Anal. Chem. 2005, 77, 983-990
Chasing Equilibrium: Measuring the Intrinsic Solubility of Weak Acids and Bases Martin Stuart* and Karl Box
Sirius Analytical Instruments Ltd., Riverside, Forest Row Business Park, Forest Row, East Sussex, RH18 5DW, United Kingdom
A novel procedure is described for rapid (20-80 min) measurement of intrinsic solubility values of organic acids, bases, and ampholytes. In this procedure, a quantity of substance was first dissolved at a pH where it exists predominantly in its ionized form, and then a precipitate of the neutral (un-ionized) species was formed by changing the pH. Subsequently, the rate of change of pH due to precipitation or dissolution was monitored and strong acid and base titrant were added to adjust the pH to discover its equilibrium conditions, and the intrinsic solubility of the neutral form of the compound could then be determined. The procedure was applied to a variety of monoprotic and diprotic pharmaceutical compounds. The results were highly repeatable and had a good correlation to available published values. Data collected during the procedure provided good diagnostic information. Kinetic solubility data were also collected but provided a poor guide to the intrinsic solubility. Many methods have been described for the measurement of solubility. These methods can usually be separated into kinetic (turbidimetric) type assays versus thermodynamic assays. The traditional approach is to use a saturation shake-flask method: drug is added to a standard buffer solution until saturation occurs, followed by shaking for 24 h-7 days, removal of excess undissolved solid, and analysis of the solution by HPLC with UV or MS detection.1,2 Such a method would often be used during drug development, or at the interface between discovery and development, to optimize candidates with appropriate biopharmaceutical properties. In recent years, the method has been modified to work with robotic liquid-handling systems using 96-well plates.3 Solid is removed by filtration or centrifugation before analysis. The use of plate-based spectrometers has also increased throughput by allowing parallel measurements to be performed. This has enabled the measurement of solubility to be made much earlier in the * To whom correspondence should be addressed. E-mail: martin.stuart@ sirius-analytical.com. (1) Dittert, L. W.; Higuchi, T.; Reese, D. S. J. Pharm. Sci. 1964, 53, 13251328. (2) Connors, K. A. A Textbook of Pharmaceutical Analysis; John Wiley & Sons: New York, 1982. (3) Avdeef, A. High-Throughput Measurements of Solubility Profiles. In Pharmacokinetic Optimization in Drug Research. Biological, Physicochemical, and Computational Strategies, E; Testa, B., van de Waterbeemd, H., Guy, F. R., Eds.; VCHA: Zu ¨ rich, 2001; pp 304-325. 10.1021/ac048767n CCC: $30.25 Published on Web 01/20/2005
© 2005 American Chemical Society
drug discovery program where inadequate aqueous solubility has been shown to be one of the causes of erratic drug absorption.4 The traditional shake-flask method provides thermodynamic solubility values and is often used as a standard method against which other methods can be validated. An alternative method for thermodynamic measurements of the intrinsic solubility of the neutral form of ionizable compounds is the potentiometric acidbase titration method described by Avdeef.5 The method involves titrating a basic compound from high to low pH, or an acidic compound from low to high pH, and calculating the apparent ionization constant from the pH of each point in the full titration curve. The intrinsic solubility is calculated from the shift compared to the aqueous ionization constant. The pH in regions of the titration curve where precipitated neutral compound is present should be measured under equilibrium conditions. In order for equilibrium to be established, a typical titration takes 3-10 h to complete dependent upon the solubility of the compound. The rate of data collection is slowed dramatically near the regions of complete dissolution as the time taken to dissolve additional solid increases significantly at this point.6 Nonetheless, high-quality thermodynamic results are obtained, and the method is often used in a development setting or at the interface between discovery and development. Kinetic-based methods have been introduced to meet the needs of drug discovery teams. These methods use solutions prepared from DMSO stocks and attempt to rank or classify molecules to see whether they meet some appropriate cutoff criteria. The detection system is usually based on light-scattering methods using laser light, or single-wavelength emissions, which detect precipitated particles in solution. The most common methods are based on the work of Lipinski7 or use robotics and 96-well plate nephelometers8 (e.g., OSI Pharmaceuticals and Lab Systems, GlaxoSmithKline, and BMG). Recently, a light-scattering method using flow cytometry has also been published.9 (4) Ho ¨rter, D.; Dressman, J. B. Adv. Drug Delivery Rev. 1997, 25, 3-14. (5) Avdeef, A. Pharm. Pharmacol. Commun. 1998, 4, 165-178. (6) Avdeef, A. In Lipophilicity on Drug Disposition: Practical and Computational Approaches to Molecular Properties Related to Drug Permeation, Absorption, Distribution, Metabolism and Excretion; Testa, B., van de Waterbeemd, H., Folkers, G., Guy, R., Eds.; University of Lausanne, 2000; Chapter 22. (7) Lipinski, C. A.; Lombardo, F.; Dominy, B. W.; Feeney, P. J. Adv. Drug Delivery Rev. 1997, 23 (1-3), 3-25. (8) Bevan, C. D.; Lloyd, R. S. Anal. Chem. 2000, 72 (8), 1781-1787. (9) Goodwin, J.; Sullivan, J.; Crespi, C. Aqueous solubility by flow cytometry II: new prototypes optimised for drug solubility testing. http://www.bdbiosciences.com/discovery_labware/gentest/products/pdf/SolubilityFlow.pdf.
