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If the concentration of a reactant remains constant (because it is a catalyst, or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, leading to a pseudo–first-order (or occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate ...
Consider , the exact solution to a differential equation in an appropriate normed space (, | | | |). Consider a numerical approximation u h {\displaystyle u_{h}} , where h {\displaystyle h} is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method .
[a] The bass and treble controls in a hi-fi system are each a first-order filter in which the balance of frequencies above and below a point are varied using a single knob. A special case of first-order filters is a first-order high-pass or low-pass filter in which the 6 dB per octave cut of low or high frequencies extends indefinitely.
A first-order filter's response rolls off at −6 dB per octave (−20 dB per decade) (all first-order lowpass filters have the same normalized frequency response). A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on.
In fact, however, the observed reaction rate is second-order in NO 2 and zero-order in CO, [5] with rate equation r = k[NO 2] 2. This suggests that the rate is determined by a step in which two NO 2 molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate ...
The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess y 2 ( x ) = v ( x ) y 1 ( x ) {\displaystyle y_{2}(x)=v(x)y_{1}(x)} where v ( x ) {\displaystyle v(x)} is an unknown function to be determined.
For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order. We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps.
The steady-state rate equation is of mixed order and predicts that a unimolecular reaction can be of either first or second order, depending on which of the two terms in the denominator is larger. At sufficiently low pressures, k − 1 [ M ] ≪ k 2 {\displaystyle k_{-1}[{\ce {M}}]\ll k_{2}} so that d [ P ] / d t = k 1 [ A ] [ M ...