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Limits to the Concept of Solid-State Stability
Limits to the Concept of Solid-state Stability J. T. CARSTENSEN”, K. DANJO*, S. YOSHIOKA~, AND M. UCHIYAMA~ Acce ted for publication April 23, Received November 12,1986,from the .:Schqol of.Pharmacy, Univery’ of Wisconsin, Madiso?, Wl 53706. 1987. *On sabbatical leave from Meyo University, Japan, and the Xafiona; lnsbfufe of Hygienic Sciences, 18-1 Amiyoga, 1-chome, Sefagaya-ku, Tokyo 158, Japan.
published data about the decomposition of propantheline bromide in the presence of moisture resulted in a general, two-variable,phenomological equation relating the decomposition to the relative humidity and temperature. The true curve, however, is biphasic, since complete liquefaction occurs at some time point. This article deals with the development of a theoretical equation relating decomposition time, temperature, and relative humidity during the solid-state stage (h., during the period when solid is still present in the system). %tract
0 Recently
In solid-state kinetics, it is tacitly assumed that the decomposing drug is a solid at the onset. In some systems, decomposition products are liquid. In other types of systems, the actual decomposition of the drug substance takes place in a (saturated) solution of moisture that is condensed onto the solid, and considered present as a bulk phase. The characteristic of the system is always that it is primarily solid. In certain cases, the decomposition profile may become biphasic (e.g., the system may become completely liquid a s decomposition progresses). In such cases, the system is a “solid” system aa long as the drug (or decomposing parent substance) is present in solid form. In the decomposition of pure solids, there are cases (the socalled Bawn kinetics1) where a solid decomposes to a liquid and a gas; examples iire parasubstituted benzoic acids.2.3 In this case, the model accounts for both the decomposition of the solid per se and the part which is dissolved in the liquid decomposition product. From this, a modelistic equation is derived which applies up to the point where the system becomes completely liquid. Beyond this point, the decomposition is merely solution kinetics. It is noted that in such a case the entire curve is described by two equations. It would not be possible to extend either of the two equations beyond their domain. Another example of bimodal behavior is the vapor pressure moisture content curve of a soluble solid compound. This curve will “end” at the relative humidity corresponding to the vapor pressure of a saturated solution. The reason for this is simple. Beyond this vapor pressure, the system is no longer solid in any sense of the word (i.e., the substance is completely in solution). Many pharmaceutical systems have been studied in the presence of water for the simple reason that moisture in general cannot be excluded from a dosage form. The most common situation is that described by Leeson and Mattocks.4 Here, the amount decomposed is accounted for by either simply assuming that the total amount decomposed is the amount of drug decomposed in solution in the sorbed moisture layer (the solid being stable),4 or by accounting for decomposition both in the solid and the sorbed solution layera6Several approaches have been taken. Some authors’ have used open systems in the sense that the product was placed in an infinite moisture sink (desiccator with defined relative humidity), whereas otherss have studied decomposition in a closed system, arguing that this comes closer to the actual situation where a drug is in a bottle. 548 /Journal of Pharmaceutical Sciences Vol. 76, No. 7, July 1987
Since it is of importance in solid-state stability to know both the effect of moisture and temperature, recent publications7--e have described accelerated solid-state hydrolysis a t various relative humidities of water-soluble drugs, using propantheline bromide as a model substance. The data from these investigations have been empirically fitted to an equation of the type
x = x,exp[(EJR)(1/298 - 1/T)1(P/18.2)”(t/50)n (1) where XIX, is fraction decomposed, x, is the original number of moles of intact substance present, E, is the activation energy, R is the gas constant, T is the absolute temperature, P is the water vapor pressure to which the system is exposed, t is time, and s and n are constants. Of phenomological necessity, this is a monophasic fit (except, it is tacitly assumed that x/x, must be less than unity). It is the purpose of this article to arrive at a theoretical equation applicable to the part of the decomposition curve of a water-soluble compound which is the “solid-state decomposition” part of the curve (i.e., the part in which solid is still present).
Experimental Section Propantheline bromide was obtained from Searle Research Laboratories and was used as received. Samples of propantheline bromide were placed on Petri dishes in desiccators at various relative humidities and temperatures. The weight gain was monitored as a function of time, and the point of complete liquefaction was recorded.
Results and Discussion Four of the accelerated conditions employed in earlier publications8 were used and are shown in Table I. It is noted that the length of time required for liquefaction is between 20 and 45 min. A large portion of previously reported decomposition curves of propantheline: therefore, apply to solution kinetics and not to kinetics in the solid state. If treated by the same philosophy as that used in Bawn kinetics, only the lowest parts of the curves would be considered as part of the solid kinetic scheme. With only a few points, this part of the curve cannot be characterized. It is important, in the following development, to gauge Table CMolsture Uptake and Llquefactlon Tlme of Propantheline Bromide at Different Accelerated Condltlons
Temperature’ “C 60 80 80 90
Relative
H20, Condensed,
YO
Time to Liquefaction, min
80.5 65.6 79.5 78.3
<30 <20 <30 <45
152 162 123 182
Humidity,
Rate, mglgls
mglg 5.1* 8.1’ 8.1’ 4.0’
dOrder of magnitude figures only, since the actual time of complete liquefaction is not known precisely, only that it was less than t h e stated number of minutes.
Q
0022-3549/87/0700-0548$01 .OO/O 1987, American Pharmaceutical Association
what shape the actual moisture uptake curve would have, both theoretically and experimentally. However, the substance is quite hygroscopic and the moisture uptake curve a t elevated temperature is difficult to obtain. Instead, multipoint room temperature data were generated in a desiccator over a saturated KCl solution (84.3% relative humidity). These data are shown graphically in Figure 1. It is noted, again, that complete liquefaction occurs in a relatively short period of time (40 d a t room temperature). To explain this curve, it is assumed that the physical situation is that there are ml grams of water vapor in the gas phase [considered an infinite reservoir, and m grams of water is adsorbed, at time t, on W grams (dry weight) of solid]. The true density of the solid is D g/cm3 (i.e., the dry volume of the solid is WID cm3). Assuming unit density for the adsorbed water, the total area, A , of solid plus adsorbed water is
A
=
F[Q +
4
~
(2)
3
where Q is a constant and F is a conventional shape factor (i.e., the particles are assumed to be isometric). (Further details of this are discussed in the Appendix.) The condensation rate, dmldt, is assumed to be proportional to the surface area, so that at time t it is
dmldt = qFlQ
+ m]213
(3)
where q, at a given temperature and pressure, is constant, F is a bulk shape factor (derived in the Appendix), and Q is a function of W , D1, and D (also derived in the Appendix). It should be noted that q, of course, is a function of the relative humidity of the surrounding atmosphere, and, in fact, in conventional condensation is proportional to P - P*, where P is the water vapor pressure in the atmosphere, and P* is the vapor pressure over a saturated solution of the solid, at the temperature in question. Equation 3 is rearranged to read
dm/[Q
+ mlua = qFdt
It should be pointed out that the plot fails to intersect at zero, so that the above views, a t best, are approximate. An explanation could be that the first percent of uptake is adsorption, and that what follows is bulk condensation. Another explanation is that Q = 1 is but an approximate value (albeit close to the true value). The nature of the adsorbed water initially may also differ from the bulk phase at later condensation. For kinetic purposes it is easier to approximate the curve in Figure 1 with
m
=
q ’ [ l - exp(-ut)l
(6)
where u and q’ are constants which will be commented on below. A fit of the data in Figure 1to eq 6 is fair, as shown in Figure 3. An interpretation of q’ is that this is the amount of water just necessary to liquefy the system. Hence, S = W/(Dq’) where S is solubility of the compound in water. Differentiation of eq 6 gives
dmldt = q’uexp(-ut) = [q’ - mlu
(7)
Since the condensation rate is linear in (P - P*), the last term may be interpreted as u being proportional to the pressure gradient, that is u =
u”P - P*]
(8)
where u’ is a pressure-independent (but temperature-dependent) constant. In this case, (q’ - m)would then be roughly
(4)
Equation 4 integrates to
where initial conditions have been invoked. Data are plotted in this fashion in Figure 2, assuming Q = 1, and it is seen that the fit is good (correlation coefficient of 0.95).
