Write the exponential decay law for the number of undecayed nuclei.

Write the differential equation that leads to radioactive exponential decay.

Define activity and state its SI unit.

Derive the relation between half-life and decay constant.

A sample has half-life \(6.0\,\mathrm h\). What fraction remains after \(18\,\mathrm h\)?

A sample has \(N=2.0\times10^{12}\) undecayed nuclei and \(\lambda=3.0\times10^{-6}\,\mathrm{s^{-1}}\). Find its activity.

A radioactive source has initial activity \(800\,\mathrm{Bq}\). After two half-lives, what is its activity?

If \(\lambda=0.020\,\mathrm{min^{-1}}\), find the half-life in minutes.

Show that activity follows the same exponential time dependence as \(N\).

A sample has \(12.5\%\) of its original undecayed nuclei. How many half-lives have elapsed?

Derive the mean lifetime \(\tau\) in terms of decay constant \(\lambda\).

A source has half-life \(10\,\mathrm s\). Find the probability that one nucleus decays within \(30\,\mathrm s\).

Explain why radioactive decay is random for one nucleus but predictable for a large sample.

A sample starts with \(N_0\) nuclei. Find the number that have decayed by time \(t\).

Why does exponential decay not have a fixed time at which the sample becomes exactly zero?

A source has \(A_0=4.0\times10^5\,\mathrm{Bq}\) and \(\lambda=1.0\times10^{-4}\,\mathrm{s^{-1}}\). Find its activity after \(1.0\times10^4\,\mathrm s\).

A parent isotope decays into a radioactive daughter. Why can daughter activity initially rise even while the parent decays?

For a short time interval \(\Delta t\), show that the decay probability of one nucleus is approximately \(\lambda\Delta t\).

A sample has two independent radioactive isotopes with activities \(A_1=A_{10}e^{-\lambda_1t}\) and \(A_2=A_{20}e^{-\lambda_2t}\). Why is the total activity generally not a single exponential?

Prove the exponential decay law by solving \(dN/dt=-\lambda N\).