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In this paper, we seek to introduce a new potentiometric method, called chasing equilibrium, for measuring the intrinsic solubility of ionizable compounds. Chasing equilibrium occurs in the presence of precipitated neutral solid. It provides equilibriumbased results by monitoring pH changes induced by precipitation and actively seeking an equilibrium pH where sample is neither further precipitating nor redissolving. The process is relatively quick and can determine intrinsic solubilities in 20-80 min/ sample. Data collection is rapid because after initial precipitation to ensure the presence of some solid, no further significant amount of slow precipitation or dissolution is necessary, the concentration of the neutral form instead being altered by rapid pH change and ionization equilibrium. The method is expected to be useful for preformulation and development scientists and may also be useful at the discovery/development interface. In practice, one source of error in some techniques is the need to separate the solution from the solid. This carries the risk, for instance, of fine particles passing through the filter, particles with a density similar to the solution failing to separate by centrifugation, or lipophilic compounds adhering to the plastic plate. These separation problems can lead to errors in the measurement of the concentration of the solution. This new method does not have a separation stage, instead requiring the solution to be in contact with the solid and avoiding exposure to this particular source of error. Terms and Definitions. We use a number of terms within this paper, and to avoid ambiguity and confusion, we give the following definitions for use within the current context. The intrinsic solubility is the solubility of the compound in its free acid or free base form.4 A supersaturated solution is one in which the concentration of dissolved neutral species is greater than the intrinsic solubility. This is not at equilibrium and may or may not have solid precipitate present. If there is no solid present, then it will start to precipitate after a period of time. If precipitate is present, then the neutral species will continue to precipitate until eventually equilibrium is achieved. A subsaturated solution is one in which the concentration of dissolved neutral species is less than the intrinsic solubility. If there is solid precipitate present, then solution and the solid will not be in equilibrium and the solid will gradually dissolve until eventually equilibrium is achieved or the solid is entirely dissolved. EXPERIMENTAL SECTION A number of compounds were selected to demonstrate a wide range of ionization modes (monoprotic and diprotic; acid, base, and ampholyte) and a wide range of intrinsic solubilities (from 1000 µg/mL). Reagents and Solutions. The preparation and standardization of HCl (Aldrich) and KOH (Fisher) are described elsewhere.10 KCl (Analytical grade, SureChem), diclofenac (sodium salt, Sigma), famotidine (ICN Biomedicals Inc), hydrochlorothiazide (Sigma), ibuprofen (sodium salt, Sigma), lidocaine (Sigma), propranolol (hydrochloride salt, Aldrich), and warfarin (Sigma) were used without further purification. Deionized water of resistivity >1014 Ω‚cm was used to prepare all solutions. (10) Avdeef, A.; Sofen, S. R.; Bregante, T. L.; Raymond, K. N. J. Am. Chem. Soc. 1978, 100, 5362-5370.
984 Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
Figure 1. Schematic of the titration head.
