31.5 Half-life and activity  (Page 8/16)

(a) The ${}^{\text{210}}\text{Po}$ source used in a physics laboratory is labeled as having an activity of $1.0\mu \text{Ci}$ on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus? (b) Identify some of the reasons that only a fraction of the $\alpha$ s emitted are observed by the detector.

Armor-piercing shells with depleted uranium cores are fired by aircraft at tanks. (The high density of the uranium makes them effective.) The uranium is called depleted because it has had its ${}^{\text{235}}\text{U}$ removed for reactor use and is nearly pure ${}^{\text{238}}\text{U}$ . Depleted uranium has been erroneously called non-radioactive. To demonstrate that this is wrong: (a) Calculate the activity of 60.0 g of pure ${}^{\text{238}}\text{U}$ . (b) Calculate the activity of 60.0 g of natural uranium, neglecting the ${}^{\text{234}}\text{U}$ and all daughter nuclides.

The ceramic glaze on a red-orange Fiestaware plate is ${\text{U}}_2{\text{O}}_3$ and contains 50.0 grams of ${}^{238}\text{U}$ , but very little ${}^{235}\text{U}$ . (a) What is the activity of the plate? (b) Calculate the total energy that will be released by the ${}^{238}\text{U}$ decay. (c) If energy is worth 12.0 cents per $\text{kW}\cdot \text{h}$ , what is the monetary value of the energy emitted? (These plates went out of production some 30 years ago, but are still available as collectibles.)

Large amounts of depleted uranium ( ${}^{238}\text{U}$ ) are available as a by-product of uranium processing for reactor fuel and weapons. Uranium is very dense and makes good counter weights for aircraft. Suppose you have a 4000-kg block of ${}^{238}\text{U}$ . (a) Find its activity. (b) How many calories per day are generated by thermalization of the decay energy? (c) Do you think you could detect this as heat? Explain.

The Galileo space probe was launched on its long journey past several planets in 1989, with an ultimate goal of Jupiter. Its power source is 11.0 kg of ${}^{238}\text{Pu}$ , a by-product of nuclear weapons plutonium production. Electrical energy is generated thermoelectrically from the heat produced when the 5.59-MeV $\text{α}$ particles emitted in each decay crash to a halt inside the plutonium and its shielding. The half-life of ${}^{238}\text{Pu}$ is 87.7 years. (a) What was the original activity of the ${}^{238}\text{Pu}$ in becquerel? (b) What power was emitted in kilowatts? (c) What power was emitted 12.0 y after launch? You may neglect any extra energy from daughter nuclides and any losses from escaping $\text{γ}$ rays.

Construct Your Own Problem

Consider the generation of electricity by a radioactive isotope in a space probe, such as described in [link] . Construct a problem in which you calculate the mass of a radioactive isotope you need in order to supply power for a long space flight. Among the things to consider are the isotope chosen, its half-life and decay energy, the power needs of the probe and the length of the flight.

Unreasonable Results

A nuclear physicist finds $1.0\mu \text{g}$ of ${}^{236}\text{U}$ in a piece of uranium ore and assumes it is primordial since its half-life is $2.3\times {10}^7\text{y}$ . (a) Calculate the amount of ${}^{236}\text{U}$ that would had to have been on Earth when it formed $4.5\times {10}^9\text{y}$ ago for $1.0\mu \text{g}$ to be left today. (b) What is unreasonable about this result? (c) What assumption is responsible?

Unreasonable Results

(a) Repeat [link] but include the 0.0055% natural abundance of ${}^{234}\text{U}$ with its $2.45\times {10}^5\text{y}$ half-life. (b) What is unreasonable about this result? (c) What assumption is responsible? (d) Where does the ${}^{234}\text{U}$ come from if it is not primordial?

Unreasonable Results

The manufacturer of a smoke alarm decides that the smallest current of $\text{α}$ radiation he can detect is $1.00\mu \text{A}$ . (a) Find the activity in curies of an $\text{α}$ emitter that produces a $1.00\mu \text{A}$ current of $\text{α}$ particles. (b) What is unreasonable about this result? (c) What assumption is responsible?