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The kinetics of dissolution of calcium sulphate dihydrate in water
Journal of Crystal Growth 35 (1976) 79—88 © North-Holland Publishing Company
THE KINETICS OF DISSOLUTION OF CALCIUM SULPHATE DIHYDRATE IN WATER JØrgen CHRISTOFFERSEN and Margaret R. CHRISTOFFERSEN Medicinsk-Kemirk Institut, KØbenhavns (Jniversitet, Râdmandsgade 71, 2200 N Copenhagen, DK Denmark Received 5 February 1976; revised manuscript received 26 March 1976
A method for analysing data from dissolution experiments during which there is a large variation in the sizes and in the number of crystals is developed. If the rate dependence on concentration is the same for all the crystals and they maintain their Original geometries, it is possible to determine the dependence of the rate of dissolution on concentration independent of the size distribution of the crystals. Applying the method to dissolution data for CaSO 4 2H~0 shows the rate of dissolution is proportional to (c5 — c), where c5 is the solubility. Analysis of the variation in the size distribution shows dr/dt a hr. These results indicate that the rate controlling diffusion. The rate constant a valueof of 2/s. The solubility,process the ionispair association constant and is theequivalent solubility to product CaSO the diffusion coefficient of 4 X 10—6 cm 4~2H20 have been redetermined.
1. Introduction
These assumptions facilitate the calculations enormously. They may in principle be checked by microscopic examination of samples taken from the system during the experiment. In practice these assumptions are never exactly fulfilled; the question is rather how serious the errors introduced by these assumptions are. Even when the assumptions (a) and (b) are poor, it is still possible to describe a crystal by means of a single parameter if the following weaker assumption is made: (c) At any time t > 0 the geometrical shape of a crystal is similar to the geometry of the crystal at time zero, except for smaller changes at the corners and edges which may occur, but only to such an extent that any characteristic linear dimension, r, at time zero remains a characteristic dimension and can be expressed as a monotonic function of m, the mass of the crystal. In the next section we show that the assumption (c), together with the assumption that the dependence of the rate of dissolution on concentration is the same for all crystals, can lead to a simple method of finding a function which expresses this concentration dependence even when the size and number of crystals varies considerably during an experiment. Knowledge of the rate dependence on concentration is of course not
For the investigation of the dissolution of a solid in a liquid it is often necessary or of interest to study systems containing a very large number of particles, which may or may not be of equal size and shape. The progress of the reaction is often followed by measuring a parameter which is associated with the bulk concentration in the liquid, e.g. the electric conductance of an electrolyte solution. Almost any theory for the kinetics of the dissolution process will primarily describe the change in size of the individual particles with time. This theoretical rate is normally expressed in terms of the concentration of the bulk solution and the size and shape of the particles. The aim of studying the overall rate of dissolution is usually to find the kinetic law followed by the individual particles, which is in priciple impossible unless some auxiliary assumptions are introduced, this being a crucial step in the analysis of the data. The following assumptions are often made: (a) At any time during an experiment all crystals are identical, i.e. they all have the same size and the same geometry (outer shape). The size will of course vary during an experiment. (b) The geometry of the crystals does not change during an experiment, 79
J. Christoffersen, M.R. Christoffersen / Dissolution of calcium sulphate dihydrate in water
80
enough to determine the rate-controlling process. Further information from other studies, e.g. of the variation of the size distribution and of the number of crystals during dissolution, are needed for the elucidation of rate the dissolution In cases where an estimated constant forprocess. a suggested rate-determining process is found to agree with an empirically determined rate constant, important information about the rate-controlling process is obtained. This method is applied to dissolution experiments in which seed crystals of calcium sulphate dthydrate are dissolved in water. These crystals have important technical applications [1,21, but preparations normally result in polydisperse samples for which the assumption (a) and (b) are doubtful.
lated by r~1= hj(mij)
m~1= h7’(rij)
,
(4) 1 are increasing
where h1 and the inverse function h7 functions. For all known mechanisms for dissolution of crystals which can be described by one parameter (assumption (c)), the rate of dissolution of a single crystal can be expressed as di~/dt= kf(r)~c) wheref(r) is a monotonic function of r;g(c) is a monotonic function of c, the concentration in the solution; f(r) has often been found to be proportional to r’1 and g(c) proportional to (c~ c)P where q and p are constants and c~is the concentration of a saturated solution. We shall assume —
2. A general method for analysing data from dissolution experiments
dr~ 1/dt=f1(r~1)g(c)
(5)
.
We describe a polydisperse sample of crystals, all of the same geometry, as a sample which at time t contains N1 N2, N1,... crystals of sizes r1 (t), r2(t), r1(t) We define time zero to be the time when the sample is added to the liquid. The total number of crystals in the sample is thus ,
...,
...,
N(t)
N~
=
,
(1)
where for simplicity we define Ni = 0 for r1(t) = 0. If the sample contains crystals of different geometries, index! is used to distinguish between the different geometries, each crystal thus having time independent values of i andj. In this case (1) has to be replaced by
It is important that the rate of dissolution expressed by (5) is separated into a product of two functions, one dependent on rand the other dependent on c. The assumption that all crystals of a particular geometry dissolve according to the same kinetic expression (5) is not important, one may consider that all crystals have different geometries. The restrictive assumption is that the rate dependence on c is the same for all crystals. For r~t> 0 the integrated form of (5) can be expressed as
f g~(t’)}dt’ f~ t
~
0
N(t) = ~
(2)
ii
i,’ =
For crystals of geometry type!, N~1is the number of crystals of size r~1(t).For a representative sample of crystals from the stock we have N~1= m0k~1
(3)
where m0 is the mass of the crystals at time zero and is the number of crystals of the type (i,j) per unit mass of the sample, i.e. ki! is a constant characteristic for the stock. For each type of geometry we choose a characteristic linear dimension, r. For any crystal of geometry, sayj, the size Ti! and the mass m~1are re-
E1(r,1)
—
E1(rg)
(6)
where the superscript 0 refers to time zero. For a suspension of crystals all dissolving, the functions E1 are monotonic functions of r~1,all increasing or all decreasing. Assumingg(c) >0 ensures that the functions E1 are all decreasing functions of r11. Without limitations,a we maylifetime assume in theancrystals with the size r11 to have longer unsaturated solution than any other crystal present. Applying (6) to the crystals with sizes r and r11 gives E1(r,1) E1(r11)+ ~ ,
(7)
J. Christoffersen, MR. Christoffersen /Dissolution of calcium sulphate dihydrate in water
where cI,~,defined by ~
0)
mE
—
Ei(r?i)
(8)
,
1(r
is independent of time. From (7) we obtain
81
where n is the amount of undissolved crystalline substance, can according to (16) be expressed as dn I drn dV÷~ (18) dt = M dt = C dt dt
r~ 1=E7’ [El(rhl)
(9)
+ cb~1]
where £71 is the inverse function ofE,. From (9) is seen that r,1 is an increasing function of r1 i.e. the sizes of all the individual crystals are given by the size of one of them, independent of the way the concentration has varied with time. From (4) and (9) we condude that the mass rn~1can be expressed as an inc~reasingfunction of rn11,i.e.
