Question

Define the term decay constant of radioactive metals. two metal P.Q have equal number of atoms at t=0. their half-lives are 3 hours and 9 hours respectively. Compare their rates of disintegration after 18 hours from the start.

Answer 1

Decay constant of radioactive element. The reciprocal of time during which number of atoms left in simple radius to $\frac{1}{e}$ times. The number of atoms in the origin sample.

Given that Two number $P$ and $Q$

$P$ and $Q$ have equal number of atoms at $t=0$

half time of $P=3$

half time of $Q=9$ and $t=18$

$\therefore$ Number of half times of $P$ in $18$ hours $\frac{6}{T_{1/2}}=\frac{18}{3}=6$

Number of radius $P$ left decayet after $6$ half time in

$N_1=N{\left(\frac{1}{2}\right)}^6$

Number of half time of $Q$ in $18$ hours $=\frac{t}{T_{1/2}}=\frac{18}{9}=2$

Number of nucle of $Q$ left undecayed after $2$ half time in

$N_2=N{\left(\frac{1}{2}\right)}^2$

$\therefore \frac{R_1}{R_2}=\frac{{\lambda }_1{N}_1}{{\lambda }_2{N}_2}=\frac{T_2{N}_1}{T_1{N}_2}[\because R=\lambda NabT=\frac{0.693}{h}]$

$\frac{R_1}{R_2}=\frac{9}{3}\times \frac{N{\left(\frac{1}{2}\right)}^6}{N{\left(\frac{1}{2}\right)}^3}=\frac{3}{16}$

$\therefore R_1{R}_2=3:16$