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J. Phys. Chem. B 2001, 105, 1212-1215
Direct Determination of Equilibrium Thermodynamic and Kinetic Parameters from Isothermal Heat Conduction Microcalorimetry Anthony E. Beezer,* Andrew C. Morris, Michael A. A. O’Neill, Richard J. Willson,† Andrew K. Hills, John C. Mitchell, and Joseph A. Connor Medway Sciences, NRI UniVersity of Greenwich, Medway UniVersity Campus, Chatham Maritime, Kent ME4 4TB, U.K., and SmithKline Beecham Pharmaceuticals, New Frontiers Science Park (South), Harlow, Essex CM19 5AW, U.K. ReceiVed: September 28, 2000
Calorimetry is recognized1-8 as a noninvasive, nondestructive method for the determination of both the thermodynamic and the kinetic parameters associated with chemical reactions. The most immediate applications of the technique have been found in the determination of long-term stability data particularly for pharmaceuticals. However, the methods proposed required that calorimetric data (thermal power, Φ, watts recorded as a function of time) be analyzed iteratively to obtain the order of the reaction n, the rate constant k, and the enthalpy change of reaction ∆RH. A necessary assumption in this process was that all of the sample placed into the calorimeter would react. This is obviously a severe constraint for the flexibility and application of the method. This paper reports a significant extension of the procedure that allows direct calculation of all of the above parameters. Moreover, the equations that are developed permit the determination of the actual quantity of the sample placed into the calorimeter that will react. Indeed, for successful determination of the desired kinetic, thermodynamic, and equilibrium parameters, it is not necessary, in principle, to have any knowledge about a sample other than its total mass. It is possible to determine, in addition to n, k, and ∆RH, the equilibrium constant K for the reaction studied together with the associated values of the Gibbs function and entropy changes ∆RG and ∆RS. Moreover, because the reaction is to be studied over a temperature range, the activation energy Ea is also accessible.
Introduction proposed1-8
that isothermal heat conducSome time ago we tion microcalorimetry could be used to determine both kinetic and thermodynamic parameters for reacting systems. These studies have subsequently been followed by those of Selzer et al.9,10 (these latter papers have been followed by a commentary by one of us).11 The advantages cited for the proposed procedure, which was applied mainly for the determination of the long-term stability of pharmaceutically related materials, were rapidity (only 50 h of observation is required, in principle, to discriminate between the first-order rate constants of 1 × 10-11 and 2 × 10-11 s-1), direct observation on the sample, whatever its form, nondestructive and noninvasive methods, and experimental simplicity. A disadvantage is that the method, as previously described,1-9 requires an iterative procedure to determine the target parameters (n, the order of the reaction; k, the rate constant; and ∆RH, the reaction enthalpy change). The explicit assumption in applying the method was that the whole sample placed into the calorimeter would react. That is, the reaction proceeded to completion and no equilibrium state involving the coexistence of reactants and products was achieved. This assumption is a severe constraint on the flexibility and application of the method. It would be useful, over a wide range of applications, to be able to determine the extent of reaction for a sample (for both defined and heterogeneous samples) placed into the calorimeter. This is an essential requirement to determine the equilibrium * To whom correspondence should be addressed. † SmithKline Beecham Pharmaceuticals.