P
f:
1.00
0
TOTAL LIQUEFACTION O.1°
1
I
20 Time (days)
i
I
40
30
Flgure 2-The uptake rate profile in Figure 1 treated according to eq 5.
-2
0.00 ! 0
I
10
1
I
I
1
10
20 Tima (&PI
30
40
Flgure 1-Moisture adsorbed onto propantheline bromide as a function of time of exposure to 80.5% relative humidity.
1
y=
- 2.699 - 0.0484~R
-
0.93
=B \ . I
-5
0
I
I
I
a
10
20
30
40
Time (drya)
Flgure *The
uptake rate profile in Figure 1 treated according to eq 6. Journal of Pharmaceutical Sciences / 549 Vol. 76, No. 7, July 1987
linearly related to area, which is feasible. It is again emphasized that eq 5 is an approximation. It is now possible to employ Leeson-Mattocks kinetics, and the rate of total decomposition (Mdenoting amount of intact drug) is given by
-dM/dt = -mkS = -q’kS[l - exp(-ut)l
(9)
where k is the first-order solution rate constant. This (invoking initial conditions) integrates to
M
=
[kSq’/ul- kSq’t - [kSq’/ulexp(-ut) (10)
For small values of u, the last term dominates the previous term and the profile will appear to be first order. The rate constant, u, is given by eq 8. Here, u’ would be of the typical Arrhenius type, and P* would adhere to a Clausiudlapeyron equation, so that
u = Zexp[-E/{RT)l [P
- P,exp[-H/{RT)l
(11)
where H is a constant related to enthalpy of evaporation. Equation 11 bears some resemblance to the phenomological equation, eq 1.
Appendix Equation 2 is correct, but some elaboration is in order. Shape factors relate to individual particles, not to a population of particles as a whole. If it is assumed that the sample consists of N particles with diameter d (cm; prior to water condensation) and true density D (g/cm3),then
N(7r/6)d3= WID
(All
and area, A , (cm2),can be expressed aslo
A,
=
NGP
=
NG(?rd3/6)Y3
(A21
This assumes the particles to be isometric,ll where G is the true shape factor. At time t, the amount of water condensed is m/D1, where D1 is the density of water a t the temperature in question. The volume of water condensed on each particle is, therefore
d(ND1) = mDd3/[D1W61
(A3)
The total particle volume (assuming ideality, that is, additivity of volumes), hence, is
V
=
[(7r/6)d31+ [ d d / ( D l W 6 ) 1
(A41
The area per particle, hence, is a = G[?r/6)+ (mDd/(D16W)IY3d2 Hence, the total area is
(A51
Glossary total surface area (an2)of drug + water at time t initial surface area (cm2) of drug a area per particle (cm7 D true density (g/cm3) of solid drug D1 true density (g/cm3) of water = particle diameter (cm) d E = activation energy (kJ/mol) F = bulk shape factor (dimensionless) G = individual shape factor (dimensionless) H = enthalpy of evaporation (kJ/mol) k = dissolution constant (cm/d) M = mass of undecomposed drug (g) N = number of particles rn = mass of adsorbed water a t time t ml = mass of water present in vapor phase at time t n = Yoshioka exponent P = water vapor pressure ( a h ) P* = saturated water vapor pressure ( a h ) Po = pre-exponential term (atm) in ClausiugClapeyron equation (eq 11) q = condensation rate constant (g/s/cm2) q‘ = saturation amount of water sorbed Q = constant (g) = DiW/D R = gas constant (kJ/deg/mol) S = solubility of drug in water (g/cm3) t = time (e.g., d) T = absolute temperature C‘K) = condensation rate constant (time - I ; eq 6 ) u u’ = proportionality constant (time-’/atm) relating condensation rate constant to pressure gradient (eq 8) V = (cm3) volume of particles + water (eq A4) W = dry weight (g) of solid x = amount of drug decomposed = amount of initial moles of drug present x, 2 = pre-exponential factor (day-’) for condensation rate constant, u A A,
= = = = =
References and Notes 1. Bawn, C. E. H. In Chemistry of the Solid State; Garner, W . E.; Butterworths: London, 1955; 254. 2. Carstensen, J.T.; Musa, M. F! J. Pharm. Sci. 1972, 61, 1112. 3. Carstensen, J. T.; Kothari, R. J. Pharm. Sci. 1981, 70, 1095. 4. Leeson, L.; Mattocks, A. J.Am. Pharm. Assoc., Sci. Ed. 1958,47, 302.
