A typical kinetic energy for a nucleon in a middle-mass nucleus may be taken as $4.00 \mathrm{MeV}$. To what effective nuclear temperature does this corresponds, based on the assumptions of the collective model of nuclear structure?

Figure shows the decay of parents in a radioactive sample. The axes are scaled by $N_{\mathrm{s}}=2.0\times {10}^6$ and $t_{\mathrm{s}}=10.0 \mathrm{s}$. What is the activity of the sample at $t=27.0 \mathrm{s}$?

The isotope ${}^{238}\mathrm{U}$ decays to ${}^{206}\mathrm{Pb}$ with a half-life of $4.47\times {10}^9 \mathrm{y}$. Although the decay occurs in many individual steps, the first step has by far the longest half-life. Therefore, one can often consider the decay to go directly to Lead. That is,

${}^{238}\mathrm{U}\to {}^{206}\mathrm{Pb}+$ various decay products.

A rock is found to contain $4.20 \mathrm{mg}$ of ${}^{238}\mathrm{U}$ and $2.135 \mathrm{mg}$ of ${}^{206}\mathrm{Pb}$. Assume that the rock contained no Lead at formation, so all the Lead present later arose from the decay of Uranium. How many atoms of (a) ${}^{238}\mathrm{U}$ and (b) ${}^{206}\mathrm{Pb}$ does the rock contain later? (c) How many atoms of ${}^{238}\mathrm{U}$ did the rock contain at formation ? (d) What is the age of the rock?

Under certain circumstances, a nucleus can decay by emitting a particle more massive than an alpha particle. Consider the decays

${\mathrm{Ra}}^{223}\to {\mathrm{Pb}}^{209}+{\mathrm{C}}^{14}$ and ${\mathrm{Ra}}^{223}\to {\mathrm{Rn}}^{219}+{\mathrm{He}}^4$

Calculate the $Q$-value for the (a) first and (b) second decay, and determine that both are energetically possible. (c) The Coulomb barrier height for the alpha-particle emission is $30.0 \mathrm{MeV}$. What is the barrier height for ${}^{14}\mathrm{C}$ emission? (Be careful about the nuclear radii). The needed atomic masses are:

${\mathrm{Ra}}^{223}=223.018 50 \mathrm{u}$
${\mathrm{Pb}}^{209}=208.981 07 \mathrm{u}$.
${\mathrm{Rn}}^{219}=219.009 48 \mathrm{u}$
${\mathrm{C}}^{14}=14.003 24 \mathrm{u}$
${\mathrm{He}}^4=4.002 60 \mathrm{u}$

Some radionuclide decay by capturing one of their own atomic electrons, a $K$-shell electron, say. An example is

${}^{49}\mathrm{V}+{\mathrm{e}}^{-}\to {}^{49}\mathrm{Ti}+v$, $T_{1/2}=331 \mathrm{d}$.

Show that the disintegration energy $Q$ for this process is given by
$Q=\left(m_{\mathrm{V}}-m_{\mathrm{Ti}}\right)c^2-E_K$,
where $m_{\mathrm{V}}$ and $m_{\mathrm{Ti}}$ are the atomic masses of ${}^{49}\mathrm{V}$ and ${}^{49}\mathrm{Ti}$, respectively, and $E_K$ is the binding energy of vanadium $K$-shell electron. (Hint: Put $m_{\mathrm{V}}$ and $m_{\mathrm{Ti}}$ as the corresponding nuclear masses and then add in enough electrons to use the atomic masses).