If the half-life of a radioactive element is $12.5 \ h$,then the time taken to disintegrate $256 \ g$ of the substance into $1 \ g$ is (in hours)

$A$ radioactive sample of $U^{238}$ decays to $Pb$ through a process for which the half-life is $4.5 \times 10^9$ years. Find the ratio of the number of nuclei of $Pb$ to $U^{238}$ after a time of $1.5 \times 10^9$ years (given $2^{1/3} = 1.26$).

$A$ radioactive element of mass $1 \,kg$ after $N$ years is left with only $125 \,g$. If the half-life of the element is $12.5 \,y$, then the value of $N$ is:

If there is $0.1 \ mg$ of radioactive $Th^{234}$, how much of it will remain undecayed after $120 \ \text{days}$, given its half-life is $24 \ \text{days}$? (Answer in $\mu g$)

If the decay or disintegration constant of a radioactive substance is $\lambda$,then its half-life and mean life are respectively:

$A$ radioactive sample at any instant has its disintegration rate $5000$ disintegrations per minute. After $5$ minutes,the rate is $1250$ disintegrations per minute. Then,the decay constant (per minute) is (in $, \ln 2$)