Free volume, definition mechanism

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The study of fluid mechanics is facilitated by understanding and using the relationship between a system and a control volume. By definition, a system is a certain mass of fluid, that can move about in space. Moreover the system is free to deform as it moves. As a result it is practically impossible to follow and account for a particular mass of fluid in a flowing process. Nevertheless, because many of the basic physical laws are written in terms of a system (e.g., F = m ), it is convenient and traditional to take advantage of the notion of a system. 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

These ideas can be formalised in terms of statistical mechanics to some extent, and an outline of the main ideas is given in the following section. We remark parenthetically that there are profound difficulties confronting the definition of an aggregate. The nature of the hydrophobic free energy of transfer of a hydrocarbon from water to the hydrophobic core of a micelle can be measured, but its temperature dependence is not understood because it depends on water, an unknown quantity. For the same reasons, solution theory, does not even tell us whether mole fractions or mole volumes are the correct ratios to use to determine entropy. However, provided certain assumptions are allowed [62-65], then simple rules emerge. The rules are if v is the hydrocarbon chain volume, a the head-group area, and 1 of an optimal... 

Values for all of the kinetic and thermodynamic parameters have been reported in the literature. They were usually obtained from experiments in which caprolactam and definite amounts of water were heated in closed systems for various periods of time. From the mechanism and the corresponding rate equation, it is readily seen that for a given temperature the concentration of water is the principal process parameter. It affects both the rate and the attainable degree of polymerization. If in the kinetic experiment, therefore, any free reactor volume (vapor space) is not essentially eliminated (which may pose some experimental problems), then the effective initial water concentration is lower and consequently a lower rate of polymerization will result. This may be one reason for certain differences in values reported by different investigators. Another reason may entail different analytical approaches. Table 2.2 and Table 2.3 show the kinetic and thermodynamic parameters as reported by two different groups [52,53] for the three principal equilibrium reactions. 

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