Calorimetric signal equation

If the assumptions made above are not valid, and/or information about the rate constants of the investigated reactions is required, model-based approaches have to be used. Most of the model-based measurements of the calorimetric signal are based on the assumption that the reaction occurs in one single step of nth order with only one rate-limiting component concentration in the simplest case, this would be pseudo-first-order kinetics with all components except one in excess. The reaction must be carried out in batch mode (Vr = constant) in order to simplify the determination, and the general reaction model can, therefore, be written as Equation 8.14 with component A being rate limiting ... 

The integral of equation (27) provides a relationship between power and time, the calorimetric signal ... 

In this method, first used by Navarro et al. [230, 231] and by van Bokhoven [232, 233], the convolution recorded in the frequency domain is accepted as the transformation equation [Eq. (2.8)] of the calorimeter. For the determination of an unknown heat effect, it is assumed that the spectrum transmittance H(jheat effect Px(t) is generated and the calorimetric signal Tx(t) is measured. After determination of the spectrum transmittance H(jcalorimeter response Tft), the thermokinefics Px(t) is obtained as the inverse Fourier transform... 

In Equation (5), Q is the enthalpy change for the reaction and V is the volume of the solution. Integration of Equation (5) gives Equation (6) and shows the relationship between the calorimetric signal and time, for solution phase reactions. 

Having found the values for k, AH and , a simulated calorimetric signal can be constructed for the reaction over any time interval and the extent of reaction determined. The reaction half life (ti ) can be found from Equation (7) ... 

As a purely academic exercise, calorimetric data can be simulated using Equation (6). To this, different degrees of error can be added representing noise in the calorimetric signal. An investigation can then be made as to the degree of... 

In this equation, Wq and W[r are weighting factors that express the importance of the residuals obtained in the calorimetric and infrared determinations, respectively. The definition of these weightings is crucial for the results that are obtained, but is not at all straightforward. Recently, an approach to this problem based on an automated sensitivity analysis has been reported [17]. Besides tackling this problem of mathematically combining the evaluation of two different signals measured for the same experiment, we shall demonstrate in Section 8.3 that the application of both measurement techniques in parallel has synergistic effects for the clarification of the physical and chemical processes that are involved in the one experiment. 

It is not trivial to state how good a certain type of calorimeter or a certain type of calorimetric measurement is in terms of precision (or accuracy), as many factors (room temperature stability, reference balance, etc.) influence this. A problem is also that different errors influence different types of measurements in different ways. For example, an error in the calibration coefficient contributes the same error to 1 and 7 days measured heats but an error in the baseline is much worse for the 7 days heat as the thermal powers are low in the end of measurement. For a short (30 min) measurement of initial reactions, the baseline error is normally of no importance as very high thermal powers are measured more important is the time constant and how well the Tian equation can deconvolute the rapidly changing signal. 

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