Apparatus. The apparatus used to perform the solubility determinations was a GLpKa titrator and a D-PAS spectrometer controlled from a computer running RefinementPro and CheqSol software; the instrument and software have been updated to support this new method and are available commercially (Sirius Analytical Instruments Ltd.). All titrations were performed in 0.15 M KCl solution under argon atmosphere, at 25 ( 1 °C, using standardized 0.5 M HCl and 0.5 M KOH solutions. The purpose of the inert gas is to exclude atmospheric carbon dioxide. A schematic of the titration head is shown in Figure 1. The pH electrode (combination Ag-AgCl, Sirius Analytical Instruments Ltd.) was connected to a custom-designed pH sensing circuit (1015 Ω impedance) for measuring pH and calibrated titrimetrically in the pH range 1.8-12.2. An overhead stirrer was connected to a motor whose speed of rotation was controlled by the computer program. The dispenser tips were made from narrow polyimideclad quartz capillary (0.5-mm inside diameter) tubes that were connected to precision dispensers that were capable of delivering small, reproducible aliquots of liquid of known volume. The temperature probe monitored the temperature during the course of the titration. A bifurcated fiber-optic dip probe (Hellma) with path length of 1 cm was connected to a D-PAS (Sirius Analytical Instruments Ltd.) ultraviolet spectrometer. Method. A quantity of substance was selected to ensure that when fully neutral it would be expected to have a concentration above its intrinsic solubility and would therefore be in a position to precipitate once the pH was adjusted appropriately. If the substance failed to precipitate during the experiment then the method could be repeated using a larger quantity. For more soluble substances, the required quantity could be quite large. There appeared to be a lower limit of around 2-5 mg, below which the method did not operate reliably in the experimental volume of 10 mL regardless of the solubility and ionization properties of the substance. The selected amount was then accurately weighed into the titration vessel. The form of the substance used was the free acid, base, or ampholyte (e.g., lidocaine) or a salt (e.g., sodium salt of diclofenac). The experiment proceeded in five stages, called dissolution, seeking precipitation, additional precipitation, chasing equilibrium, and redissolution.
Dissolution. A measured volume (10 mL) of KCl solution was added to the titration vessel; the solution was then stirred throughout the experiment. A measured volume of either acid or base titrant was added to adjust the solution to a pH at which the solute was fully dissolved in its ionized form. If the sample was an acid, the pH was adjusted by adding base titrant. If the sample was a base, acid titrant was added. The solution was stirred until the entire sample dissolved. Complete dissolution was not strictly necessary unless the starting material was a salt. In that case, the entire sample had to be fully dissolved so that the concentration of the counterion was known. Seeking Precipitation. The solution of ionized solute was back-titrated by adding measured aliquots of base or acid titrant until the solution became cloudy, which indicated that the poorly soluble neutral species had precipitated. For example, acid titrant was added to acids and base titrant was added to bases to induce precipitation. The volumes added during this stage were calculated to achieve a fast titration without overshooting the precipitation point by more than ∼1 pH unit. The occurrence of precipitation was detected using the spectroscopic dip probe. A wavelength was chosen at which the dissolved solute absorbs little or no light in any of its ionized or neutral species. The first appearance of precipitate was detected by noting the sharp reduction in the amount of light transmitted at that wavelength, caused by the absorption and scattering of light by the precipitate. The use of the dip probe also made it possible to automate the solubility analysis, as there was no need for a person to watch the experiment, and also to determine a turbidimetric solubility value. Additional Precipitation. Additional aliquots of the same titrant were added, until the pH had changed by a further predefined increment (e.g., 0.1 pH unit) or until a fixed time had elapsed (e.g., 60 s) should the precipitation cause the pH to spontaneously readjust as quickly as titrant was added. The purpose of the additional precipitation stage was to ensure that sufficient precipitation was present for the next stage of the experiment. Chasing Equilibrium. After precipitation of the neutral species had occurred, the solution was repeatedly changed from supersaturated to subsaturated and back again by changing the pH. This stage was repeated until around five to eight saturationstate changes had been measured (see Figures 2 and 4). In contrast to a pH-stat experiment, where changes in pH are counteracted by the addition of titrant to return the pH to the required value, the operation of chasing equilibrium was to actively encourage pH changes by adding titrant that changed the pH even further in the direction that it was already changing as it spontaneously proceeded toward equilibrium. By noting the small pH changes due to gradual precipitation or dissolution of the sample, the direction toward equilibrium could be assessed. This pH gradient was used to determine the type of titrant to be added to move the pH in that direction more quickly than it would have proceeded unaided. The rapid pH change due to titrant addition changed the ionization of the sample, altering the concentration of the neutral (un-ionized) form, and taking the system closer to, or beyond, equilibrium. When the system had been taken beyond
Figure 2. Change of pH gradient while chasing equilibrium of the weak acid diclofenac, showing five changes from supersaturation to subsaturation or vice versa.
