Chapter 41

Using the uncertainty principle and the radius of a nucleus, estimate the minimum possible kinetic energy of a nucleon in, say, iron. Ignore relativistic corrections. [Hint: A particle can have a momentum at least as large as its momentum uncertainty.

The nuclide \({ }_{76}^{191}\) Os decays with \(\beta^{-}\) energy of 0.14 MeV accompanied by \(\gamma\) rays of energy \(0.042 \mathrm{MeV}\) and \(0.129 \mathrm{MeV}\) (a) What is the daughter nucleus? (b) Draw an energy-level diagram showing the ground states of the parent and daughter and excited states of the daughter. (c) To which of the daughter states does \(\beta^{-}\) decay of \({ }_{76}^{191}\) Os occur?

The nuclide \(\frac{191}{76} \mathrm{Os}\) decays with \(\beta^{-}\) energy of 0.14 MeV accompanied by \(\gamma\) rays of energy 0.042 MeV and 0.129 MeV. \((a)\) What is the daughter nucleus? \((b)\) Draw an energy-level diagram showing the ground states of the parent and daughter and excited states of the daughter. (c) To which of the daughter states does \(\beta^{-}\) decay of \(\frac{191}{76} \mathrm{Os}\) occur?

Determine the activities of \((a) 1.0 \mathrm{~g}\) of \({ }_{53}^{131} \mathrm{I}\left(T_{\frac{1}{2}}=8.02\right.\) days \()\) and \((b) 1.0 \mathrm{~g}\) of \({ }_{92}^{238} \mathrm{U}\left(T_{\frac{1}{2}}=4.47 \times 10^{9} \mathrm{yr}\right)\)

\(\begin{array}{l}{\text { Determine the activities of }(a) 1.0 \mathrm{g} \text { of } \frac{131}{53} \mathrm{I}\left(T_{\frac{1}{2}}=8.02 \text { days) }\right.} \\ {\text { and }(b) 1.0 \mathrm{g} \text { of }_{92}^{238} \mathrm{U}\left(T_{1}=4.47 \times 10^{9} \mathrm{yr}\right)}\end{array} \)

(a) Show that the mean life of a radioactive nuclide, defined as $$\tau=\frac{\int_{0}^{\infty} t N(t) d t}{\int_{0}^{\infty} N(t) d t}$$ is \(\tau=1 / \lambda .\) (b) What fraction of the original number of nuclei remains after one mean life?

If the mass of the proton were just a little closer to the mass of the neutron, the following reaction would be possible even at low collision energies: $$\mathrm{e}^{-}+\mathrm{p} \rightarrow \mathrm{n}+\nu$$ (a) Why would this situation be catastrophic? \((b)\) By what percentage would the proton's mass have to be increased to make this reaction possible?

What is the ratio of the kinetic energies for an alpha particle and a beta particle if both make tracks with the same radius of curvature in a magnetic field, oriented perpendicular to the paths of the particles?

A 1.00-g sample of natural samarium emits \(\alpha\) particles at a rate of \(120 \mathrm{~s}^{-1}\) due to the presence of \({ }_{62}^{147} \mathrm{Sm} .\) The natural abundance of \({ }_{62}^{14} \mathrm{Sm}\) is \(15 \% .\) Calculate the half-life for this decay process.

Almost all of naturally occurring uranium is \({ }_{92}^{238} \mathrm{U}\) with a half-life of \(4.468 \times 10^{9}\) yr. Most of the rest of natural uranium is \({ }_{92}^{235} \mathrm{U}\) with a half-life of \(7.04 \times 10^{8} \mathrm{yr} .\) Today a sample contains \(0.720 \%{ }_{92}^{235} \mathrm{U} .\) ( \(a\) ) What was this percentage 1.0 billion years ago? (b) What percentage of \({ }_{92}^{235} \mathrm{U}\) will remain 100 million years from now?

Almost all of naturally occurring uranium is \(\frac{238}{92} \mathrm{U}\) with a half-life of \(4.468 \times 10^{9}\) yr. Most of the rest of natural uranium is \(\frac{235}{92} \mathrm{U}\) with a half-life of \(7.04 \times 10^{8} \mathrm{yr.}\) Today a sample contains0.720\(\%_{0} \frac{235}{92} U\) (a) What was this percentage 1.0 billion years ago? (b) What percentage of\(\frac{235}{92} \mathrm{U}\) will remain 100 million years from now?

Some radioactive isotopes have half-lives that are larger than the age of the universe (like gadolinium or samarium). The only way to determine these half- lives is to monitor the decay rate of a sample that contains these isotopes. For example, suppose we find an asteroid that currently contains about \(15,000 \mathrm{~kg}\) of \({ }_{64}^{152}\) Gd (gadolinium) and we detect an activity of 1 decay/s. What is the half-life of gadolinium (in years)?

(I) A laboratory has a \(1.80-\mu \mathrm{g}\) sample of radioactive \(\frac{13}{7} \mathrm{N}\) whose decay constant \(\lambda=1.16 \times 10^{-3} \mathrm{s }^{-1} .\) Calculate the initial number of nuclei, \(N_{0},\) present in the sample. Use the radioactive decay law, \(N=N_{0} e^{-\lambda t},\) to determine th number of nuclei \(N\) present at time \(t\) for \(t=0\) th 30 minutes \((1800\) s) in steps of 0.5 \(\min (30\) s). Make a grap of \(N\) versus \(t\) and from the graph determine the half-life the sample.