Using (11), (3), (4), (5) and (14) together with (18) gives = —(rn0/~~g(c)F(m/rn0) (19) ,
~,
rnjj = G,1(rn11)
m~1>0
,
(10)
-
From (3) and (10) is seen that the total mass, rn, of the crystals can be expressed as rn
=
~N~1rn~1
=
(11)
m0G(rn11),
with G(m11) defined by G(rn11) =
k~~G11(m11)
~
,
r•1 >0
,
(12)
and where rn0 is the mass of the seed crystals at time zero. From (11) we obtain 1(rn/m rn11 =G 0), (13) 1 is the inverse function of G. From (10) where G— and (13) we obtain 1 (rn/rn rn11 = G11[G 0)] (l4P -
where dm k~1 fj(r~1). (20) Expressing the empirical rate in the form of (19) has the advantage that the rate dependence of any one
F(rn/rn0)
=
~i,i
~
of the variables rn0, c and rn/rn0 can be determined for constant values of the two other parameters, without assuming how these parameters influence the rate of dissolution. In order to test that the relative sizes of the crystals at a given time are independent of the variation of c in the preceding time interval, two types of experiments are discussed. In the first type the volurne of the solution is constant and the concentration varies with time. In the other type the concentration is kept constant by sufficient addition of water and thus the volume varies with time. For both types of experiments the rate of dissolution has been tabulated as a function of m0,rn/m0 andc. If the assumptions on g(c) for(19) a constant of rn/rna plot of J/rn0 against which is basedvalue are correct, 0 gives a straight line. The possible functionsg(c) can hereby be determined except for a constant factor. If for example the function g(c) is proportional to (c5 c)P, the value of p can be determined from a plot of log(J/rn0) against log(c5 c). For constant value of rn/rn0 each experiment contributes only one point to such a plot. When the function g(c) is known, a plot of J/g(c)rn0 against F(rn/rn0) gives a straight line. The possible functions F(rn/rn0) can hereby be determined. If, for example, the function F(rn/rn0) is proportional to (rn/rn0)°,the value of a can be determined from a plot of log[J/g(c)rn0] against log(rn/rn0). From such plots a value of a can be determined for each experiment. —
From (14) is seen that the mass of any of the crystals can be expressed as an increasing function of rn/rn0. From the mass balance rn + c VM = rn0 + c0 V0M,
(15) where V is the volume of the solution; c0 and V0 the concentration and the volume of the solution at time zero and M the molar mass of the dissolving substance, we obtain ldrn
dV
dc (16)
The overall rate of dissolution, J, defined by J~—dn/dt
,
(17)
—
3. Experimental Analytical grade chemicals and deionized destilled water were used. Seed crystals were prepared by drop-
J. C’/,rjstofjersen, MR. Christojfersen /Dissolution of calcium sulphate dihydrate in water
ioo~
100 pm A
—
W
___
I
~
~
—~ _1— ~
(a)
_
__
H—~
lOPm
(I
4~
---~
~1II;i T (c)
Hg. I. Stereo electron micrographs of( aSO urated solution for which ~n~/ I -
=
(d)
I
4 2H20. (a) Needles from the stock. (b) Plate from the stock. (c) Crystals from a sat4.5 g/l. (d) Needle from the same solution as (c).
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
83
wise addition of 2.0 1 of 0.60 M calcium chloride to 2.0 1 of 0.60 M sodium sulphate solution. The crystals were washed with water until free of chloride, after
periments crystals (not always in excess) were added to water and the volume changed from 0.13 to 0.23 1. The ratio n0/V0 was in the range
which the crystals were dried at 36°Cfor two days. The dried crystals were analysed for calcium by means of a Perkin—Elmer 305A atomic absorption spectrophotometer. The water of crystallization was determined by heating crystals to about 1000°Cfor 2 h. The crystals decreased in weight by 20.97% (calculated for CaSO4 2H~O:20.93%). All experiments were made using crystals from a single preparation. Two forms of crystals were present, as can be seen in fig. 1. 93% (by number) of the crystals were needles of longest side 50—300 pm. 97% of these crystals were smaller than 200 pm. The other type of crystal (“plates”) had the form of two plates joined to form a wedge. The longest dimension of these crystals was about 300 pm. All experiments were made at 25.0 ±0.1°Cand except for a few experiments made to investigate the effect of the rate of stirring on the reaction rate, a constant stirring rate of 6.7 s~was used. The conductance, L, of the suspension was measured using a conductivity cell and a Radiometer CDM3 conductivity meter. In the first type of experiment the conductance of the suspension was recorded against time. In order to improve the sensitivity of the measuring technique, the dc signal from the conductivity meter was fed into a Radiometer pHM64 pHmeter which was connected to a Hewlett—Packard digital analog converter, which could back out parts of the hereby signal. A in concentration of 3 X 10—6theM could be change detected. From these experiments rate of dissolution was determined as J = V dc/dt. mitial concentrations ranged from 0—0.7 c and n 0/V was in the range (6 X l0~—4x 10_2)nol/l. There was not always an excess of crystals to saturate the solution. The rate was measured 2--l.4X 102M.in the concentration range 0.2X 10 In the other type of experiment the signal from the conductivity meter was fed into a Metrohm E473 potentiostat which controlled a Metrohm E41 5 autoburette (Dosimat) containing water. The volume of water added was recorded as a function of time. Thus the conductance of the suspension could be kept constant at a predetermined value by the automatic addition of water. From these experiments the rate of dissolution was determined as J = cd V/dt. In these cx-
(9 X l0—~—5X 10—2) mol/l. The rate was measured in the concentration range 0.5 X 102_l .45 X 10—2 M.
.
4. Results and discussion The conductance, L~,of a saturated solution contaming crystals depends slightly on the amount of crystals present per unit volume. Experimentally we obtained L~ L5(1 where kcr = 1
—
—
13n/V) =Lskcr
(21)
,
(22)
jln/V,
and L5 is no thecrystals. conductance of a<0.06 saturated solution where containing For n/V mol/l, = 0.12 1/mol. The conductance, Lcr, of an unsaturated solution containing crystals is assumed to be similarly given by Lcr kcrL
(23)
where L is the conductance of the solution without crystals. For experiments at constant volume, n/Vwas kept so low (less than 0.03 mol/l) that the differences between ~cr/~k’ dL/dc dLcr/dt and 1l~crIdt)L are lessand than 1%. and Suchbetween errors are negligible (C in comparison with the general reproducibility of the experiments (approx. 10%). For these experiments we approximate the overall rate of dissolution by V-~-~ = V dc dl~cr dc thJ~cr dt V-~(_-~_) (24) L For experiments at constant conductance, the effect of crystals interfering with the conductance measurements causes the concentration to vary slightly during an experiment. A first order approximation gives = + V dc dV I Lj3(n/V + c) dt ~ c(L~+ kcr dLI~) (25) =
~
.
—
The value of n/V was kept so low (less than 0.03 mol/l)
.1. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
84
that the approximation J~cdV/dt
(26)
less than 0.3%. In the following we assume that p = and conclude that the function g(c) is proportional to Cs
introduces an error of less than 1% in J. The variation in c during an experiment can also be neglected cxdept in the term c~ c when c c5. Points for which a correction of c was not negligible compared with the reproducibility of experiments are indicated in fig. 2 by parentheses. Due to the uncertainty in c5 c, these points were omitted in the calculation of the best straight lines, The are experimental results of bothlog(J/rn types of experiments shown in fig. 2 in which 0) is plotted against log(c5 c) for four values of rn/rn0. Experiments were not to analysed <0.03 g. The solubility, c~, is taken be 1.52for X rn 10—2 M (see appendices A—C). For identical experiments, points corresponding to the mean values of J are plotted. For constant values of rn/rn 0 straight lines with the slope p, 1 .00
—
—
—
C.
