10.7 Half-life and activity  (Page 7/15)

(a) Calculate the activity $R$ in curies of 1.00 g of ${}^{\text{226}}\text{Ra}$ . (b) Discuss why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.

Show that the activity of the ${}^{\text{14}}\text{C}$ in 1.00 g of ${}^{\text{12}}\text{C}$ found in living tissue is 0.250 Bq.

Mantles for gas lanterns contain thorium, because it forms an oxide that can survive being heated to incandescence for long periods of time. Natural thorium is almost 100% ${}^{\text{232}}\text{Th}$ , with a half-life of $1\text{.}\text{405}\times {\text{10}}^{\text{10}}\text{y}$ . If an average lantern mantle contains 300 mg of thorium, what is its activity?

Cow’s milk produced near nuclear reactors can be tested for as little as 1.00 pCi of ${}^{\text{131}}\text{I}$ per liter, to check for possible reactor leakage. What mass of ${}^{\text{131}}\text{I}$ has this activity?

(a) Natural potassium contains ${}^{\text{40}}\text{K}$ , which has a half-life of $1\text{.}\text{277}\times {\text{10}}^9$ y. What mass of ${}^{\text{40}}\text{K}$ in a person would have a decay rate of 4140 Bq? (b) What is the fraction of ${}^{\text{40}}\text{K}$ in natural potassium, given that the person has 140 g in his body? (These numbers are typical for a 70-kg adult.)

There is more than one isotope of natural uranium. If a researcher isolates 1.00 mg of the relatively scarce ${}^{\text{235}}\text{U}$ and finds this mass to have an activity of 80.0 Bq, what is its half-life in years?

${}^{\text{50}}\text{V}$ has one of the longest known radioactive half-lives. In a difficult experiment, a researcher found that the activity of 1.00 kg of ${}^{\text{50}}\text{V}$ is 1.75 Bq. What is the half-life in years?

You can sometimes find deep red crystal vases in antique stores, called uranium glass because their color was produced by doping the glass with uranium. Look up the natural isotopes of uranium and their half-lives, and calculate the activity of such a vase assuming it has 2.00 g of uranium in it. Neglect the activity of any daughter nuclides.

A tree falls in a forest. How many years must pass before the ${}^{\text{14}}\text{C}$ activity in 1.00 g of the tree’s carbon drops to 1.00 decay per hour?

What fraction of the ${}^{\text{40}}\text{K}$ that was on Earth when it formed $4\text{.}5\times {\text{10}}^9$ years ago is left today?

A 5000-Ci ${}^{\text{60}}\text{Co}$ source used for cancer therapy is considered too weak to be useful when its activity falls to 3500 Ci. How long after its manufacture does this happen?

Natural uranium is 0.7200% ${}^{\text{235}}\text{U}$ and 99.27% ${}^{\text{238}}\text{U}$ . What were the percentages of ${}^{\text{235}}\text{U}$ and ${}^{\text{238}}\text{U}$ in natural uranium when Earth formed $4\text{.}5\times {\text{10}}^9$ years ago?

The ${\beta }^{-}$ particles emitted in the decay of ${}^3\text{H}$ (tritium) interact with matter to create light in a glow-in-the-dark exit sign. At the time of manufacture, such a sign contains 15.0 Ci of ${}^3\text{H}$ . (a) What is the mass of the tritium? (b) What is its activity 5.00 y after manufacture?

World War II aircraft had instruments with glowing radium-painted dials. The activity of one such instrument was $1.0\times {\text{10}}^5$ Bq when new. (a) What mass of ${}^{\text{226}}\text{Ra}$ was present? (b) After some years, the phosphors on the dials deteriorated chemically, but the radium did not escape. What is the activity of this instrument 57.0 years after it was made?

(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.