Aquat Geochem (2011) 17:21–50 DOI 10.1007/s10498-010-9105-0 ORIGINAL PAPER

Generic Issues of Batch Dissolution Exemplified by Gypsum Rock Victor W. Truesdale

Received: 24 February 2010 / Accepted: 2 June 2010 / Published online: 16 June 2010 Ó Springer Science+Business Media B.V. 2010

Abstract Recent work has emphasized that the empirical rate equation for batch dissolution of a solid consists of a forward term involving the surface area minus a back reaction term involving surface area and concentration of dissolved solid. Integrated forms exist for use at extremes of high under-saturation and of very heavy solid loadings which lead to saturation. A middle condition allows for significant decrease in solid supply and simultaneous arrival at saturation. This study tests the three approaches simultaneously to the batch dissolution of gypsum, thereby demonstrating a consistent applicability of the aforementioned rate equation. Previously, some mineral dissolutions have displayed so-called nonlinear kinetics and hence have not appeared to conform to this rate equation. This paper provides a template for future investigation of those situations; dissolution experiments are not easy to perform, and instances of the so-called nonlinear kinetics may represent experimental artefact. The relationship between this empirical approach and that of Transition State Theory used in mineral dissolution is discussed, and a new, linear proof for the applicability of the ‘middle ground’ equations is demonstrated. Stirring experiments highlight the difference between the conditions in fluidized bed and laminar flow reactors. Gypsum dissolution is found to be transport limited at all but very vigorous laboratory stirring conditions, although the relationship between the rate of shrinkage of gypsum particles and stirring seems to be relatively simple. A stirring factor is applied to the rate equation overall to allow for differences in reactor design, and it is suggested that this should also be applicable to laminar flow reactors. The link between batch and chemo-stat dissolutions is stressed, together with a need to contour dissolution data on a new graph of particle size versus stirring rate. Keywords Batch dissolution  Dissolution kinetics  Shrinking object  Shrinking sphere  Gypsum dissolution  Carbonate dissolution  Silica dissolution  Geo-engineering  Transition State Theory  Biogenic silica  Gypsum  Stirring rate  Common ion

V. W. Truesdale (&) School of Life Sciences, Oxford Brookes University, Gipsy Lane, Headington, Oxford OX3 0BP, UK e-mail: [email protected]

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1 Introduction Greenwood et al. (2001), Truesdale et al. (2005a, b) and Truesdale (2007, 2008, 2009, 2010) studied batch (closed system) dissolution partly to understand the dynamics of biogenic silica dissolution in the oceans (e.g., Yool and Tyrell 2003; Kamatani 1982; Tre´gur et al. 1989), but increasingly for its own sake. Hence, these dynamics are of interest in the dissolution of a vast range of substances as well as silica, involved for example in industrial and other applications, e.g., mineral extraction (Sohn and Wadsworth 1979), radioactive-waste disposal (Ichenhower et al. 2008) and drug delivery (Balbach and Korn 2004). Even within environmental chemistry itself, dissolution dynamics are central to understanding the carbonate system of seawater (Berner 1980; Busenberg and Plummer 1986) and freshwater (e.g., Svensson and Dreybrodt 1992), the stability of submarine salt deposits, and globally in mineral weathering (Helgeson et al. 1984; Holdren and Berner 1979), including natural volcanic glasses (Wolff-Boenisch et al. 2004). Further, specifically within the context of gypsum Raines and Dewers (1997) point out that evaporite deposits, including gypsum-bearing formations, underlie 35–40% of the area of the 48 most southerly states of the US, where they exert profound effects upon water quality. Finally, futuristic geo-engineering projects designed to control excess atmospheric carbon dioxide, which causes global warming and hence climate change (Lovelock and Rapley 2007), need to be underpinned by good understandings of the dissolution rates of silica and carbonate particles in the oceans. In studies of the dissolution of biogenic silica (diatom frustules) in seawater in the late twentieth century, the convention was to base the integrated rate equation upon an exponential increase in dissolved silica concentration (e.g., Kamatani 1982; Tre´gur et al. 1989). This was also the case in the hugely more extensive and general field of mineral weathering, stemming from Transition State Theory (TST) as applied by Lasaga (1981) and Aagaard and Helgeson (1982). Interestingly, Kamatani et al. (1980) broke with tradition, addressing the shrinkage of biogenic silica particles during dissolution by the mathematical relationship between the volume and the surface area of a sphere. In this model, dissolution stops either when the supply of solid runs out or when the solution becomes saturated. However, although not identified by Kamatani et al. (1980), there was confusion between these two conditions which allowed the exponential model to be applied incorrectly to the under-saturated condition for biogenic silica. After illuminating this, Truesdale et al. (2005a, b) confirmed that the exponential approach only applies to dissolving loads which are in excess of saturation and where loading remains essentially constant throughout dissolution; that is, the condition generally used with investigations based upon TST. Meanwhile, through deeper study of Kamatani et al.’s (1980) shrinking sphere model, Truesdale et al. (2005a) and Truesdale (2007, 2008, 2010) derived equations for the batch dissolution of either mono-dispersed or poly-dispersed particles (populations of particles of either one size or many sizes, respectively) at high under-saturation. From this, it could be seen that the earlier progress by Kamatani et al. (1980) probably stalled because the populations of diatom frustules which they studied would have been poly-, not mono-dispersed. Truesdale’s (2007) routine sieving of particles removed much of that complexity. Truesdale (2008, 2009) extended this work to dissolutions of particles of mixed-size, and generally to shrinking objects rather than just shrinking spheres. A key understanding of this (Truesdale 2010) is that the shrinking object model is more simply posited as an object shrinking at a constant rate in one of its linear dimensions. Truesdale (2010), concerned that ‘nonlinear kinetics’ can so easily be the result of inadequate

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experimental design, suggested an experiment that could be adopted as a universal prerequisite to determine their presence or absence in any dissolution system. This paper develops the recent improvements to modelling of the dynamics of batch dissolution even further by testing the equations on gypsum about which, anyway, much is already known from other approaches (e.g., Barton and Wilde 1971; Christoffersen and Christoffersen 1976; Liu and Nancollas 1971; Gobran and Miyamoto 1985; Lebedev and Lekhov 1990; Raines and Dewers 1997; Jeschke et al. 2001; Kuechler et al. 2004). Two particular advantages of studying gypsum are that it dissolves relatively quickly and that conductimetry can be used, thereby avoiding more cumbersome wet-chemical analysis. Some substances take weeks to dissolve, and it is much more efficient to try out new methodologies on more rapidly dissolving substances. Also, conductimetry provides virtually continuous analytical output with time. As gypsum is a natural mineral, it provides a suitable working material with which to test Truesdale’s (2010) assertion that working at very low solid loadings with the Truesdale (2007) equation should be preferable to using the high loadings conventional in the TST approach. This would certainly circumvent problems with secondary mineral precipitation (Tole et al. 1986). TST is fundamentally a thermodynamic approach to chemical kinetics, whereas the O’Connor and Greenberg (1958) equation, and its derivatives, the Kamatani et al. (1980) and Truesdale (2007) models, are essentially empirical. Indeed, if TST were to be applied correctly, it would provide the thermodynamic justification for the more empirical models. Not surprisingly then, Aagaard and Helgeson’s (1982) derivation of a TST dissolution equation includes surface area, just as does O’Connor and Greenberg (1958). Practically, however, actual engagement with surface area through the TST approach has been minimal, the opportunity being taken instead to use high loadings of solid so as to retain a constant surface area by means of what is really, the Ostwald isolation condition. Therefore, there is little to choose between the integration of O’Connor and Greenberg (1958) and that of Aagaard and Helgeson (1982) for constant surface area, and problems in either, implies the same in the other. This is how the so-called problem of nonlinear kinetics has arisen in both domains whereby dissolution behaviour at high under-saturation appears to be different to that at near saturation (e.g., Nagy and Lasaga 1992; Van Cappellen and Qiu 1997; Jeschke et al. 2001; Rickert et al. 2002; Truesdale et al. 2005a). The condition is perhaps easily understood from the O’Connor and Greenberg (1958) model, where, in violation of the equation, the net rate of dissolution does not decrease linearly with the increasing background concentration of the dissolved solid (see later). Reasons for this effect, which potentially threatens the viability of both the TST and O’Connor and Greenberg (1958) approaches, have been sought within the solution chemistry (Truesdale et al. 2005a) and within the nature of the solid surface (see, for example: Jeschke and Dreybrodt 2002; Lu¨ttge et al. 2003; Colombani and Bert 2007; Dove et al. 2005; Zhang and Lu¨ttge 2009). In marked contrast then, Kamatani et al. (1980) and Truesdale (2007) deliberately investigated systems in which there is a significant loss of surface area during dissolution. The real prospect offered by TST seems to have been that it would deliver a standard and unified approach to weathering study which would require only one or two parameters to characterize each mineral’s dissolution; weathering could be reduced to one, neat table of data. Unfortunately, as many minerals did not seem to obey the TST equation, workers were forced to adopt various empirical modifications of it (e.g., Nagy and Lasaga 1992; Van Cappellen and Qiu 1997; Jeschke et al. 2001; Rickert et al. 2002), with the loss of the unified approach. The progress made here promises a way around the nonlinear kinetics; it offers a second opportunity for the unified approach as well as a way to discover why the problems were encountered in the first place. The paper also demonstrates how to

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validate the Kamatani et al. (1980) equation with a linear fitting, thus making its proof that more certain. Before moving on, therefore, it is perhaps important to stress that this paper is principally about testing equations, and any information gained about gypsum is almost incidental.

