In one average lifetime of a radioactive nucleus,

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 $= 6.023 \times 10^{23}$).

$A$ sample originally contained $10^{20}$ radioactive atoms,which emit $\alpha$-particles. The ratio of $\alpha$-particles emitted in the third year to that emitted during the second year is $0.3$. How many $\alpha$-particles were emitted in the first year?

Half-life of a substance is $10$ years. In what time,it becomes $\frac{1}{4}$ of the initial amount?

The half-life of a radioactive substance is $3.6$ days. How much of $20\, mg$ of this radioactive substance will remain after $36$ days?

The radioactivity of a sample is $R_1$ at a time $T_1$ and $R_2$ at a time $T_2$. If the half-life of the specimen is $T$,the number of atoms that have disintegrated in the time $(T_2 - T_1)$ is proportional to