The half-life of radium is $1620$ years and its atomic weight is $226 \, g/mol$. What is the number of atoms decaying per second in a $1 \, g$ sample? $[N_A = 6.023 \times 10^{23} \, \text{atoms/mol}]$

The half-life of a radioactive sample is $20 \ days$. This means that:

Decay constants of two radioactive samples $A$ and $B$ are $15x$ and $3x$ respectively. They have an equal number of initial nuclei. The ratio of the number of nuclei left in $A$ and $B$ after a time $t = \frac{1}{6x}$ is

Ten percent of a radioactive sample has decayed in $1$ day. After $2$ days,the decayed percentage of nuclei will be ...... $\%$

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:

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. $A$ rock initially contained $68 \times 10^{-6} \text{ g}$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$,then what is the value of $Z$?