equilibrium, the pH gradient reversed and chasing equilibrium then proceeded by adding the opposite titrant. The rate of change of pH was measured once it had settled to a sustained response (see Figure 5). An example of equilibrium chasing is shown in Figure 2 for a weak acid. Adding base titrant increases the pH and reduces the average protonation level of the weak acid, thus reducing the concentration of neutral species without having to wait for more of it to precipitate. Similarly, adding acid titrant rapidly increases the concentration of the neutral species without having to wait for more dissolution. Redissolution. After sufficient data had been collected to calculate a solubility result, the pH was adjusted to a value at which the sample became fully ionized, and the solution was held at that pH while the sample dissolved. The purpose of this stage was to ensure that no crystals or solid sample remained on the apparatus that may have impaired its performance in subsequent assays. After redissolution, the probes were washed before any further actions took place. A patent application has been made for this procedure. Data Processing. Before the assay began, the following were known: (i) the formula weight of the sample. This was different from the molecular weight when the sample was introduced as a salt; (ii) the weight of sample added; (iii) the concentration of the acid and base titrants; (iv) the ionization constant(s) (pKas) of the sample, both the value and type (acid or base). These were measured using the same GLpKa instrument. At each point, the volume of added water, acid, and base titrant was known and the pH was known. Standard techniques were used to calibrate the measured pH to the concentration pH (p[H]).11 (1) The concentration of free hydrogen ions was determined from the p[H].
[H+] ) 10-p[H] Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
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(2) The total volume (vt) of the assay was determined from the sum of water (vw), acid (va), and base (vb) titrants added.
v t ) vw + va + v b
(2)
(3) The concentration of free hydroxide ions was determined from the free hydrogen ion concentration and the water dissociation constant (Kw).
[OH-] ) Kw/[H+]
to use accurately measured pKa values. The use of calculated in silico pKa values may also lead to errors in solubility measurement by this technique. (8) Each time the measured sustained pH gradient changes direction, the intrinsic solubility is bracketed by the neutral species concentration at the points on either side of the change. The change in concentration between these two points is small, and so both points give a value close to the intrinsic solubility. For example, for a monoprotic acid
(3) S ≈ [HA]
The value of the Kw was corrected for temperature and ionic strength. (4) The concentration of free positive ions (e.g., K+) was determined from the amount of base titrant (e.g., KOH) plus any positive ions in the original sample if it was a salt.
[K+] )
vbcb + mszs+/fw vt
(4)
where cb is the calibrated concentration of base titrant, ms is the sample weight of test compound in grams, zs+ is the charge of any positive salt counterions, zero if not introduced as a salt, and fw is the formula weight of the test compound. (5) The concentration of free negative ions (e.g., Cl-) was determined from the amount of acid titrant (e.g., HCl) plus any negative ions in the original sample if it was a salt.
[Cl-] )
vaca + mszs-/fw vt
(5)
where ca is the calibrated concentration of acid titrant, ms is the sample weight of test compound in grams, zs- is the charge of any negative salt counterions, zero if not introduced as a salt, and fw is the formula weight of the test compound. (6) The concentration of all the ionized species of the sample was then determined from a charge balance equation. For example, for a monoprotic acid, this was simply the concentration of the deprotonated species A-.
[A-] ) [H+] - [OH-] + [K+] - [Cl-]
(6)
(7) From the free hydrogen ion concentration [H+] and the sample ionization constant (pKa), the concentration of neutral species in solution was determined from the concentration of the ionized species. For example, for a monoprotic acid,
[HA] ) [A-][H+]/10-pKa
(7)
The value of the pKa was corrected for ionic strength. The accuracy of the values used for ionization constants directly affects the accuracy of the neutral species concentration calculation. Each unit of error in the value used for the pKa closest to the neutral species gives an error of the order of one log unit in the intrinsic solubility result. It is therefore extremely important (11) Avdeef, A.; Bucher, J. J. Anal. Chem. 1978, 50, 2137-2142.
