Ind. Eng. Chem. Res. 2005, 44, 7057-7061
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KINETICS, CATALYSIS, AND REACTION ENGINEERING Non-isothermal Dehydration Kinetics of Ceftriaxone Disodium Hemiheptahydrate Chun-tao Zhang,* Jing-kang Wang, and Yong-li Wang School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China
The non-isothermal dehydration behavior of ceftriaxone disodium (CTXN) hemiheptahydrate, investigated by thermogravimetry-derivative thermogravimetry (TG-DTG), proceeds in two steps: (1) the loss of two and one-half water molecules and (2) the loss of one water molecule. The dehydration kinetics were achieved from the TG-DTG curves by both the Achar method and the Coats-Redfern method with 21 frequently cited basic kinetic models. The dynamic dehydration processes for steps 1 and 2 are best expressed by the Zhuralev-Lesokin-Tempelman equation and the Avrami-Erofeyev equation (n ) 1), suggesting a three-dimensional diffusioncontrolled mechanism and a nucleation-controlled mechanism, respectively. The corresponding kinetic compensation effects (KCEs) for the two steps are found to be: ln A ) 0.354 54 × 10-6.5733 and ln A ) 0.297 66 × 10-5.47166, respectively. 1. Introduction At least one-third of the solid crystalline substances listed in the European Pharmacopoeia have been reported to form associations with water in different states during processing and storage; in ∼20% of these associations, the hydrate is the official crystal form.1 The presence of the water molecules influences the intermolecular interactions (affecting the internal energy and enthalpy) and the crystalline disorder (entropy) and, hence, influences the free energy, thermodynamic activity, solubility, stability, bioavailability, and in vivo performance of the dosage form.2 To control the state of hydration of the active ingredient, it is, therefore, important and necessary to understand the kinetics and mechanisms of hydration and dehydration processes under the appropriate conditions. Ceftriaxone disodium (CTXN) hemiheptahydrate, C18H16N8Na2O7S3‚3.5H2O, is a third-generation, semisynthetic, broad-spectrum cephalosporin antibiotic, and its structural formula is given in Figure 1. The bactericidal activity of ceftriaxone results from inhibition of cell wall synthesis, and it has a high degree of stability in the presence of beta-lactamases, both penicillinases and cephalosporinases of gram-negative and grampositive bacteria.3 Since it first came into the market in 1982, CTXN has become fashionable all around the world and one of the most important parenterally applied antibiotics. Despite their biological and pharmacological significance, the thermal analysis of antibiotics has received little attention. Martin-Gil et al.4 applied differential thermal analysis (DTA) to study the beta-lactam antibiotics (ceftriaxone acid included) and presented a tentative correlation between the thermal characteristics and the structures. Differential scanning calorim* To whom correspondence should be addressed. Tel.: 8622-27405754. Fax: 86-22-27374971. E-mail:
[email protected].
Figure 1. Molecular structure of CTXN hemiheptahydrate.
etry (DSC) was employed to examine the thermal decomposition stability of CTXN hemiheptahydrate, and the decomposition activation energy was determined by the Kissinger equation and the Ozawa equation.5 The survey of the literature indicates that very little work has been carried out on the thermal dehydration process of CTXN hemiheptahydrate. The aim of the present work on the non-isothermal dehydration kinetic analysis of the thermogravimetry-derivative thermogravimetry (TG-DTG) data by the Achar method6 and the CoatsRedfern method7 is to achieve the most probable kinetic model which gives the best description of the studied dehydration process and allows the calculation of reliable values of the kinetic parameters E and A. 2. Experimental Apparatus and Methods 2.1. Materials. A white crystalline powder of CTXN hemiheptahydrate was supplied by a pharmaceutical company (Shandong province, China) and recrystallized in the State Research Center of Industrial Crystallization and Technology with a purity >99.5%. The product crystals were vacuum-dried at ambient temperature for 24 h. Later, the crystals were ground in an agate mortar and then fractionated into the particle size range of 57-65 µm with USP standard sieves.