6 . Carstensen, J.T.; Pothisiri, P. J. Pharm. Sci. 1975, 64, 37. 6. Carstensen, J. T.; Attarchi, F.; Hou, X-P. J. Pharm. Sci. 1985, 74, 741. 7. Yoshioka, S.; Shibazaki, T.; Ejima, A. Chem. Pharm. Bull. 1982, 30, 3734. 8. Yoshioka, S.; Uchiyama, M. J . Pharm. Sci.1986, 75, 92. 9. Yoshioka, S.; Uchiyama, M. J. Pharm. Sci. 1986, 75, 459. 10. Short, M. P.; Sharkey, P.; Rhodes, C. T. J. Pharm. Sci. 1972,61, 1733.
11. Carstensen J. T. In Solid Pharmaceutics, Mechanical Properties and Rate Phenomena; Academic: NY, 1980; p 52.
where
and
so that the total area, Na, is of the form shown in eq 1.
550 /Journal of Pharmaceutical Sciences Vol. 76, No. 7, July 1987
Acknowledgments This work was su ported by grants from Sandoz Research Institute, E. Hanover, NJ, and Merrell-Dow Research Institute, Cincinnati, OH.
published data about the decomposition of propantheline bromide in the presence of moisture resulted in a general, two-variable,phenomological equation relating the decomposition to the relative humidity and temperature. The true curve, however, is biphasic, since complete liquefaction occurs at some time point. This article deals with the development of a theoretical equation relating decomposition time, temperature, and relative humidity during the solid-state stage (h., during the period when solid is still present in the system). %tract
0 Recently
In solid-state kinetics, it is tacitly assumed that the decomposing drug is a solid at the onset. In some systems, decomposition products are liquid. In other types of systems, the actual decomposition of the drug substance takes place in a (saturated) solution of moisture that is condensed onto the solid, and considered present as a bulk phase. The characteristic of the system is always that it is primarily solid. In certain cases, the decomposition profile may become biphasic (e.g., the system may become completely liquid a s decomposition progresses). In such cases, the system is a “solid” system aa long as the drug (or decomposing parent substance) is present in solid form. In the decomposition of pure solids, there are cases (the socalled Bawn kinetics1) where a solid decomposes to a liquid and a gas; examples iire parasubstituted benzoic acids.2.3 In this case, the model accounts for both the decomposition of the solid per se and the part which is dissolved in the liquid decomposition product. From this, a modelistic equation is derived which applies up to the point where the system becomes completely liquid. Beyond this point, the decomposition is merely solution kinetics. It is noted that in such a case the entire curve is described by two equations. It would not be possible to extend either of the two equations beyond their domain. Another example of bimodal behavior is the vapor pressure moisture content curve of a soluble solid compound. This curve will “end” at the relative humidity corresponding to the vapor pressure of a saturated solution. The reason for this is simple. Beyond this vapor pressure, the system is no longer solid in any sense of the word (i.e., the substance is completely in solution). Many pharmaceutical systems have been studied in the presence of water for the simple reason that moisture in general cannot be excluded from a dosage form. The most common