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

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

Two different radioactive elements with half-lives $T_1$ and $T_2$ have undecayed atoms $N_1$ and $N_2$ respectively present at a given instant. The ratio of their activities at that instant is

$A$ heavy nucleus $Q$ of half-life $20 \text{ minutes}$ undergoes alpha-decay with a probability of $60 \%$ and beta-decay with a probability of $40 \%$. Initially,the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

The half-life of a radioactive sample undergoing $\alpha$-decay is $1.4 \times 10^{17} \; s$. If the number of nuclei in the sample is $2.0 \times 10^{21}$,the activity of the sample is nearly:

$A$ radioactive substance has decay constants $\lambda_{\alpha}$ and $\lambda_{\beta}$ for $\alpha$ and $\beta$ emission,respectively. If the substance emits both $\alpha$ and $\beta$ particles,find the effective half-life of the substance.