Rate equation for first order reaction

 First order reaction :The rate of the reaction is proportional to the first power of the cocentration of the reactant R is called first order reaction .

The rate of first order reaction.

The rate of first order reaction ∝ [R]

Where ,reaction R →P is first order reaction.

The differential rate expression of first order reaction is as under

Rate =−d[R]/dt=k[R]..........(i)

∴d[R]/[R]=-kdt.......(ii)

Intergrating this equation both side ,

In [R] =-kt+1 .......(iii)

Here,I=Constant of integration

When ,t= 0, [R]=[R]°

 .Where [R]0 is the initial concentration of the reactant.Put this value in eq. (iii)

In [R]°=−k×(0)+I

∴In[R]0=I.......(iv)

If we put I =In [R]° in equation (iii),we get

In [R]=-kt+In [R]°......(VA)

∴

 kt =In [R]°-In [R]

∴kt=In[R]°/[R]

∴k=1/t×In[R]°/[R].......(VB)

Equation V(A) and V(B) both are integrated rate equation (VA) and (VB) can written in the form

log [R]=−kt/2.303+log[R]°......(ViA)

K=2.303/t xlog [R]°/[R].....(VIB)