2 A Summary of the Equations Used The symbols used in the equations are listed in Table 1. 2.1 The O’Connor and Greenberg (1958) rate equation The rate equation for biogenic silica dissolution is dc 1 ¼  ðk1  S  k2  S  cÞ dt V

ð1Þ

where k1 and k2 are the forward and back reaction rate constants, respectively, S is the surface area of the solid, and c is the concentration of dissolved silica at time, t. The inverse V term on the RHS is introduced here for the first time to gain dimensional

Table 1 The symbols used in the paper Symbol

SI units

Identity

c

mol l-1 (M)

Concentration of dissolved solid

cT

mol l-1 (M)

Total loading of solid in the reaction mixture; equal to the final dissolved solid concentration in the dissolutions conducted here as they remain under-saturated

csat (ce)

mol l-1 (M)

Dissolved concentration at saturation

-2

-1

k1

mol m

k2

l m-2 s-1

s

Back reaction rate constant of O’Connor and Greenberg (1958) model

S

m2

Surface area

aKa

M-2/3 s-1

Modulus of the gradient of Kamatani et al. (1980) equation (Eq. 2)

aTr

M1/3 s-1

Modulus of gradient of Truesdale (2007) equation (Eq. 4)

aRKa

M-1 s-1

Rationalized aKa; aKa/c1/3 T

-1

Forward rate constant of O’Connor and Greenberg (1958) model

aRTr

s

Rationalized aTr; aTr/c1/3 T ; gradient of Eq. 6

N

1

Number of particles

V

l

Volume of reaction mixture in litres

MB

kg mole-1

Molar mass of B

Q

kg m3

Density of solid

Kip

M-1

Association constant for the ion pair CaSO04 (Eq. 12)

r, r0

m

Radius of dissolving sphere at time t and 0, respectively (Eq. 11)

u

m s-1

Rate of shrinkage of sphere radius (Eq. 10)

fu

1

Fraction of solid undissolved at any time in an under-saturated dissolution (Eq. 6)

H

lm s-1

Hyperbolic coefficient (Fig. 3)

Ksp

M2

Solubility product

x

s-1

Stirring rate

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Table 2 Linear regression data for the fittings of the Truesdale (2007) model (Eq. 6) to dissolutions at high under-saturation (0.0500 g l-1; 2.9 9 10-4 M) Gradient (:aRTr) (±std.error)/s-1

Intercept (±std.error)

r2

d.f. (n - 2)

106

(1,710 ± 3) 9 10-6

0.996 ± 0.001

0.999

29

180

(1,104 ± 4) 9 10-6

0.991 ± 0.002

0.999

25

212

(944 ± 2) 9 10-6

0.994 ± 0.001

0.999

19

250

(713 ± 5) 9 10-6

0.996 ± 0.003

0.998

40

355

(528 ± 4) 9 10-6

0.990 ± 0.003

0.997

44

425

(380 ± 5) 9 10-6

0.914 ± 0.007

0.994

16

500

(257 ± 5) 9 10-6

0.991 ± 0.007

0.994

16

1,000

(155 ± 1) 9 10-6

0.879 ± 0.002

0.999

21

Sieve size (lm)

The whole curve was included for sieve sizes 106–355 lm, whereas only the later part (two-thirds) was used for sizes 425 lm and above, to establish the finishing time of the run. This, together with the starting point, was used to calculate the rationalized Shrinking Rate constant, aRTr, used in Fig. 3

consistency between Eq. 1 and its later derivatives. Without this, Eq. 1 should really express the flux, dq dt ; of dissolving solid emanating from the surface. In practice, no numerical difference is experienced when the concentrations are expressed, per litre. 2.2 The Kamatani et al. (1980) Shrinking Sphere Model Analytic integration of Eq. 1 to N identical spheres yields the equation: 2 0 1 1 1    3 3 1 1 csat 63 BðcT  cÞ þðcsat  cTÞ C ln ln þ @ A 24 1 1 2 ðcsat  cÞ ðcsat  cT Þ3 2 c3T þ ðcsat  cT Þ3 0 11 !3 1 1 1 3 3 3 ðc 3 2 ð c  c Þ  c Þ 2  c  ð c  c Þ 1 1 T sat T sat T T A  32  tan1 5 þ32  tan1 @ 1 1 1 1 32  ðcsat  cT Þ3 32  ðcsat  cT Þ3   2 4pN 3ðMSiO2 ÞV 3 t ¼ k2 V 4pNq

ð2Þ

where csat and cT are the final concentrations either at saturation or when all the solid has been expended, and N, MSiO2 , V and q are the number of particles present, the molar mass of silica (0.0600843 kg mole-1), the volume of solution, and the density of the solid, respectively. It is useful to define the constant:   2 4pN 3ðMSiO2 ÞV 3 ð3Þ aKa ¼ k2  V 4pNq with units of M-2/3 s-1, and where the subscript, Ka, identifies it with the Kamatani et al. (1980) model. Truesdale (2009) showed that the entire Kamatani et al. (1980) derivation applies to other shapes, e.g., a cube, tetrahedron, whence other constants analogous to aKa and k2 are defined using analogues of Eq. 3 appropriate to particular shaped particles. Note the V under 4pN in the first square bracket of the right-hand side of Eq. 2.

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Greenwood et al. (2001) and Truesdale (2010) accidentally omitted this, but it follows through from the comment made earlier about the O’Connor and Greenberg model. Truesdale (2009, 2010) named aKa, ‘the shrinking sphere rate constant’ of the Kamatani et al. (1980) model. Now, it seems better to demote it to ‘the shrinking sphere rate parameter’ as it is not universally constant as first imagined, and retain the original title for aRKa, which is rationalized aKa, equal to aKa/c1/3 T . As will be shown this is universally constant; it has units of M-1 s-1. Of course, Eq. 2 can be linearized by transforming each set of concentration versus time data. Each concentration value is used to calculate a corresponding value for the whole of the left-hand side of Eq. 2, and these are then plotted against time. This may well be the way in which Kamatani et al. (1980) analysed their runs although that was never clear. 2.3 The Truesdale (2007) Model for Highly Under-Saturated Dissolution When, for a sphere, the second term of Eq. 1 is neglected, integration yields:  2 k1  4pN 3ðMSiO2 ÞV 3 1=3 t cT  ðcT  cÞ1=3 ¼  3V 4pNq

ð4Þ

where aTr ¼

 2 k1  4pN 3ðMSiO2 ÞV 3 3V 4pNq

ð5Þ

in which aTr is defined analogously to aKa above (Sect. 2.2) as ‘the shrinking sphere rate parameter’ of the Truesdale (2007) model, with units of M1/3 s-1. Division through by c1/3 T (Truesdale 2007) gives a rationalized linear form involving, fu, the fraction of solid undissolved (f1/3 u versus t):    1=3 c 1=3 1 1 ¼ aTr  t þ 1 ð6Þ cT cT where the gradient is aRTr. Eq. 6 can be further developed to give a cubic in time: 1=3

2=3

c ¼ a3Tr  t3  3a2Tr  cT  t2 þ 3aTr  cT t:

ð7Þ

As with aKa of Eq. 3, the mathematical form of Eqs. 4, 6 and 7 is retained with other regular objects (Truesdale 2009), although an analogy to Eq. 5 has to be derived to define an equivalent to aTr for each object. 2.4 The Exponential Approach to Saturation With Heavy Loadings of Solute With a great excess of solute, which will leave the surface area of solute effectively constant, integration of Eq. 1 leads to:  ð8Þ c ¼ csat  1  ek2 St where csat = k1/k2. It is the mis-application of this well-known integration which confused much of the literature dealing with biogenic silica dissolution during the late twentieth century (Truesdale et al. 2005a, b).

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2.5 Quantifying the Role of Stirring in Dissolutions Experimentation by Truesdale (2010) strongly suggested that all of the shrinking object equations and the O’Connor and Greenberg (1958) would apply to silica gel after multiplication by a Reynolds number, W, which would characterize the vigorous stirring regime used. Thus, for example, Eqs. 4 and 5 would yield: 1=3

cT  ðcT  cÞ1=3 ¼ W  a  t:

ð9Þ

The further proof of this exciting possibility became one of the main objectives of this study, with the realization that W would complement the fact that the shrinking object model is actually underpinned by the more fundamental physical statement; the shrinkage of a regular object at a constant linear velocity, u (Truesdale 2009). For example, for a sphere of radius, r: dr=dt ¼ u;

u\0

ð10Þ

so that: r ¼ ro þ u  t:

ð11Þ

In effect then, stirring merely enhances u which is itself a component of both aKa and aTr as it is embedded within k2 or k1, respectively (Truesdale 2009).

