Q:

How can radioactivity change depending on the number of nuclei?

I came across this problem. Please help me solve it.

A radioactive sample S1 having an activity of $5\ \mathrm{\mu Ci}$ has twice the number of nuclei as another sample S2 which has an activity of $10\ \mathrm{\mu Ci}$. The half lives of S1 and S2 can be
(A) 20 years and 5 years, respectively
(B) 20 years and 10 years, respectively
(C) 10 years each
(D) 5 years each

All through out my high school i was never introduced to the idea that radioactivity could change depending on the number of nuclei. Can someone throw some light on this?

A:

The decay rate is
$$\left|\frac{\mathrm{d}N}{\mathrm{d}t}\right| = \lambda N.$$
Half-life is
$$\tau_{1/2}=\frac{\ln 2}{\lambda}.$$
I think you can figure it now.

