Question

One gram of radium is reduced by $2mg$ in $5$ years by $\alpha$- decay. The half life of radium is:
(log${}_{10}998=$ $2.9991$ )

• A. $1520.4$ years
• B. $1671.7$ years
• C. $1617.5$ years
• D. $1732.86$ years

Answer 1

The correct option is D $1732.86$ years

$N_0=$ One gram $=$ initial amount

$N=$ Final amount $=998gm$

$t=5yrs$

$\frac{dN}{dt}=-\lambda N$

$ln\frac{N}{N_0}=-\lambda t$

$ln\frac{1000}{998}=\lambda \times 5$

$⟹\lambda =4\times {10}^{-4}$

$t_{1/2}=\frac{0.693}{4}\times {10}^4$

$t_{1/2}=1732.86yrs$