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Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
(8)
Kinetic Solubility. A “kinetic solubility” value can also be determined from the data. Unlike the intrinsic solubility, which is an equilibrium value, kinetic solubility values are strongly time dependent and this dependence cannot be related in a single concentration value. These kinetic solubility values could be used as additional information relating to the degree of supersaturation that occurred but would not be expected to be reproducible between different kinetic methods. The kinetic solubility values reported here are equivalent to the concentration of the neutral species at the point in the titration where precipitation is first detected. This value gives an indication of how much of the solution remains supersaturated shortly after precipitation begins. Bjerrum Analysis. A useful visualization can be obtained from the Bjerrum function (or difference curve or formation curve), which shows the average number of bound protons (hydrogen ion binding capacity) versus pH.12 Such a graph clearly reveals values of pKas for the portions of the curve where the sample remained fully in solution and may be useful diagnostically in correction for such factors as substance purity, acidity error, or carbonate content. The Bjerrum curve for ibuprofen shown in Figure 3 shows that the experimental data fit well to the theoretical Bjerrum curve for fully dissolved ibuprofen. The sample precipitated out at around pH 5.7, after which the Bjerrum curve no longer matched this theoretical curve. Had the sample remained in solution, the curve would be expected to follow the theoretical curve and cross the half-bound proton intercept at the pKa of 4.35. Once the sample has precipitated, the chasing equilibrium procedure quickly brings the solution close to equilibrium with the precipitate and then oscillates between supersaturation and subsaturation with very small changes to the neutral species concentration. The data points collected during this period should all lie close to a Bjerrum function that can be easily calculated from the known data and the intrinsic solubility result, by assuming that all the solid is in the form of the neutral species. The theoretical Bjerrum function in the presence of precipitation can be calculated from the following equation for monoprotic acids
Bj ) 1 - SKa/[Xtotal][H+]
(9)
(12) Avdeef, A.; Kearney, D. L.; Brown, J. A.; Chemotti, A. R., Jr. Anal. Chem. 1982, 54, 2322-2326.
Table 1. Individual Zero pH Gradient Crossing Points for the Intrinsic Solubility of Diclofenaca sample wt (mg) 3.39 3.57 5.48 5.96 6.20 7.69 8.63 11.41 16.53 23.80 a
concn of the neutral species at zero pH gradient crossing pts (µg/mL) 0.87 0.75 0.88 0.98 1.05 0.82 0.92 0.88 0.86 1.03
0.74 0.71 0.85 0.95 0.98 0.84 0.94 0.87 0.89 1.03
0.65 0.75 0.82 0.95 1.01 0.83 0.89 0.87 0.87 1.03
0.65 0.72 0.81 0.90 0.94 0.83 0.92 0.86 0.87 1.02
0.63 0.82 0.81 0.91 1.01 0.81 0.87 0.84 0.87 1.03
0.63 0.77 0.80 0.89 0.96 0.82 0.93 0.84 0.88 1.02
0.63 0.77 0.80 0.92 1.02 0.82 0.87 0.84 0.88 1.03
0.67 0.76 0.78 0.89 0.98 0.80 0.94 0.83 0.87 1.03
mean of 8 crossing pts
SD
0.68 0.76 0.82 0.93 0.99 0.82 0.91 0.86 0.87 1.03
0.08 0.03 0.03 0.03 0.04 0.01 0.03 0.02 0.01 0.003
The intrinsic solubility determined in each assay is equivalent to the mean concentration of the crossing points.
Figure 3. Bjerrum curve for ibuprofen. The experimental data fit well to the theoretical curve before precipitation (dashed) and lie close to the theoretical curve after precipitation (dot-dashed).
and from the following equation for monoprotic bases
Bj ) S[H+]/[Xtotal]Ka
(10)
where Bj is the average number of bound protons per molecule, S is the intrinsic solubility of the sample (molar), [Xtotal] is the sample concentration (total weight of sample divided by the total volume) (molar), [H+] is the concentration of hydrogen ions (molar), and Ka is the equilibrium constant of the sample (molar). Multiprotic compounds have similar curves, with ampholytes having a curve at both sides of the neutral region showing their behavior as either an acid or a base. RESULTS The assays for diclofenac are used here to illustrate the results obtained from chasing equilibrium. Diclofenac is a monoprotic acid with a pKa of 3.99.13 The results of 10 titrations, each changing the direction of the pH gradient eight times, are summarized in (13) Sirius Technical Application Notes. Vol. 2, Sirius Analytical Instruments Ltd., Forest Row: E. Sussex, U.K., 1996 (ISBN 1 901125 05 X).
Figure 4. Near-equilibrium pH gradients for 7.69 mg of diclofenac.