3 Experimental 3.1 Laboratory Work 3.1.1 Preparation of Gypsum Particles Natural gypsum rock was obtained from British Plaster Board (BPB), Newark, UK, where it is analysed as part of routine quality control to ensure that it loses its theoretical water content (20.9%) upon heating. Lumps (*2-cm cubes) were placed in a polythene bag, broken with a hammer against a wooden block, and shaken (Endecotts EFL-MK3 mechanical shaker) through a stack of 21-cm brass sieves (Endecotts, London) to give a good spread of the required crystal sizes. Occasional use of Analar CaSO42H2O (Hopkins and Williams, UK) was made in the solubility experiments. 3.1.2 The Dissolution Vessel and Associated Equipment Dissolutions at high under-saturation were performed in a vigorously stirred (Grant Instruments s-2 axial-flow stirrer; Cambridge, UK) 1-l glass beaker held in a Grant SS40 temperature-controlled water bath set at 20.0 (±0.1)°C. Dissolutions reaching saturation were performed in a 300-ml beaker. In both cases, the gypsum was dispensed from a weighing boat by ‘tipping and tapping’, with a final weighing to compensate for any residue. The salt thereby entered the reaction mixture within about 1 s. The variation in salt concentration was monitored using a Phillips PW9505, 10-scale conductivity bridge connected to a Kipp and Zonen (Holland) BD11e flat-bed chart recorder, each scale offering a 0.1 volt output. Readings from the chart were manually fed into a spreadsheet. The 93 range conversions (e.g., 1 and 3 mS cm-1) built into the instrument were validated

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empirically. Platinized platinum electrodes were used in the study. The electrodes were protected within a 1-cm-diameter cylindrical glass body perforated by two small holes to encourage solution to circulate. A Heidolph RZR1 stirrer was used in experiments demanding variable stirring speeds, as well as in most of the saturation studies. (The latter demanded very high loadings (26.6 g l-1) and a smaller volume of mixture (300 ml) was economical; this required a slower stirring rate to prevent solution slopping out). The instrument includes a manual screw-rheostat providing up to 30 mm of travel, i, along a linear scale, and a second, slower shaft geared to the main one at 8.5:1. The main axle speed was calibrated in 5-mm increments of rheostat travel after fitting the stirrer chuck to the slow-speed axle, and gearing it down by a further 3:1. Rotations of the final shaft were counted against a stopwatch over a period of about 1.5 min, and the speed of rotation of the fast shaft calculated from the combined gear ratios. The calibration between rheostat travel and axle rate, x, was linear:  x s1 ¼ 0:92ð0:01Þ  iðmmÞ þ 5:0ð0:2ÞðrpsÞ; where the figures in brackets are standard errors for the gradient and intercept from linear regression with 8 d.f., whence the coefficient of determination (r2) for the fitting was 0.999. Typically then, the stirrer delivered speeds between about 5 and 35 rps. It was equipped with a 32-mm-diameter, three-bladed, plastic propeller of the type conventionally used to stir a 20-l water bath. The stainless steel stirrer shaft of each stirrer was insulated from the stirrer chuck using a short sleeve of fairly tightly fitting plastic tubing. Without this, stopping the stirrer in mid-dissolution led to a reduction in the signal on the chart recorder by as much 20%. Even though there never was any cause to do this, the effect was nevertheless eliminated as a precautionary measure. The open beakers used in this study were satisfactory for short dissolution runs (\1 h). However, prolonged exposure to evaporation of saturated mixtures provoked super-saturation. Over an 8-h test, evaporation from the open and vigorously stirred 1-l beaker in a laboratory at about 18°C led to a steady increase in concentration of 7.0 9 10-5 M h-1 above saturation, while no increase in concentration was observed in a control flask where the stirrer projected into a screw-capped bottle. Apparently, the approach to equilibrium from the super-saturated state is slow. If not appreciated for what it is, this effect appears as a continuing dissolution which, of course, would entirely confound the modelling. In future, dissolution vessels should be covered routinely. 3.1.3 The Various Calibration Curves Required Although the conductivity of solutions containing either sodium or calcium chloride or sodium sulphate were found to vary linearly with concentration over all the ranges used (0–100 lS cm-1 to 0–3 mS cm-1), that for calcium sulphate did not, showing instead an increased progressive loss of signal as concentration increased. (Interestingly, a similar calibration effect was found with calcium sulphate but again, not with calcium chloride, when determining calcium concentration by flame photometry.) This appears to reflect classical increased association between Ca2? and SO42- ions (Crow 1994) in which the zero-charge on an ion pair effectively robs the solution of conductivity that would otherwise be manifested from the associated Ca2? and SO42- ions. Therefore, we have:

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29 0 Ca2þ þ SO2 4 CaSO4

ð12Þ

where the equilibrium constant for the ion pair, Kip, is defined as:

 CaSO0 Kip ¼  2þ  4 2   SO4 Ca

ð13Þ

and where the total calcium concentration in solution, T, consists of both free ions and ion pairs, so that:  2þ  2 

 Ca þ SO4 þ CaSO04 T¼ 2  2þ  2  



 Ca þ SO4 : þ Kip  Ca2þ  SO2 ¼ 4 2 After allowing for charge balance in which [Ca2?] = [SO42-]:

2

 Kip  Ca2þ þ Ca2þ  T ¼ 0 Now, as conductivity, j, is proportional to [Ca2?], this quadratic equation solves as:  p ð14Þ j ¼ v  1 þ 1 þ 4  Kip  T = 2  Kip

Chart response / %

where v is added as a fitting parameter ultimately related to the sum of the molar ion conductances of calcium and sulphate. This interpretation was supported by the close fit found between conductivity measurements for the 0–3 and 0–1 mS cm-1 ranges (the former exemplified in Fig. 1) and the model, with Kip = 75 M-1 and v for the 1 mS cm-1 range was 9,920. The fitting was made using a least squares approach incorporating the facility SolverTM within the spreadsheet Excel. The equation was re-arranged to give concentration and applied in the spreadsheet to each run. Liu and Nancollas (1971) used an unspecified ion-pair approach in which Kip = 169 M-1, and Christoffersen and Christoffersen (1976) report values of about 200 M-1. However, both sets of authors corrected for variation in activity coefficient. For practical purposes, an accurate value for Kip is actually unnecessary here as it is really an interpolation tool and, in any case, in the fittings v and Kip are not entirely independent. The use of an empirical calibration curve based upon the power function:

100 80 60 40 20 0 0

0.005

0.01

0.015

0.02

[Gypsum] / M Fig. 1 Examples of two independent calibration curves for natural gypsum obtained in different months using the 1-mS cm-1 scale. The blue and red continuous lines are derived from the ion-pair model in which Kip = 75 M-1 and v * 9,920 (Eq. 14). The change in gradient at 0.015 M marks saturation. From this, Ksp is estimated to be 2.25 9 10-4 M2

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jð0100 chart divisionsÞ ¼ u  ½CaSO4 v

ð15Þ

is also possible. For the 0–3 mS cm-1 range, u and v were typically 1,171 (chart divisions) and 0.6974, respectively, for concentrations between 0 and 0.015 M gypsum. Of course, the empirical power function lacks the mechanistic basis of the ion-pair model. 3.2 Modelling Modelling of dissolution data was performed in the spreadsheet ExcelTM. Construction of the linear version of the Kamatani et al. (1980) model was surprisingly similar to the nonlinear one (Truesdale 2010), but required actual data to be substituted for simulated concentration settings in the calculation of the left-hand side of Eq. 2. These calculated values were then matched up with their corresponding times and the graph drawn.

4 Results During this study, a straight line proof of the Kamatani et al. (1980) equation was discovered. Consequently, all three batch dissolution models can now be proved in this way: the Truesdale (2007) equation in highly under-saturated solutions, the Kamatani et al. (1980) equation up to and just above saturation, and the simple exponential model for solutions loaded grossly beyond saturation. The key primary kinetic variables to be determined then are the shrinking object rate parameters, aTr and aKa, associated, respectively, with the Truesdale (2009) and Kamatani et al. (1980) models, and the exponential rate constant for the grossly loaded condition. Sometimes, it is convenient to use the raw rate parameters aKa and aTr emanating from plots of Eqs. 2 and 4, respectively, while at other times the rationalized rate parameters, aRTr and aRKa, which are calculated by dividing by the third power of the loading, e.g., aTr/c1/3 T (Eq. 6), are more useful. The description of the results follows this overall plan but only after the experiment which defined the concentration for gypsum saturation has been dealt with. 4.1 Gypsum Saturation at 20°C The saturation concentration for both Analar CaSO42H2O and natural gypsum samples at 20.0°C was determined to be 0.015 (±0.001) M from calibration graphs obtained on two separate occasions with the two sets of electrodes used. Below saturation, the calibration curves fitted the ion-pair model (Sect. 3), but as saturation was approached the gradient of the curves decreased rapidly towards a plateau (Fig. 1). The solution remained clear up to the point where the gradient changed, but then rapidly became milky as further small increments of gypsum were added. Since the solubility product, Ksp, is the product of the saturation concentrations of calcium and sulphate ions, Ksp = 2.25 9 10-4 M2. 4.2 Dissolution in Highly Under-Saturated Dissolutions 4.2.1 The Dissolution of 106-lm Particles at Concentrations of 0–2.9 9 10-3 M Experiments showed that the Truesdale (2007) equation applied to loadings of 106-lm gypsum particles between 0–2.9 9 10-4 M. Thus, over a period of days, two separate runs with 1.44 9 10-4 and 2.90 9 10-4 M loadings yielded essentially identical robust straight

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31

(1-C /C T)

