Abstract
The quantities r k α, R k α, R k , \(\tilde{R}_{k}\), \(\bar{R}_{k}\) or \(\bar{\tilde{R}}_{k}\) that appear in the species mass transport equations (3.50), (3.51), (3.248) and (4.71) and those of Tables 3.5, 3.7, 3.9 3.11 and 4.6 represent rates of production of mass of chemical species k due to chemical reactions occurring within a phase α, termed as homogeneous reactions, or between two or more phases, termed as heterogeneous reactions.
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Notes
- 1.
Referring to radioactive decay processes the decay rate \(\vartheta _{k}\) is frequently expressed in terms of a reaction half-life t 1∕2k of species k, which is a specific solution of the reaction equation
$$\displaystyle{\frac{dC_{k}} {\mathit{dt}} = -\vartheta _{k}C_{k}}$$applied to a simple chemical batch reaction (without diffusion/dispersion and advection). Its analytical solution yields
$$\displaystyle{C_{k} = C_{0k}{e}^{-\vartheta _{k}t}}$$The time t for the concentration C k to decrease from the initial concentration C 0k to its half value \(\tfrac{1} {2}C_{0k}\) corresponds to the half-life t 1∕2. From above it results
$$\displaystyle{t_{1/2k} = \frac{\ln 2} {\vartheta _{k}}}$$where ln2 = 0. 693 is the natural logarithm of 2. Accordingly, the decay rate \(\vartheta _{k}\) can be expressed by
$$\displaystyle\begin{array}{rcl} \vartheta _{k} = \frac{\ln 2} {t_{1/2k}}& & {}\\ \end{array}$$where the half-life t 1∕2k has to be specified for a given (radioactive) species k.
- 2.
Considering the Michaelis-Menten reaction rate in the form \(\hat{R}_{A} = v_{m}C_{A}^{l}/(K_{m} + C_{A}^{l})\):
-
(i)
If C A l is large compared to K m then \(C_{A}^{l}/(K_{m} + C_{A}^{l}) \approx 1\) and the reaction rate becomes
$$\displaystyle{\hat{R}_{A} \approx v_{m}}$$ -
(ii)
If \(C_{A}^{l} = K_{m}\) then \(C_{A}^{l}/(K_{m} + C_{A}^{l}) = \tfrac{1} {2}\) and the reaction rate gives
$$\displaystyle{\hat{R}_{A} = \tfrac{1} {2}v_{m}}$$ -
(iii)
If If C A l is small compared to K m then \(C_{A}^{l}/(K_{m} + C_{A}^{l}) \approx C_{k}^{l}/K_{m}\) and it is
$$\displaystyle{\hat{R}_{A} = \frac{v_{m}} {K_{m}}C_{k}^{l}}$$
-
(i)
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Diersch, HJ.G. (2014). Chemical Reaction. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_5
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