Table 1. Each titration used a different weight of sample ranging from 3 to 24 mg. The mean solubility result is 0.87 µg/mL with a standard deviation of 0.10, which is comparable to the reported literature constant of 0.82 µg/mL.14 An example of a chasing equilibrium graph is shown in Figure 4. A kinetic solubility value of ∼45 µg/mL was found, a factor 50 times the equilibrium concentration. Summary of Results. The results from the assays of all compounds studied are summarized in Table 2. Final resulting values are the mean of the results from 6 to 10 assays. The increased time taken for the lidocaine assays can be largely explained by the dissolution time. The sample was introduced as large crystals of the free base and took ∼20 min to fully dissolve. The instrument automatically detected the change in pH as the sample dissolved, and the titration toward precipitation was delayed until the sample had fully dissolved. This delay could be reduced by taking measures to increase the dissolution rate, for example, by using a smaller particle size or introducing the sample in the form of a more soluble salt. DISCUSSION pH Response after Titrant Addition. The pH gradient values plotted in Figures 2 and 4 were obtained by analyzing the change (14) Avdeef, A.; Berger, C, M.; Brownell, C. Pharm. Res. 2000, 17 (1), 85-89.
Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
987
Table 2. Summary of Results from the Studya
pKa diclofenac (acid) famotidine (ampholyte) hydrochlorothiazide (diacid) ibuprofen (acid) lidocaine (base) propranolol (base) warfarin (acid)
3.99 6.77 11.01 8.75 9.88 4.35 7.95 9.54 4.94
sample weight (mg)
time taken (min)
ionic strength (M)
temp (°C)
kinetic solub (µg/mL)
intrinsic solub (µg/mL)
intrinsic solub (lit.) (µg/mL)
3.4-24 10 assays 102-123 6 assays 50-58 6 assays 6.2-51 10 assays 96-280 10 assays 10-19 6 assays 10-12 6 assays
33 ( 2
0.154 ( 0.002
25.2 ( 0.2
45 ( 6
0.9 ( 0.1
0.8 ( 0.2
61 ( 5
0.171 ( 0.003
25.0 ( 0.05
5900 ( 650
740 ( 40
1100 ( 200
52 ( 5
0.171 ( 0.004
24.9 ( 0.02
2400 ( 500
630 ( 9
700 ( 90
43 ( 4
0.161 ( 0.007
25.0 ( 0.2
180 ( 10
50 ( 4
49 ( 2
79 ( 7
0.184 ( 0.02
24.9 ( 0.1
4600 ( 900
3500 ( 100
3810 ( 20
60 ( 7
0.157 ( 0.001
25.0 ( 0.05
340 ( 20
81 ( 6
70 ( 20
60 ( 9
0.153 ( 0.000
24.9 ( 0.02
120 ( 1
5.3 ( 0.2
5.6 ( 0.3
a Values shown with an error of ( one standard deviation. The pK s were measured on a GLpKa at 0.15 M ionic strength and 25 °C.13,15 a Literature values were all measured at 25 °C, and at 0.15 M ionic strength,14 except lidocaine16 and warfarin,17 where the ionic strength was not reported.
Figure 5. pH response in a precipitating solution of diclofenac after the addition of base titrant.
in measured pH that occurred after the addition of titrant while chasing equilibrium. Figure 5 illustrates how a pH gradient value was obtained when base titrant was added to a supersaturated solution of diclofenac with precipitate already present. An aliquot of base titrant was added at time 0 in Figure 5, and the pH was measured and recorded once per second until the change in pH had reached a sustained response showing a settled direction and a stable rate of change. In stage 1 of the curve, the pH electrode is shown responding to the free base titrant added to the solution. In stage 2, the pH electrode shows a multivariate response, governed by ionization of the dissolved solute, the precipitation of the solute, the absorption/emission of carbon dioxide, the protonation/deprotonation of carbonic acid, and the response time of the pH electrode. The reactions governing stage 1 and stage 2 reach a steady state after ∼60 s. The shape of the response in the first two stages is unaffected by the solute, being the same for a solution of weak acid or weak base or even for no compound at all. 988 Analytical Chemistry, Vol. 77, No. 4, February 15, 2005
In stage 3, the pH electrode responds predominantly to the increase in pH that occurs as the sample precipitates or dissolves. In Figure 5, the neutral species concentration is greater than the intrinsic solubility and so the pH gradient is positive as the acid precipitates. When the linear fit of a number of the final pH points (e.g., 30 points) of this sustained response curve is better than a requested value (e.g., correlation coefficient >0.9), the final pH and the gradient of the line are recorded. The first two stages of the response are seen when base titrant is added, regardless of whether the neutral species concentration is greater than or less than the intrinsic solubility. The third stage changes to a negative slope when the solution becomes subsaturated, the pH falling as more of the solid acid is dissolved. In a similar manner, the pH response after addition of acid titrant goes through the same stages, but with the first two stages being inverted. The third stage is still dominated by the precipitation or dissolution of the sample. When the gradient in the third stage is very large, the first two stages may not be apparent, the entire response being dominated by the precipitation or dissolution of the sample. When the sample compound is a base, the same stages are seen but the direction of the pH gradient is reversed; e.g., a negative gradient indicates supersaturation. When no sample is present, the same stages are seen, but the third stage has a very small gradient and is usually dominated by the carbon dioxide equilibrium or electrical noise. Effects of Carbon Dioxide. When carbon dioxide is dissolved in water it forms carbonic acid (H2CO3), which has two protons that dissociate with pKas at 6.1 and 9.9 (at 25 °C and 0.15 M ionic strength). Carbon dioxide dissolves in water according to Henry’s law with a constant of ∼0.03 M/bar, and with the partial pressure of carbon dioxide in the atmosphere usually being ∼4 × 10-4 bar, the equilibrium concentration of carbon dioxide in water would be ∼10-5 M.18 If the concentration of neutral carbonic acid is greater than this, carbon dioxide gas is gradually evolved (15) Sirius Technical Application Notes. Vol. 1, Sirius Analytical Instruments Ltd., Forest Row: E. Sussex, U.K., 1995 (ISBN 901125 00 9). (16) Powell, M. F. In Analytical Profiles of Drug Substances; Florey, K., Ed.; Academic Press: San Diego, 1986; Vol. 15, pp 761-779
Figure 6. pH gradient due to carbon dioxide while titrating from low pH to high pH, measured 2 min after titrant addition. The gradient reduced when exclusion of the atmosphere was improved.
to the atmosphere. Conversely, if the concentration is lower, carbon dioxide gas is gradually absorbed from the atmosphere. When the pH of water in a vessel exposed to the atmosphere is changed during a titration by adding acid or base titrant, the concentration of dissolved neutral carbon dioxide changes as the carbonic acid becomes more or less ionized. Consequently, there is a change in the rate at which the gas is absorbed from the atmosphere or evolved into it. This gradual change in the total concentration of carbonic acid changes the pH, and this can be measured as a pH gradient, as shown in Figure 6. A constantly refreshed blanket of an inert gas will exclude atmospheric carbon dioxide and remove any evolved carbon dioxide, reducing the partial pressure of carbon dioxide above the solution and so reducing the effects of carbon dioxide absorption on the measured pH gradient. A high-quality seal around the top of the titration vessel has been developed for the Sirius GLpKa to minimize the mixing of ambient air into the inert gas blanket. Using degassed water and acid reagents reduces the concentration of predissolved carbon dioxide that is introduced to the solution. Carbonic acid is fully ionized at high pH so degassing the base titrant is ineffective and other measures were used, such as preventing atmospheric carbon dioxide ever coming into contact with the base titrant by preparing it in an atmosphere with reduced carbon dioxide and then using a scrubber column to prevent atmospheric carbon dioxide entering the titrant storage vessel. Titrations were performed to measure the effects of carbon dioxide, by adding acid titrant (0.5 M HCl) to ionic strength adjusted water (0.15 M KCl) to reach pH 2. Base titrant (0.5 M KOH) was then added to raise the pH in steps until pH 12 was reached. At each point, the rate of pH change was measured. These were performed first with a standard GLpKa and second with an improved seal and degassed reagents, both with an argon gas blanket. Typical results are shown in Figure 6. The pH (17) Bergstro ¨m, C. A. S.; Strafford, M.; Lazorova, L.; Avdeef, A.; Luthman, K.; Artursson, P. J. Med. Chem. 2003, 46, 558-570. (18) Butler, J. N. Ionic equilibrium: Solubility and pH calculations; John Wiley & Sons: New York, 1998.
