Relationships between time and concentration

The rate law(equation) relates rate to concentration of reacting species (such as substance A).

Second order rate equation
reaction rate = –	Δ[A]	= k[A]²
	Δt

First order rate equation
reaction rate = –	Δ[A]	= k[A]
reaction rate = –	Δt	= k[A]

Zero order rate equation
reaction rate = –	Δ[A]	= k
reaction rate = –	Δt	= k

The relationship between time and concentration of substance A is also important because time is more easily measured than rate. This can be established by integrating each rate law. Integration is a mathematical process.

Integrated rate equations have a linear form (y = mx + b) where, as shown below, y is a function of [A] that depends on the order in A, x is time, m (the slope) is either +k or –k depending on the order and b is a function of the initial concentration of A.

Zero order
integrated rate equation:
[A] = –kt + [A]₀ y = [A]
x = t
m = slope = -k

First order
integrated rate equation:
ln[A] = -kt + ln[A]₀

y = ln[A]
x = t
m = slope = -k

Second order
integrated rate equation:

1	=	kt	+	1
[A]				[A]₀

y = 1/[A]
x = t
m = slope = k

[A] against t is linear (constant slope), the reaction is zero order in A.
ln[A] against t is linear, the reaction is first order in A.
1/[A] against t is linear, the reaction is second order in A.
The rate constant k can be determined from the slopes of the lines.

Thus if a plot of