For each constant value of rn/rn0, i.e. for each set of points determining the straight lines in fig. 2, the mean value (J/rn0(c5 c)) of .J/rn0(c5 c) has been determined. In fig. 3 log(J/rn0(c5 c)~is plotted against log(rn/rn0). The straight line obtained in fig. 3 has the slope 1.16 ±0.01. The vertical lines through the points in fig. 3 have lengths equal to twice the standard deviations. From fig. 3 we conclude1-16. that the function F(rn/m0) is proportional to (rn/rn0) Assuming 1.16 (27) g(c) F(rn/rn0) ~ (c~ c) (rn/rn0) a rate constant k has to be introduced into (19). From the intercept of the line in fig. 3 with the line log(rn/rn 3 1/s g for the 0) = 0, k = (6.4 ±0.1) X i0 —
—
—
,
—
stirring ratecan of 6.7 s~is obtained. Thebyoverall rate of dissolution therefore be described the function 116 (28) ,
J = km13(c~— c) (rn/rn0)
which is an expression of the type (19). A doubling of the stirring rate caused a 30% increase in k. The rate constant k may depend on the size distribution of
I
-4-5
the stock crystals.
-5.0 I CI
~ I’a IE CI
2
-2.5 2O,~ -
-6.0
____
-
1
-3.0
-2.5
-2.0
-30V CI
2
cs-c log ~-~‘ Fig. 2. Plot of log(J/mo) against log(c 5 — C) for four values of rn/mo. Circles: rn/mo = 0.65; squares: rn/rn = 0.40; triangles: rn/rn0 = 0.25;hexagons: rn/mo = 0.158. Open symbols refer to experiments with V constant, c0 = 0; vertical lines through symbols refer to experiments with V constant, co ~ 0; horizontal lines through symbols refer to experiments with c constant.
_____________________________ -1.0
-0.5 0 log (rn/rn0) Fig. 3. Mean value of log[J/mo(c5 — c)] for each of the four values of rn/mo, see fig. 2, plotted against log(m/mo). The length of the vertical lines through a point is equal to twice the standard deviation.
J. Christoffersen, MR. Christoffersen /Dissolution of calcium sulphate dihydrate in water
85
5. Size distribution
corresponding to expt. 2 were calculated using (5) with f1.(r~1)cx:r11q and q = 0 and q = —1 and taking in-
Two experiments were made in order to investigate the change in the size distribution as crystals dissolve: 0.72 g of crystals were suspended in 0.086 1 of water (expt. 1) and 0.72 g of crystals were suspended in 0.181 of water (expt. 2). After the suspensions had reached saturation, the number of crystals present in each suspension was determined by counting the crystals in a blood cell counting chamber viewed under a light microscope. In expt. 1 there were 1.01 X l0~ crystals, 15.8% of which were plates. In expt. 2 there were 3.1 X 106 crystals, 45% of which were plates. The number of plates was thus approximately the same in both experiments, whereas the number of needles in expt. 2 was only 20% of the number in expt. 1. From this we conclude that the number of needles decreases significantly in our dissolution cxperiments. For both experiments samples were photographed and the size distribution of the needles was determined by measuring the longest these crystals a Carl Zeiss TGZ3 particleside sizeofanalyzer with 48using equally large size intervals, which covered the size-range 0—350 pm. Curve 1 in fig. 4 shows for expt. 1 the percentage of needles in a size interval plotted against the length corresponding to the mid-point of the interval. Curve 2 in fig. 4 shows for expt. 2 the number of needles in a size interval expressed as the percentage of the total number of needles in expt. 1. From the distribution function of expt. 1, distribution functions
to account that 80% of the needles dissolve completely. The results of these calculations are also shown in fig. 4. It can be seen that the distribution calculated with q = 0 does not agree with the experimental distribution curve 2, but that q = —lisa reasonable assumption. This means J ~x 1/r. In order to calculate the apparent diffusion coefficient, Dapp, the radius, r, of a sphere circumscribing a crystal is used as the size parameter. For the needles the longest side is approximately 2r. The diffusion equation has not yet been solved for bodies with geometry as complicated as that of calcium sulphate crystals. We therefore for simplicity assume the diffusion flux out of a sphere circumscribing a crystal to depend only on r and not on the geometry of the crystals. Together with the steady state approximation of diffusion controlled dissolution [3] this assumption leads to ~ (29) dn dt 4irDapp(Cs c) J= = If the rate of dissolution is diffusion controlled, the apparent diffusion coefficient can be found from (28) and (29) if the distribution function is known. Using the distribution function determined in expts. I and 2 for the needles and assuming the value of r for the plates to be equal to the maximum value of r for the needles in expt. 1, we calculate Dapp = 3 X 10—6 cm2/s from expt. I and Dapp = 4 X 10—6 cm2/s from expt. 2.
I
I
I
8
—
~
——
6. The dissolution of crystals mounted in a gel The dissolution of crystals mounted in an agarose-
6-
gel placed on the lower side of a cover glass could be studied through a light microscope by placing the
4,
cover glass on top of a Plexiglass chamber through
—~.
20
~
100
2r/jjifl
5°
Fig. 4. Size distributions of needles. Curve 1: obtained from expt. 1 in which rn 0/V 8.4 g/l; curve 2: obtained from expt. 2 in which mo/V = 4.0 g/l; (X) calculated for expt. 2 with q = —1; (.) calculated for expt. 2 with q = 0.
graphs which water of the dissolving passed. crystals A intervalometer Paillard-Bolex were taken Hl6 with camera suitcontrolled by Shackman TU1 was connected towas aa Leitz Wetzlar microscope and photoable constant time intervals. Films showing the dissolution process at rates 4—80 times the natural rate were studied. From these films we find for the individual crystals of both types that dr/dr increases as r decreases. If
86
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
two or more crystals are close together, relatively little substance dissolves from adjacent parts of the crystals compared with the amount of substance which dissolves from the free parts of the crystals. A hole often appears in each of the two ends of a needle. Except for formation of such holes, the geometry of well-dispersed needles appears to be approximately constant. Nuclei on the surfaces of the plates disappear and
(b) aje not valid, a value of ~Nir~cannot be obtained without knowledge of the size distribution of the crystals and therefore a value of D cannot be obtained.
Appendix A The solubility product of CaSO
4 21120 at 25°C
holes are often formed in their surfaces.
7. Conclusion Taking into account that the assumption on which (29) is based is expected to lead to a value of Dapp which is less than the real diffusion coefficient, D, the values of Dapp obtained for calcium sulphate dthydrate are in reasonable agreement with the diffusion coefficients of magnesium sulphate and zinc sulphate a 2/sin[4]). similar concentration range (D = 7 X 106 cm The value of Dapp together with the proportionality of the rate to (c 5 c), and the behaviour of dissolving crystals mounted in a gel indicate that the rate-controlling process is diffusion. That the rate of dissolution depends slightly on the stirring rate indicates that the rate-controlling process is diffusion (in a stagnant medium) combined with convective-diffusion [3,5a]. The influence of convection on the absolute value of the rate appears to be negligible, Using a rotating disc method, Barton and Wilde [Sb] found the rate of dissolution of CaSO4 2H20 to agree with diffusion-controlled dissolution with 2/s. D Liu 8 Xand 106 cm Nancollas [6] have followed the dissolution of calcium sulphate dthydrate of size 80—120 pm using a calcium electrode. They found the rate of dissolution to be proportional to rn 2/3(c 0(rn/rn0) 5 c) and concluded, using assumptions (a) and (b) that the rate-controlling process is diffusion through a diffusion layer of constant thickness. If the size of crystals changes during an experiment, it is doubtful if the diffusion layer is of constant thickness. Liu and Nancolla’s results can, however, still lead to the conclusion of diffusion-controlled dissolution without assuming a constant diffusion layer, but not together with assumptions (a) and (b), which in fact appear to be doubtful. In their case the ~ appears to be 2/3.term If assumptions (a) and proportional to (rn/rn0) —
.