constant for a reacting system and would greatly improve the interpretation of, for example, solid-state reaction data, compatibility data, and perhaps surface analysis data. The developed procedures should, for maximum utility, enable direct calculation of the required parameters (n, k, and ∆RH, that is, no requirement for iterative procedures). The calculations should result in values of the parameters listed above, and if it is possible to calculate the equilibrium constant K, then, because ∆RH is known, the Gibbs function change ∆RG and the entropy change ∆RS should be calculable. (Note hereafter these parameters are identified, for simplicity, as H, G, and S.) The methods described in this paper require that the reacting system be studied over a range of temperature, and as a result, the availability of k as f(T) allows, in addition, access to the activation energy Ea. Methods for constructing the calorimetrically based equations from classical kinetic expressions have been well documented1-8,12-14 for nth-order reactions and for both sequential and parallel reaction systems. First-order kinetic reactions form a special case and, hence, both general solutions and solutions specific to the first-order case will be presented. The equations that are developed below have been tested against simulated calorimetric data and have been shown to be successful in allowing the determination of the parameters listed above. Development of the Equations. The development of the equations will be exemplified through a simple reaction system (although as is shown later more complex reactions can also be treated):
ASB
10.1021/jp003539s CCC: $20.00 © 2001 American Chemical Society Published on Web 01/19/2001
Determination of Equilibrium Parameters
J. Phys. Chem. B, Vol. 105, No. 6, 2001 1213
Table 1a rate constant (k) initial signal (real) polynomial order initial signal (calculated) goodness of fit rate constant (k) initial signal (real) polynomial order initial signal (calculated) goodness of fit
0.002 705 3 0.002 296 5 1 0.002 01
2 0.002 23
3 0.002 28
4 0.002 29
5 0.0023
6 0.0023
7 0.0023
8 0.0023
9 0.0023
0.948 64 0.997 97 0.999 93 1 1 1 1 1 1 2.7053 × 10-6 2.2962 × 10-6 1 2 3 4 5 6 7 8 9 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 2.30 × 10-6 1
1
1
1
1
1
1
1
1
-1
a
The table shows that for a reaction with a first-order rate constant of 0.002 705 3 s the value of Φ0 is found from a 4th-order polynomial onward and for a reaction with a first-order rate constant of 2.7053 × 10-6 s-1 a first-order polynomial provides the value of Φ0.
where, for convenience, A and B refer to amounts of reactant and products [either moles or grams; it is simple to convert these equations into those described by concentrations (mol dm-3); however, in the final equations, quantities must be used because H is a per mole parameter which is essentially independent of concentration]. A commentary on the simple extension of the equations to more complex reaction systems is made below. Selzer et al.9 note that, following the identical earlier treatment of Willson et al.,1 it is possible to write that the calorimetric output Φ0 at time t ) 0 is given, for the general case, by
Φ0 ) kHATn and so, for the first-order case, by
Φ0 ) kHAT where AT is the load placed into the calorimeter. This means that AT ) A {reacting amount of sample} + (AT - A) {nonreacting amount of sample}. Here, as a subsequent development will show, AT is the total sample quantity that could, in principle, react. Thus, a plot of ln Φ0 vs ln AT will be linear, with the slope equal to the order of the reaction n. There is, however, a simple and direct test1 for a first-order process that consists of simply plotting ln Φ vs t. A linear plot results only from a first-order reaction, and the slope of this plot is -k, the first-order rate constant. It is appropriate to identify AT as the sample quantity loaded into the calorimeter (or the total mass of a quantitatively uncharacterized sample placed in the calorimeter) because, if there is an incomplete reaction (i.e., an equilibrium exists between reactants and products), then not only is AT the only known experimental parameter but it is true that the same fraction of any sample quantity loaded into the calorimeter will react (for a constant reaction mechanism) at the given isothermal and environmental (T, RH, pO2, pH, etc.) conditions. This has the consequence that the slope of the ln Φ0 vs ln AT plot will always remain equal to n but the intercept value could vary (for varying extents of reaction). However, the intercept value is not required in the subsequent development. In general, the value of Φ0 is not measured directly because the experimental procedures for the operation of isothermal heat conduction calorimeters (such as the thermal activity monitor produced by Thermometric, Jarfalla, Sweden) require an initial period in which calorimetric measurements are not made. This period incorporates the preparation of the sample and a necessary calorimetric equilibration period, and these may total between 1 and 2 h. To obtain the appropriate value of Φ0, it is necessary to develop a method that uses the data recorded over the