situation is that described by Leeson and Mattocks.4 Here, the amount decomposed is accounted for by either simply assuming that the total amount decomposed is the amount of drug decomposed in solution in the sorbed moisture layer (the solid being stable),4 or by accounting for decomposition both in the solid and the sorbed solution layera6Several approaches have been taken. Some authors’ have used open systems in the sense that the product was placed in an infinite moisture sink (desiccator with defined relative humidity), whereas otherss have studied decomposition in a closed system, arguing that this comes closer to the actual situation where a drug is in a bottle. 548 /Journal of Pharmaceutical Sciences Vol. 76, No. 7, July 1987
Since it is of importance in solid-state stability to know both the effect of moisture and temperature, recent publications7--e have described accelerated solid-state hydrolysis a t various relative humidities of water-soluble drugs, using propantheline bromide as a model substance. The data from these investigations have been empirically fitted to an equation of the type
x = x,exp[(EJR)(1/298 - 1/T)1(P/18.2)”(t/50)n (1) where XIX, is fraction decomposed, x, is the original number of moles of intact substance present, E, is the activation energy, R is the gas constant, T is the absolute temperature, P is the water vapor pressure to which the system is exposed, t is time, and s and n are constants. Of phenomological necessity, this is a monophasic fit (except, it is tacitly assumed that x/x, must be less than unity). It is the purpose of this article to arrive at a theoretical equation applicable to the part of the decomposition curve of a water-soluble compound which is the “solid-state decomposition” part of the curve (i.e., the part in which solid is still present).
Experimental Section Propantheline bromide was obtained from Searle Research Laboratories and was used as received. Samples of propantheline bromide were placed on Petri dishes in desiccators at various relative humidities and temperatures. The weight gain was monitored as a function of time, and the point of complete liquefaction was recorded.
Results and Discussion Four of the accelerated conditions employed in earlier publications8 were used and are shown in Table I. It is noted that the length of time required for liquefaction is between 20 and 45 min. A large portion of previously reported decomposition curves of propantheline: therefore, apply to solution kinetics and not to kinetics in the solid state. If treated by the same philosophy as that used in Bawn kinetics, only the lowest parts of the curves would be considered as part of the solid kinetic scheme. With only a few points, this part of the curve cannot be characterized. It is important, in the following development, to gauge Table CMolsture Uptake and Llquefactlon Tlme of Propantheline Bromide at Different Accelerated Condltlons
Temperature’ “C 60 80 80 90
Relative
H20, Condensed,
YO
Time to Liquefaction, min
80.5 65.6 79.5 78.3
<30 <20 <30 <45
152 162 123 182
Humidity,
Rate, mglgls
mglg 5.1* 8.1’ 8.1’ 4.0’
dOrder of magnitude figures only, since the actual time of complete liquefaction is not known precisely, only that it was less than t h e stated number of minutes.