—
Using a Radiometer Calcium Selectrode, a calomel electrode, and a Radiometer pHM64 pH-meter, we found 2~) pCa5 —log [(Ca 5/M]= 2.305 ±0.005, 2~) where (Ca 5is the activity of calcium ions in a saturated solution. The electrodes were calibrated by making similar measurements of calcium chloride solutions and assuming the liquid junction potentials of asolution calciumofchloride and of a calcium sulphate similar solution pCa are identical. The activity coefficient of calcium ions, YCa’ in calcium chloride solutions was calculated using Guggenheim and Stokes’ expression [7] for the mean activity coefficient, .‘~‘
z.~2
A
—logy+
=
jl/2
____________ 1
+
—
0.028 z÷z I I
(30)
I .536 j1,’2
(where A is a Debye—HUckel constant, I the ionic strength and z the ionic charge), and Bates and Guggenheim’s expression [8] for the activity coefficient of chloride ions,yCl, —logy~ 2/(l + 1.511/2), (31) 1 A1’/ together with the relationship
=
(YCaYC1)’
(32)
In the activity range of interest, 3.54 > pCa> 1.89, the values calculated for pCa for calcium chloride solutions differ by less than 0.005 from the values recommended by Bates [9] and[10], from the values calculated using Davies’ equation
logy1 =
o.s Z1
(
jl/2 + jl/2
—
0.3 i).
(33)
Assumingactivity that calcium ions and sulphate sulphate ions havesoluidentical coefficients in calcium
J. Chr.toffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
tions we obtain pK5 = 4.61 ±0.01 or K5 = 2 for the solubility product of (2.45 ±0.05) X l0~ M calcium sulphate dihydrate, K 5. Liu and Nancollas [11] obtained K5 = measurements (2.58 ±0.15) X 105 M2 from similar potentiometric using (33) for calibration. Gardner and Glueckauf [12] calculated K 2 from Marshall and 5 = 2.49measurements X 10~M [13] and from Slusher’s solubility Brown and Prue’s measurements of osmotic coefficients [14] and Bjerrum’s electrostatic theory [15].
87
termined a function d, the distance closest approach of as “free” ions ofofopposite charge of [141. Results
ford1 = 9.2 A and d2 = 5.2 A are given in table 1. 1 Gardner Glueckauf [12] calculatedKa = 258 M— ford = 9 and A. Liu and Nancollas [11] found Ka = 169 ±15 M~with activity coefficients calculated from (33) which corresponds to (35) with d 5 A. Ainsworth [16] found Ka = 204 ±4 M1 ford = 5 A from potentiometric measurements in a cell without liquid junction. Hanania and Israelian [17] obtained ka = 234 ±8 M1 by a method corresponding to (35) withd’1~4.6A.
Appendix B The ion pair association constant of CaS0 4~21120 at
Appendix C
25°C The solubility ofCaSO4 21120 at 25°C From atomic absorption spectrometry we find the total concentration of calcium in a saturated solution to be c 5 = (1.53 ±0.02) X 10—2 M. From the potentiometric measurements we calculate c~= (1.50 ±0.02) X 10—2 M from (34) and (35). From EDTA titrations using the jump, we calcium find c electrode to register the potential 5 = (1.52 ±0.02) X 10—2 M. In this paper we have used c~= 1.52 X 10—2 M. Other[13], determinations c~include Slusher 1.51 X 10—2ofM; Liu and Marshall Nancollasand [6], 1.48 X 10—2 M; Madgin and Swales [18], 1.51 X 10—2 M;Bock [19], 1.52X 10—2 M;Moreno and Osborn [20], 1.51 X 10_2 M;Nakayama and Rasnick [21], 1.47 X 10—2 M. In the last two references K 5 and Ka were calculated with the assumption of individual activity coefficients for calcium and sui-
2~)determined at 25.0°Cby poTable 1 gives (Ca tentiometric measurements and the total concentration of calcium sulphate, c, made up by weight and checked by atomic absorption spectrometry for solutions of calcium sulphate dthydrate. From these data the association constant Ka, 2~)(SO~), (34) Ka = [CaSO4]/(Ca can be determined by assuming expression for the 2~ionsanand of SO~ions. activity coefficients of Ca Using —logy 4
A z+z_1I1/2 2/a0 1 + dI’/
(35
where a°is a Debye—Huckel constant, Ka can be deTable I Ion paii association constant of CaSO4 2H20;d1 2c 102(Ca~)i02[Ca2~j 10 1 (M) (M) (M)
=
9.2 A,d2
=
5.2 A
2[Ca2’I 10
2
Kaa 1)
Kai (M’)
(M)
(M
1.3516
0.462
0.797
260
0.909
207
1.2286
0.434
0.738
260
0.836
208
0.9975
0.380
0.628
256
0.701
205
0.6650 0.4988
0.288 0.235 0.176
0.451 0.354 0.253
258 263 258
0.491 0.378 0.266
210 219
0.3325
mean: 259
± 10
215 mean: 211
± 10
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
88
phate
and the values of K5 and Ka are not directly comparable with values quoted here. ions
Acknowledgements We thank Professor Arne E. Nielsen for stimulating discussions. Aalborg Portland Concrete Research Laboratory, Karlstrup is thanked for providing the stereo electron micrographs. Statens Naturvidenskabelige Forskningsr~dis thanked for granting the Carl Zeiss TGZ3 particle size analyzer and the Perkin Elmer 305A atomic absorption spectrophotometer.
References [1] W. Lerch, Res. Lab. Bull. 12 (Portland Cement Assoc., 1946). [2] H.H. Steinour, Res. Lab. Bull. 98 (Portland Cement Assoc., 1958). [3] A.E. Nielsen, J. Phys. Chem. 65 (1961) 46. [4] R.A. Robinson and RH. Stokes, Electrolyte Solutions (Butterworths, London, 1970) p. 513. [5] (a) V.G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, N.J., 1962) p. 81. (b) A.F.M. Barton and N.M. Wilde, Trans. Faraday Soc. 67 (1971) 3590.
16] Sung-Tsen Lio and G.H. Nancollas, J. Inorg. Nucl. Chem. 33 (1971) 2311. [7] E.A. Guggenheim and R.H. Stokes, Trans. Faraday Soc. 54 (1958) 1646. R.G. Bates and E.A. Guggenheim, Pure Appl. Chem. 1 (1960) 163. [9] R.G. Bates and M. Alfenaar, Ion-selective Electrodes (Nat!. Bur. Std. (US) Spec. Publ. 314) p. 202. [10]C.W. Davies, Ion Association (Butterworths, London, [11]Sung-TsenLiu and G.H. Nancollas, J. Crystal Growth 6 (1970) 281. [12] A.W. Gardner and E. Glueckauf, Trans. Faraday Soc. 66 (1970) 1081. [13]W.L. Marshall and R. Slusher, 3. Phys. Chem. 70 (1966)
181
[14]P.G.M. Brown and J.E. Prue, Proc. Roy. Soc. (London) A 232 (1955) 320. [15] N. Bjerrum, Kgl. Danske Vidensk. Selskab Mat.-Fys. Skrifter 7 (1926). [161 R.G. Ainsworth, J. Chem. Soc. Faraday Trans. I 69 (1973) 1028. [17] G.I.H. Hanania and S.A. Israe!ian, J. Soln. Chem. 3 (1974) 57. [18] W.M. Madgin and D.A. Swales, J. Appl. Chem. 6 (1956) 482. [19]E. Bock, Can. J. Chem. 39(1961)1746. [20] E.C. Moreno and G. Osborn, Soil Sci. Soc. Am. Proc. 27 (1963) 614. [21] F.S. Nakayama and B.A. Rasnick, Anal. Chem. 39 (1967) 1022.