1/3

1 0.8 0.6 0.4 0.2 0 0

200

400

600

800

t/s Fig. 2 Fittings of the Truesdale (2007) model to runs with gypsum loadings of 1.435 9 10-4 M (blue) and 2.327 9 10-4 M (red). These correspond to saturations of 1 and 16%, respectively. Points on the red plots are displaced to the right by 200 s. The points are experimental results, and the solid lines are model representations. With the red line, the model is fitted to the first few points

line plots with coefficients of determination of 0.9995 and 0.9999, respectively. Taken in the same order, the gradients (aRTr), with standard errors, on the plot of (1 - c/cT)1/3 versus time were 1.62 (±0.01) 9 10-5 s-1 and 1.68 (±0.01) 9 10-5 s-1, respectively, while the intercepts (with standard errors) were, 0.998 and 0.996 (both ± 0.001). The former of the two runs is illustrated by the blue plot in Fig. 2. In contrast, a loading of 2.32 9 10-3 M gypsum introduced significant curvature (Fig. 2) and precluded the use of the Truesdale (2007) equation; the Kamatani et al. (1980) approach needs to be invoked with the higher loading. These experiments used the Grant s-2 stirrer in the 1-l beaker, in combination with the 100- and 300-lS cm-1 conductivity scales. 4.2.2 The Effect of Particle Size Upon Dissolution Rate of Gypsum The hyperbolic plot proved by Truesdale (2009) to exist when rationalized shrinking object rate parameter, aRTr, is plotted against mean particle size is demonstrated for gypsum by the trend in the points of Fig. 3. The results were obtained using gypsum loadings close to the 2.9 9 10-4 M of the experiments in the preceding Sect. 4.2.1. Whereas particles from sieves with meshes of 355 lm or less yielded robust straight line plots (r2 [ 0.998) as (1 - c/cT)1/3 versus time, those from sieves with meshes wider than 425 lm yielded curves more akin to the two-particle dissolution described by Truesdale (2008). This is understandable as the sieving system used accommodated a wider spectrum of particle sizes as sieve size increases; this is displayed in Fig. 3 by the uncertainty bars. As a consequence, therefore, throughout this work particle sizes were generally restricted to less than 425 lm. However, specifically in constructing the hyperbola of Fig. 3, runs for the larger particles were used because Truesdale (2008) has proved that a two-component dissolution terminates at the same time as that of a dissolution of only the slow-dissolving component. Accordingly, in these specific cases the shrinking object rate parameter was simply calculated as the gradient of the straight line between the start and finish of the reaction on the rationalized plot (note that this argument is reasonable here, but not in the previous section, where the curvature arises for different reasons). Two model fittings to the points in Fig. 3 were obtained by linear regression of the experimentally determined shrinking object rate parameter, aRTr, against reciprocal particle size, 1/d. Regression of all eight points yielded a gradient and intercept (±std. errors) of 0.25 ± 0.01 lm s-1 and -10 (±3) 9 10-5 s-1, respectively, with a coefficient of determination of 0.993. The small negative intercept is discernible in Fig. 4 and suggests a

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0.001

a

R Tr

/s

-1

0.002

0 0

200

400

600

800

1000

1200

d / μm Fig. 3 Variation in Rationalized Shrinking Object rate constant, aRTr, with particle size, d, of gypsum. The points represent experimental points and the red and blue continuous lines represent model derived data fitted according to Fig. 4 for hyperbolic coefficient, H, values equal to 0.227 and 0.250 lm s-1, respectively. The uncertainty bars depict the sieving intervals

mismatch between the point for the highest rate and those for the lower ones. Indeed, regression of only the first seven points increased the coefficient of determination to 0.999, with gradient and intercepts (±std. errors) of 0.227 ± 0.004 lm s-1 and -5 (±1) 9 10-5 s-1, respectively. When represented as a rectangular hyperbola (Fig. 3), both fittings demonstrate at this stage of the investigation good agreement with the experimentally generated data, with the lower value of the gradient giving the better fit. Moreover, the mismatch is easily seen to be consistent with the aforementioned bias in the measurement of shrinking object rate parameter towards the slower dissolving component in the fractions prepared with the larger sieve sizes. The level of proof inherent in Fig. 3 is satisfactory for this study, but in future work it should be possible to reduce the uncertainties for the large sieve sizes by coupled sievings, in which two similar sized sieves, e.g., 1,000 and 1,020 lm, are used together. 4.2.3 The Effect of Stirring Rate Upon Under-Saturated Dissolutions Plots of (1 - c/cT)1/3 versus time remained linear (r2 [ 0.999; Fig. 5) when stirring rates in the 1-l beaker were increased stepwise from 10 to 40 rps. This shows that the shrinking object model (Eq. 6) continued to be effective over this stirring range. Meanwhile, the rationalized shrinking object rate constant, aRTr, was found to increase exponentially from about 2.5 9 10-4 to 1.4 9 10-3 s-1 (Fig. 6; Table 3) with increase in stirring rate. Replicate dissolutions of *2.9 9 10-4 M loadings of 106-lm crystals were used in this experiment and with the 0–100-lS cm-1 scale on the conductivity bridge the reading on the recorder reached a maximum of *56 chart divisions. The continuous line in Fig. 6 was fitted by inverting the graph and fitting a straight line log plot. The fitting routine, Solver,

R Tr

/s

-1

0.0016

a

Fig. 4 The variation of Shrinking Object rate parameter, aRTr, with reciprocal of particle size. The two lines yield the two fittings in Fig. 3

0.0012 0.0008 0.0004 0 -0.0004

0

0.002

0.004 6

0.006 -1

1/d / 10 m

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33

(1-c /c T)

1/3

1 0.8 0.6 0.4 0.2

99.2%

0 0

1000

2000

3000

4000

t/ s Fig. 5 Shrinking object plots (Eq. 6) for six runs under various stirring rates. In each case, a robust straight line as obtained extending over a dissolution period equal to the point where the descending line intercepts the time axis. The dashed line at 0.2 ordinate units represents 99.2% of the dissolution; points below this line are characteristically more uncertain

Laminar flow over static bed

0.0008

R Tr

/s

-1

0.0012

a

Fluidised bed

0.0004

0.0000 0

10

20

30

ω/ s

40

-1

Fig. 6 Variation in dissolution rate of gypsum with increase in stirring rate, x. The points represent measured values of the rationalized shrinking object rate constant (Eq. 6) from the curves in Fig. 4. The continuous line is an exponential model fitted to the data. The peak value of aRTr is estimated to be 1.41 9 10-3 s-1

Table 3 The quality of the fittings of the Truesdale (2007) model (Eq. 6) at various stirring rates of Fig. 3 Gradient (:aRTr) (±std.error)/s-1

Intercept (±std.error)

r2

d.f. (n - 2)

9.6

(242 ± 1) 9 10-6

0.980 ± 0.001

0.999

45

11.4

(559 ± 2) 9 10-6

0.998 ± 0.001

0.999

36

14.2

(807 ± 2) 9 10-6

0.996 ± 0.001

1.009

34

18.8

(112 ± 1) 9 10-6

0.996 ± 0.002

0.999

29

28

(131 ± 1) 9 10-6

1.000 ± 0.001

0.999

25

35.4

(137 ± 1) 9 10-6

0.990 ± 0.003

0.998

28

Grant s-2 (1 l)

(145 ± 1) 9 10-5

0.996 ± 0.002

0.999

25

Heidolph (300 ml)

(165 ± 1) 9 10-5

0.996 ± 0.001

1.000

21

x/s-1

Supplementary runs

The mixture contained *0.500 g of 106-lm gypsum particles. Supplementary runs include a comparison of dissolution with the fixed speed Grant s-2 stirrer in the 1-l beaker and the Heidolph variable speed stirrer in a 300-ml beaker, at proportionately identical loadings

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available in the spreadsheet Excel was used to find the plateau value of Fig. 6 which maximized the value of the coefficient of determination (r2) for the fitting between ln(aTr/ C1/3 T ) and stirring rate, x. The gradient and intercept (±std. errors) were -0.138 ± 0.003 and -6.79 ± 0.05, respectively, and the coefficient of determination (r2) was 0.998 for 3 d.f; the corresponding plateau value was 0.00141 s-1. A run performed with the Grant s-2 fixed-rate stirrer in the 1-l beaker instead of the Heidolph variable rate stirrer, but otherwise under the same conditions as used in the previously described experiment, yielded a rationalized shrinking object rate constant, aRTr, of 0.00145 s-1 (Table 3). This figure vindicated the casual observation, based upon the swirling within the 1-l beaker, that the Grant s-2 stirrer was slightly more vigorous than the Heidolph stirrer set at its maximum speed. 4.2.4 The Effect of Selected Salts Upon the Under-Saturated Dissolution of Gypsum In an under-saturated dissolution, the presence of concentrations of sodium chloride comparable to that of the dissolving gypsum was not found to interfere significantly. Two runs, both with 2.9 9 10-4 M loadings of 250-lm gypsum particles, but the first in de-ionized water and the second in 0.0200 g l-1 (3.42 9 10-4 M) sodium chloride, yielded rationalized shrinking object rate constants, aRTr, of (739 ± 2) 9 10-6 and (721 ± 3) 9 10-6 s-1, respectively. The straight line fittings to the runs were robust; the coefficient of determination (r2) for each curve was greater than 0.999 for 35 and 38 d.f., respectively; in both cases, the intercept was indistinguishable from 1.00 (p = 0.05). The plot for the run in de-ionized water attained a maximum deflection on the chart recorder of 52%. The run with sodium chloride was carried out over the same conductivity range (100 lS cm-1) but ran between 30 and 83% on the chart, thereby taking account of the extra background conductivity from the sodium chloride. By the same approach as above but in two separate runs, the presence of concentrations of either calcium chloride dihydrate (0.0309 g l-1; 2.1 9 10-4 M) or sodium sulphate (0.0293 g l-1; 2.06 9 10-4 M) were also found not to impose a significant effect upon dissolution of 250-lm gypsum particles at these high under-saturations. Taken in the same order, rationalized shrinking object rate constants, aRTr, of (677 ± 3) 9 10-6 and (695 ± 3) 9 10-6 s-1 were obtained. In both cases, the conductivity ranged between approximately 37 and 90 lS cm-1, similarly to that in the above-mentioned experiment with sodium chloride. Once again, the straight line fittings to the runs were robust; the coefficient of determination (r2) for each curve was greater than 0.999 for 35 and 40 d.f., respectively; in both cases, the intercept was indistinguishable from 1.00 (p = 0.05). Note that there is a residual difference of about 6% between the measurements of rationalized shrinking object rate constant, aRTr, not accounted for between the two sets of runs; in the context of this study, this is small enough to be ignored. 4.3 Dissolutions with Loadings up to and Just Exceeding Saturation The Kamatani et al. (1980) model provided good by-eye nonlinear fits to separate dissolution runs with 106-lm particles of gypsum below saturation (Fig. 7) and for 250-lm particles above it (Fig. 8). Values of the Kamatani shrinking object rate parameter, aKa, obtained in this way are listed in Table 4. Meanwhile, linear fittings were found to be highly satisfactory (Fig. 9) with coefficients of determination (r2) routinely [0.997. Given that the fittings with the 106-lm particles cover two orders of magnitude change in loadings (between 1.4 9 10-4 and 1.3 9 10-2 M), they strongly support full

