Equations, mathematical calorimetry

The kinetic and thermodynamic characterisation of chemical reactions is a crucial task in the context of thermal process safety as well as process development, and involves considering objectives as diverse as profit and environmental impact. As most chemical and physical processes are accompanied by heat effects, calorimetry represents a unique technique to gather information about both aspects, thermodynamics and kinetics. As the heat-flow rate during a chemical reaction is proportional to the rate of conversion (expressed in mol s 1), calorimetry represents a differential kinetic analysis method [ 1 ]. For a simple reaction, this can be expressed in terms of the mathematical relationship in Equation 8.1 ... 

This contrast between Idealized and real systems Is a recurring and natural feature and we shall not shun it. A certain emphasis on the model systems is motivated by the consideration that such studies are the basis for further understanding. At the same time a certain prudence against fitting a limited number of equations to a limited number of data is advised. Rarely do such procedures lead to unique solutions. Even a perfect fit does not imply that the underlying model applies, although It may describe the adsorption empirically. The rule remains that for a proper characterization preferably further measurements should be carried out, say adsorption at more temperatures and/or with different adsorptives, or adsorption in conjunction with (mlcro)-calorimetry. Once such experiments have been done and found to concur, the mathematical confidence limit may become as high as the physical one. 

Besides steady state measurements, there is probably good reason to use flow micro calorimetry for the study of non-steady-state behavior in systems with immobilized bio catalysts. Here, the mathematical description is more complicated, requiring the solution of partial differential equations. Moreover, the heat response can evolve non-specific heats, like heat of adsorption/desorption or mixing phenomena. In spite of these complications, the possibility of the on-line monitoring of the enzyme reaction rate can provide a powerful tool for studying the dynamics of immobilized biocatalyst systems. 

The glass transition temperature is measured using differential scanning calorimetry (DSC) (179,284), by which a polymer sample is heated, and its enthalpic changes are measured in response. The temperature at which the heat capacity of the polymer drops is the glass transition temperature. Dynamic mechanical spectroscopy (DMS) is also used to determine the glass transition temperature. Certain mathematical equations, such as the Fox equation, relate the copolymer composition to the glass transition temperature. 

Linear differential equation of first order called the heat balance equation of a simple body, has found wide application in calorimetry and thermal analysis as mathematical models used to elaborate various methods for the determination of heat effects. It is important to define the conditions for correct use of this equation, indicating all simplifications and limitations. They can easily be recognized from the assumption made to transform the Fourier-Kirchhoff equation into the heat balance equation of a simple body. 

A review of the published papers leads to the conclusion that mathematical models which are particular forms of the general heat balance equation of second or higher order are frequently used in isoperibol and DSC calorimetry. When the mathematical model is known, the inverse filter method is used to determine the thermokinetics. 

Thermodynamics is too wide ranging and complex to be dealt with in detail here. We shall therefore confine ourselves to an outline of essential aspects relevant to calorimetry. Some basic mathematical knowledge is, however, needed to understand the thermodynamic background and the equations presented. For a more thorough study and a better understanding necessary for calorimetry, readers are referred to textbooks of thermodynamics or physical chemistry (Falk and Ruppel, 1976 Adkins, 1983 Zemansky and Dittman, 1997 Callen, 1985 Keller, 1977 Lebon, Jou, and Casas-Vazquez, 2008). 

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