10.1021/ie050210q CCC: $30.25 © 2005 American Chemical Society Published on Web 07/27/2005
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Table 1. Instrument and Scan Parameters of XPRD for CTXN Hemiheptahydrate radiation goniometer divergence slits receiving slits inst. profile breadth
Cu KR (0.15405 nm) Rigaku D/Max-2500 1° 0.15 mm 0.078°(2θ)Si(311)
scan range scan speed scan step temp. power sett.
our experience, the rational expression of Senum and Yang10 gives sufficiently accurate results:
4.78° ∼ 50° (2θ) 1 s/step 0.02° (25 ( 1) °C 40 kV, 100 mA
2.2. Non-isothermal Dehydration Studies. The TG-DTG experiments were performed using the Netzsch TG209 system, under a dynamic atmosphere of dry nitrogen at a flow rate of 18 mL/min, over the temperature range of ambient to 200 °C at a heating rate of 10 °C /min. The runs were conducted using an alumina crucible containing ∼5.080 mg of the sample and an R-Al2O3 crucible as reference material. Several runs were carried out, and identical results were obtained. 2.3. X-ray Powder Diffraction (XRPD). The X-ray powder diffraction data of the CXTN hemiheptahydrate sample were collected by Rigaku D/Max-2500. The experimental conditions and XRPD spectra are shown in Table 1 and Figure 2, respectively.
π(x) )
x3 + 18x2 + 88x + 96 x + 20x3 + 120x2 + 240x + 120 4
3.1. Differential Method using the Achar Equation. This method applies generally to all reaction mechanisms where the rate can be expressed as a function of R. The data are plotted for each of the selected models and the linearity is compared to determine the best model. The Achar equation6 is
ln
[
ln
Most reactions studied by thermal analysis techniques can be described by the following equations8
(4)
[ ] [ G(R) T2
) ln
)]
2RT E AR 1βE E RT
(
(5)
and, assuming (2RT/E) , 1, leads to
d(R) A -x ) e f(R) dT β
(1) ln
or
dR AE -x ) e ∫0R f(R) βR
]
d(R)/dT A E ) ln β RT f(R)
The slope and the intercept of the plots of ln[(d(R)/dT)/ f(R)] vs 1/T allow the calculation of the activation energy E and the preexponential constant A. 3.2. Integral Method using the Coats-Redfern Equation. The Coats-Redfern equation7 is
3. Theory
G(R) )
(3)
[ ] π(x) x
(2)
where R is the fraction dehydrated, T is the absolute temperature, A is the preexponential constant, β is the constant heating rate, E is the activation energy, R is the gas constant, and the functions f(R) and G(R) represent the mathematical expression of the kinetic model in differential form and integral form, respectively. The most frequently cited basic kinetic models9 are summarized in Table 2. The reduced activation energy is x ) E/RT, and π(x) is an approximation of the temperature integral. There are various expressions of the π(x) term available in the literature. According to
[ ] [ ] G(R) T2
) ln
E AR βE RT
(6)
The slope and the intercept of the plots of ln[G(R)/T2] vs 1/T allow the calculation of the activation energy E and the preexponential constant A. 3.3. Kinetic Compensation Effects (KCEs). The KCEs are widely reported in many thermally stimulated processes, especially in the solid-state thermal decompositions. The following mathematical expression is generally used to describe the KCEs8
ln A ) aE + b
(7)
where a ) 1/RTp and b ) ln[-βxp/Tpf′(Rp)] are the kinetic compensation constants, and Tp, xp, and Rp represent the absolute temperature, the reduced activation en-
Figure 2. X-ray powder diffraction spectra of CTXN hemiheptahydrate.
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Figure 3. Typical TG-DTG plot for the dehydration of CTXN hemiheptahydrate. Table 2. Kinetic Functions f(r) and G(r) Used for the Present Analysis kinetic function no.
kinetic function name
f(R)
G(R)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Mampel power (n ) 2) Valensi Jander (n ) 2) Ginstiling-Brouns Jander (n ) 1/2) reverse Jander Zhuralev-Lesokin-Tempelman Avrami-Erofeyev (n ) 1) Avrami-Erofeyev (n ) 2/3) Avrami-Erofeyev (n ) 1/2) Avrami-Erofeyev (n ) 4) Avrami-Erofeyev (n ) 2) Avrami-Erofeyev (n ) 3) reaction order (n ) 1/2) reaction order (n ) 1/3) reaction order (n ) 1/4) Mampel power (n ) 1) Mampel power (n ) 3/2) reaction order (n ) 2) reaction order (n ) 2) reaction order (n ) 2/3)
1/(2R) -[ln(1 - R)]-1 3(1 - R)2/3[1 - (1 - R)1/3]-1/2 3[(1 - R)-1/3 - 1]-1/2 4(1 - R)1/2[1 - (1 - R)1/2]1/2 3(1 + R)2/3[(1 + R)1/3 - 1]-1/2 3(1 - R)4/3[(1 - R)-1/3 - 1]-1/2 1-R 3(1 - R)[-ln(1 - R)]1/3/2 2(1 - R)[-ln(1 - R)]1/2 (1 - R)[-ln(1 - R)]-3/4 (1 - R)[-ln(1 - R)]-1/2 (1 - R)[-ln(1 - R)]-2/3 2(1 - R)1/2 3(1 - R)2/3 4(1 - R)3/4 1 2R-1/2/3 (1 - R)2 (1 - R)2 2(1 - R)3/2
R2 (1 - R)ln(1 - R)+ R [1 - (1 - R)1/3]2 (1 - 2R/3) - (1 - R)2/3 [1 - (1 - R)1/2]1/2 [(1 + R)1/3 - 1]2 [(1 - R)-1/3 - 1]2 -ln(1 - R) [-ln(1 - R)]2/3 [-ln(1 - R)]1/2 [-ln(1 - R)]4 [-ln(1 - R)]2 [-ln(1 - R)]3 1 - (1 - R)1/2 1 - (1 - R)1/3 1 - (1 - R)1/4 R R3/2 1/(1 - R) 1/(1 - R)-1 (1 - R)-1/2
ergy, and the fraction dehydrated at the maximum of the TG curve, respectively.
Table 3. Kinetic Parameters of CTXN Hemiheptahydrate for Step 1 Dehydration Process
4. Results and Discussion
E no. (kJ/mol)
4.1. Thermal Dehydration Processes. The representative curves of TG and DTG for the sequence of dehydration steps in the CTXN‚3.5H2O powdered sample are shown in Figure 3. It is observed that two DTG maxima occur at mean temperatures of 70.3 °C and 133.9 °C. The mass loss measurements suggest the sequence of dehydration as 2.5 mol and 1 mol at these mean temperatures. For a sample mass of 5.080 mg, the observed loss in mass corresponds to a liberation of three and one-half water molecules. The dynamic TG measurements above suggest the following steps of dehydration in CTXN hemiheptahydrate with respect to temperatures:
1 38.4798 6.95560 -0.89076 59.8846 14.53059 -0.96951 2 50.8061 10.98312 -0.95803 66.4592 16.47822 -0.98101 3 66.3093 15.37842 -0.98879 75.0813 18.39408 -0.96171 4 56.1937 11.52704 -0.97397 69.2798 16.09536 -0.98524 5 1.7584 -5.88078 -0.24594 13.4665 -2.08334 -0.97607 6 30.3248 1.50719 -0.83422 54.1637 9.95675 -0.96144 7 96.6560 26.93257 -0.99041 95.3145 26.33543 -0.99893 8 36.0555 7.26470 -0.97172 39.4400 8.28170 -0.99618 9 21.0231 1.76839 -0.92167 24.4076 2.71096 -0.99618 10 13.5069 -1.06474 -0.83407 16.8914 -0.20256 -0.99526 11 171.3475 54.46869 -0.99747 174.7320 55.58788 -0.90768 12 81.1528 23.23041 -0.99371 84.5374 24.31668 -0.99745 13 126.2502 38.90844 -0.99664 129.6347 40.01678 -0.99761 14 20.8822 0.79447 -0.93347 32.5903 4.78257 -0.98456 15 25.9400 2.31470 -0.96065 34.7120 5.25201 -0.98992 16 28.4688 2.98986 -0.96670 35.8327 5.42418 -0.99214 17 5.7088 -4.28945 -0.40903 27.1137 3.18629 -0.96223 18 22.0943 1.39197 -0.80781 43.4992 8.93496 -0.96731 19 66.4022 18.81884 -0.95432 24.6894 4.09487 -0.86004 20 66.4022 18.81884 -0.95432 57.4604 15.49150 -0.99046 21 51.2289 12.34862 -0.96405 9.5161 -2.63559 -0.79649
Achar method
70.3 °C
Step 1: CTXN‚3.5H2O 98 CTXN‚H2O + 2.5H2O 133.9 °C
Step 2: CTXN‚H2O98CTXN + H2O 4.2. Non-isothermal Kinetics for Steps 1 and 2. On the basis of 21 frequently used kinetic functions in both differential and integral forms,9 the kinetic analy-
ln A (s-1)
Coats-Redfern method r
E (kJ/mol)
ln A (s-1)
r
sis was carried out using a linear least-squares method from the TG-DTG data. The calculated kinetic parameters (E and A) and correlation coefficients (r) are summarized in Tables 3 and 4.
