# Methods for Determination of Order of Reaction

## Methods for Determination of Order of a Reaction

Following methods can be use to determine the order of reaction-1. Differential Method or Initial Rate Method

2. Graphical Method

3. Half Life Method

4. Van't Hoff Differential Method

## Differential Method

It is also called initial rates method. In this method concentration of one reactant varies while others are kept constant concentration and initial rate of reaction is determined. Suppose if three reactants A, B and C are taking part in the reaction then in this method we keep vary concentration of one reactant (for example reactant A) while concentration of other reactants such B and C constant.

## Graphical Method

This method can be used when there is only one reactant.

a. If the plot of log [A] vs time 't' is a straight line, the reaction follows first-order.

b. If the plot of 1/[A] vs time 't' is a straight line, the reaction follows second order.

c. If the plot of 1/[A]^{2} is a straight line , the reaction follows third order.

d. Generally, for a reaction of nth order, a graph of 1/[A]^{n-1} vs time 't' must be a straight line.

## Half Life Method

This method is used only when the rate law involved by only one concentration term.

t_{1/2} ∝ a^{1 − n}

or, t_{1/2} = K. 1/a^{n − 1}

or, log t_{1/2} = logK + (1 − n)a

Graph of *log t _{1/2} vs log a*, gives a straight line with slope (1-n) , where 'n' is the order of the reaction.Determining the slope we can find the order 'n' of reaction.

## Van't Hoff Differential Method

The differential rate equations for different order of reactions are-

1. dx/dt = k(a-x)^{0} for zero order reactions.

2. dx/dt = k(a-x)^{1} for first order reactions.

3. dx/dt = k(a-x)^{2} for second order reactions.

4. dx/dt = k(a-x)^{3} for third order reactions.

So for nth order reaction, the rate equation is-

dx/dt = k(a-x)^{n}

Let (a-x) = 'c' at any instant

− dx/dt = kc^{n}

For two different concentrations c_{1} and c_{2} of the reactants, we have-

− dc_{1}/dt = kc_{1}^{n} -----equation-1

− dc_{2}/dt = kc_{2}^{n} -----equation-2

taking log on both sides we get-

log(− dc_{1}/dt) = logk + nlog c_{1} -----equation-3

log(− dc_{2}/dt) = logk + nlog c_{2} -----equation-4

subtracting equation-4 from equation-3 we get-

nlog c_{1} − nlog c_{2} = log(− dc_{1}/dt) − log(− dc_{2}/dt)

n(log c_{1} − log c_{2}) = log(− dc_{1}/dt) − (− dc_{2}/dt)

n = [log(− dc_{1}/dt) − (− dc_{2}/dt)]/(log c_{1} − log c_{2})

The rates of reactions at two different concentrations can be calculated from the slopes of 'c' vs 't' plots. Substituting these values in the above equation, the order of the reaction can be determined.