Q
0022-3549/87/0700-0548$01 .OO/O 1987, American Pharmaceutical Association
what shape the actual moisture uptake curve would have, both theoretically and experimentally. However, the substance is quite hygroscopic and the moisture uptake curve a t elevated temperature is difficult to obtain. Instead, multipoint room temperature data were generated in a desiccator over a saturated KCl solution (84.3% relative humidity). These data are shown graphically in Figure 1. It is noted, again, that complete liquefaction occurs in a relatively short period of time (40 d a t room temperature). To explain this curve, it is assumed that the physical situation is that there are ml grams of water vapor in the gas phase [considered an infinite reservoir, and m grams of water is adsorbed, at time t, on W grams (dry weight) of solid]. The true density of the solid is D g/cm3 (i.e., the dry volume of the solid is WID cm3). Assuming unit density for the adsorbed water, the total area, A , of solid plus adsorbed water is
A
=
F[Q +
4
~
(2)
3
where Q is a constant and F is a conventional shape factor (i.e., the particles are assumed to be isometric). (Further details of this are discussed in the Appendix.) The condensation rate, dmldt, is assumed to be proportional to the surface area, so that at time t it is
dmldt = qFlQ
+ m]213
(3)
where q, at a given temperature and pressure, is constant, F is a bulk shape factor (derived in the Appendix), and Q is a function of W , D1, and D (also derived in the Appendix). It should be noted that q, of course, is a function of the relative humidity of the surrounding atmosphere, and, in fact, in conventional condensation is proportional to P - P*, where P is the water vapor pressure in the atmosphere, and P* is the vapor pressure over a saturated solution of the solid, at the temperature in question. Equation 3 is rearranged to read
dm/[Q
+ mlua = qFdt
It should be pointed out that the plot fails to intersect at zero, so that the above views, a t best, are approximate. An explanation could be that the first percent of uptake is adsorption, and that what follows is bulk condensation. Another explanation is that Q = 1 is but an approximate value (albeit close to the true value). The nature of the adsorbed water initially may also differ from the bulk phase at later condensation. For kinetic purposes it is easier to approximate the curve in Figure 1 with
m
=
q ’ [ l - exp(-ut)l
(6)
where u and q’ are constants which will be commented on below. A fit of the data in Figure 1to eq 6 is fair, as shown in Figure 3. An interpretation of q’ is that this is the amount of water just necessary to liquefy the system. Hence, S = W/(Dq’) where S is solubility of the compound in water. Differentiation of eq 6 gives
dmldt = q’uexp(-ut) = [q’ - mlu
(7)
Since the condensation rate is linear in (P - P*), the last term may be interpreted as u being proportional to the pressure gradient, that is u =
u”P - P*]
(8)
where u’ is a pressure-independent (but temperature-dependent) constant. In this case, (q’ - m)would then be roughly
(4)
Equation 4 integrates to
where initial conditions have been invoked. Data are plotted in this fashion in Figure 2, assuming Q = 1, and it is seen that the fit is good (correlation coefficient of 0.95).
P
f:
1.00
0
TOTAL LIQUEFACTION O.1°
1
I
20 Time (days)
i
I
40
30
Flgure 2-The uptake rate profile in Figure 1 treated according to eq 5.
-2
0.00 ! 0
I
10
1
I
I
1
10
20 Tima (&PI
30
40
Flgure 1-Moisture adsorbed onto propantheline bromide as a function of time of exposure to 80.5% relative humidity.
1
y=
- 2.699 - 0.0484~R
-
0.93
=B \ . I
-5
0
I
I
I
a
10
20
30
40
Time (drya)
Flgure *The
uptake rate profile in Figure 1 treated according to eq 6. Journal of Pharmaceutical Sciences / 549 Vol. 76, No. 7, July 1987
linearly related to area, which is feasible. It is again emphasized that eq 5 is an approximation. It is now possible to employ Leeson-Mattocks kinetics, and the rate of total decomposition (Mdenoting amount of intact drug) is given by
-dM/dt = -mkS = -q’kS[l - exp(-ut)l
(9)
where k is the first-order solution rate constant. This (invoking initial conditions) integrates to
M
=
[kSq’/ul- kSq’t - [kSq’/ulexp(-ut) (10)
For small values of u, the last term dominates the previous term and the profile will appear to be first order. The rate constant, u, is given by eq 8. Here, u’ would be of the typical Arrhenius type, and P* would adhere to a Clausiudlapeyron equation, so that
u = Zexp[-E/{RT)l [P
- P,exp[-H/{RT)l
(11)
where H is a constant related to enthalpy of evaporation. Equation 11 bears some resemblance to the phenomological equation, eq 1.
