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The second form of the inequality is the most useful, as the difference in $\mathcal{J}(\Delta t)$ and $\mathcal{L}(\Delta t)^2$ measures the variation in the rate coefficient. When the difference between these two quantities is zero, the rate coefficient is constant.   This inequality is derived from the first order rate law and survival function. In traditional kinetics, irreversible decay is only dependent on one rate coefficient, $\omega$, and the mechanism of the reaction. The rate law of a first order reaction of A irreversibly decaying into B is   \begin{equation}  \frac{-dA}{dt}=\omega[A] \frac{-dC_A(t)}{dt}=\omega[A]  \end{equation}  The time dependent rate coefficient, k(t), is determined by integrating the rate law of the reaction and forming a survival function from the integrated rate law. The survival function is simply a measure of a the cocentration of species at some time compared to its initital concentration.