A detector records \(120\,\mathrm{s^{-1}}\) with background \(35\,\mathrm{s^{-1}}\). Find the net count rate.

A detector has efficiency \(0.25\) and net count rate \(80\,\mathrm{s^{-1}}\). Estimate the source activity.

A source count rate falls from \(1600\,\mathrm{s^{-1}}\) to \(200\,\mathrm{s^{-1}}\) in \(9.0\,\mathrm h\), after background correction. Find the half-life.

A sample has remaining parent fraction \(f=0.30\) and decay constant \(\lambda\). Write its age.

A carbon sample has \(25\%\) of the original \(^{14}\mathrm C\) ratio. If \(t_{1/2}=5730\,\mathrm{yr}\), estimate its age.

A source has count rate \(500\,\mathrm{s^{-1}}\), background \(100\,\mathrm{s^{-1}}\), and efficiency \(0.40\). Find the activity.

A count is taken for \(60\,\mathrm s\) and records \(3000\) counts. Background is \(15\,\mathrm{s^{-1}}\). Find the net count rate.

Given molar mass \(M\), sample mass \(m\), and decay constant \(\lambda\), write the activity of a pure radioactive sample.

A \(2.0\,\mathrm{mg}\) pure sample has molar mass \(200\,\mathrm{g\,mol^{-1}}\) and \(\lambda=1.0\times10^{-5}\,\mathrm{s^{-1}}\). Find its activity.

Explain why count-rate dating must use background-corrected count rates.

A source has net count rate \(C_0\) initially and \(C\) later. Derive the age in terms of half-life.

A sample's corrected count rate is \(0.18\) of its original value. Express its age in half-lives.

Why does detector efficiency affect inferred activity but not the measured half-life if efficiency stays constant?

A detector has a dead-time loss that becomes important at high rates. How would this affect a half-life measurement from early high-rate data?

A parent isotope has half-life \(T\). After what time is \(90\%\) of it decayed?

A sample contains a long-lived isotope and a short-lived contaminant. How can a plot of \(\ln C\) versus \(t\) reveal the mixture?

A decay curve is plotted as \(\ln(C-C_b)\) against \(t\). What is the slope, and how does it give half-life?

In radiometric dating, why is a closed-system assumption necessary?

A parent decays to a stable daughter with no initial daughter present. Express the daughter-to-parent ratio in terms of \(\lambda t\).

Derive the age formula \(t=(1/\lambda)\ln(1+D/N)\) for parent-daughter dating with no initial daughter.