THE KINETICS OF DISSOLUTION OF CALCIUM SULPHATE DIHYDRATE IN WATER JØrgen CHRISTOFFERSEN and Margaret R. CHRISTOFFERSEN Medicinsk-Kemirk Institut, KØbenhavns (Jniversitet, Râdmandsgade 71, 2200 N Copenhagen, DK Denmark Received 5 February 1976; revised manuscript received 26 March 1976
A method for analysing data from dissolution experiments during which there is a large variation in the sizes and in the number of crystals is developed. If the rate dependence on concentration is the same for all the crystals and they maintain their Original geometries, it is possible to determine the dependence of the rate of dissolution on concentration independent of the size distribution of the crystals. Applying the method to dissolution data for CaSO 4 2H~0 shows the rate of dissolution is proportional to (c5 — c), where c5 is the solubility. Analysis of the variation in the size distribution shows dr/dt a hr. These results indicate that the rate controlling diffusion. The rate constant a valueof of 2/s. The solubility,process the ionispair association constant and is theequivalent solubility to product CaSO the diffusion coefficient of 4 X 10—6 cm 4~2H20 have been redetermined.
1. Introduction
These assumptions facilitate the calculations enormously. They may in principle be checked by microscopic examination of samples taken from the system during the experiment. In practice these assumptions are never exactly fulfilled; the question is rather how serious the errors introduced by these assumptions are. Even when the assumptions (a) and (b) are poor, it is still possible to describe a crystal by means of a single parameter if the following weaker assumption is made: (c) At any time t > 0 the geometrical shape of a crystal is similar to the geometry of the crystal at time zero, except for smaller changes at the corners and edges which may occur, but only to such an extent that any characteristic linear dimension, r, at time zero remains a characteristic dimension and can be expressed as a monotonic function of m, the mass of the crystal. In the next section we show that the assumption (c), together with the assumption that the dependence of the rate of dissolution on concentration is the same for all crystals, can lead to a simple method of finding a function which expresses this concentration dependence even when the size and number of crystals varies considerably during an experiment. Knowledge of the rate dependence on concentration is of course not
For the investigation of the dissolution of a solid in a liquid it is often necessary or of interest to study systems containing a very large number of particles, which may or may not be of equal size and shape. The progress of the reaction is often followed by measuring a parameter which is associated with the bulk concentration in the liquid, e.g. the electric conductance of an electrolyte solution. Almost any theory for the kinetics of the dissolution process will primarily describe the change in size of the individual particles with time. This theoretical rate is normally expressed in terms of the concentration of the bulk solution and the size and shape of the particles. The aim of studying the overall rate of dissolution is usually to find the kinetic law followed by the individual particles, which is in priciple impossible unless some auxiliary assumptions are introduced, this being a crucial step in the analysis of the data. The following assumptions are often made: (a) At any time during an experiment all crystals are identical, i.e. they all have the same size and the same geometry (outer shape). The size will of course vary during an experiment. (b) The geometry of the crystals does not change during an experiment, 79
J. Christoffersen, M.R. Christoffersen / Dissolution of calcium sulphate dihydrate in water
80
enough to determine the rate-controlling process. Further information from other studies, e.g. of the variation of the size distribution and of the number of crystals during dissolution, are needed for the elucidation of rate the dissolution In cases where an estimated constant forprocess. a suggested rate-determining process is found to agree with an empirically determined rate constant, important information about the rate-controlling process is obtained. This method is applied to dissolution experiments in which seed crystals of calcium sulphate dthydrate are dissolved in water. These crystals have important technical applications [1,21, but preparations normally result in polydisperse samples for which the assumption (a) and (b) are doubtful.
lated by r~1= hj(mij)
m~1= h7’(rij)
,
(4) 1 are increasing
where h1 and the inverse function h7 functions. For all known mechanisms for dissolution of crystals which can be described by one parameter (assumption (c)), the rate of dissolution of a single crystal can be expressed as di~/dt= kf(r)~c) wheref(r) is a monotonic function of r;g(c) is a monotonic function of c, the concentration in the solution; f(r) has often been found to be proportional to r’1 and g(c) proportional to (c~ c)P where q and p are constants and c~is the concentration of a saturated solution. We shall assume —
2. A general method for analysing data from dissolution experiments
dr~ 1/dt=f1(r~1)g(c)
(5)
.
We describe a polydisperse sample of crystals, all of the same geometry, as a sample which at time t contains N1 N2, N1,... crystals of sizes r1 (t), r2(t), r1(t) We define time zero to be the time when the sample is added to the liquid. The total number of crystals in the sample is thus ,
...,
...,
N(t)
N~
=
,
(1)
where for simplicity we define Ni = 0 for r1(t) = 0. If the sample contains crystals of different geometries, index! is used to distinguish between the different geometries, each crystal thus having time independent values of i andj. In this case (1) has to be replaced by
It is important that the rate of dissolution expressed by (5) is separated into a product of two functions, one dependent on rand the other dependent on c. The assumption that all crystals of a particular geometry dissolve according to the same kinetic expression (5) is not important, one may consider that all crystals have different geometries. The restrictive assumption is that the rate dependence on c is the same for all crystals. For r~t> 0 the integrated form of (5) can be expressed as
f g~(t’)}dt’ f~ t
~
0
N(t) = ~
(2)
ii
i,’ =
For crystals of geometry type!, N~1is the number of crystals of size r~1(t).For a representative sample of crystals from the stock we have N~1= m0k~1
(3)
where m0 is the mass of the crystals at time zero and is the number of crystals of the type (i,j) per unit mass of the sample, i.e. ki! is a constant characteristic for the stock. For each type of geometry we choose a characteristic linear dimension, r. For any crystal of geometry, sayj, the size Ti! and the mass m~1are re-
E1(r,1)
—
E1(rg)
(6)
where the superscript 0 refers to time zero. For a suspension of crystals all dissolving, the functions E1 are monotonic functions of r~1,all increasing or all decreasing. Assumingg(c) >0 ensures that the functions E1 are all decreasing functions of r11. Without limitations,a we maylifetime assume in theancrystals with the size r11 to have longer unsaturated solution than any other crystal present. Applying (6) to the crystals with sizes r and r11 gives E1(r,1) E1(r11)+ ~ ,
(7)
J. Christoffersen, MR. Christoffersen /Dissolution of calcium sulphate dihydrate in water
where cI,~,defined by ~
0)
mE
—
Ei(r?i)
(8)
,
1(r
is independent of time. From (7) we obtain
81
where n is the amount of undissolved crystalline substance, can according to (16) be expressed as dn I drn dV÷~ (18) dt = M dt = C dt dt
r~ 1=E7’ [El(rhl)
(9)
+ cb~1]
where £71 is the inverse function ofE,. From (9) is seen that r,1 is an increasing function of r1 i.e. the sizes of all the individual crystals are given by the size of one of them, independent of the way the concentration has varied with time. From (4) and (9) we condude that the mass rn~1can be expressed as an inc~reasingfunction of rn11,i.e.
Using (11), (3), (4), (5) and (14) together with (18) gives = —(rn0/~~g(c)F(m/rn0) (19) ,
~,
rnjj = G,1(rn11)
m~1>0
,
(10)
-
From (3) and (10) is seen that the total mass, rn, of the crystals can be expressed as rn
=
~N~1rn~1
=
(11)
m0G(rn11),
with G(m11) defined by G(rn11) =
k~~G11(m11)
~
,
r•1 >0
,
(12)
and where rn0 is the mass of the seed crystals at time zero. From (11) we obtain 1(rn/m rn11 =G 0), (13) 1 is the inverse function of G. From (10) where G— and (13) we obtain 1 (rn/rn rn11 = G11[G 0)] (l4P -
where dm k~1 fj(r~1). (20) Expressing the empirical rate in the form of (19) has the advantage that the rate dependence of any one
F(rn/rn0)
=
~i,i
~
of the variables rn0, c and rn/rn0 can be determined for constant values of the two other parameters, without assuming how these parameters influence the rate of dissolution. In order to test that the relative sizes of the crystals at a given time are independent of the variation of c in the preceding time interval, two types of experiments are discussed. In the first type the volurne of the solution is constant and the concentration varies with time. In the other type the concentration is kept constant by sufficient addition of water and thus the volume varies with time. For both types of experiments the rate of dissolution has been tabulated as a function of m0,rn/m0 andc. If the assumptions on g(c) for(19) a constant of rn/rna plot of J/rn0 against which is basedvalue are correct, 0 gives a straight line. The possible functionsg(c) can hereby be determined except for a constant factor. If for example the function g(c) is proportional to (c5 c)P, the value of p can be determined from a plot of log(J/rn0) against log(c5 c). For constant value of rn/rn0 each experiment contributes only one point to such a plot. When the function g(c) is known, a plot of J/g(c)rn0 against F(rn/rn0) gives a straight line. The possible functions F(rn/rn0) can hereby be determined. If, for example, the function F(rn/rn0) is proportional to (rn/rn0)°,the value of a can be determined from a plot of log[J/g(c)rn0] against log(rn/rn0). From such plots a value of a can be determined for each experiment. —
From (14) is seen that the mass of any of the crystals can be expressed as an increasing function of rn/rn0. From the mass balance rn + c VM = rn0 + c0 V0M,
(15) where V is the volume of the solution; c0 and V0 the concentration and the volume of the solution at time zero and M the molar mass of the dissolving substance, we obtain ldrn
dV
dc (16)
The overall rate of dissolution, J, defined by J~—dn/dt
,
(17)
—
3. Experimental Analytical grade chemicals and deionized destilled water were used. Seed crystals were prepared by drop-
J. C’/,rjstofjersen, MR. Christojfersen /Dissolution of calcium sulphate dihydrate in water
ioo~
100 pm A
—
W
___
I
~
~
—~ _1— ~
(a)
_
__
H—~
lOPm
(I
4~
---~
~1II;i T (c)
Hg. I. Stereo electron micrographs of( aSO urated solution for which ~n~/ I -
=
(d)
I
4 2H20. (a) Needles from the stock. (b) Plate from the stock. (c) Crystals from a sat4.5 g/l. (d) Needle from the same solution as (c).
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
83
wise addition of 2.0 1 of 0.60 M calcium chloride to 2.0 1 of 0.60 M sodium sulphate solution. The crystals were washed with water until free of chloride, after
periments crystals (not always in excess) were added to water and the volume changed from 0.13 to 0.23 1. The ratio n0/V0 was in the range
which the crystals were dried at 36°Cfor two days. The dried crystals were analysed for calcium by means of a Perkin—Elmer 305A atomic absorption spectrophotometer. The water of crystallization was determined by heating crystals to about 1000°Cfor 2 h. The crystals decreased in weight by 20.97% (calculated for CaSO4 2H~O:20.93%). All experiments were made using crystals from a single preparation. Two forms of crystals were present, as can be seen in fig. 1. 93% (by number) of the crystals were needles of longest side 50—300 pm. 97% of these crystals were smaller than 200 pm. The other type of crystal (“plates”) had the form of two plates joined to form a wedge. The longest dimension of these crystals was about 300 pm. All experiments were made at 25.0 ±0.1°Cand except for a few experiments made to investigate the effect of the rate of stirring on the reaction rate, a constant stirring rate of 6.7 s~was used. The conductance, L, of the suspension was measured using a conductivity cell and a Radiometer CDM3 conductivity meter. In the first type of experiment the conductance of the suspension was recorded against time. In order to improve the sensitivity of the measuring technique, the dc signal from the conductivity meter was fed into a Radiometer pHM64 pHmeter which was connected to a Hewlett—Packard digital analog converter, which could back out parts of the hereby signal. A in concentration of 3 X 10—6theM could be change detected. From these experiments rate of dissolution was determined as J = V dc/dt. mitial concentrations ranged from 0—0.7 c and n 0/V was in the range (6 X l0~—4x 10_2)nol/l. There was not always an excess of crystals to saturate the solution. The rate was measured 2--l.4X 102M.in the concentration range 0.2X 10 In the other type of experiment the signal from the conductivity meter was fed into a Metrohm E473 potentiostat which controlled a Metrohm E41 5 autoburette (Dosimat) containing water. The volume of water added was recorded as a function of time. Thus the conductance of the suspension could be kept constant at a predetermined value by the automatic addition of water. From these experiments the rate of dissolution was determined as J = cd V/dt. In these cx-
(9 X l0—~—5X 10—2) mol/l. The rate was measured in the concentration range 0.5 X 102_l .45 X 10—2 M.
.
4. Results and discussion The conductance, L~,of a saturated solution contaming crystals depends slightly on the amount of crystals present per unit volume. Experimentally we obtained L~ L5(1 where kcr = 1
—
—
13n/V) =Lskcr
(21)
,
(22)
jln/V,
and L5 is no thecrystals. conductance of a<0.06 saturated solution where containing For n/V mol/l, = 0.12 1/mol. The conductance, Lcr, of an unsaturated solution containing crystals is assumed to be similarly given by Lcr kcrL
(23)
where L is the conductance of the solution without crystals. For experiments at constant volume, n/Vwas kept so low (less than 0.03 mol/l) that the differences between ~cr/~k’ dL/dc dLcr/dt and 1l~crIdt)L are lessand than 1%. and Suchbetween errors are negligible (C in comparison with the general reproducibility of the experiments (approx. 10%). For these experiments we approximate the overall rate of dissolution by V-~-~ = V dc dl~cr dc thJ~cr dt V-~(_-~_) (24) L For experiments at constant conductance, the effect of crystals interfering with the conductance measurements causes the concentration to vary slightly during an experiment. A first order approximation gives = + V dc dV I Lj3(n/V + c) dt ~ c(L~+ kcr dLI~) (25) =
~
.
—
The value of n/V was kept so low (less than 0.03 mol/l)
.1. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
84
that the approximation J~cdV/dt
(26)
less than 0.3%. In the following we assume that p = and conclude that the function g(c) is proportional to Cs
introduces an error of less than 1% in J. The variation in c during an experiment can also be neglected cxdept in the term c~ c when c c5. Points for which a correction of c was not negligible compared with the reproducibility of experiments are indicated in fig. 2 by parentheses. Due to the uncertainty in c5 c, these points were omitted in the calculation of the best straight lines, The are experimental results of bothlog(J/rn types of experiments shown in fig. 2 in which 0) is plotted against log(c5 c) for four values of rn/rn0. Experiments were not to analysed <0.03 g. The solubility, c~, is taken be 1.52for X rn 10—2 M (see appendices A—C). For identical experiments, points corresponding to the mean values of J are plotted. For constant values of rn/rn 0 straight lines with the slope p, 1 .00
—
—
—
C.