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Table 4. Kinetic Parameters of CTXN Hemiheptahydrate for Step 2 Dehydration Process Achar method E no. (kJ/mol) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
158.1719 206.5807 263.1365 225.9831 8.8271 120.6837 374.5966 140.3811 85.0473 51.55128 673.3603 318.0409 495.7006 84.65108 103.2278 112.5161 28.9210 93.5465 251.8413 251.8413 196.1112
Coats-Redfern method
ln A (s-1)
r
E (kJ/mol)
ln A (s-1)
r
41.90297 55.96943 71.68563 60.37169 -2.75991 28.15077 105.6275 37.49566 20.76774 10.59472 196.1287 90.60439 143.4254 19.83161 25.08311 27.62391 3.55384 22.78730 71.43748 71.43748 53.77343
-0.96538 -0.98817 -0.95779 -0.93300 -0.84306 -0.94191 -0.99930 -0.99858 -0.99447 -0.97428 -0.99930 -0.99980 -0.9953 -0.99406 -0.99886 -0.99930 -0.77919 -0.94342 -0.99011 -0.99011 -0.99418
249.4237 267.6536 288.1512 274.4658 64.47529 228.2176 331.7146 151.3093 98.63749 72.30159 625.3555 309.3247 467.3401 135.6567 140.7226 143.3123 121.3588 185.3913 58.6390 186.7039 25.9664
68.59332 73.54729 78.37968 74.14976 13.52298 59.77838 91.82250 40.20882 24.38395 16.37485 180.2007 87.11486 133.7185 34.62500 35.80415 36.32585 30.83510 49.77772 13.02295 51.21890 2.23184
-0.98676 -0.98975 -0.99254 -0.99077 -0.98982 -0.98384 -0.99645 -0.99447 -0.99420 -0.99392 -0.90483 -0.99472 -0.99480 -0.99083 -0.99217 -0.99279 -0.98601 -0.98652 -0.98177 -0.99869 -0.97708
The most probable mechanism for each dehydration step was determined following the commonly employed method: The values of E and ln A obtained by both the Achar method and the Coats-Redfern method must not only be approximately equal but must also correspond to a higher linear correlation coefficient. The results in Table 3 clearly show that the values of E and ln A obtained from both the Achar method and the Coats-Redfern method are very close to each other, and the linear correlation coefficients (r) by the two
methods are both better when kinetic function No. 7 was taken as the most probable mechanistic function for the step 1 dehydration process of ceftriaxone sodium hemiheptahydrate. Kinetic function No. 7 is the ZhuralevLesokin-Tempelman equation, G(R) ) [(1 - R)-1/3 - 1]2 and f(R) ) 3(1 - R)4/3[(1 - R)-1/3 - 1]-1/2, which belongs to a three-dimensional diffusion-controlled mechanism. Homoplastically, the results in Table 4 suggest that kinetic function No. 8 is the most probable mechanistic function for the step 2 dehydration process. Kinetic function No. 8 is the Avrami-Erofeyev equation (n ) 1), G(R) ) -ln(1 - R) and f(R) ) 1 - R, which belongs to a nucleation-controlled mechanism (n ) 1). By virtue of the above obtained kinetic models, the fraction of dehydrated R can be expressed as a function of absolute temperature through eq 2.
For step 1: G(R) ) [(1 - R)-1/3 - 1]2 ) R(T) ) 1 For step 2:
( ) ( ))
AE -x π(x) e w βR x 1/2 AE -x π(x) e +1 βR x
[(
( ) (
]
-3
AE -x π(x) e w βR x AE -x π(x) e R(T) ) 1 - exp βR x
G(R) ) -ln(1 - R) )
( ))
(8)
(9)
Figure 4 shows the comparison of experimental data (full lines) and calculated R(T) (points) using eqs 8-9.
Figure 4. Non-isothermal TG curves of CTXN hemiheptahydrate: (a) step 1 and (b) step 2. (O) indicates data calculated by the Achar method, (g) indicates data calculated by the Coats-Redfern method, and full lines were the experimental data.
Figure 5. KCEs of CTXN hemiheptahydrate: (a) step 1 and (b) step 2.