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35

0.012

[Gypsum] / M

0.01 0.008 0.006 0.004 0.002 0 0

500

1000

1500

t / s Fig. 7 Typical dissolution runs for gypsum at below saturation loadings at 20°C. The points represent laboratory measurements, while the continuous lines are model derived fittings. The next to lowest graph (green) is that for the lowest loading after an eightfold expansion; otherwise, it was indistinguishable from the abscissa 0.015

[Gypsum] / M

Fig. 8 Typical fitting between experimental points (red) and data generated by the Kamatani et al. (1980) model (continuous blue line) for dissolution of 250lm gypsum particles at abovesaturation loadings (note the red points are so numerous in places to appear as a red line)

0.01 0.005 0 0

500

1000

1500

2000

t/s Table 4 The variation in Kamatani Shrinking Object rate parameter, aKa, with loading of particles

Loading (M)

aKa (M-2/3 s-1)

r2 for the linear fitting

Nonlinear Linear (a). 106-lm particles (below saturation) 1.44 9 10-4

0.015

0.0155 ± 0.0001 0.998

2.33 9 10-3

0.042

0.0398 ± 0.0003 0.998

5.89 9 10-3

0.066

0.061 ± 0.001

0.997

9.87 9 10-3

0.084

0.074 ± 0.001

0.998

1.28 9 10-2

0.091

0.079 ± 0.001

0.999

1.56 9 10-2

0.038

0.0321 ± 0.0001 0.999

2.33 9 10-2

0.039

0.0379 ± 0.0001 0.999

3.49 9 10-2

0.043

0.0415 ± 0.0001 0.999

–

0.078 ± 0.001

(b). 250-lm particles (above saturation)

(c). 250-lm particles (26.6 g l-1; i.e., well above saturation) 1.55 9 10-1

0.996

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Table 5 The increasing divergence of aTr, with increased gypsum loading of 106-lm particles, from its value predicted from aKa and the solubility of gypsum Gypsum loading (M)

aKa measured (M-2/3 s-1)

aTr predicted (M1/3 s-1)

aTr measured (M1/3 s-1)

0

0

0

0

0

1.44 9 10-4

0.015

7.60 9 10-5

7.60 9 10-5

0

2.33 9 10-3

0.042

2.13 9 10-4

1.95 9 10-4

8

0.066

-4

2.70 9 10-4

19

-3

5.89 9 10

3.34 9 10

Divergence from prediction (%)

LHS of Eq.2

100 80 60 40 20 0 0

500

1000

1500

2000

2500

Time / s Fig. 9 The quality of the fittings obtainable through linearization of the Kamatani et al. (1980) model. Top to bottom: 0.01558, 0.02331 and 0.03491 M loadings of 250-lm gypsum particles

0.1

a Ka / M

-2/3

s

-1

compliance of the Kamatani et al. (1980) model with the rigorous tests advocated by Bunnett (1974) for rate constant evaluation in a kinetics investigation. Thus, when checking for first-order behaviour in solution kinetics, Bunnett (1974) stressed that not only must each individual curve fit the model closely, but they must yield essentially the same rate constant over at least one order of magnitude change in reactant concentration. Figure 10 shows this essentially up to saturation since the Kamatani et al. (1980) shrinking object rate parameter varies directly with the third power of the gypsum loading, as it is predicted to do theoretically by the fact that Eq. 3 is a function of N1/3. Linear regression gave a line with r2 = 0.995, with gradient (aRKa) and intercept (±std. errors) of 0.34 ± 0.01 M s-1 and -0.002 ± 0.002 M-2/3 s-1, respectively, for 4 d.f., showing that the small intercept measured is not statistically different from zero at the 95% level. The ability to overcome the slight disparities between the linear and nonlinear fittings of the Kamatani et al. (1980) shrinking object rate parameter, aKa, at the three highest loadings

0.05

sat 0 0

0.1

(Gypsum loading)

0.2 1/3

/ M

1/3

Fig. 10 The variation of the Kamatani et al. (1980) Shrinking Object rate parameter (aKa) with the third power of gypsum loading of 106-lm particles. The red squares arise from linear fittings and the blue diamonds from nonlinear ones. The continuous line arises from linear regression of the red squares against the abscissa values

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37

(Fig. 10) provided a tangible increase in objectivity, thereby demonstrating the advantages of the linear fitting. The results from a separate experiment with 250-lm gypsum involving particle loadings of between 1.56 and 3.49 9 10-2 M (Table 4) are consistent with the above-mentioned experience with 106-lm gypsum particles. Linear regression of the plot between aKa and c1/3 T yielded a gradient and intercept (±std. error) of 0.14 ± 0.01 M s-1 and 0.001 ± 0.003 M-2/3 s-1, respectively, with a coefficient of determination of 0.986 for 2 d.f. Hence, the intercept was not significantly different from zero, and aRKa = 0.14 is equal to the gradient. 4.4 Dissolution with Loadings of Gypsum Far in Excess of Saturation 4.4.1 Scaling Dissolution Down to a 300-ml Beaker The highest loadings of gypsum consumed so much gypsum (26.6 g l-1) that it was desirable to scale the experiment down from 1 l to 300 ml. By trial and error, the Heidolph variable stirrer was adjusted to the maximum speed which would not also force solution out of the 300-ml beaker as did the overly vigorous Grant s-2 stirrer. Then, two parallel dissolutions were conducted with 106-lm particles, one using the Grant s-2 stirrer in the 1-l beaker, the other using the Heidolph stirrer in the 300-ml beaker. This revealed (Table 3) that the hydrodynamic condition within the 300-ml beaker with the Heidolph stirrer was, in fact, more slightly vigorous than was that with the more powerful Grant s-2 stirrer in the 1-l beaker. 4.4.2 The Effect of Background Dissolved Gypsum Concentration Upon the Rate of Approach to Saturation Truesdale (2010) recommended an experiment, which should be performed as a first step in any dissolution study, to establish whether the so-called nonlinear kinetics were likely to be encountered (see Sect. 1). In this, the same high-loading of solid would be applied to a series of mixtures with increasing initial dissolved solid concentrations. Truesdale (2010) showed that a plot of the concentration after any standard dissolution time versus initial background concentration would be linear for a system free of ‘nonlinear kinetics’. This experiment offers the distinct advantage of reducing cross-sample variability by tightening the time control. Unfortunately, here with gypsum the dissolutions were too rapid for this design, and so it was defaulted to one involving full runs on a series of solutions. The approach to saturation at a constant loading of 26.66 g l-1 of 250-lm gypsum crystals (0.155 M) was studied using a series of eleven solutions of initial saturation of between 0 and 83%. This level of loading (109 the solubility) ensured that the amount of solid which actually dissolved was insignificant in comparison with the loading. The mixtures were prepared from different proportions of a saturated solution of gypsum and de-ionized water, and by adding 8.0020 ± 0.0020 g of gypsum to each 300-ml aliquot of solution (correction for the small variations in loading had an inconsequential effect). The mixtures retained a milky appearance throughout the experiment; there was no evidence of particles remaining on the beaker bottom through lack of stirring. In every case (Table 6), the curves closely followed an exponential approach to equilibrium as the coefficient of determination (r2) for the log plots was consistently above 0.988; the worst fits were in runs closest to saturation where, of course, the range of concentration change was least. The experiment showed generally close agreement

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Table 6 The fit of a simple exponential model to the approach to saturation in solutions containing various background dissolved concentrations of gypsum, but with constant gypsum loadings of 26.6 g l-1 of 250-lm particles (*109 saturation)

Average = 0.0199 s-1; s = 0.0015 s-1

Initial saturation (%)

Exponential rate constant/s-1 (±std. error)

r2

d.f.

