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Rate Laws for Common Reaction Orders

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Nomenclature

[A]t	concentration of reactant A at any time t
[A]0	initial concentration of A
k	rate constant

Details

Graphs of concentration, natural logarithm of concentration, and 1 over concentration as a function of time can be used to find the order of a reaction with respect to a reactant A. This method is based on integrated rate laws. Calculus is necessary to derive integrated laws from the rate laws relating rate and concentration. However, calculus is not required in order to apply the integrated rate laws. For a first-order reaction, the integrated rate law is:

(Eq1)

ln [A]t [A]0  =  –kt

The integrated rate law for a first-order reaction can also be written in the form:

ln [A]_t = –kt + ln [A]₀

Eq2 is of the form y = mx + b; that is, it is the equation of a straight line of slope –k and y-intercept, ln [A]₀. The following table summarizes information about rate laws for the common reaction orders:

Order with Respect to A	Rate Law	Integrated Rate Law	Graph of following quantity vs. t is a straight line	Slope of linear graph equals
0	rate = k	[A]t = –kt + [A]0	[A]t	–k
1	rate = k[A]	ln [A]t = –kt + ln [A]0	ln [A]t	–k
2	rate = k[A]2	1 [A]t  =  kt + 1 [A]0	1 [A]t	k