Derive an integrated rate equation for the rate constant of a first-order reaction.

Explanation

Consider a general first order reaction
`R rarr P`
The differential rate equation rate equation for above reaction can be written as
`"Rate "=-(d[R])/(R )=K[R]^(1)`
`=(d[R])/(R )=-Kxxdt`
Integrating on both sides of the above equation
`int(d[R])/(R )=-k intdt`
`lnR=-Kt+1" ...(1)"`
Where `1rarr` Intigration constant
At `t=0` the concentration of reactant `R=[R]_(0)` where `[R]_(0)` is initial concentration of reactant substituting in equation (1) we get
`ln[R]_(0)=(-Kxx0)+I`
`ln[R]_(0)=1" ...(2)"`
Substitute I value in equation (1)
`In[R]=-Kt+ln[R]_(0)`
`Kt=ln[R]_(0)-lnR`
`KT=ln""(R_(0))/(R )`
`or K=(2.303)/(t)log""([R]_(0))/(R )`