For each constant value of rn/rn0, i.e. for each set of points determining the straight lines in fig. 2, the mean value (J/rn0(c5 c)) of .J/rn0(c5 c) has been determined. In fig. 3 log(J/rn0(c5 c)~is plotted against log(rn/rn0). The straight line obtained in fig. 3 has the slope 1.16 ±0.01. The vertical lines through the points in fig. 3 have lengths equal to twice the standard deviations. From fig. 3 we conclude1-16. that the function F(rn/m0) is proportional to (rn/rn0) Assuming 1.16 (27) g(c) F(rn/rn0) ~ (c~ c) (rn/rn0) a rate constant k has to be introduced into (19). From the intercept of the line in fig. 3 with the line log(rn/rn 3 1/s g for the 0) = 0, k = (6.4 ±0.1) X i0 —
—
—
,
—
stirring ratecan of 6.7 s~is obtained. Thebyoverall rate of dissolution therefore be described the function 116 (28) ,
J = km13(c~— c) (rn/rn0)
which is an expression of the type (19). A doubling of the stirring rate caused a 30% increase in k. The rate constant k may depend on the size distribution of
I
-4-5
the stock crystals.
-5.0 I CI
~ I’a IE CI
2
-2.5 2O,~ -
-6.0
____
-
1
-3.0
-2.5
-2.0
-30V CI
2
cs-c log ~-~‘ Fig. 2. Plot of log(J/mo) against log(c 5 — C) for four values of rn/mo. Circles: rn/mo = 0.65; squares: rn/rn = 0.40; triangles: rn/rn0 = 0.25;hexagons: rn/mo = 0.158. Open symbols refer to experiments with V constant, c0 = 0; vertical lines through symbols refer to experiments with V constant, co ~ 0; horizontal lines through symbols refer to experiments with c constant.
_____________________________ -1.0
-0.5 0 log (rn/rn0) Fig. 3. Mean value of log[J/mo(c5 — c)] for each of the four values of rn/mo, see fig. 2, plotted against log(m/mo). The length of the vertical lines through a point is equal to twice the standard deviation.
J. Christoffersen, MR. Christoffersen /Dissolution of calcium sulphate dihydrate in water
85
5. Size distribution
corresponding to expt. 2 were calculated using (5) with f1.(r~1)cx:r11q and q = 0 and q = —1 and taking in-
Two experiments were made in order to investigate the change in the size distribution as crystals dissolve: 0.72 g of crystals were suspended in 0.086 1 of water (expt. 1) and 0.72 g of crystals were suspended in 0.181 of water (expt. 2). After the suspensions had reached saturation, the number of crystals present in each suspension was determined by counting the crystals in a blood cell counting chamber viewed under a light microscope. In expt. 1 there were 1.01 X l0~ crystals, 15.8% of which were plates. In expt. 2 there were 3.1 X 106 crystals, 45% of which were plates. The number of plates was thus approximately the same in both experiments, whereas the number of needles in expt. 2 was only 20% of the number in expt. 1. From this we conclude that the number of needles decreases significantly in our dissolution cxperiments. For both experiments samples were photographed and the size distribution of the needles was determined by measuring the longest these crystals a Carl Zeiss TGZ3 particleside sizeofanalyzer with 48using equally large size intervals, which covered the size-range 0—350 pm. Curve 1 in fig. 4 shows for expt. 1 the percentage of needles in a size interval plotted against the length corresponding to the mid-point of the interval. Curve 2 in fig. 4 shows for expt. 2 the number of needles in a size interval expressed as the percentage of the total number of needles in expt. 1. From the distribution function of expt. 1, distribution functions
to account that 80% of the needles dissolve completely. The results of these calculations are also shown in fig. 4. It can be seen that the distribution calculated with q = 0 does not agree with the experimental distribution curve 2, but that q = —lisa reasonable assumption. This means J ~x 1/r. In order to calculate the apparent diffusion coefficient, Dapp, the radius, r, of a sphere circumscribing a crystal is used as the size parameter. For the needles the longest side is approximately 2r. The diffusion equation has not yet been solved for bodies with geometry as complicated as that of calcium sulphate crystals. We therefore for simplicity assume the diffusion flux out of a sphere circumscribing a crystal to depend only on r and not on the geometry of the crystals. Together with the steady state approximation of diffusion controlled dissolution [3] this assumption leads to ~ (29) dn dt 4irDapp(Cs c) J= = If the rate of dissolution is diffusion controlled, the apparent diffusion coefficient can be found from (28) and (29) if the distribution function is known. Using the distribution function determined in expts. I and 2 for the needles and assuming the value of r for the plates to be equal to the maximum value of r for the needles in expt. 1, we calculate Dapp = 3 X 10—6 cm2/s from expt. I and Dapp = 4 X 10—6 cm2/s from expt. 2.
I
I
I
8
—
~
——
6. The dissolution of crystals mounted in a gel The dissolution of crystals mounted in an agarose-
6-
gel placed on the lower side of a cover glass could be studied through a light microscope by placing the
4,
cover glass on top of a Plexiglass chamber through
—~.
20
~
100
2r/jjifl
5°
Fig. 4. Size distributions of needles. Curve 1: obtained from expt. 1 in which rn 0/V 8.4 g/l; curve 2: obtained from expt. 2 in which mo/V = 4.0 g/l; (X) calculated for expt. 2 with q = —1; (.) calculated for expt. 2 with q = 0.
graphs which water of the dissolving passed. crystals A intervalometer Paillard-Bolex were taken Hl6 with camera suitcontrolled by Shackman TU1 was connected towas aa Leitz Wetzlar microscope and photoable constant time intervals. Films showing the dissolution process at rates 4—80 times the natural rate were studied. From these films we find for the individual crystals of both types that dr/dr increases as r decreases. If
86
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
two or more crystals are close together, relatively little substance dissolves from adjacent parts of the crystals compared with the amount of substance which dissolves from the free parts of the crystals. A hole often appears in each of the two ends of a needle. Except for formation of such holes, the geometry of well-dispersed needles appears to be approximately constant. Nuclei on the surfaces of the plates disappear and
(b) aje not valid, a value of ~Nir~cannot be obtained without knowledge of the size distribution of the crystals and therefore a value of D cannot be obtained.
Appendix A The solubility product of CaSO
4 21120 at 25°C
holes are often formed in their surfaces.
7. Conclusion Taking into account that the assumption on which (29) is based is expected to lead to a value of Dapp which is less than the real diffusion coefficient, D, the values of Dapp obtained for calcium sulphate dthydrate are in reasonable agreement with the diffusion coefficients of magnesium sulphate and zinc sulphate a 2/sin[4]). similar concentration range (D = 7 X 106 cm The value of Dapp together with the proportionality of the rate to (c 5 c), and the behaviour of dissolving crystals mounted in a gel indicate that the rate-controlling process is diffusion. That the rate of dissolution depends slightly on the stirring rate indicates that the rate-controlling process is diffusion (in a stagnant medium) combined with convective-diffusion [3,5a]. The influence of convection on the absolute value of the rate appears to be negligible, Using a rotating disc method, Barton and Wilde [Sb] found the rate of dissolution of CaSO4 2H20 to agree with diffusion-controlled dissolution with 2/s. D Liu 8 Xand 106 cm Nancollas [6] have followed the dissolution of calcium sulphate dthydrate of size 80—120 pm using a calcium electrode. They found the rate of dissolution to be proportional to rn 2/3(c 0(rn/rn0) 5 c) and concluded, using assumptions (a) and (b) that the rate-controlling process is diffusion through a diffusion layer of constant thickness. If the size of crystals changes during an experiment, it is doubtful if the diffusion layer is of constant thickness. Liu and Nancolla’s results can, however, still lead to the conclusion of diffusion-controlled dissolution without assuming a constant diffusion layer, but not together with assumptions (a) and (b), which in fact appear to be doubtful. In their case the ~ appears to be 2/3.term If assumptions (a) and proportional to (rn/rn0) —
.
