118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure.This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point.These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility.The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure.This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point.These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility.The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order.The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No.13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility. The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure.This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13.118 ELECTRICAL THEORY OF ADBORPTTON The writer considers the double layer as consisting of a swface of rigidly fixed atoms under continuous bombardment of positively and negatively charged ions, any particular point on the rigid surface becoming in turn negative, neutral and positive, these conditions arisdg in any order. The observed contact difference is the average effect of these conditions. Where several kinds of atoms are present in the solution the average number of any one of them at the surface will depend on their concentbration, valency and mobility.The variation of contact Werence from negative to neutral and positive was observed with cotton and aluminium sulphate near the neutral point. These variations occurred during the same experiment, the readings being direct measurements of E.1I.F.s developed by filtration under pressure. This point would be covered by putting n2 = 1 and = 2 or 3 in Mukherjee’s equation No. 13. AN INVESTIGATION OF SMOLUCHOWSKI’S EQUATION AS APPLIED TO THE COAGULATION OF GOLD HYDROSOL. BY LEONARD ANDERSON, B.Sc., PH.D. (A Papey read before T H E FARADAY SOCIETY, Monday, November 1 2 t 4 1923, SIR ROBERT ROBERTSON, K.B.E., F.R.S., in fh chi^.) Received June 2 7 fh, I g 2 3. INTRODUCTION. As regards the kinetics of coagulation of colloidal solutions the Smoluchowski equation is the only one which has a theoretical basis, Freundlich,l working with aluminium hydroxide sols, deduced an equation, based on measurements of the variation of the viscosity of a sol during co- agulation, which gives the relationship between the time and the amount of precipitated colloid. I t has the drawback, however, that no definite relationship is known to exist between the size of particles and the viscosity of a colloidal solution.Smoluchowski 2 was led to the theoretical treatment of the problem of coagulation from a consideration of certain experiments of Zsigmondy on colloidal gold. Zsigmondy found that, on coagulating a gold sol by means of electrolyte, the speed of coagulation increased with increasing electrolyte concentration until a maximum speed was obtained.This maximum speed is independent of all further increase in electrolyte concentration. This fact formed the basis of Smoluchowski’s theory of “ rapid coagulation.” I n the absence of electrolyte, the electrical double layer on the particles prevents coalescence taking place on impact of one or more particles. Addition of electrolyte to the colloid system diminishes the electrical double layer on the particles, and a force of attraction comes into play. The region of ‘‘ rapid coagulation,” as formulated by Smoluchowski, corre- sponds to a state of complete electrical discharge of the particles and con- sequently of maximum attractive forces between them. Smoluchskis Theory.3-According to this theory each particle in a homogenous sol is supposed to possess a sphere of attraction R, within which the attraction is so strong that any other particle, whose centre enters this sphere, is firmly held.In an originally uniform sol, whose particles have been completely discharged, the initial number of particles whose centres are less than R apart is vanishingly small. In course of time, Brownian movement brings the particles into all possible configurations. In consequence of Brownian movement and of Freundlich, Trans. Fuuaday SOL, 9, 66, 1913. Smoluchowski, Zecfschr. physikai Chcm., 9, 129, 1917. 