Ind. Eng. Chem. Res., Vol. 44, No. 18, 2005 7061 Table 5. Kinetic Compensation Parameters for CTXN Hemiheptahydrate calculated by linear least-squares method
calculated by definition
step
a (mol/kJ)
b
r
a (mol/kJ)
b
1 2
0.35454 0.29766
-6.5733 -5.47166
0.99703 0.99971
0.350208 0.29549
-6.84859 -4.07546
There is quite good agreement between the experimental data and the prediction of the Zhuralev-LesokinTempelman model for step 1. Some discrepancies between the experimental data and the prediction of the Avrami-Erofeyev model (n ) 1) are observed for step 2. This behavior is consistent with the facts that the consistency of the values of E and ln A obtained between the Achar method and the Coats-Redfern method for step 1 is remarkable, while the discrepancies of the values of E and ln A obtained between the Achar method and the Coats-Redfern method for step 2 are larger than those for step 1. 4.3. Kinetic Compensation Effect. By inspecting the values in Tables 3 and 4, parallel variation of ln A versus E is observed (see Figure 5). This means the presence of a compensation effect. The KCE parameters, a and b, obtained by the linear least-squares method, are summarized in Table 5. According to the obtained mechanistic functions and E, a and b can also be calculated by a ) 1/RTp and b ) ln[-βxp/Tpf′(Rp)]. The parameters (a and b) obtained by the two methods are almost the same (see Table 5), which can imply that the mechanism functions and E achieved in Section 4.2 are reasonable and reliable. The above results indicate that the kinetic parameters E and A are mutually correlated. Any change in the activation energy is therefore “compensated” by the change in ln A as expressed in eq 6. This compensation behavior is rather a rule in the nonthermal kinetics as a consequence of the same dehydration mechanism of the series. 5. Conclusions It is concluded that CTXN hemiheptahydrate exhibits two dehydration steps:
The first step: 70.3 °C
CTXN‚3.5H2O 98 CTXN‚H2O + 2.5H2O
The second step: 133.9 °C
CTXN‚H2O 98 CTXN + H2O The non-isothermal dehydration kinetics for steps 1 and 2 are best expressed by the Zhuralev-LesokinTempelman equation and the Avrami-Erofeyev equation (n ) 1), suggesting a three-dimensional diffusioncontrolled mechanism and a nucleation-controlled mechanism, respectively. The kinetic parameters and the mechanistic functions of the dehydration processes are
d(R) A -E/RT 3 ) e (1 - R)4/3[(1 - R)-1/3 - 1]-1 dT β 2 E ) 96.6560 kJ/mol, A ) 4.97 × 1011 s-1
Step 1:
d(R) A -E/RT ) e (1 - R) dT β E ) 140.3811 kJ/mol, A ) 1.92 × 1016 s-1
Step 2:
The corresponding kinetic compensation effects (KCEs) for two steps are found to be ln A ) 0.354 54 × 10-6.5733 and ln A ) 0.297 66 × 10-5.47166, respectively. Literature Cited (1) Han, J.; Gupte, S.; Suryanarayanan, R. Applications of pressure differential scanning calorimetry in the study of pharmaceutical hydrates. II. Ampicillin trihydrate. Int. J. Pharm. 1998, 170, 63. (2) Khankari, R. K.; Grant, D. J. W. Pharmaceutical hydrates. Thermochim. Acta. 1995, 248, 61. (3) British Pharmacopoeia Commission; British Pharmacopoeia 2000; The Stationery Office: London, 2000; Vol. II, pp 1350-1353. (4) Martin-Gil, J.; Martinez Villa, F.; Ramos-Sanchez, M. C.; Martin-Gil, F. J. Studies on beta-lactam antibiotics. J. Therm. Anal. 1984, 29, 1351. (5) Jie, Z.; Jianfang, Z. Application of differential scanning calorimetry to the study of thermal stability of cephalosporins in solid state. Acta Pharm. Sin. 1987, 22, 278. (6) Narahari Achar, B. N.; Bridley, G. W.; Sharp, J. H. Thermal decomposition kinetics of some new unsaturated polyesters. Proc. Int. Clay Conf., Jerusalem 1966, 1, 67. (7) Coats, A. W.; Redfern, J. P. Kinetic parameters from thermogravimetric data. Nature 1964, 201, 68. (8) Ma¨lek, J. The kinetic analysis of nonthermal data. Thermochim. Acta 1992, 200, 257. (9) Gao, X.; Dollimore, D. The thermal decomposition of oxalates. Part 26: A kinetic study of the thermal decomposition of manganese(II) oxalate dihydate. Thermochim. Acta. 1993, 215, 47. (10) Senum, G. I.; Yang, R. T. Rational approximations of the integral of the Arrhenius function. J. Therm. Anal. 1997, 11, 445.
Received for review February 21, 2005 Revised manuscript received June 8, 2005 Accepted June 24, 2005 IE050210Q
