The half-life of $Ra^{226}$ is $1620$ years. Calculate the number of atoms that decay in one second in $1 \ g$ of radium (Avogadro number $= 6.023 \times 10^{23}$).

$A$ radioactive material is reduced to $1/8$ of its original amount in $3$ days. If $8 \times 10^{-3} \, kg$ of the material is left after $5$ days,the initial amount of the material is $....... \, g$.

$A$ radioactive element emits $200$ particles per second. After $3$ hours,$25$ particles per second are emitted. The half-life period of the element will be .......... $minutes$.

The half-life of a radioactive sample is $T$. The fraction of the initial mass of the sample that decays in an interval $T / 2$ is

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620$ years and $810$ years. The time (in years) after which one-fourth of the material remains is:

Two radioactive materials $Y_1$ and $Y_2$ initially contain the same number of nuclei. Their decay constants are $9 \lambda \ s^{-1}$ and $6 \lambda \ s^{-1}$ respectively. The time after which the ratio of the number of undecayed nuclei of $Y_1$ and $Y_2$ becomes $\frac{1}{e}$ is: