Question

Derive the mathematical expression for law of radioactive decay for a sample of radioactive nucleus

Answer 1

Answer: Radioactive decay follows first order kinetics:

$A\rightarrow Products$

$rate=k[A]$     (1)

also $rate=-\frac{d[A]}{dt}$    (2)

equating 1 and 2

$k[A]=-\frac{d[A]}{dt}$  

$k\times dt=-\frac{d[A]}{[A]}$  

Integrating both sides and putting the limits for time as t=0 to t=t and for concentration from A=$[a_0]$to A=a

$k(t-0)={ln[A_0]-ln[A]$

Rate law expression for first order kinetics is given by:

$k=\frac{2.303}{t}\log\frac{[a_0]}{[a]}$

where,

k = rate constant

t = time taken for decay process

$a_0$ = initial amount of the reactant

$a$ = amount left after decay process