# Integrated Rate Equations

The integrated rate equations are different for the reactions of different reaction orders. We shall determine these equations only for zero and first order chemical reactions.

## Zero Order Reaction

The rate of the reaction is proportional to zero power of the concentration of reactants.

As any quantity raised to power zero is unity

Integrating both sides

Where, I is the constant of integration.

At t = 0, the concentration of the reaction R = [R]_{0} , where [R]_{0} is initial concentration of the reaction.

Substituting in equation,

Substituting the value of I

Example:

- The decomposition of gaseous ammonia on a hot platinum surface at high pressure.

- Thermal decomposition of HI on a gold surface.

## First Order Reaction

The rate of the reaction is proportional to the first power of the concentration of the reactant R.

Integrating this equation, we get

Again, I is the constant of integration and its value can be determined easily.

When t= 0, R= [R]_{0}, where [R]_{0} is the initial concentration of the reactant.

Therefore, equation can be written as

Substituting the value of I in equation

Rearranging this equation

At time t_{1} from above eq.

Where, [R]_{1} and [R]_{2} are the concentrations of the reactants at time t_{1} and t_{2} respectively.

Subtracting

Comparing equation (2) with y = mx +c, if we plot In[R] against t, we get a straight line with slope = -k and intercept equal to ln[R]_{0}

The first order rate equation (3) can also be written in the form

Example:

- Hydrogenation of ethane,

- Decomposition of N
_{2}O_{5}and N_{2}O