3 Ibid. 623624 AN INVESTIGATION OF SMOLUCHOWSKI’S EQUATION AS the existence of ‘‘ spheres of attraction ” an irreversible state of coagulation is finally set up. It would be futile to merely give the deduction of Smoluchowski’s equation; that will be found in the original paper.Smoluchowski, combining probability considerations and the laws of diffusion, derives a series of equations which give the rate of disappearance of the particles in a colloid solution during “ rapid ” coagulation. The following equation which gives the rate of disappearance of primary particles is the one used for the following investigation : VO (zqp where yo is the number of primaries at zero time, v1 is the number of primaries at a time t, and /3 is a constant equivalent to ~TDRv,, where D is the ditrusion coefficient and R is the radius of the sphere of attrac- tion. Smoluchowski also attempts to extend his theory to slow coagulation. In this case owing to incomplete discharge of the electric layers, the attractive forces between the particles are not at their maximum and hence only a fraction of the collisions result in union.A probability factor c is therefore introduced to allow for this. The resulting equations obtained by Smoluchowski are identical in form with those obtained for rapid coagulation except that the term /3 is now replaced by cP. Thus the vo The probability factor equation v1 = is assumed by Smoluchowski to be constant throughout the course of ( I + €/3t)2’ vo becomes v1 = ( I + pry - . coagulation. A fundamental assumption of Smoluchowski’s theory is that the rate of disappearance of primaries is greater than that of a simple K g bimolecular ” reaction. As a corrollary to the above it would follow that if coagulation were treated as a bimolecular process, the bimolecular velocity ‘( constant ” kbi should always increase with time for both slow and rapid coagulation.In the case of slow coagulation, however, a rapid fall in the Smoluchowski constant and also in kbi is obtained experimentally, although theoretically the former should remain constant and the latter should rise. PREVIOUS INVESTIGATIONS OF SMOLUCHOWSKI’S EQUATION. Smoluchowski’s equations have been tested by Zsigrnondy, by Westgren and Reitstiitter, and more recently by Kruyt and Arkel, and by Mukherjee and Papaconstantinou. These investigators, with the exception of the last named, used an ultramicroscopic method and followed the coagulation by making a count of the number of particles present at various intervals of time.The experiments of Zsigmondyl gave reasonable constants for /3 (variation of about 5 0 per cent.) when rapid coagulation was studied. Westgren and Reitstiitter worked with coarse gold sols, and found the ratio of i.e. the radius of the sphere of attraction, divided by the radius R of the primary particles. The values of -, which of course should be constant, varied in some cases by IOO per cent. Y ’ 1 Zsigmondy, Zeitschr. physikal Chcm., 92, 600. 1917. a Westgren and Reitstotter, Zeitsthr. physikal Chem., 92, 750, 1917.APPLIED TO THE COAGULATION OF GOLD HYDROSOL 625 Using the data of Westgren and Reitstotter,' the values of /3 have been calculated by the present writer using the equation DATA OF WESTGRBN AND RBITST~TTBR. Serirs I.swim 2. t Sea. 0 60 I20 240 420 600 900 1320 (Re%ve). 10'0 8-70 8-36 7-51 6*29 5 46 5-06 9-46 B (Calcuia ted). - 0'149 0-098 0'083 0.083 0.083 0.065 0.057 t Secr. 0 30 60 I80 300 420 600 120 L V (Relative). 10'0 6 *63 5'45 3 '92 3.12 2'3 9 I%+ I '42 B (Cdculatcd). - 1'0 0.834 0'775 0'73s 0'645 0'635 0.604 It will be observed that the tendency of /3 to fall is quite marked in spite of the fact that in series 2 the speed of coagulation is such that the total number of particles is halved in the first minute. Kruyt and Arkel,2 working with selenium sol, find that Smoluchowski's equation holds in the rapid region but that fl falls rapidly for coagulation at inter- mediate speed. Mukherjee and Papaconstantinou s measured the varia- tion in the optical absorption coefficient, which accompanied the change of colour of colloidal gold in presence of electrolyte.If t,, t,, and ts are the times required by the coagulating sol to reach the same absorption coefficient, using a different concentration of electrolyte in each case, it can be deduced from Smoluchowski's equation that where Mukherjee and Papaconstantinou obtained data which gave reasonable constants for the ratio of T1 : T, : T,. The act of coagulation of colloidal gold involves a change of colour from red to blue. The red is supposed to be due to primary particles. On the assumption that the percentage of red remaining is proportional to the change in the absorption coefficient, the values of /3 were calculated by the writer from Mukherjee and Papaconstantinou's data, using the YO ( I + / 3 t ) Z * equation v1 = In the case of potassium chloride and potassium nitrate as electrolyte, the value of p calculated on the above basis showed good constancy €or p..However, in the case of barium chloride (the speed of coagulation being slower than in the two previous cases) the value1 of /3 calculated 1 Ibid. 2 Kruyt and Arkel, Rec. Trav. Ckim. Pays Bas., s, 656,1920. Mukherjee and Papaconstantinou, Phil. Mag., 44, 305,1922. VOL. XIX-T2&626 .4N INVESTIGATION OF SMOLUCHOWSKI'S EQUATION AS Time in Minutes. 0 I 2 4 5 7 9 I3 16 - from the data of Mukherjee and Papaconstantinou was found to fall as coagulation proceeded. This is shown in the following table :- DATA OF MUKHERJEE AND PAPACONSTANTINOU. Absorption Per Cent. Red Coefficient. (Calc.).0'0453 JOO'O 0*1603 71.56 0'2007 61.5 7 0.2687 44'76 35'73 0'3237 31-15 0'3527 23'93 0'3732 18-91 0'4497 0'3051 Conc. BaC12. 100'0 40 25 IS 5 0.00075 N - 0.380 0'333 0.287 0.280 B (Calc.). - 0.182 0.137 0'123 0.134 0-1 13 0.116 0'096 0.081 - I n the last column the value of p falls by more than one-half. I t is certain that the alteration in the value of p is real, as will be shown later. In addition to the above investigations, attention must be drawn to those of Hatschek,l who devised a colorimetric method of testing the applicability of Smoluchowski's equation. The details of the method will be found in the original paper. The principle of the method is as follows : A rectangular cell divided by an oblique partition contains, in one half red gold sol and in the other half blue sol in suspension stabilised by gelatin.This cell when viewed from the front shows a colour range varying from IOO per cent. red to 100 per cent. blue. A second cell, similar to the first, is placed on top and into it is poured the coagulating sol which is being examined. The colour of the sol in the upper cell varies gradually, with time, from IOO per cent. red to blue. By direct aomparison of the tints in the upper and lower cells, the percentage of red remaining at any instant can be estimated directly. Using this method, Hatschek tested the applicability of the equation VO 'l = ( I + Pt)Z where yo = IOO per cent. red. Y1 = the percentage of red at time f . The following values are taken from Hatschek's paper :- DATA OF HATSCHEK.0.0075 N 0 2'0 3'0 4'5 8'0 14.0 28 '0 Per Cent. Red. 100'0 60 50 40 30 20 I0 8. 0.145 0.138 0.130 0.103 0.088 0.077 Conc. HCI. 0.00826 N Time in Minutes. 0 1'5 3'0 5 '5 16.0 ' Per Cent. Red. 1 B* I- -___ I I Hatschek, I'raus. Faraday Soc., 17, 499, 1921.APPLIED TO THE COAGULATION OF GOLD HYDROSOL 627 The constancy of /3 is not satisfactory although in the faster reaction the fall in p is much less marked. Hatschek also gives instances in which coagulation does not go to completion, Le. no further colour change takes place even after many hours. I t follows from this that p ultimately be- comes zero and consequently a fall of p from the initial value is inevitable. The work to be recorded in the present paper consists of a further and more detailed examination of the Smoluchowski equation using the method of Hatschek.It is convenient to divide the various cases encountered into three divisions, namely :- (i) Rapid coagulation. (ii) Intermediate speed of coagulation. (iii) Slow coagulation. PREPARATIVE. Preparation of CoZoidaZ GoZd.-The water used for this purpose was conductivity water which had been redistilled from a silver-lined copper vessel. All standard solutions used (electrolytes, etc.) were made up with this redistilled water. The gold sols themselves were made by three methods. ( a ) Sodium Citrate Mt.thod.-To 240 C.C. of redistilled water are added 2.5 C.C. of a 0.6 per cent. solution of gold chloride and the whole heated to boiling. Three cubic centimetres of a I per cent. solution of sodium citrate are then added.The solution turns a clear port red and then a further 2 - 5 C.C. of gold chloride are added and the solution again boiled. (6) Method of Hatschek.-3oo C.C. of redistilled water are placed in a flask together with 0.06 grams of white dextrin and 2 C.C. of a normal caustic soda solution. Five C.C. of a 0.6 per cent. solution of gold chloride are added and the mixture slowly heated to boiling. Between 95' C. and 100' C. the whole turns ruby red. (c) T h Formaldehyde Method.-480 C.C. of redistilled water are brought to boiling-point and 10 C.C. of a 0.6 per cent. gold chloride solution to- gether with 14 C.C. of o.18N potassium carbonate solution are added. When the solution is boiling vigorously a 0.3 per cent. solution of formalde- hyde is slowly added as recommended by El1iott.I The addition of formal- dehyde is continued until no further change in colour occurs. The sols prepared were dialysed in collodion dialysers against distilled water.The specific conductivity of the dialysed sols ranged from 0-8 x 10 - to 2 x 10 - mhos. The dialysed gold sols contained 56 milligrams of gold per litre. Method of Procedure.-All glass and quartz vessels were cleansed, for each fresh sol, with chromic acid and aqua regia, washed out with distilled water and finally with steam from redistilled conductivity water. The con- centration of colloid in the comparison sol was in all cases identical with that of the coagulating sol. In all cases the sol was poured into the electrolyte and rapidly mixed. Hydrochloric acid, potassium chloride, barium chloride and aluminium chloride were used as coagulating agents and the comparison sol was made by using the electrolyte, which was being examined .The whole is heated. Elliott, '3!wrn. Iitdiis. and En:. CIum., 13, 699, 192 1. Note.-The present investigation was already partly completed before the writer became aware of the results obtained by Mukherjee and Papaconstantinou by the absorp- tion coefficient method, It is entirely accidental, therefore, that the same precipitating electrolytes were used in both cases.628 AN INVESTIGATION OF SMOLUCHOWSKI'S EQUATION AS 100'0 25.6 19.2 12.8 6.4 EXPERIMENTAL RESULTS. In the following tables, selected from numerous results, /3 was calculated, using the equation - 1-54 1-25 1.06 1-17 VO "l = ( I + /3t)2' - 2'34 2.48 2-15 1-98 where vo = IOO per cent.red, and v1 = the per cent. red at time f. In several cases the value of kbi has been calculated assuming that the primaries (i.e. the percentage red) disappear simply by union with each other. In the results recorded below, only those experiments which gave concordant results, after two or more repetitions have been utilised. - 7'0 12.4 7-10 11*8 A. CONDITIONS APPROXIMATING TO RAPID COAGULATION. Time. (!ha.). I . EZech.olyte-Uyd~oc~Zoric Acid. Per Cent. 8. Red. 1 Sol 12A (Formaldehyde). HC1 = 0*00833 N. Sol rzA. HCl =O'OI N. Time (Secs.). 0 35 57 95 150 Per Cent. I Red. 1 B. t I- Time (Stcr.). 0 25 35 50 90 Per Cent. Red. --- 100'0 25.0 16'6 12.8 6.4 2. Blectro&fe- Potassium ChZoride. Sol 45 (Formaldehyyde).-KCl = 0.033 N.Time. Per Cent. (Secs.). 1 Red. 1 0 17 45 60 84 160 240 I 100'0 ~ 60.0 40.0 29'3 12.9 6.4 20'0 - ' 0.928 0'748 0'848 0'884 0.930 0'737 3. ElectroZyte-Barium Chlode. Sol 22 (Fomaldchyde). Sol 19 (Formaldehyde). BaC1, = 0.00332 N. BaCI, = o*ooxM N. I I 0 I5 25 45 80 130 100.0 46'6 33'3 20 '0 9'3 4'3 - 1.860 1-760 1.650 1.710 1-850 Time (Secs.). Per Cent Red. 100'0 53'3 38'6 29'3 24.0 I 6'0 10'6 6.6 B. - 1'11 0'915 0.925 0'777 0'782 0.828 (3.913APPLIED TO THE COAGULATION OF GOLD HYDROSOL 629 Time. Per Cent. The constancy of p in the above tables is good, especially in the case of barium chloride as electrolyte. Analogous behaviour was observed with aluminium chloride. The results may be taken as indicating that the Smoluchowski equation is holding in the region observed.In fact, the values of p in the case of barium ion are apparently more concordant than have been previously obtained. 1 B. B. CONDITIONS OF INTERMEDIATE SPEED OF COAGULATION. EZeciroZyte- Hydroch Zovic Acid. Sol 7 (Formaldehyde). HCl = 0.00847 N. Sol 10 (Fmaldchyde). HCI = 0*0053 N. I--1 100'0 30 67.9 60 59.0 5x-3 I80 44'9 25'6 360 540 I I0 20.5 1 Time. (secs.). - I 0.428 0.302 0'212 0'164 0'162 , 0.134 > Per Cent. Red. 100'0 56 '4 38'4 25'5 19'4 12.7 6.4 (Secs.). 1 Red. - 0.662 0.526 0'490 0.282 0.258 0'211 B. -____ 4 1'5 1-26 0.87 0'50 I *08 1'11 EZectro@#e- Potassiu m Chloride. Sol 45 (Formaldehyde). KCI = 0.03166 N. Time. (Secs.). 0 30 so 130 I80 360 Per Cent. Red. 100'0 60 '0 33'3 26'0 22 06 '3'3 I kbi X 102. - 1'33 1-60 1-07 0'70 I '50 EZectroZyte-Barium Ch Zoride.Sol 19 (Formaldehyde). 3aC1, = 0.00134 N. Time. (Secr.). Per Cent. 1 Red. 0 30 55 560 I00 220 1080 100'0 73'3 60.0 46'6 33 '3 13'3 3 '3 k - 0.336 0.317 0.279 0'200 0'187 - kbi x Id. - 0.72 0.72 0'63 0'43 0'60 - In the above tables it will be observed that the value of /3 falls con- tinuously as coagulation proceeds, indicating that the Srnoluchowski equation is not depicting the true rate of coagulation.630 AN INVESTIGATION OF SMOLUCHOM'SKI'S EQUATION AS c. SLOW COAGULATION. Electroote- HydrochZo?-ic Acid. Sol 7 (Citrute Method).-HC1 = 0*00508 N. Time (Minutes). 0 I 2 4 6 8-25 11'5 24 35 Per Cent. 1 Red. .I 100'0 76.0 66'7 53'3 44'0 40'0 35'9 26-7 22.5 B* - 0.146 0-117 0.08 j 0.070 0.060 0.039 0.031 0'1 I2 - 3'1 1.8 1.8 2'0 1'0 0.88 0'77 0'63 The change to 100'0 per cent.blue was incomplete after 2 hours. Electrody te-Potassium Chloride. Sol 45 (Formaldehyde Method).-KCl = 0,0266 N. Time I Per Cent. ~ kbi x 102. (Secs.). Red. I I I 100'0 73'3 60.0 50.8 44'0 37'3 30'7 - 0-214 0.1 66 0.147 0.113 0.067 0'100 - 0.4 8 0'34 0'26 0.17 0'22 0'10 EZectro(yte-Barium ChZoride. Sol 19 (Formaldehyde Method).