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The steady state approximation, [1] occasionally called the stationary-state approximation or Bodenstein's quasi-steady state approximation, involves setting the rate of change of a reaction intermediate in a reaction mechanism equal to zero so that the kinetic equations can be simplified by setting the rate of formation of the intermediate equal to the rate of its destruction.
Moreover, if the system is given a fixed, finite input (i.e., a step), then any resulting oscillations in the output will decay at an exponential rate, and the output will tend asymptotically to a new final, steady-state value.
One of the simplest examples of such a system is the case of a bathtub with the tap open but without the bottom plug: [dubious – discuss] after a certain time the water flows in and out at the same rate, so the water level (the state variable being Volume) stabilizes and the system is at steady state. Of course the Volume stabilizing inside ...
The rate equation for the rate of formation of product P may be obtained by using the steady-state approximation, in which the concentration of intermediate A* is assumed constant because its rates of production and consumption are (almost) equal. [8] This assumption simplifies the calculation of the rate equation.
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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.
Steady-states can be stable or unstable. A steady-state is unstable if a small perturbation in one or more of the concentrations results in the system diverging from its state. In contrast, if a steady-state is stable, any perturbation will relax back to the original steady state. Further details can be found on the page Stability theory.
Consider the simple example where the catalyst associates with substrate A, followed by reaction with B to form product, P and free catalyst. Regardless of the approximation applied, multiple independent parameters (k 1, k −1, and k 2 in the case of steady-state; k 2 and K 1 in the case of pre-equilibrium) are required to define the system ...