—
Using a Radiometer Calcium Selectrode, a calomel electrode, and a Radiometer pHM64 pH-meter, we found 2~) pCa5 —log [(Ca 5/M]= 2.305 ±0.005, 2~) where (Ca 5is the activity of calcium ions in a saturated solution. The electrodes were calibrated by making similar measurements of calcium chloride solutions and assuming the liquid junction potentials of asolution calciumofchloride and of a calcium sulphate similar solution pCa are identical. The activity coefficient of calcium ions, YCa’ in calcium chloride solutions was calculated using Guggenheim and Stokes’ expression [7] for the mean activity coefficient, .‘~‘
z.~2
A
—logy+
=
jl/2
____________ 1
+
—
0.028 z÷z I I
(30)
I .536 j1,’2
(where A is a Debye—HUckel constant, I the ionic strength and z the ionic charge), and Bates and Guggenheim’s expression [8] for the activity coefficient of chloride ions,yCl, —logy~ 2/(l + 1.511/2), (31) 1 A1’/ together with the relationship
=
(YCaYC1)’
(32)
In the activity range of interest, 3.54 > pCa> 1.89, the values calculated for pCa for calcium chloride solutions differ by less than 0.005 from the values recommended by Bates [9] and[10], from the values calculated using Davies’ equation
logy1 =
o.s Z1
(
jl/2 + jl/2
—
0.3 i).
(33)
Assumingactivity that calcium ions and sulphate sulphate ions havesoluidentical coefficients in calcium
J. Chr.toffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
tions we obtain pK5 = 4.61 ±0.01 or K5 = 2 for the solubility product of (2.45 ±0.05) X l0~ M calcium sulphate dihydrate, K 5. Liu and Nancollas [11] obtained K5 = measurements (2.58 ±0.15) X 105 M2 from similar potentiometric using (33) for calibration. Gardner and Glueckauf [12] calculated K 2 from Marshall and 5 = 2.49measurements X 10~M [13] and from Slusher’s solubility Brown and Prue’s measurements of osmotic coefficients [14] and Bjerrum’s electrostatic theory [15].
87
termined a function d, the distance closest approach of as “free” ions ofofopposite charge of [141. Results
ford1 = 9.2 A and d2 = 5.2 A are given in table 1. 1 Gardner Glueckauf [12] calculatedKa = 258 M— ford = 9 and A. Liu and Nancollas [11] found Ka = 169 ±15 M~with activity coefficients calculated from (33) which corresponds to (35) with d 5 A. Ainsworth [16] found Ka = 204 ±4 M1 ford = 5 A from potentiometric measurements in a cell without liquid junction. Hanania and Israelian [17] obtained ka = 234 ±8 M1 by a method corresponding to (35) withd’1~4.6A.
Appendix B The ion pair association constant of CaS0 4~21120 at
Appendix C
25°C The solubility ofCaSO4 21120 at 25°C From atomic absorption spectrometry we find the total concentration of calcium in a saturated solution to be c 5 = (1.53 ±0.02) X 10—2 M. From the potentiometric measurements we calculate c~= (1.50 ±0.02) X 10—2 M from (34) and (35). From EDTA titrations using the jump, we calcium find c electrode to register the potential 5 = (1.52 ±0.02) X 10—2 M. In this paper we have used c~= 1.52 X 10—2 M. Other[13], determinations c~include Slusher 1.51 X 10—2ofM; Liu and Marshall Nancollasand [6], 1.48 X 10—2 M; Madgin and Swales [18], 1.51 X 10—2 M;Bock [19], 1.52X 10—2 M;Moreno and Osborn [20], 1.51 X 10_2 M;Nakayama and Rasnick [21], 1.47 X 10—2 M. In the last two references K 5 and Ka were calculated with the assumption of individual activity coefficients for calcium and sui-
2~)determined at 25.0°Cby poTable 1 gives (Ca tentiometric measurements and the total concentration of calcium sulphate, c, made up by weight and checked by atomic absorption spectrometry for solutions of calcium sulphate dthydrate. From these data the association constant Ka, 2~)(SO~), (34) Ka = [CaSO4]/(Ca can be determined by assuming expression for the 2~ionsanand of SO~ions. activity coefficients of Ca Using —logy 4
A z+z_1I1/2 2/a0 1 + dI’/
(35
where a°is a Debye—Huckel constant, Ka can be deTable I Ion paii association constant of CaSO4 2H20;d1 2c 102(Ca~)i02[Ca2~j 10 1 (M) (M) (M)
=
9.2 A,d2
=
5.2 A
2[Ca2’I 10
2
Kaa 1)
Kai (M’)
(M)
(M
1.3516
0.462
0.797
260
0.909
207
1.2286
0.434
0.738
260
0.836
208
0.9975
0.380
0.628
256
0.701
205
0.6650 0.4988
0.288 0.235 0.176
0.451 0.354 0.253
258 263 258
0.491 0.378 0.266
210 219
0.3325
mean: 259
± 10
215 mean: 211
± 10
J. Christoffersen, MR. Christoffersen / Dissolution of calcium sulphate dihydrate in water
88
phate
and the values of K5 and Ka are not directly comparable with values quoted here. ions
Acknowledgements We thank Professor Arne E. Nielsen for stimulating discussions. Aalborg Portland Concrete Research Laboratory, Karlstrup is thanked for providing the stereo electron micrographs. Statens Naturvidenskabelige Forskningsr~dis thanked for granting the Carl Zeiss TGZ3 particle size analyzer and the Perkin Elmer 305A atomic absorption spectrophotometer.
References [1] W. Lerch, Res. Lab. Bull. 12 (Portland Cement Assoc., 1946). [2] H.H. Steinour, Res. Lab. Bull. 98 (Portland Cement Assoc., 1958). [3] A.E. Nielsen, J. Phys. Chem. 65 (1961) 46. [4] R.A. Robinson and RH. Stokes, Electrolyte Solutions (Butterworths, London, 1970) p. 513. [5] (a) V.G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, N.J., 1962) p. 81. (b) A.F.M. Barton and N.M. Wilde, Trans. Faraday Soc. 67 (1971) 3590.
16] Sung-Tsen Lio and G.H. Nancollas, J. Inorg. Nucl. Chem. 33 (1971) 2311. [7] E.A. Guggenheim and R.H. Stokes, Trans. Faraday Soc. 54 (1958) 1646. R.G. Bates and E.A. Guggenheim, Pure Appl. Chem. 1 (1960) 163. [9] R.G. Bates and M. Alfenaar, Ion-selective Electrodes (Nat!. Bur. Std. (US) Spec. Publ. 314) p. 202. [10]C.W. Davies, Ion Association (Butterworths, London, [11]Sung-TsenLiu and G.H. Nancollas, J. Crystal Growth 6 (1970) 281. [12] A.W. Gardner and E. Glueckauf, Trans. Faraday Soc. 66 (1970) 1081. [13]W.L. Marshall and R. Slusher, 3. Phys. Chem. 70 (1966)
181
[14]P.G.M. Brown and J.E. Prue, Proc. Roy. Soc. (London) A 232 (1955) 320. [15] N. Bjerrum, Kgl. Danske Vidensk. Selskab Mat.-Fys. Skrifter 7 (1926). [161 R.G. Ainsworth, J. Chem. Soc. Faraday Trans. I 69 (1973) 1028. [17] G.I.H. Hanania and S.A. Israe!ian, J. Soln. Chem. 3 (1974) 57. [18] W.M. Madgin and D.A. Swales, J. Appl. Chem. 6 (1956) 482. [19]E. Bock, Can. J. Chem. 39(1961)1746. [20] E.C. Moreno and G. Osborn, Soil Sci. Soc. Am. Proc. 27 (1963) 614. [21] F.S. Nakayama and B.A. Rasnick, Anal. Chem. 39 (1967) 1022.