A free neutron decays according to the equation $n\to p+e^{-}+\nu$. If the neutron-hydrogen atomic mass difference is $840 \mathrm{\mu u}$, what is the maximum kinetic energy $K_{\max }$ possible for the electron produced in a neutron decay?

The radionuclide ${\mathrm{C}}^{11}$ decays according to
${\mathrm{C}}^{11}\to {\mathrm{B}}^{11}+{\mathrm{e}}^{+}+v$, $T_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=20.3 \min$

The maximum energy of the emitted positron is $0.960 \mathrm{MeV}$.
(a) Show that the disintegration energy $Q$ for this process is given by,
$Q=\left(m_{\mathrm{C}}-m_{\mathrm{B}}-2m_{\mathrm{e}}\right)c^2$
where, $m_{\mathrm{C}}$ and $m_{\mathrm{B}}$ are the atomic masses of ${\mathrm{C}}^{11}$ and ${\mathrm{B}}^{11}$, respectively, and $m_{\mathrm{C}}$ is the mass of a positron.

(b) Given the mass values, $m_{\mathrm{C}}=11.011434 \mathrm{u}$, $m_{\mathrm{B}}=11.009305 \mathrm{u}$ and $m_{\mathrm{e}}=0.0005486 \mathrm{u}$, calculate $Q$ and compare it with the maximum energy of the emitted positron given above. (Hint: Let $m_{\mathrm{C}}$ and $m_{\mathrm{B}}$ be the nuclear masses and then add in enough electrons to use the atomic masses)

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?