Consider the first order reaction $R \to P$. The fraction of molecules decomposed in the given first order reaction can be expressed as:

For a first-order reaction $A \rightarrow \text{product}$, the rate constant is $k = 2 \times 10^{-2} \text{ s}^{-1}$. If the initial concentration of $A$ is $[A]_0 = 1.0 \text{ mol dm}^{-3}$, find the value of $\log \frac{1}{[A]_t}$ after $t = 100 \text{ s}$.

The following results are obtained in one pseudo first order reaction:

Time $(s)$	$0$	$30$	$60$	$90$
Concentration $(mol \ L^{-1})$	$0.551$	$0.312$	$0.173$	$0.085$

$(a)$ Calculate the average rate of reaction between $30$ and $60$ seconds.
$(b)$ Calculate the rate constant $(k)$ of this first order reaction.

Mathematical representation for $t_{1/4}$ life for a first-order reaction is given by:

Derive the half-life $t_{1/2}$ of a first-order reaction.

Assertion : For a first order reaction,$t_{1/2}$ is independent of initial concentration.
Reason : For a first order reaction,rate constant $k \propto [R]$.