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The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation; this assumes that the reaction goes to completion. For example, the integrated rate law for a first-order reaction is
In the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given. A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example:
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
that can be solved once we know the solution of the leading order equation. But we know that in this case we can use the adiabatic approximation. When () does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. Indeed, in this case we introduce the unitary transformation
Therefore, as q increases from 0 to 1, the solution U(x; q) of the zeroth-order deformation equation varies (or deforms) from the chosen initial guess u 0 (x) to the solution u(x) of the considered nonlinear equation. Expanding U(x; q) in a Taylor series about q = 0, we have the homotopy-Maclaurin series
Zero order reaction. Zero-order process (statistics), a sequence of random variables, each independent of the previous ones; Zero order process (chemistry), a chemical reaction in which the rate of change of concentration is independent of the concentrations; Zeroth-order approximation, an approximation of a function by a constant
It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation. When used as a method for advection equations, or more generally hyperbolic partial differential equations, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache.
It is also called propositional logic, [2] statement logic, [1] sentential calculus, [3] sentential logic, [4] [1] or sometimes zeroth-order logic. [ b ] [ 6 ] [ 7 ] [ 8 ] Sometimes, it is called first-order propositional logic [ 9 ] to contrast it with System F , but it should not be confused with first-order logic .