-BaCl! = O ~ O I N. Time (Minutes). 0 I 2-33 3 '75 5 *66 24-50 3 6.0 67'5 107.0 Per Cent. Red. I 100'0 86.7 73'3 66.6 60'0 46.6 33'3 221.0 17'3 - 0*07+ 0.071 0'060 0.05 I 0.018 0.013 om08 0'020 - 1-53 1-58 0.96 0.86 0'74 0'18 0.056 0'20 I t will be seen in the above tables that p and even kbi fall rapidly-the more so the slower the speed of coagulation. Although in nearly all the results quoted the sols used were made by the formaldehyde method, experiments were conducted using sols made by the other methods mentioned previously.for a given concentration of electrolyte varies with the mode of preparation of the sol. A lesser degree of variation occurs when sols are prepared in an identical manner. Probably this is a question of difference in size of the particles and possibly also of their structure. The results obtained show that the value ofAPPLIED TO THE COAGULATION OF GOLD HYDROSOL 631 In addition to a fall in p, it is possible to arrange the electrolyte con- centration so that the sol never reaches IOO per cent. blue after standing many hours. I t would perhaps be of interest to compare the values of /3 for slow coagulation with hydrochloric acid as electrolyte, with those obtained by other workers cited in the Introduction.I n series I of Westgren’s data the value of p fell from 0.149 at the end of the first minute to 0-057 after 2 2 minutes. I n the recalculation of Mukherjee and Papaconstantinods data, using barium chloride as electrolyte, the value of p fell from 0.182 after the first minute to 0.081 after 16 minutes. In the first table for slow coagulation by the author, given above, the value of p fell from 0.146 after the first minute to 0.060 after I 1.5 minutes. Although the data are obtained by three totally different methods, the rate of fall of p is about the same in each case. This is evidence in favour of the general applicability of the various experimental methods employed.DISCUSSION OF RESULTS. The experiments indicate that for the coagulation of gold sols, by means of the electrolytes chosen, there is a “rapid” region in which Smoluchowski’s equation holds reasonably well. The constancy of /3 is quite good, especially in the case of barium chloride as electrolyte. In a region of smaller electrolyte concentration than the above, an excessive slowing down in the speed of coagulation with time is observed. This is in agreement with the data of Kruyt and Arkel and also with the one case (incidentally the slowest speed) of Mukherjee and Papaconstantinou in which the value of p, calculated by the writer, was shown to fall. The general conclusion arrived at by the writer is that the Smoluchow- ski equation is strictly limited in its application.Smoluchowski asserts that the curves depicting slow and rapid coagu- lation should have a similar form, the only varying factor being the proba- bility that an impact will give union. For so-called “ rapid ” coagulation this probability factor is unity and for non-rapid coagulation it is 6, where r < I. This factor z is assumed by Smoluchowski to be constant through- out the course of any one coagulation, but this is not found to be the case. The factor depends upon, and must be some function of, the residual nett charge on the particle. When two charged particles unite, the surface density of the charge on the complex is different from that on the original particle and therefore different repulsive forces come into play. I t is therefore very probable that the factor for union between a charged primary and a charged complex, is less than the factor z for union between two primaries.Let us assume that each primary particle has a radius r and possesses a nett charge E. When two primaries approach one another, a force of repulsion comes into play, reaching a maximum value of E? F = - @K’ where K is the dielectric constant of the medium. that this force of repulsion is overcome. Before two such particles can unite, their relative velocity must be such632 AN INVESTIGATION OF SMOLUCHOWSKI’S EQUATION AS If W,, is the work done in overcoming this force of repulsion, the probability that any two particles will possess the necessary critical velocity is given by This probability factor P,, is the factor Z.Let us now consider a union between a primary and a secondary par- If the density is The ticle. constant, the radius of the secondary will be Y viand its charge 2E. force of repulsion between a primary and a secondary will now be For simplicity consider the latter to be a sphere. The work Wl, to be done before union can take place is now greater and the probability of union becomes - Wl!. P,, = z kT L- W1,. 