The Dried Product - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 153

where the left-hand side is the heat flow per second, the first term on the
right-hand side is the number of mols reacting per second, and DH is the
enthalpy of the reaction.
For a general reaction, such as aA þ bB

- cC, the heat flow can be

expressed as

(dq/dt) ¼ k DH[A -(aq/DH)] m [B -(aq/DH)] n

By plotting power (dq/dt) against heat flow (q), the constants DH, k, m, n, A, B
can be determined by a process of iteration.
For processes with DH 40 kJ mol 1 , microcalorimeters can detect

reaction rates corresponding to 6 mmol month 1 . Alternatively, for first-
order rate constants of 10 10 s 1 , half-lives of 200 years can be conveniently
measured (equivalent to reaction rates of 0.3% per year). For examples of
various aspects of isothermal microcalorimetry, see Refs. 146-148.
As discussed for DSC earlier, disadvantages of microcalorimetry in-
clude its chemical non-specificity and also a possibility that unpredictable
physical changes might lead to non-Arrhenius behaviour. Such changes
include unexpected phase transformations, water transfer between excipi-
ents and drugs, improperly defined rate constants, and parallel reactions
of the same order but with different activation energies.

D

11.7 Dynamics in Supersaturated Solutions

Before discussing the role of any residual water that remains within the
solid solution after freeze-drying, we shall consider the dynamics of
water and other molecules within supersaturated solutions, such as are
formed in aqueous PHC solutions during any drying process. A ground-
breaking study has revealed that, although completely miscible on a
macroscopic scale, extreme complexities arise in the diffusional motions
of water, relative to those of the PHC molecules, as their concentration
increases beyond the saturation value. 149 We report here the major
results and conclusions of this study because of their high relevance to
an understanding of the molecular-scale processes that accompany the
concentration and temperature changes during drying. z
The study encompassed measurements of rotational and translational
diffusion rates of different parts of PHC molecules (by 13 C NMR) and
-OD, -CH 2 OD and D 2 O ''hydration phase'' (by 2 H NMR) over a range

z Some of the contents of the thesis have since been published in English. 150,151,156 Although the
main experimental details and data are recorded in the published papers, their discussion in terms of
the molecular dynamic details is hardly touched upon; they can only be found in the thesis itself.
The translation and condensation to a 15-page English summary has been performed by F. Franks
and can be obtained from him at [e-mail].