31.5 Half-Life and Activity

When ${}_{84}^{215}{\text{P}} \text{o}$ decays, the product is ${}_{82}^{211}{\text{P}} \text{b}.$ The half-life of this decay process is 1.78 ms. If the initial sample contains 3.4 x 10¹⁷ parent nuclei, how many are remaining after 35 ms have elapsed? What kind of decay process is this (alpha, beta, or gamma)?

Data from the appendices and the periodic table may be needed for these problems.

An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of $^{\text{14}} \text{C}$. Estimate the minimum age of the charcoal, noting that $2^{\text{10}} = \text{1024}$.

Data from the appendices and the periodic table may be needed for these problems.

(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.

Data from the appendices and the periodic table may be needed for these problems.

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 $1 . \text{405} \times \text{10}^{\text{10}} \text{y}$. If an average lantern mantle contains 300 mg of thorium, what is its activity?

Data from the appendices and the periodic table may be needed for these problems.

(a) Natural potassium contains $^{\text{40}} \text{K}$, which has a half-life of $1 . \text{277} \times \text{10}^{9}$ 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.)

Data from the appendices and the periodic table may be needed for these problems.

$^{\text{50}} \text{V}$ 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?

Data from the appendices and the periodic table may be needed for these problems.

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?

Data from the appendices and the periodic table may be needed for these problems.

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?

Data from the appendices and the periodic table may be needed for these problems.

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?

Data from the appendices and the periodic table may be needed for these problems.

(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…

Data from the appendices and the periodic table may be needed for these problems.

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…

Data from the appendices and the periodic table may be needed for these problems.

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 $\alpha$ particles emitt…

Data from the appendices and the periodic table may be needed for these problems.

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