E.3 Radioactive decay：練習問題

A student plots a graph of the natural logarithm of activity, \(\ln(A)\), against time \(t\) in minutes for a radioactive sample. The plot is a straight line with a gradient of \(-0.023\text{ min}^{-1}\) and a vertical axis intercept of \(4.60\). Determine the initial activity \(A_0\) and the half-life \(T_{1/2}\) of the sample.

A radioactive source has a decay constant of \(\lambda = 0.0125\text{ s}^{-1}\). What is the approximate half-life of this source?

A sample initially contains only radioactive isotope X, which decays into a stable isotope Y. The half-life of X is \(T\). If at a later time \(t\), the ratio of the number of nuclei of Y to the number of nuclei of X is \(7:1\), what is the value of \(t\)?

A radioactive sample has an initial activity of \(1600\text{ Bq}\). Given that the half-life of the isotope is \(4\text{ hours}\), what will be the activity of the sample after \(12\text{ hours}\)?

For a radioactive sample, a student plots a graph with Activity \(A\) on the vertical axis and the number of undecayed nuclei \(N\) on the horizontal axis. Which statement correctly identifies the feature of this graph?

A sample of a radioactive isotope has a half-life of $$2.0$$ hours. If the initial activity of the sample is $$1600\,Bq$$, what will its activity be after $$6.0$$ hours?

A sample of organic material is found to have a carbon-14 activity of \(0.240\text{ Bq}\) per gram of carbon. Given that the activity in living tissue is \(0.255\text{ Bq}\) per gram and the half-life of carbon-14 is \(5730\text{ years}\), calculate the age of the sample in years.

A sample contains Phosphorus-32 (\({}^{32}_{15}\text{P}\)), which undergoes beta-minus decay with a half-life of 14.3 days.
(a) Define the decay constant, \(\lambda\).
(b) Calculate the decay constant for \({}^{32}\text{P}\) in units of \(\text{s}^{-1}\).
(c) If the initial activity of the sample is \(4.0 \times 10^6\text{ Bq}\), determine the number of radioactive nuclei remaining after 30 days.

A pure sample of polonium-210 (\({}^{210}_{84}\text{Po}\)) has an initial mass of \(2.0\text{ mg}\). Polonium-210 has a half-life of \(138\text{ days}\).
(a) Calculate the initial activity of the sample in Becquerels (Bq).
(b) Determine the time required for the activity to fall to \(10\%\) of its initial value.
(c) Identify the daughter nucleus produced when polonium-210 undergoes alpha decay.