The atomic mass of $14C$ is $14.003242$ u. Show that the ${\beta }^{-}$ decay of $14C$ is energetically possible, and calculate the energy released in the decay.

The unstable isotope ${}^{40}K$ is used for dating rock samples. Its half-life is $1.28\times {10}^9$ y.

(a) How many decays occur per second in a sample containing $1.63\times {10}^{-6}$ g of ${}^{40}K$?

(b) What is the activity of the sample in curies?

The common isotope of uranium, ${}^{238}U$, has a half-life of $4.47\times {10}^9$ years, decaying to ${}^{234}Th$ by alpha emission.

(a) What is the decay constant?

(b) What mass of uranium is required for an activity of $1.00$ curie?

(c) How many alpha particles are emitted per second by $10.0$ g of uranium?

Radioactive isotopes used in cancer therapy have a 'shelf-life,' like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of ${}^{60}Co$ is $5000$ Ci. When its activity falls below $3500$ Ci, it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these ${}^{60}Co$ sources in your inventory was manufactured on October 6, 2011. It is now April 6, 2014. Is the source still usable? The half-life of ${}^{60}Co$ is $5.271$ years.

What particle (a particle, electron, or positron) is emitted in the following radioactive decays?

(a) ${}_{14}^{27}Si{\to }_{13}^{27}Al$

(b) ${}_{92}^{238}U{\to }_{90}^{234}Th$

(c) ${}_{33}^{74}As{\to }_{34}^{74}Se$

What nuclide is produced in the following radioactive decays?

(a) $\alpha$ decay of ${}_{94}^{239}Pu$

(b) ${\beta }^{-}$ decay of ${}_{11}^{24}Na$

(c) ${\beta }^{+}$ decay of ${}_8^{15}O$

(a) Is the decay $n\to p+{\beta }^{-}+\overline{v_e}$ energetically possible? If not, explain why not. If so, calculate the total energy released.

(b) Is the decay $n\to p+{\beta }^{+}+\overline{v_e}$ energetically possible? If not, explain why not. If so, calculate the total energy released.

The most common isotope of uranium, ${}_{92}^{238}U$, has atomic mass $238.050788$ u. Calculate (a) the mass defect; (b) the binding energy (in MeV); (c) the binding energy per nucleon.

Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. What magnetic-field magnitude is required for this transition to be induced by photons with frequency $22.7$ MHz?