictance resulting vhen a ground po~vder ]vas i i w l c8oiild he practically eliminated by proper filtering. Cohen and Blekkingh do not give any particle-size measurements and their paper liy Halarew (2) appearing at results are not conclusive for thi- reason. the same time as that of C'ohen and Hlekkingh substantiates the conclusion that no increase in the soliibility of hariiim sulfate is obserretl for particle? 0.1 micron in diameter. l:XPCRI\IL\-T
\ L : I FFI.,C'I' O F bI'L( I F I C h l - I I F \ C E T-I'OS
LIZ \ D
HRO\I \TF. IS
0.1
I IIL 5 O L U H I L I T l
OF
PEIICHLORIC I C I D IOLT7TIO\-
-1s pointetl ont I n - ('ohen and Hlekkingh ( 5 ) ,the difficiilty involved in ii\ing cwiductimetric ineaiiirement~to determint) the effect of particle iize upon t h e diihilitv of :I ciihstnn~elie- in thc fact that an incrcaqe i n thc (vndti(*tancedoes
842
D. R. M.%T AS'D I. 11. KOLTHOFF
not necessarily represent an increase in the solubility. -1true determination of the increase in solubility can be obtained only if the concentrations, or activities, of the ions of the substance are measured. Tn the present study lead chromate was used. It has been established by Rolthoff and Eggertson (15) that lead chromate can be precipitated as a fine powder which ages rapidly with a large increase in its specific surface. In the present study the concentrations of lead and chromate ions in solution were determined by voltametric measurements. -1lthough the concentration of lead chromate in Jyater is too small to lie measured voltametrically, in 0.1 M perchloric acid the concentration is sufficiently large to lir mcasurrd in this manner. In the experiments to follow, 0.1 3 J perchloric acid vas used as solvent and solubility measurements tyere made at various stages of aging of lead chromate. Speciiic-surface measurements were :ilso made and correlated with 'the solubility data. dnaly1ical nzelhod~f o r chroma[e and lend l k o separate methods were used for the determination of both lead and chromate in acid solution. Chromatc concentration was determined by amperometric titration with ferrous ammonium sulfate, using a rotating microelectrode of platinum. 'The lead concentration was determined by diffusion-current measurement- ivith a dropping mercury electrode after the chromate had been reduced to the chronic. state with hydro.;ylamine hydrochloride Chromate and lead \rere also determined together polarographically, by measuring the diffusion currents at potentials of 0 and -0.6 volt. The amperometric- determination of small concentrations of chromate has been described in a previous publication (16). Chromate (in acid solution) can be titrated rapidly by amperometric titration with ferrous iron, using the rotating platinum-wire microelectrode as the indicator electrode. The potential of the indicator electrode is maintained at f1.0 volt vs. the saturated calomel electrode, which is used as the reference electrode. The method is accurate and is precise ;1.ichromate. t o within 0.3 per cent at cwiicentrations as miall as 1 to 2 X The half-wave potential of lead in acid solution is at approyimately -0.4 volt us. a saturated calomel electrode. If chromate is also present in an acid lead solution two waves are obtained, chromate at a positive potential and lead a t -0.4 volt. The current-voltage measurements of such a solution are shown in figure 1, curve 1. This curve represents the current-voltage measurements after the removal of oxygen from a solution 1.33 x 10-4 Ji in lead nitrate arid 1.33 X 10-4 Jf in potassium chromate in 0.1 *Ifperchloric acid. The solution was made by mising equal volumes of 2.66 x 10-1 -Iflead nitrate and 2.GG X lop4 N' potassium cahromate, both in 0.1 perchloric acid. The chromate wave may hr eliminated by the addition of hydrosylaminc hydrochloride. CurVt' 2 in figure I represents the current-voltage curvc of the same lead chromate solution as c u n ~ 1 h t after the addition of one drop of 20 per cent hydroxylamine hydrochloride to 20 ml. of solution. In curves 1 and 2 a wave is ohserved at ahout -0.8 volt. The proliability that thib n-ave representi the reduction
SOLTBILITT +\S F Y S C T I O S O F I’.kRTICLE
8 13
SIZE:
of chromic ion t o chromous is demonstrated by current-voltuge measurements using 2 x 10-4 chromium chloride in 0.1 -11 perchloric acid, Tvhich are shown by curve 3. This w a ~ edoes not interfere with the lead diffusion-current measurements at -0.6 volt. Curve 4 represents current-voltage measurements of 0.1 M perrhloric acid and of 0.1 11 perchloric acid containing one drop of hydroxylamine hydrochloride pw 20 ml. The difference betu-een the diffusion currents of curves
I
I I I -0.4 -0.6 -0.8 Potsntisl vs. saturated calomel ulootrode I
4.2
I -1.0
II -1.2
FIG 1. Current-voltage measurements x i t h a dropping mercury elect] ode of lead, chroinate, and chromic solutions a t 25°C. Curve 1: 1.33 X 10-4 .lf potassiuni chromate and 1.33 X 10-4 If lead nitrate in 0.1 .lf perchloric acid. Curve 2 . -4s curve 1, but one drop of 20 per cent hydroxylamine hydrochloride added. Curve 3 : 2 X loe4 ‘11chromic chloride in 0.1 -11‘ perchloric acid. Curve 4 : 0.1 J1 perchloric acid containing 1 drop of 20 per cent hydroxylamine hydrochloride.
