The half-life of ${}^{198} {Au}$ is $3 \, \text{days}$. If the atomic weight of ${}^{198} {Au}$ is $198 \, \text{g/mol}$, then the activity of $2 \, \text{mg}$ of ${}^{198} {Au}$ is ..... $\times 10^{12} \, \text{disintegration/second}$.

The half-life of a sample of a radioactive substance is $1 \text{ hour}$. If $8 \times 10^{10}$ atoms are present at $t = 0$,then the number of atoms decayed in the duration $t = 2 \text{ hours}$ to $t = 4 \text{ hours}$ will be

$A$ piece of bone of an animal from a ruin is found to have $^{14}C$ activity of $12$ disintegrations per minute per gm of its carbon content. The $^{14}C$ activity of a living animal is $16$ disintegrations per minute per gm. How long ago nearly did the animal die? (Given half-life of $^{14}C$ is $t_{1/2} = 5760$ years)

The radioactivity of a sample at time $T_1$ is $R_1$ and at time $T_2$ is $R_2$. If the mean life of the sample is $T$,then the number of nuclei disintegrated in the time interval $(T_2 - T_1)$ is:

$A$ sample of a radioactive element has a mass of $10 \, g$ at an instant $t = 0$. The approximate mass of this element in the sample after two mean lives is .......... $g$.

What is the relationship between the decay constant $\lambda$ and the half-life $\tau_{1/2}$?