observation period. We have concluded that the application of successively higher order polynomial fits to, say, the first 20 h of any experimental data set allows for the successful extrapolation of the data for each experiment back to t ) 0. Table 1 (from data simulated via Mathcad software)5,12-14 shows that for a range of values of k (over 10 orders of magnitude in k were examined, but for illustrative purposes, only data for a relatively fast reaction and for a moderately fast reaction are displayed in the table) the correct value of Φ0 is found as the limiting value of the first (i.e., the constant) term in the polynomial fit equations. Unsurprisingly, excellent fits are achieved more directly for slower reactions, and as is shown in the table, an excellent fit is found from a first-order polynomial for a reaction with a moderately slow rate constant (half-life of around 7 days). Thus, n is easily found in all cases. It is also possible, as is noted earlier,5 to determine the order of the reaction for thermal power-time curves that display significant curvature from the ratio of two times at specified percentage points of the initial signal value (ref 5 details the methods and a protocol for the preparation of appropriate tables of order vs time ratios for orders ranging from 0.8 to 2.5). In the most general case, the total area under the Φ vs t curve from t ) 0 to ∞ (here designated Q and associated with A, that quantity of the load that can react) can be calculated from an observation period of 50 h or so. This value, Q, clearly represents the number of joules involved in the complete reaction of all reactable material. This is expressed in this way to make clear the distinction between the amount which will react, A (associated with Q joules) and the number of joules QT, which would be observed if all of the sample, AT, had reacted (Q e QT). Q can be calculated as follows: From ref 1, it is shown that Φ1 ) kH(A - x)n; here x is the number of moles of product formed at time t1. With the recognition that A ) Q/H and that x ) q1/H, where q1 is the area under the thermal power-time curve from t ) 0 to t1, then this equation becomes
Φ1 ) kH1-n(Q - q1)n Writing this equation again for t ) t2 and dividing one equation into the other produces
[Φ1/Φ2]1/n ) (Q - q1)/(Q - q2) If [Φ1/Φ2]1/n is set equal to R, then Q can be calculated from
Q ) (q1 - Rq2)/(1 - R) For the first-order case, Φ ) kH(A - x) ) k(Q - q), and so because k, Φ, and q are known as f(t), then Q can be determined.
1214 J. Phys. Chem. B, Vol. 105, No. 6, 2001 Because Q can be determined at any temperature, then a test for the complete reaction of all of the load, AT, placed in the calorimeter is that Q should remain constant for equal-value AT loads (i.e., the value of Q ) QT, normalized to unit quantity (mol, g)) studied over a range of temperatures. If this is found, then Q/AT will yield H directly. Now that n, Φ, q, t, QT, AT, and H are known, then k is determined from the equations given above. Modeled data for illustrative purposes shows that the correct values of QT are found for values of H ranging from 10 to 100 kJ mol-1 and of the equilibrium constant K, ranging from 0.5 to 50. Thus, there appear to be no significant problems associated with the determination of QT over a wide range of H and K (a wider range of values for K has not been systematically investigated, but those calculations that have been done suggest that there is no reason to suspect that problems will arise). If Q is found to vary with temperature, then an equilibrium exists, and it is convenient to calculate the equilibrium constant and the enthalpy as described below before the rate constant k is determined. Calculation of the Equilibrium Constant. For the reaction A S B, the equilibrium constant can be written as
K ) [B]/[A] (square brackets are not meant to imply a concentrationdependent equilibrium constant but are simply a general form for an equilibrium constant expression). Because here [B] is equivalent to x, the number of moles of product formed at time t used in the above kinetic equations, then it is clear that, at equilibrium, [B] ) Q/H and that [A] ) (AT - [B]) ) (QT/H Q/H). Thus