Figure 7. Near-equilibrium graph taking time to establish equilibrium.
gradient is significantly reduced, although not entirely eliminated, with the improved precautions. In practice, the pH gradients caused by carbon dioxide while chasing equilibrium are not as large as shown here, the titrant aliquots being much smaller and so adding less carbon dioxide. When measuring solubility by chasing equilibrium, the pH gradient due to carbon dioxide is superimposed on the gradient due to precipitation and dissolution of the solid acid or base. If carbon dioxide is not sufficiently excluded, this can result in poor determination of the solubility. The measures taken here to limit the effects of carbon dioxide have so far proved sufficient to allow the measurement of the solubility of several compounds, down to the level of 0.1 µg/mL. DIAGNOSTICS Zero-Gradient Crossing Points. The results in Table 1 and the example in Figure 4 show just how tightly spaced the data is within each assay. In occasional experiments, it takes several attempts at chasing equilibrium from both sides before the final result is homed in upon. This is shown in Figure 7 for an assay using ibuprofen. In this example, the nonequilibrium crossing points represent the data collected while the system was approaching equilibrium. It is readily identifiable when the equilibrium gradient curve has found equilibrium. Hence, unlike in many other methods, where equilibrium is assumed after a certain time period, these plots are useful diagnostic tools for identifying when equilibrium had been established. Salt Precipitation. The assumption is made in the chasing equilibrium method that the precipitate is the free acid, base, or ampholyte and not a salt. For monoprotic compounds, the validity of this assumption can be confirmed by the direction of the pH gradient immediately after precipitation. A salt with a nonprotogenic counterion (e.g., K+, Cl-) precipitating will exhibit a gradient in the direction opposite to that expected from precipitation of the free form. For example, when a free monoprotic acid precipitates (HA f HAsolid), some of the neutral species that is lost to solid is replaced by the action of the ionization equilibrium with the charged species (H+ + AAnalytical Chemistry, Vol. 77, No. 4, February 15, 2005
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f HA), reducing the concentration of free protons and so increasing the pH. When a potassium salt of this acid precipitates (K+ + A- f KAsolid), some of the lost acid ions (A-) are replaced as the neutral species dissociates to maintain ionization equilibrium (HA f H+ + A-), resulting in an increase in the concentration of free protons and a decrease in pH. In this study, all the monoprotic compounds, which includes the least soluble compound in the study (diclofenac), exhibited an initial slope in the direction expected for precipitation of the neutral species, indicating that the precipitate was not a salt. This simple test does not necessarily hold true for multiprotic compounds or salts with protogenic counterions. An in-depth treatment of this field is beyond the scope of the current study and is the subject of ongoing research. Kinetic Solubility. The kinetic solubility value obtained is highly dependent on experimental conditions, such as the speed of titration (which affects the degree of supersaturation before precipitation begins) and the sensitivity of the precipitation detector. For the samples in this study, the kinetic solubility value was higher than the equilibrium solubility value. If the sample undergoes a high degree of supersaturation, then the value may be many times higher. Some compounds show a marked difference between the kinetic approximation of solubility and the intrinsic solubility, as shown in Table 2. For example, diclofenac precipitated at more than 50 times the intrinsic solubility. The kinetic solubility cannot be used as a reliable guide to the intrinsic solubility of a compound. CONCLUSIONS This new procedure gives highly repeatable values for the intrinsic solubility of weak acids, bases, and ampholytes. These experimental results are in good agreement with literature values, and the errors also compare well with the range of errors typically encountered in other methods for intrinsic solubility determination over this order of magnitude of results (1-4000 µg/mL).
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In contrast to traditional techniques, it does not have to wait passively for equilibrium to establish itself. The point of equilibrium is actively sought by changing the concentration of the neutral form via a change in pH. This can drastically reduce the time taken for an individual assay and also removes the need for the presumption that the time allowed was sufficient for equilibrium to be actually achieved. It provides excellent diagnostic information for the experiment. The Bjerrum (difference) graph gives rich information that can confirm that the experiment has proceeded correctly. The nearequilibrium pH gradient versus concentration graph gives positive indication that the equilibrium point has been found and crossed repeatedly. Traditional techniques rely on waiting a long time and assuming that equilibrium will have been reached. The time required for this is often determined by the judgment and experience of the individual scientist. It can be used over a wide range of solubility values, down to 1 µg/mL or lower. The lower end of the measurable solubility range seems to be largely determined by the effects of carbon dioxide on the pH gradient. The procedure also provides a measurement of the kinetic solubility. Comparing the kinetic solubility to the intrinsic solubility gives a rough indication of the level of supersaturation that could be expected. It also demonstrates the unreliability of using kinetic solubility data when high-quality solubility results are required. ACKNOWLEDGMENT We thank John Comer (Sirius) and Derek Reynolds (REYTEK Limited) for helpful discussions. We thank colleagues at Sirius for their contributions toward the development of software for the method, in particular Roger Allen and Andy Latham.
Received for review August 18, 2004. Accepted November 11, 2004. AC048767N