0

-0.0217 (±0.0001)

0.999

22

8.33

-0.0186 (± 0.0003)

0.994

26

16.66

-0.0189 (±0.0001)

0.999

22

25

-0.0222 (±0.0002)

0.998

24

33.33

-0.0206 (±0.0002)

0.997

22

40.66

-0.0193 (±0.0002)

0.998

26

50

-0.0178 (±0.0004)

0.996

26

59

-0.0195 (±0.0003)

0.996

26

66.66

-0.0205 (±0.0002)

0.997

21

75.66

-0.0185 (±0.0004)

0.992

16

83.33

-0.0215 (±0.0005)

0.988

21

between the rate constants, with a mean of 0.0199 s-1 and a coefficient of variation of only 7.3%. As no trend was discernible in the concentration ordered results, variation between the rate constants is consistent with random experimental uncertainty, and within this level of uncertainty, the so-called nonlinear kinetics were absent. 4.4.3 The Effect of Pre-existing Concentrations of Either Calcium or Sulphate Upon the Approach to Saturation In two separate experiments, the exponential rate constant for the approach to saturation for 250-lm gypsum crystals was found to be essentially unaffected by the presence initially in the solution of either the ‘common ions’, calcium (0–0.0132 M) or sulphate (0–0.028 M), added as chloride and sodium salts, respectively. The first run consisted of the dissolution of gypsum at a loading of 0.155 M in deionized water (Fig. 11), that is with no background calcium or sulphate. The 3-mS cm-1 scale was used as saturation occurred at about 30% fsd, and that left the further 70% of chart space for the later runs. In these, increased initial loadings of CaCl22H2O or anhydrous Na2SO4 moved the starting point on the chart recorder progressively to higher deflections. Each run therefore consisted of a set, initial background coupled to a

[Gypsum] / M

0.016 0.012 0.008 0.004 0 0

50

100

150

200

250

300

350

t/s Fig. 11 The variation in approach to saturation in mixtures with a constant, 0.155 M loading of 250-lm natural gypsum particles at initial calcium chloride concentrations of (from top to bottom) 0.00, 0.0030, 0.0050, 0.0096 and 0.0132 M. The total amount of gypsum dissolving decreases with increase in the concentration of calcium chloride, according to the common-ion effect

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Aquat Geochem (2011) 17:21–50 Table 7 The essential lack of variation in the exponential rate constant for approach to saturation as the background concentration of either [Ca2?] or [SO42-] was varied in dissolutions of 250-lm particles of gypsum (A different sample of gypsum was used in the second experiment)

39

Initial [Ca2?] or [SO42-] (M)

Exponential rate constant/s-1 (±std. error)

r2

d.f.

a. with CaCl22H2O 0

-0.0207 (±0.0001)

0.999 22

0.0061

-0.0194 (±0.0002)

0.997 22

0.0100

-0.0229 (±0.0002)

0.999 20

0.0191

-0.0228 (±0.0003)

0.996 22

0.0263

-0.0198 (±0.0002)

0.997 24

b. with Na2SO4

a: Average = 0.021 s-1; std. dev. = 0.002 s-1; b: Average = 0.019 s-1; std. dev = 0.001 s-1

0

-0.0181 (±0.0002)

0.996 22

0.0041

-0.0185 (±0.0002)

0.998 24

0.0099

-0.0190 (±0.0002)

0.997 21

0.0146

-0.0189 (±0.0002)

0.998 22

0.0224

-0.0206 (±0.0002)

0.996 23

0.0275

-0.0180 (± 0.0002)

0.998 22

Fig. 12 Variation in the amount of gypsum dissolved with initial calcium chloride concentration— the ‘common-ion effect’. The points are determined experimentally, while the continuous line represents model generated data

Sat [Gypsum] / M

progressively increasing signal due to dissolving gypsum. The starting concentrations were double-checked, as gravimetric additions and as concentrations derived from the starting points and the two separate calibration curves for the salts. The raw chart data was written into the spreadsheet, and the initial set signal removed from the run. Only then was the chart data compared with a conductivity calibration curve (Eq. 14) so as to yield dissolved gypsum concentrations. The data were then transformed logarithmically to give the rate constant, k2S (Eq. 8). Mean rate constants (with std. dev.) of 0.021 (s = 0.002) and 0.019 (s = 0.001) were obtained (Table 7) in the calcium and sulphate experiments, respectively. Two separate sievings of 250-lm gypsum particles were used as each experiment was selfcontained; even so, the closeness of the results indicates that there is no reason to suspect any marked difference between them. Of course, the ‘common-ion effect’ restricts the amount of gypsum that can dissolve (Fig. 11). The overlay of the points for the calcium and the sulphate experiments in Fig. 12 confirms the expected behaviour for two salts, each one producing one mole of common ion per mole of salt. However, the actual dissolutions of gypsum were consistently lower than that predicted by the common-ion effect (Fig. 12). The latter were based upon an initial calcium concentration of p M calcium ions allowing the dissolution of a further q M of gypsum, so that at saturation the calcium and sulphate concentrations are (p ? q) and q

0.015

0.01

0.005

0 0

0.01

0.02 2+

Initial [Ca ] or

2[SO4 ]

0.03

/M

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M, respectively. Substitution of these into the solubility product gives a quadratic equation in q: q2 þ pq  Ksp ¼ 0

ð16Þ

whose positive root is: p 2 ð17Þ q ¼ p þ p þ 4  Ksp =2: 2It is a trivial matter to show that in the parallel equation for SO4 ions, p and q of Eq. 16 are swopped over. Further experiments showed that the discrepancies between observed and predicted behaviours in Fig. 12 relate to the calibration curve used. However, while the amounts of gypsum dissolved are in error, the reported rate constants (Table 7) are unaffected. Thus, the calibration curve generally used here (Eq. 14) was found to overlay very well over 90% of that for the solutions containing 0.011 and 0.021 M sodium sulphate initially, after these had been translated to begin at the origin. In effect then, over-use of the general calibration curve underestimates the concentrations in each of the runs by factors of 0.715 and 0.600, respectively. However, the rate constant of an exponential curve is independent of any proportional expansion or contraction to all its component parts. 4.5 Collating the Various Kinetics Measurements The robustness of the relationship between aKa and c1/3 T proved in Sect. 4.3 (Table 4) is majorly enhanced when combined with the later saturation experiment conducted with 250-lm particles (Table 6). Thus, the run used to calculate the first point of Table 6 for the exponential rate constant for approach to saturation with 250-lm particles was re-fitted to the linear version of the Kamatani et al. (1980) model. The result is displayed in Table 4, below other fittings for 250-lm particles. Thereafter, linear regression of aKa upon c1/3 T for this enlarged data set yielded a gradient and intercept (±std. error) of 0.14 (± 0.01) M s-1 and 0.001 (±0.002) M-2/3 s-1, respectively, with a coefficient of determination, r2, of 0.992. As with the shorter data set then (Sect. 4.3), the intercept for the enlarged set is still not significantly different from zero, and the gradient (=aRKa) is unchanged. The span of abscissa values in the enlarged set is almost twice that in the shorter set, proving the applicability of the Kamatani et al. (1980) model over a further order of magnitude of concentration.

5 Discussion 5.1 The Findings for Gypsum Dissolution This paper reports unprecedented consistency in the modelling and practical investigation of the batch dissolution of gypsum rock, while at the same time proving that the dissolution is not affected by ‘nonlinear kinetic effects’; that is, the kinetics follow the O’Connor and Greenberg (1958) model (Eq. 1). The above-mentioned claim rests upon rigorous testing of three derivatives of the O’Connor and Greenberg (1958) model, which is now summarized as seven main points. First, this study was free of the difficulties encountered in earlier ones, e.g., Jeschke and Dreybrodt (2002) and Gobran and Miyamoto (1985), where the lack of a proper rate equation hampered interpretation of gypsum dissolution. The inventive approach of

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41

Jeschke and Dreybrodt (2002) in which the plot for each kinetic run was split arbitrarily into two parts was therefore unnecessary. In terms of precision and accuracy, the approach used here is very much more akin to homogeneous solution kinetics where the rate equation is, of course, well defined. Secondly, this work has used, probably for the first time, the linearized version of the Kamatani et al. (1980) model, with its higher level of proof than the nonlinear version (Truesdale 2007, 2008, 2009, 2010). Thirdly, the Truesdale (2007) equation for dissolution at high under-saturation (Eq. 6) has been shown to apply consistently to gypsum particles from sieves of between 106 and 1,000 lm. Particles of 355 lm or smaller gave robust straight line plots which provided values of the shrinking object rate constant (Table 2). Under these circumstances of constant vigorous stirring, consistent with theory (Truesdale 2009) the rate of dissolution of gypsum varies inversely with particle size (Fig. 3). This was similarly proved with sucrose and silica gel (Truesdale 2009, 2010), although in the latter study two separate hyperbolae were required to explain the data. Therefore, interpretation of the gypsum results, like those for sucrose, did not require an assumption of more than one type of solid, as did the silica gel. Meanwhile, gypsum particles larger than 425 lm were slightly problematic and were suspected of having contaminating ‘crystal fines’ from the crushing of the rock, adhering to their surfaces. This effect is ubiquitous in crushed mineral dissolution study (Petrovich (1981) and is probably the main primary obstacle to reliable data (e.g., Schott et al. 1981; Tole et al. 1986; Burch et al. 1993). Nevertheless, it was still possible to estimate the shrinking object rate constant for the slowest dissolving particles within these samples by interpreting the runs as mixed particle dissolutions (Truesdale 2008). The results were then consistent with those obtained with the \355-lm particles (Fig. 3). Fourthly, a new experiment has shown that maximal turbulent flow in a laboratory beaker (that is, with no slopping) imposed a maximum dissolution rate for gypsum at 20°C and high under-saturation (Table 3), with the shrinking object rate constant, aRTr, decreasing exponentially towards lower stirring rates (Fig. 6). Fig. 6 can be relied upon because it is consistent with earlier experience with silica gel (Truesdale 2010), a substance chemically different to gypsum. The silica study was a prototype stirring experiment, comparing the effects of shaking and stirring rather than different degrees of stirring. Nevertheless, it showed that at high under-saturation, the shrinking object model applied to dissolution under radically different levels of agitation, thereby suggesting that a single multiplying factor might well compensate for change in stirring rate. As argued later, this is potentially of immense importance to dissolution work. Fifthly, in highly under-saturated conditions (Sect. 4.2.4), sodium chloride and the common ions calcium and sulphate, were found to exert an insignificant effect upon dissolution rates. Similarly, the rate constant for approach to saturation at very high loadings of gypsum (Sects. 4.4.2, 4.4.3) was insensitive to the presence of the ‘common’ ions, although of course, less gypsum dissolved the greater their initial concentration (Figs. 11, 12). This contradicts statements made by Barton and Wilde (1971) and by Gobran and Miyamoto (1985) who were only able to deploy, at best, a semi-quantitative interpretation. Therefore, given the newer, rigorous experimental design adopted here, gypsum dissolution kinetics seem to be much simpler than thought hitherto, particularly ensuring an absence of ‘nonlinear’ effects. Sixthly, in comparison with any previous fittings, those of the Kamatani et al. (1980) model to experimental dissolutions in the middle ground between high under-saturation and very high gypsum loadings (Figs. 6, 7) are unsurpassed, especially when the new, linearized version of the model is also considered (Fig. 9). Moreover, the derived values

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showed a consistency not hitherto witnessed, whereby the rationalized Kamatani et al. (1980) rate parameter, aRKa, was constant (at 0.35 ± 0.01 M-1 s-1) across an exceptionally wide range of loadings of 106-lm particles (Fig. 10; Table 4a). Similarly, for 250-lm particles (Table 4b, c) where aRKa was 0.14 ± 0.01 M-1 s-1. Judged against earlier findings, such precise numerical definition of a dissolving substance over extended ranges of solid loadings is remarkable. Seventhly, with loadings well in excess of saturation (26.6 g l-1) gypsum dissolution followed the exponential model with an exponential rate constant of 0.0199 s-1 at 20°C (Tables 6, 7). Overall then, there is ample evidence within these experiments to conclude that gypsum behaves ideally in dissolution according to the O’Connor and Greenberg (1958), Kamatani et al. (1980) and Truesdale (2007) models. The absence of ‘nonlinear kinetic behaviour’ is consistent with an existing collation of findings (Colombani 2008). It is imperative, however, that mixtures approaching saturation should not be allowed to evaporate and become super-saturated, thereby giving the impression of a continued, slow dissolution (Sect. 3.1.2); of course, this effect would be enhanced at higher temperatures. 5.2 The Wider Implications of the Stirring Experiment Conducted Here 5.2.1 General Impact Overall, this work can be expected to transform dissolution study generally. This is not only with batch, but with the chemo-stat, too, since ultimately that is a fluidized-bed reactor, too. It will impinge upon other approaches, e.g., the rotating disc (Barton and Wilde 1971; Sjo¨berg and Rickard 1983; Raines and Dewers 1997; Dreybrodt and Gabrovsek 2000), since it throws the various approaches into sharp relief. Finally, the work presages a new round of dissolution work to replace much of that characteristic of the late twentieth century which was dogged by ‘nonlinear kinetics’, as outlined in the Introduction. Indeed, new observations of minerals which were problematic in the earlier round, but made with the approach tested here, ought to shed new light upon the origin of those difficulties. The key hope of the TST approach, that of tabulating the dissolution characteristics of minerals with but a few measured parameters, might well be resurrected. These claims are expounded in more detail in the following paragraphs. 5.2.2 A Personal Reflection A wide literature upon mixing exists, but one nevertheless, that is often quite opaque. Dissolution is a subject of crucial practical interest to many workers who, however, do not have an easy grasp of the subtleties of hydrodynamics. Moreover, much of the literature is rooted in special cases, and re-application to other substances is not necessarily easy. Consequently, there is a need for an entry into the subject that will appeal to the laboratory experimenter as much as the theorist. It is argued below then that Fig. 6 presents an entirely new opportunity to understand the role of stirring in dissolution. 5.2.3 The Impact of the Stirring Experiments The stirring experiments (Sect. 4.2.3) show very clearly (Fig. 6) that the shrinking object model holds over a wide range of stirring velocities in what is labelled here as the

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fluidized-bed zone. Evidently, dissolution is controlled here jointly by transport, particle shrinkage and an inherent chemical reaction at the surface. This adds further weight to Truesdale’s (2010) suggestion (from silica dissolution) that the O’Connor and Greenberg (1958) model (Eq. 1) can be multiplied by a hydrodynamic factor, W, so that: dc W ¼  ðk1  S  k2  S  cÞ dt V

ð18Þ

where W increases exponentially with stirring rate towards the plateau value according to Fig. 6. Of course, stirring rate, as used here, is entirely relative to the experimental rig used; it is not absolute, and great care should be taken if applying it to another rig. Once again, the quality and simplicity of the results in Fig. 6 are very encouraging, especially given the hydrodynamic complexity within a swirling and turbulent reactor. More than this though, Fig. 6 eloquently differentiates turbulent flow in the fluidized bed, where the particles are carried about in fluid, from that of laminar flow over a static bed which will occur at some lower stirring rate at which the particles, starting with the larger ones, can be expected to fall out of suspension. At even lower rates, all the particles will lie on the bottom and the stirrer will merely force liquid over the bed, and at sufficiently low stirring rates this will be laminar flow. Ichenhower et al. (2008) describe what appears to be just such a reactor design—the single pass flow-through reactor. Interestingly, the spinning disc approach can be seen to apply to the intimate surroundings of particles in both the laminar flow and the fluidized bed; it is just a matter of replicating the interfacial conditions. Of course, dissolution in the turbulence of the fluidized bed will be faster than with either laminar flow proper or any intermediate, mixed mode. Such a marked change in mechanism between these two extreme zones can be expected to demand some change, perhaps a total one, in mathematical models. Nevertheless, a point of inflexion can be anticipated just below the lowest of the points shown in Fig. 6 (x * 10 s-1), thereby making the curve sigmoid, overall. This curve can be expected to intercept the ordinate at a slight positive value as, even without stirring, diffusion will allow some dissolution to take place. To the right, beyond the fluidized bed at much higher stirring rates, the mechanism of dissolution can be expected to change again as, inevitably, shear at the propeller becomes much greater, causing excessive cavitation in the mixture, and perhaps even shearing of the solid particles. The axle angular velocity, x, measured here to characterize turbulent mixing must also be a relative measure; the effectiveness of stirring will obviously depend upon a combination of the axle velocity, the shape and size of both the propeller and the vessel, and the number, size and shape of obstructions to flow, for example, the conductivity probe introduced into the reaction vessel. All of this means that absolute values for dissolution rates cannot be ascertained until optimal designs of dissolution baths are selected. It follows that within environmental science including, for example, mineral dissolution in geochemistry, particle dissolution in oceanography, there is a need for some standardization of reactor vessels. Only this will allow results obtained with a given solid by one worker, to be compared with those of others. This will apply as much to the chemo-stat, the flow-through reactor and the spinning disc approaches as to batch dissolution; ultimately, there is a need for consistency within each approach, and across all approaches. In environmental science, numerical statements about dissolutions need to reflect global truth. Accordingly, they will need to be underpinned by internationally robust standards of experimental design probably not required within more local activities, e.g., industry.

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5.3 The Consistency of Results from the Three Integrations of the O’Connor and Greenberg (1958) Equation 5.3.1 Why Simultaneous Application of the Three Integrations is Desirable A prime intention of this work was to test all three integrated forms of the O’Connor and Greenberg (1958) model, the Truesdale (2007) model for under-saturated mixtures (Eqs. 4, 6); the Kamatani et al. (1980) model for mixtures both above and below saturation (Eq. 2), and the general exponential approach to equilibrium (Eq. 8), by applying them simultaneously to a single substance. This will not be needed for all substances, but it needs to be proved with a good sample to generally validate the approach, and specifically to identify the cause of the ‘nonlinear kinetics’ in mineral dissolution generally (e.g., Schott et al. 1981; Nagy and Lasaga, 1992; Burch et al. 1993), and specifically, earlier with biogenic silica (Van Cappellen and Qiu (1997) and Rickert et al. (2002) with chemo-stat dissolution; Truesdale et al. (2005a) with batch dissolution). First, there is a great need to establish whether this is a real effect or an experimental artefact (Truesdale 2010). Dissolution experiments are not generally easy to do, and the margin for mistake is wide. Secondly, it needs to be known whether some substances behave ideally in this respect while others do not. Finally, the two main reasons given so far for the effect relate either to solution chemistry or the morphology of the solid surface, particularly the distribution of surface defects (Jeschke and Dreybrodt 2002; Lu¨ttge et al. 2003; Dove et al. 2005). It could be very wasteful of resources to pursue either of these assumptions without firm evidence that one is the cause of ‘nonlinear kinetics’. Incidentally, an even finer study with the shrinking object model than this current one should be capable of identifying change in the essential shape of substances during dissolution (Truesdale 2009). Gypsum was chosen for this study so as to decrease analytical uncertainties below that evident in the work with silica gel (Truesdale 2010). A similar earlier shift to salts (and sugar) during development of the Truesdale (2007) equation itself proved to be valuable—with their electrical conductivity, salts offer unsurpassed ease of collection of continuous experimental data with which to minimize analytical uncertainty. 5.3.2 The Consistency Found Here Between the Truesdale (2007) and Kamatani et al. (1980) Models at High Under-Saturation Consistency between the Kamatani et al. (1980) and Truesdale (2007) equations in undersaturated mixtures (*2.9 9 10-4 M) was evident in the first five dissolution runs with 106–355-lm particles listed in Table 2. Thus, the five ratios of aTr to aKa yielded a mean and standard deviation of 0.0150 and 0.0003 M, respectively, which is equal to the solubility of gypsum, as explained by Truesdale (2010). Really, this test checks the compatibility of the two spreadsheets used, since the value for the solubility at saturation has to be inserted into the Kamatani et al. (1980) model. It is the fact that good straight lines were obtained in all the applications (r2 [ 0.999) that really shows that both equations apply. That the Truesdale (2007) equation is restricted to high under-saturation is evident from a comparison of aKa and aTr values for the first three runs of Table 4. Hence, Table 5 shows that the correct gradient is only given by the first run, with progressively greater deviation as the loading is increased, such that with a loading of 5.89 9 10-3 M gypsum, the gradient is 19% too low. This emphasizes that once the O’Connor and Greenberg (1958) equation is known to hold for a substance, dissolution at high under-saturation can be characterized by application of either the Truesdale (2007) or the Kamatani et al. (1980)