2 and 1 at -0.6 volt corresponds to the lead difYusion current. Csing measurements of the current a t -0.6 volt an accurate determination of lead in solutions of small concentrations was obtained. The accuracy of the lead-concentration measurements with known solutions of lead nitrate and potassium c.hromate in 0.1 M perchloric acid \vas 1-2 per cent for lead concentrations of 1t o 2 X 10-4 M . The difference between the current a t 0 volt of an acid lead chromate solution (curve I) and the residual current at the same potential (curve 4) can be used t o determine the diffusion current of chromate. The difference between the
844
1).
R. 1 I A Y AKD I. >I. KOL'PHOFF
(urient at -0 (iIolt (riiive 1) and the residual current a t that potential
(CLI~VC
4) can be used to determine the sum of the diffusion currents for chromate and lead. By subtracting the diffusion current of chromate from the sum of the diffusion currents for chromate and lead a value can be obtained representing the lead diffusion current. T'sing such measurpmants accuracieq of 2 per cent for both lead and chromatr were obtained n i t h knoun bolution:, of 1 t o 2 X 10-4 M concentrations of lead and chromate in 0.1 -11 perchloric. acid. The acid solution of chromate formed a thin film on the mercury by chemical intera(tion of the chromate n it11 mercury. This tended to rorrupt the tip of the capillary when the dropping elwtrodt. \\-ab in contact with the solution for several minutes. The r e d t w t + ii decrease in the drop time. To eliminate this difficulty the electrode u a \ placed in the wlution after the air was removed and imniediately beforc making diffiision-c~ui,ientmeasurementh .Wer each set of measurements the tip of the capillary -,I 215 cleaned hy dipping in concentrated iiitiic acid and iirising n-it11 distilled n - a t p i . 'l'he formation of chromate film g a w n o measiirabk depletion in the chi uni:itr (*onwritrationof the holution even with :I pool of n i c w i i i ~in~ the hottom of t h e cell for LL period of an hour. 13y using ;L saturated (~:iIoniclrcferenrcl elec~i~odc a. an outside elcctrode instead of a mercury pool and in\erting the mercwy c~l(~c~trodc iminediately 1)efow 11-e, iriterf e r e ~ ~was ( ~ elirniriatid * It ihoi~ldlw inentionrd nt t h i h point that no rnisinium SUpprCam' \\-a. ncc try f 01 cliffiiiion-cw.1cnt meabui-emmts of lead chromatc solution5 of ion- concentration.. Except at concentrations higher than those used in thi. 1 ~ 0 1 -nku nia\imuni o(~41irietl. In c.oncentrations of IO-? Jf or greater both chromatr :ind lead gi\ c maximil There IKX\ no appi-eciahlr differencc in the diffusion cwrent5 of the fre>hly niadc cdloidal iolution and the completely coagulated wlutiorib of ltaacl caliromate :it potmtials bet\\-een 0 ancl - l . G voltb. Thus measurement- of lead ;incl c1iroin:itc ut potriitiiils of 0 :ind -0.G volt should be free of inarcuiwicl\ diic to colloidnl nintnixl.
Particle-six? rnrasurenients Xethods for thc tleteiniiiiatiori of the specific i i i r f a ~ ~ofc ~lcncl c.hrom:tte ha\ v beer1 deqcribed by T
