 # Second Order Reaction

The sum of power of concentration of reactants in the rate law expression is called the order of that chemical reaction. Reactions can be first order reaction, second order reaction, pseudo first order reaction etc. depends on the concentration of the reactants. In this article we will discuss about second order reactions in detail.

Suppose a reaction is – aA + bB 🡪 cC + dD

Rate according to rate law expression = k [A]x [B]y

Where x and y are concentrations of A and B respectively.

So, order of reaction will be = x + y

We can say x is the order of reaction with respect to A and y is the order of reaction with respect to B.

Now if suppose x=1 and y = 1 then the reaction will be 2nd order reaction. Reactions in which reactants are identical and form a product can also be second order reactions.

Many reactions such as decomposition of nitrogen dioxide, alkaline hydrolysis of ethyl acetate, decomposition of hydrogen iodide, formation of double stranded DNA from two strands etc. can be explained by second order kinetics.

## What is Second Order Reaction?

A reaction is called a second order reaction when the overall order is two. Suppose if the reaction is as follows –

A + A 🡪 P

Or 2A 🡪 P

In these reactions rate is proportional to the square of the concentration of one reactant. The differential rate law for the above second order reaction can be written as follows –

Rate of such reactions can also be written as r = k[A]2

Here k is rate constant for second order reaction. Unit of reaction rate (r) is moles per liter per second (mol.L-1.s-1) and the unit of second order rate constant is M-1.s-1 (M is molarity which can be expressed as mol/L).

If both the reactants are different in the reaction –

A + B 🡪 P

Rate for the above reaction can be written as follows –

R = k[A]x[B]y

Where the sum of x and y is equal to two.

## Examples of Second Order Reactions

Few examples of second order reaction are given below –

• Nitrogen dioxide decomposes into nitrogen monoxide and oxygen. Reaction is given below-

2NO2 🡪 2NO + O2

• Decomposition of hydrogen iodide – Hydrogen iodide breaks down into iodine and hydrogen. Reaction is given below –

2HI 🡪 I2 + H2

• Decomposition of nitrosyl bromide –

2NOBr 🡪 2NO + Br2

• Hydrolysis of an ester in presence of a base –

CH3COOC2H5 + NaOH 🡪 CH3COONa + C2H5OH

• Combustion Reaction –

O2 + C 🡪 O + CO

## Integrated and differential Rate Equation for Second Order Reactions

We are considering here that equation where chemical reaction can be represented as follows –

A + A 🡪 P _ _ _ _ _ (1)

Generally, polymerization reactions follow the same as in them two monomer units combine and form a polymer.

The differential rate law equation for the chemical equation (1) can be written as follows –

(Image to be added soon)_ _ _ _ _ (2)

On rearranging the above equation (2), we get –

(Image to be added soon) _ _ _ _ _ (3)

On integrating the above equation (3) considering that concentration of the reactant changes between time 0 and time t, we get –

(Image to be added soon) _ _ _ _ _(4)

Applying the power rule of integration in equation (4), we get –

On simplifying equation (5), we get –

Equation (6) is the required integrated rate expression of second order reactions.

## Second Order Reaction Graph

On rearranging the equation (6), we get –

On comparing equation (7) with straight line equation or linear equation y = mx + c, we can write –

Y = 1/[A]t (on y-axis)

X = t (on x-axis)

m = k (Slope)

c = 1/[A]0 (Intercept)

so, graph can be drawn as follows –

It is clear from the graph that slope is equal to the value of rate constant k.

## Half life of Second Order Reactions

The amount of time required by reactant/s in a reaction for undergoing decay by half is called half life of that reaction. In the same way the amount of time required by reactant/s to undergo decay by half in second order reaction is called half life of second order reaction. So, while calculating the half life of a reaction t becomes t1/2 and as t=t1/2 then [A]t becomes [A]0/2.

Now putting the values of t and [A] in equation (6), we get –

On solving equation (8), we get –

On simplifying equation (9), we get –

Equation (11) is the equation for half life of second order reaction.

As we can see t1/2 is inversely proportional to the concentration of the reactant in second order reactions. Graph is given below for half life of second order reactions which is drawn between [A] and t.