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Chemistry; Biology; Geology ... Exponential response formula; ... stable if their convergence is bounded by exponential decay. Exponential stability is a form of ...
The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations.
In chemistry, chemical stability is the thermodynamic stability of a chemical system, in particular a chemical compound or a polymer. [ 1 ] Thermodynamic stability occurs when a system is in its lowest energy state , or in chemical equilibrium with its environment.
An exponential decay can be described by any of the following four equivalent formulas: [6]: 109–112 = / = / = = where N 0 is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles , number of atoms, etc.),
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form / =, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part.
The exponential function in parentheses corresponds to the fraction of total change that has been achieved as time passes and the difference between C ss and C 0 equals the total amount of change. Finally, at steady state, the concentration is expected to equal the rate of synthesis, production or infusion divided by the first-order elimination ...
where A is the pre-exponential factor for the reaction, R is the universal gas constant, T is the absolute temperature (usually in kelvins), and k is the reaction rate coefficient. Even without knowing A, E a can be evaluated from the variation in reaction rate coefficients as a function of temperature (within the validity of the Arrhenius ...
All chemical transformations pass through an unstable structure called the transition state, which is poised between the chemical structures of the substrates and products. The transition states for chemical reactions are proposed to have lifetimes near 10 −13 seconds, on the order of the time of a single bond vibration. No physical or ...