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model to a run; either approach will do. Now that the Kamatani et al. (1980) model has been linearized, the choice is unrestricted. 5.3.3 The Kamatani et al. (1980) Model as an Extension of the Truesdale (2007) Model, or Vice versa The results in Sect. 4.3 (Fig. 10) show how the Kamatani et al. (1980) model goes well beyond the limitation to highly under-saturated conditions implicit in the Truesdale (2007) model. The proof of this applicability of the Kamatani et al. (1980) model is greatly enhanced by the linearized version devised here (Fig. 9). These runs with loadings of 250lm gypsum particles also prove the model slightly above saturation. 5.3.4 Comparing Results Obtained With the Kamatani et al. (1980) Model With Those From Saturation Study (Sect. 4.4) Although not immediately apparent, the impressive and unprecedented validation of the Kamatani et al. (1980) model from the combined experiments with 106-lm particles and 250-lm gypsum particles at loadings of between 0 and 0.155 M (Sect. 4.3) is entirely consistent with the saturation studies of Sect. 4.4. Thus, the apparent difference in magnitude between the two measured parameters, the exponential rate constant obtained from use of Eq. 8 (0.022 s-1), and that of aKa (of 0.078 s-1; Table 4) obtained with a 26.6 g l-1 loading of gypsum, is not fundamental; they merely express the same thing. The two key statements derived from Eq. 1 (Truesdale 2007) are: dc 1 ¼   ðk2  SÞ  ðcsat  cÞ dt V dc ¼ aKa  ðcT  cÞ2=3 ðcsat  cÞ: dt

ð19Þ ð20Þ

So, when cT  csat it follows that cT  c and the first bracketed term of the right-hand side of Eq. 20 is constant, whence, comparing coefficients of Eqs. 19 and 20: 1 2=3  ðk2  SÞ ¼ aKa  cT ¼ 0:078  0:290 ¼ 0:022 s1 : V 5.3.5 The Specific Surface Area of the 250-lm Gypsum Rock Particles The specific surface area (surface area per g) of the 250-lm gypsum rock particles has been calculated to be 103 cm2 g-1 by combining the exponential rate constant for approach to saturation with the shrinking object rate parameter from the Kamatani et al. (1980) model. Thus, for the 26.6 g l-1 loading (Table 4c), aKa = 0.078 s-1 and substitution of this into Eq. 3 gives k2 = 0.081 l m-2 s-1. The number of particles present has also to be substituted into aKa, and this was calculated to be 1.136 9 106 from the density of gypsum and the geometric volume of the particles, assuming each is a sphere with a radius of 125 lm. Final substitution into: 1  ðk2  SÞ ¼ 0:022 V estimates the surface area of the 26.6 g of gypsum in the reactor, S, as 2,730 cm2. This divided by the weight yields the 103 cm2 g-1 specific surface area, which seems

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reasonable. An investment in further investigations of gypsum using BET measurements together with the equations validated here would seem now to be particularly worthwhile. 5.4 A Key Role for Figs. 3 and 6 in Dissolution Study 5.4.1 The Significance of Fig. 3 On its own, Fig. 3 presents an important validation of the Truesdale (2007) model. This is the third such test reported, the others covering sucrose and silica gel. For any dissolving substance, this diagram is the very important bridge between batch and chemo-stat dissolutions. Thus, for a given reactor under given physical conditions of temperature, turbulence, etc., the curve provides a value of the rationalized shrinking rate parameter, aRTr, which can be used together with a solid loading, cT, to define aTr for the Truesdale (2007) cubic equation (Eq. 7) for the solid’s dissolved concentration, c, during dissolution. In contrast, the chemo-stat approach takes measurements of the rate of dissolution under various conditions. Hence, each of these is the gradient at a certain point in time along a curve described by Eq. 7. Note that a chemo-stat investigation need not follow just one batch dissolution; it could cross two or more batch runs. Note also that, as already stated, generally the chemo-stat approach has not been applied to dissolutions that encounter changes in surface area, the condition that Eq. 7 defines. Nonetheless, that is a practical deficiency in the use of the chemo-stat approach rather than anything fundamental. 5.4.2 Figure 3 in Relation to Fig. 6 Interestingly, Fig. 3 is a vertical section across Fig. 6, at a particular stirring rate within the zone of the fluidized bed (Fig. 6). Thus, Fig. 6 for 106-lm particles could actually accommodate a family of curves such that, for example, the curve for 250-lm particles would have a similar shape to that for 106 lm, but lie closely below it since, for a given stirring rate, a 250-lm particle will present a lower aRTr. Similarly, smaller particle sizes will produce curves above the 106-lm curve in Fig. 6. By similar argument, Fig. 3 can also accommodate a family of rectangular hyperbolae, each one relating to a given setting of stirring speed. Indeed, with families of curves either of these two figures will completely map out the variation of aRTr over both stirring rate and particle size. Finally then, it also follows that it is also possible to merge all of this information into a plot of particle size versus stirring rate, upon which aRTr values are contoured. Figure 13 presents a prototype plot of particle size versus stirring rate. The steps A–E show how, in principle, horizontal shifts A ? B and C ? D to the left, in which stirring rate and therefore aRTr is reduced, can be exactly compensated by downwards shifts B ? C and D ? E in which particles size is diminished, and hence aRTr is increased. The precise curvature and crowding of the contours has yet to be determined. As a contour is followed from the right of the diagram to its left to left, it must enter the region of mixed fluidized bed and laminar flow conditions, where a radical change in curvature can be anticipated. A further change is anticipated in the far left of the diagram, in the region of laminar flow, proper. Again interestingly, it seems possible that the shrinking object model might operate here, too, in reactors of the type described by Ichenhower et al. (2008) but, of course, with an entirely different hydrodynamic weighting, W, in Eq. 18. It is also anticipated that in Fig. 13, the fluidized-bed zone will occupy the area close to the origin because even at very low stirring rates, very fine particles will not settle in the reactor.

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47 LOW

B

d / μm

1000

D

A C

FB

LF 500

E

HIGH

0 0

10

20

ω / s

30

40

-1

Fig. 13 Schematic of the graph of particle size, d, versus stirring rate, x, with contours of aRTr between high values to the bottom right and low ones to the top left. The pattern of contouring is not expected to be uniform. A–E represents alternate lowering of stirring rate and then particle size to maintain a constant value of aRTr. The pink and blue straight lines demarcate an intermediate zone between laminar flow (LF) and fluidized bed (FB)

5.5 On the Likely Applicability of the Shrinking Object Model to Systems in Laminar Flow The idea that there may be two domains in which the shrinking object model might hold gains support from Raines and Dewers’ (1997) discussion of the effect upon dissolution rate of varying the flow of water past a gypsum surface. From their rotating disc experiments, they conclude that gypsum dissolution occurs by a mixed mechanism involving both surface reaction and transport control, such that, in general, a faster flow of water past a gypsum surface will increase the dissolution rate because the transport rate rises. They also recognize that the impact of faster flows could be explained as the thinning of the hydro-dynamical boundary layer adjacent to the gypsum surface. Whatever, the condition at the solid-solution boundary within the hypothesized laminar flow zone must differ from that in its counterpart, the turbulent fluidized bed (Fig. 6). Simple observation of the latter reveals that the particles are carried around bodily in the solution, with the boundary condition probably caused by rotation and tumbling that occurs within the immediate fluid, as well as turbulent exchange of solution in the immediate vicinity of a particle with that in the bulk of the solution which is progressively increasing in dissolved solid concentration. Indeed, when the rotating disc is adopted, to retain laminar flow the component of the total process that leads to exchange of dissolved solid with the bulk solution—the convective effect–has to be managed well (Raines and Dewers 1997). This analysis suggests then that although the scale of the processes is very different in the two cases, the shrinking object model might still apply, but with a different value u in Eq. 10. 5.6 The Hydrodynamic Approach to Dissolution Study This project (Greenwood et al. 2001; Truesdale et al. 2005a, b; Truesdale 2007, 2008, 2009, 2010), of which this paper is but one part, began with experiments on biogenic silica but progressed into the fundamentals of dissolution modelling after it took up the Kamatani et al. (1980) Shrinking Sphere model. Progress has been made under the shadow of TST, the approach adopted mainly in mineral dissolution, while up-ahead it is anticipated that there will be a need to rationalize findings with the considerable work on the hydrodynamic approach. As the latter is a sizeable task, it has deliberately being left for a later paper. Until then, such partial mention that has been made of the hydrodynamical approach should not be taken as a measure of the author’s lack of appreciation of its importance.

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5.7 Gypsum Concentration at Saturation The 0.015 (±0.001) M solubility of gypsum measured here (Sect. 4.1) is in good agreement with Christoffersen and Christoffersen (1976) and the several literature values collated by Colombani (2008) for working temperatures of 20–25°C. Interestingly though, none of these is consistent with the figure of (1.0 or 2.4) 9 10-5 M2 often quoted for Ksp at 25°C in the older literature (e.g., Bard 1965; Chang 1994; Weast 1976). Acknowledgments The author thanks Professor Linda King, Dean of the School of Life Sciences, Oxford Brookes University, for her continued hosting of this work during the author’s retirement. Thanks also go both to Stephen Casterton, Regional Technical Service Manager, BPB (Newark, UK) for generously supplying the gypsum rock, and to Dr Karl Morten of Oxford University Obstetrics department, for the invaluable loan of the Heidolph stirrer. Thanks go to two referees who suggested very useful alterations to the manuscript.

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