Second Order Reaction

The rate of a second order reaction is proportional to either the concentration of a reactant squared, or the product of concentrations of two reactants.

For the general case of a reaction between A and B, such that



the rate of reaction will be given by
1. Initial concentrations of the two reactants are equal:

Equation (1) can be written as: Separating the variables and integrating gives:

Provided that [A] = [A]₀ at t = 0 the constant of integration C becomes equal to 1 / [A]₀.

Thus the second order integrated rate equation is


A plot of  1 / [A]   vs   t   produces a straight line with slope   k and intercept   1 / [A]₀  . The plot should be linear up to a conversion of about 50%.


2. Starting concentrations of the two reactants are different:

If [A]₀ and [B]₀ are different the variable x is used.

Equation (1) becomes



where [A]₀ - x = [A], [B]₀ - x = [B] and x is the decrease in the concentration of A and B.

Equation (5) can be integrated after separation of the variables and partial fraction expansion. The result is:

where C is the constant of integration.

Using the condition that x = 0, when t = 0, the value of C can be found



and equation (6) becomes
If [A]₀ > [B]₀, then a plot of

against t will have a positive slope, equal ([A]₀ - [B]₀) k.

If the experimental method yields reactant concentrations rather than x, the equivalent form of equation (8) is



Because equivalent amounts of A and B are reacting, [A] can be expressed in terms of [B].

If [B] = x , [A] = [A]₀ - (x₀ - x)

Provided that the initial concentration of A is twice the initial concentration of B - equation (10) becomes




Summary

Reaction Order	Differential Rate Law	Integrated Rate Law	Linear Plot	Slope of Linear Plot	Units of Rate Constant
0	- d[A] / dt = k	[A] = [A]0 - kt	[A]  vs  t	- k	mol · L-1 · s-1
1st	- d[A] / dt = k [A]	[A] = [A]0 e - kt	ln[A]  vs  t	- k	s-1
2nd	- d[A] / dt = k [A]2	1 / [A] = 1 / [A]0 + kt	1 / [A]  vs  t	k	L · mol-1 · s-1

 

 

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