Appendix Equation 2 is correct, but some elaboration is in order. Shape factors relate to individual particles, not to a population of particles as a whole. If it is assumed that the sample consists of N particles with diameter d (cm; prior to water condensation) and true density D (g/cm3),then
N(7r/6)d3= WID
(All
and area, A , (cm2),can be expressed aslo
A,
=
NGP
=
NG(?rd3/6)Y3
(A21
This assumes the particles to be isometric,ll where G is the true shape factor. At time t, the amount of water condensed is m/D1, where D1 is the density of water a t the temperature in question. The volume of water condensed on each particle is, therefore
d(ND1) = mDd3/[D1W61
(A3)
The total particle volume (assuming ideality, that is, additivity of volumes), hence, is
V
=
[(7r/6)d31+ [ d d / ( D l W 6 ) 1
(A41
The area per particle, hence, is a = G[?r/6)+ (mDd/(D16W)IY3d2 Hence, the total area is
(A51
Glossary total surface area (an2)of drug + water at time t initial surface area (cm2) of drug a area per particle (cm7 D true density (g/cm3) of solid drug D1 true density (g/cm3) of water = particle diameter (cm) d E = activation energy (kJ/mol) F = bulk shape factor (dimensionless) G = individual shape factor (dimensionless) H = enthalpy of evaporation (kJ/mol) k = dissolution constant (cm/d) M = mass of undecomposed drug (g) N = number of particles rn = mass of adsorbed water a t time t ml = mass of water present in vapor phase at time t n = Yoshioka exponent P = water vapor pressure ( a h ) P* = saturated water vapor pressure ( a h ) Po = pre-exponential term (atm) in ClausiugClapeyron equation (eq 11) q = condensation rate constant (g/s/cm2) q‘ = saturation amount of water sorbed Q = constant (g) = DiW/D R = gas constant (kJ/deg/mol) S = solubility of drug in water (g/cm3) t = time (e.g., d) T = absolute temperature C‘K) = condensation rate constant (time - I ; eq 6 ) u u’ = proportionality constant (time-’/atm) relating condensation rate constant to pressure gradient (eq 8) V = (cm3) volume of particles + water (eq A4) W = dry weight (g) of solid x = amount of drug decomposed = amount of initial moles of drug present x, 2 = pre-exponential factor (day-’) for condensation rate constant, u A A,
= = = = =
References and Notes 1. Bawn, C. E. H. In Chemistry of the Solid State; Garner, W . E.; Butterworths: London, 1955; 254. 2. Carstensen, J.T.; Musa, M. F! J. Pharm. Sci. 1972, 61, 1112. 3. Carstensen, J. T.; Kothari, R. J. Pharm. Sci. 1981, 70, 1095. 4. Leeson, L.; Mattocks, A. J.Am. Pharm. Assoc., Sci. Ed. 1958,47, 302.
6 . Carstensen, J.T.; Pothisiri, P. J. Pharm. Sci. 1975, 64, 37. 6. Carstensen, J. T.; Attarchi, F.; Hou, X-P. J. Pharm. Sci. 1985, 74, 741. 7. Yoshioka, S.; Shibazaki, T.; Ejima, A. Chem. Pharm. Bull. 1982, 30, 3734. 8. Yoshioka, S.; Uchiyama, M. J . Pharm. Sci.1986, 75, 92. 9. Yoshioka, S.; Uchiyama, M. J. Pharm. Sci. 1986, 75, 459. 10. Short, M. P.; Sharkey, P.; Rhodes, C. T. J. Pharm. Sci. 1972,61, 1733.
11. Carstensen J. T. In Solid Pharmaceutics, Mechanical Properties and Rate Phenomena; Academic: NY, 1980; p 52.
where
and
so that the total area, Na, is of the form shown in eq 1.
550 /Journal of Pharmaceutical Sciences Vol. 76, No. 7, July 1987
Acknowledgments This work was su ported by grants from Sandoz Research Institute, E. Hanover, NJ, and Merrell-Dow Research Institute, Cincinnati, OH.