# Identification of the Kinetic Parameters of Thermal Micro-Organisms Inactivation
## 2 Basics of the Method  
In a closed system, the death of bacterial cells and their spores is considered the only irreversible process, and the rate of cell death in general is considered proportional to the actual number of living micro-organisms. Furthermore, it should be noted that the first-order kinetics of death does not always occur, especially for bacterial spores immediately after thermal exposure [2; 10; 18; 19; 22; 23]. To enhance the generality of the proposed approach, when the equation of the model is formulated, the power law of this dependence \(dF/dt\propto F^{p}\) is assumed. A mathematical model is used to describe the process of death of the micro-organisms. The model relates the rate of change \(F\) of the dimensionless (relative to the initial quantity of living cells) as a kinetic dependence of the \(p\)-th order where the inactivation rate constant is determined through the absolute temperature \(T\) in the form of an Arrhenius function  
\[\frac{dF}{dt}=-k_{d}F^{p}=-F^{p}\cdot A\cdot\exp\left(-\frac{E_{d}}{RT}\right),\;\;F(t=0)=F_{0}=1,\;\;R=8.314\;\frac{J}{K\cdot mol}, \tag{1}\]  
where: \(t\geq 0\) is time; \([t]\) = min or s;  
\(A\) is pre-exponential (frequency factor); \([A]\) = \([t]\); and  
\(E_{d}\) is the inactivation energy of the micro-organisms, determined by the relationship between the rate constant of death \(k_{d}\) against temperature \(T\) which is equivalent to the term activation energy of a chemical reaction, \([E_{d}]\) = \(]\cdot\cdot\cdot\cdot\)mol\({}^{-1}\). The initial experimental data are given in table \(\left\{t_{ij},F_{ij}\right\}\) (\(i\) = 0,..., \(N_{j}\)) as a dependence of the dimensionless number of living bacterial cells or spores in different time moments \(t_{ij}\) at a constant temperature \(T_{j}\) (\(j\) = 1,..., \(M\)) of biological culture. According to the physical meaning, the analyzed data should begin with the values \(F_{0j}\) = 1. It is worth mentioning that, according to the form of the dependence (1), it is similar to the Wigner-Polanyi equation for thermal desorption of molecules adsorbed on the solid-state surface [25; 27]. At constant temperature, the rate coefficient \(k_{d}\) = const, in such a way that, after separating the variables and integrating the right- and left-hand sides of Equation (1),  
\[\int\limits_{F_{0}}^{F}\frac{dF}{F^{p}}=-\int\limits_{0}^{t}k_{d}dt=k_{d}t, \tag{2}\]  
the time dependence for the dimensionless number of living micro-organisms is as follows:  
\[F(t)= \begin{cases}\exp(-k_{d}t),&\text{if $p=1$},\\ \left[1-(1-p)k_{d}t\right]^{1/(1-p)},&\text{if $p\
eq 1$}.\end{cases}  \tag{3}\]  
From Equation (3) can be expressed the value of the pre-exponential factor  
\[A=\frac{1}{t}\cdot\exp\left(\frac{E_{d}}{RT}\right)\cdot \begin{cases}-\ln(F), &\text{if $p=1$},\\ \left[1-F^{1-p}\right]/(1-p),&\text{if $p\
eq 1$}.\end{cases}  \tag{4}\]  
In accordance with Equation (4), for each range of values \(t_{0j}-t_{ij}\) of the initial data table \(\left\{t_{ij},F_{ij}\right\}\) (\(i\) = 0,..., \(N_{j}\)), \(T_{j}\) (\(j\) = 1,..., \(M\)), a set of constants \(A_{ij}\) can be calculated:  
\[A_{ij}=\frac{1}{t_{ij}}\cdot\exp\left(\frac{E_{d}}{RT_{j}}\right)\cdot \begin{cases} -\ln\left(F_{ij}\right),&\text{if $p=1$},\\ \left[1-F_{ij}^{1-p}\right]/(1-p),&\text{if $p\
eq 1$},\end{cases}  \tag{5}\]  
\[i=1,\ldots,N_{j}\;\;\;j=1,\ldots,M.\]It is assumed that the \"true\" values \(p,A,E_{d}\) of the triplet are reached when the set of calculated values \(A_{ij}\) has a minimum spread, since the pre-exponential factor \(A\) is a constant essentially.