K ) Q/(QT - Q) This expression can be written for studies at different temperatures, Tm (m ) 1, 2, 3, etc.), and QT (or its mass or molenormalized equivalent) will have the same value at each temperature if an equal AT is loaded into the calorimeter at each of the temperatures. For a two-state reaction, the van’t Hoff isochore relates15 K and T through H/R. Thus, it is possible to write (assuming no dependence of ∆RCP upon the temperature and/or no change in the reaction mechanism over the short temperature range involved; see below for further remarks on this point), for the reaction at two temperatures T1 and T2, ln(K1/K2) ) -H/R(1/T1 - 1/T2). This expression also holds for temperatures 2, 3, etc. Now, provided that the equality
T1T2/(T2 - T1) ) T2T3/(T3 - T2) holds {for T1 ) 293 K and T2 ) 298 K, then T3 must be equal to 303.2 K, T4 must be equal to 308.5 K, etc.}, then it is easy to demonstrate that K1/K2 ) K2/K3. From the expression for K given above, this latter relationship can be expressed in terms of Qm and QT and subsequently solved (for temperatures m ) 1, 2, and 3) for QT as
QT ) (Q22Q1 + Q22Q3 - 2Q3Q2Q1)/(Q22 - Q3Q1) Because both QT and AT are now known, then H is calculated from H ) QT/AT. Now that H is known, then it easy to calculate k, the rate constant for the given order of reaction, in the manner noted above and also to calculate A from A ) Qm/H. K is now clearly accessible, and thus, values of G and S follow from standard equations, namely, G ) -RT ln K and G ) H - TS. (Of course, the calculated values of K should also allow a check
Beezer et al. on the derived value of H because a plot of ln K vs 1/T should have a slope of -H/R.) The availability of k as f(T) clearly allows for the determination of Ea from an Arrhenius plot of ln k vs 1/T. Relatively complex formulations of the equilibrium constant, such as that relating to the (simple!) reaction A S B + C, where K ) [B][C]/[A], will result in an expression based upon Qm and QT values as K ) (Q/H)(Q/H)/(QT/H - Q/H) ) Q/H(QT - Q). Because H is a constant, then it can be taken over to make the equilibrium constant K′ ) KH, and the value of this is dependent only on known values of Qm and QT. A plot of ln K′ vs 1/T will still be linear, with a slope of -H/R, allowing H to be determined and, hence, the values of AT and A to be calculated. For more complex equilibria such as A + 2B S C + D, then, provided that the quantities of A and B are known, it will be possible to calculate the required K’s. The quantities of C and D formed can always be expressed through Qm and QT, as shown above, with H being determined on a “per mole of A” basis. Even if the quantities of A and B are not known, it will be possible to evaluate the required parameters, provided an essentially constant amount of B is present because, in this circumstance, one can formulate K′ in a manner similar to that described above, recognizing now that K′ is equivalent to K[B] (or, as necessary, some form such as KH[B]). Therefore, a plot of ln K′ vs 1/T will again (always) yield the value of H from the slope. Knowledge now of H, Qm, and QT will allow for the calculation of both A and AT, i.e., the amount of reactable material present in an undefined sample and the total amount of that material present in the sample. To check the soundness of these equations, calorimetric data (Φ vs t) were simulated5,12-14 via Mathcad software, and for values of K from 0.5 to 50 and for H values from 10 to 100 kJ mol-1, the required values of QT were always recovered. Hence, all of the target parameters were calculated. It is of interest to note that in the case where AT (say for a complex and quantitatively unspecified sample) is not known but the reaction has a determinable order the equations outlined above can still be employed. Qm values are determinable for standardized loads placed into the calorimeter and studied over a suitable temperature range and a value for QT derived. Hence, models for an equilibrium reaction can be explored, i.e., models could be fitted, and an appropriate model which allowed for the calculation of a constant QT found. By appropriate, it is meant one which yields a linear plot of ln K vs 1/T (here K is formed in an appropriate way in terms of Q and QT: the necessary assumption is that the reaction under study conforms to the requirements of the van’t Hoff isochore). A first guide to the stoichiometric form of the reaction can be had from the known order of the reaction. The consequence of this approach is that, from the van’t Hoff relationship, it will be possible to calculate the value of H. Possession of the value of H will permit the calculation of the number of moles of potentially reactable material present in the sample from AT ) QT/H and the number of moles actually reacted, A, from A ) Q/H. As always, such information does not identify the reacting material. A suitable test for a reaction system where there is either a dependence of ∆RCP on temperature or a mechanism change (i.e., multiple reactions occur simultaneously) over the temperature range is to calculate QT for temperatures where m ) 1, 2, and 3 and compare them with the value calculated for temperatures where m ) 2, 3, and 4. If there is conformity with the van’t Hoff requirements, then values of QT must be independent