8 Where W1, i3: __--- (I + 72)2 It is thus evident that the probability factor is varying continuously as the complexes become larger. The falling value of c would partly account for the filling value of /3 observed in this region, since the /3 calculated in the previous tables implicitly contains Z. However even if this modification were introduced into the Smoluchowski equation, the rate of disappearance of primaries would, theoretically, always be greater than that obtained by a bimolecular process, assuming primaries simply united with each other. The data obtained however show that in some cases the value of kh falls, i.e. coagulation is proceeding even more slowly than would be expected on the basis of a bimolecular process.Furthermore, if primaries did disappear simply by union with each other it would follow that once coagulation has commenced it should proceed until no more primaries are left. On this basis incomplete colour change from red to blue should not be possible, since any red colour remaining would indicate unchanged primaries (attributing the red colour to the latter). However, as we have seen, incomplete colour change does occur and simply depends upon the con- centration of the electrolyte present.This phenomenon admits of two ex- planations : I . I t may be due to the possibility that the rate of disappearance of primaries is counterbalanced by an opposing effect : that is, primaries are being reformed either by spontaneous disruption of complexes or by collision of complexes with each other. Such reversibility however would seem to entail a behaviour, on dialysis, of incompletely coagulated sol which has not yet been observed. 2. A more probable explanation would seem to be that the initial primary particles (giving the red colour) are unequaZZy charged. In the case of slow and eventually incomplete coagulation very small amounts of electrolyte are used and it is conceivable that the amount adsorbed is not sufficient to reduce the charge of some of the particles (which initially carry an excessive charge) below the critical limit which will permit coagulation to take place.If this conception of unequal charge is correct, the Smoluchowski equation could not be expected to be applicable in general.APPLIED TO THE COAGULATION OF GOLD HYDROSOL 633 In reviewing the whole problem of coagulation, it would appear that the Smoluchowski equation in its present form is limited in its application. Before it can be applied to all types of coagulation it apparently requires modification to allow for the two factors : (a) The decrease of the probability factor as coagulation proceeds. (b) The existence of incomplete coagulation as a consequence of unequal, and in some cases therefore, of excessive initial electrical charge on the primary particles. SUMMARY. I. A survey of investigations bearing on the Smoluchowski equation has been given. 2. Colorimetric determinations of the rate of coagulation of gold sols by hydrochloric acid, potassium chloride, barium chloride and aluminium chloride have been carried out, using the method of Hatschek. 3. I n agreement with previous investigators, a region of rapid coagula- tion is found in which Smoluchowski's equation holds fairly well. The equation holds most satisfactorily in the case of a certain concentration range of barium chloride as coagulant. 4. A slower region of coagulation is found in which the equation is inapplicable. 5. Possible explanations of (4) have been suggested. 6. I t is concluded that, on the whole, the Smoluchowski equation in its present form is strictly limited to rapid coagulation. 7. Certain of the results obtained can be interpreted as indicating con- siderable variations in magnitude of the charge on individual primary par- ticles of the sol. Addendum-Since the above paper was written the author has found that Kruyt and Arkel working with selenium sol have investigated the Smoluchowski equation, by the method of counting the particles. These observers find that the theory of Smoluchowski holds in the region in which the velocity of coagulation does not differ greatly from the velocity for completely discharged particles. At lower electrolyte concentration the coagulation takes place more slowly than the theory demands. This is in agreement with the results obtained in the above work. The investigation described in the Paper was carried out under the direction of Professor W. C. M. Lewis. 1 Kruyt and Arkel, Koll. Zeitsch., p, 29, 1923. Muspratt Laboratory of Physical and El'ectroehemisity, University of Liverpool. VOL. XIX-T24