**Integrated rate equation for ****zero order reaction**:

A reaction is said to be of zero order, if its rate is independent of the concentration of the reactants. Consider the general zero order reaction:

A ———-> Product

Let [A] be the concentration of the reactant **A** and** k** is the rate constant for the zero order reactions. For the zero order reaction, the rate of reaction is independent of the concentration of **A. **Then,

– d[A] / dt = k [A]° = k

Rearranging the above equation,

– d[A] = k dt

Integrating the above equation, we get

– ʃd[A] = k ʃdt

– [A] = kt + I —————— (1)

Where, I = integration constant.

If t = 0, then [A] = [A]_{o}. Therefore above equation becomes

– [A]_{o} = k x 0 + I

– [A]_{o} = I

Substituting the value of I in expression (1), we get

— [A] = kt – [A]_{o }

Or, kt = [A]_{o }— [A]

Above equation is called Integrated rate equation of zero order reaction Where [A]_{o} is the initial concentration of A. [A] is the concentration at time **t.**

Note: The above expression is a straight line equation, thus the value of** k **can be calculated graphically. On plotting a graph of **[A] **against** t**, we get a straight line and the slope of the line is given as Slope= – k

Fig: A plot of [A] against time (t).

**Integrated rate equation for first order reaction: **

Let us consider a general reaction of first order as-

A ———> Product

If the initial concentration of A is [A]_{o}, k is the rate constant and [A] is concentration at time** t**, then the differential form of first order reaction can be written as-

– d [A] / dt = k [A]

Rearranging the above equation, we have

– d[A]/ [A]= k dt

Integrating the above equation, we get

Or, – In [A] = kt + I ——————– (2)

Where, I = integration constant.

If t = 0, then [A] = [A]_{o}. Therefore above equation becomes

-In [A]_{o} = k x 0 + I

Or, -In [A]_{o }= I

Substituting the value of I in expression (2), we get

-In [A] = kt – In [A]_{o}

Or, kt = In [A]_{o} – In [A]

Changing it to common log, we get

Note: The value of **k **also be calculated graphically by plotting log [A] against time (t) as –