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)

The half-life of a radioactive substance is $100 \ s$. How many grams of the radioactive substance will remain after $5 \ min$ if the initial amount is $8 \ g$ (in $g$)?

$A$ radioactive substance has a half-life of $10^8$ years and an activity of $10^4 \,Bq$. The number of atoms of this substance present is

$A$ sample initially contains only $U-238$ isotope of uranium. With time,some of the $U-238$ radioactively decays into $Pb-206$ while the rest of it remains undisintegrated. When the age of the sample is $P \times 10^8$ years,the ratio of the mass of $Pb-206$ to that of $U-238$ in the sample is found to be $7$. The value of $P$ is. . . . . . [Given: Half-life of $U-238$ is $4.5 \times 10^9$ years; $\log_e 2 = 0.693$]

$1 \, mg$ of gold undergoes decay with a $2.7 \, \text{days}$ half-life period. The amount left after $8.1 \, \text{days}$ is ......... $mg$.

The half-lives of a radioactive substance are $T$ and $2T$ years for $\alpha$-emission and $\beta$-emission,respectively. The total decay constant for the simultaneous decay of the $\alpha$ and $\beta$ radioactive substance is: