Starting with differential rate law equation for a first order reaction, derive the integrated rate law equation for a first order reaction. How is it related to the rate constant?

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

R → P

\frac{- dA}{dt} = kA or \frac{- dA}{A} = k dt

On integrating, we get

\int_{}^{} - \frac{dA}{A} = k \int_{}^{} dt

or -In A = kt + C ..........1

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

When t = 0, A = [A]₀, where [A]₀ is the initial concentration of the reactant.
Therefore,above equation can be written as
ln [A]₀ = –k × 0 + C
ln [A]₀ = C
putting the value of C in equation (1), we get

or - In A = kt - In

A_{0}

\frac{A_{0}}{A}

= kt or

taking antilog both side, we get

A = A_{0}

e^{- kt}

(Units of K are s^-1).

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