Determination of Equilibrium Parameters of T. In principle, it may be expected that changing the temperature range significantly could emphasize one reaction over another, thus reducing curvature in the van’t Hoff plot, whereas the effect may not be so pronounced if the curvature was the result of the temperature dependence of ∆RCP. Conclusions The equations described above offer the opportunity to characterize reactions, through microcalorimetric study, via their equilibrium thermodynamic and kinetic parameters. The equations are applicable to (almost) any reaction system from simple well-defined systems to uncharacterized reacting systems: from pure solids through soft solids and gas/solid systems to solutionphase reacting systems. This is not an exhaustive list of reacting systems that can, in principle, be studied via the methods described in this paper. The procedures have been established and illustrated through models provided by simulated data. These models have the advantage, for the purposes of this paper, of yielding accurate and precise results for the (known in the modeling case) sought parameters (n, k, H, K, G, S, Ea, and the reactable quantities of A). It is of interest to note that for solids it would be expected that there will be a dependence of Q upon particle size (reactable surface area) and that QT would, in this case, represent the total reaction of all of the material present, as if it were at a surface. It may be possible, therefore, to use this method to characterize surfaces and their available reaction areas (through, possibly, the ratio Q/QT). This observation also suggests that solids should be studied at some specified size fraction. The test of the utility of these equations and their sensitivity to, for example, precisely specified temperatures for study will be in their application to real experimental data. We have recently proposed5 that the aqueous-solution-phase, imidazolecatalyzed hydrolysis of triacetin be considered as a test reaction
J. Phys. Chem. B, Vol. 105, No. 6, 2001 1215 for isothermal heat conduction microcalorimeters used to evaluate both thermodynamic and kinetic parameters. This reaction has been shown to be well behaved (order, rate constant, and enthalpy as determined by the earlier iterative procedure) for periods ranging from hours up to months (100 days is the maximum period of study to date). However, the data on this reaction which are currently available have not been subjected to the “constant Q as a function of T” test referred to above, nor have sufficient temperature experiments been performed to allow exploitation of the new equations described here. Such a study is in progress, and the results will be reported in a subsequent paper. References and Notes (1) Willson, R. J.; Beezer, A. E.; Mitchell, J. C.; Loh, W. J. Phys. Chem. 1995, 99, 7108-7113. (2) Willson, R. J.; Beezer, A. E.; Mitchell, J. C. Thermochim. Acta 1995, 264, 27-40. (3) Willson, R. J.; Beezer, A. E.; Mitchell, J. C. Int. J. Pharm. 1996, 132, 45-51. (4) Beezer, A. E.; Willson, R. J.; Mitchell, J. C.; Hills, A. K.; Gaisford, S.; Wood, E.; Connor, J. A. Pure Appl. Chem. 1998, 70, 633-638. (5) Willson, R. J.; Beezer, A. E.; Hills, A. K.; Mitchell, J. C. Thermochim. Acta 1999, 325, 125-132. (6) Gaisford, S.; Hills, A. K.; Beezer, A. E.; Mitchell, J. C. Thermochim. Acta 1999, 328, 39-45. (7) Beezer, A. E.; Gaisford, S.; Hills, A. K.; Willson, R. J.; Mitchell, J. C. Int. J. Pharm. 1999, 179, 159-165. (8) Beezer, A. E. Thermochim. Acta 2000, 349, 1-7. (9) Selzer, T.; Radau, M.; Kreuter, J. Int. J. Pharm. 1998, 171, 227241. (10) Selzer, T.; Radau, M.; Kreuter, J. Int. J. Pharm. 1999, 184, 199206. (11) Beezer, A. E. Int. J. Pharm. 2000, 207, 117-118. (12) Willson, R. J. Ph.D. Thesis, University of Kent, U.K., 1995. (13) Gaisford, S. Ph.D. Thesis, University of Kent, U.K., 1997. (14) Hills, A. K. Ph.D. Thesis, University of Kent, U.K., 2001, in preparation. (15) Atkins, P. W. Physical Chemistry, 6th ed.; Oxford: Oxford, U.K., 1998; pp 219-220.
