Problems & Exercises

The energy of 30.0 $\text{eV}$ is required to ionize a molecule of the gas inside a Geiger tube, thereby producing an ion pair. Suppose a particle of ionizing radiation deposits 0.500 MeV of energy in this Geiger tube. What maximum number of ion pairs can it create?

Verify that a $2\text{.}3\times {\text{10}}^{\text{17}}\text{kg}$ mass of water at normal density would make a cube 60 km on a side, as claimed in Example 31.1. (This mass at nuclear density would make a cube 1.0 m on a side.)

What is the radius of an $\alpha$ particle?

(a) Calculate the radius of ${}^{\text{58}}\text{Ni}$, one of the most tightly bound stable nuclei.

(b) What is the ratio of the radius of ${}^{\text{58}}\text{Ni}$ to that of ${}^{\text{258}}\text{Ha}$, one of the largest nuclei ever made? Note that the radius of the largest nucleus is still much smaller than the size of an atom.

What is the ratio of the velocity of a $\beta$ particle to that of an $\alpha$ particle, if they have the same nonrelativistic kinetic energy?

The detail observable using a probe is limited by its wavelength. Calculate the energy of a $\gamma$-ray photon that has a wavelength of $1\times {\text{10}}^{-\text{16}}\text{m}$, small enough to detect details about one-tenth the size of a nucleon. Note that a photon having this energy is difficult to produce and interacts poorly with the nucleus, limiting the practicability of this probe.

What is the ratio of the velocity of a 5.00-MeV $\beta$ ray to that of an $\alpha$ particle with the same kinetic energy? This should confirm that $\beta$s travel much faster than $\alpha$s even when relativity is taken into consideration. (See also Exercise 31.11.)

${\beta }^{-}$ decay of ${}^3\text{H}$ (tritium), a manufactured isotope of hydrogen used in some digital watch displays, and manufactured primarily for use in hydrogen bombs.

${\beta }^{+}$ decay of ${}^{\text{50}}\text{Mn}$.

Electron capture by ${}^7\text{Be}$.

$\alpha$ decay of ${}^{\text{210}}\text{Po}$, the isotope of polonium in the decay series of ${}^{\text{238}}\text{U}$ that was discovered by the Curies. A favorite isotope in physics labs, since it has a short half-life and decays to a stable nuclide.

${\beta }^{-}$ decay producing ${}^{\text{137}}\text{Ba}$. The parent nuclide is a major waste product of reactors and has chemistry similar to potassium and sodium, resulting in its concentration in your cells if ingested.

$\alpha$ decay producing ${}^{\text{228}}\text{Ra}$. The parent nuclide is nearly 100% of the natural element and is found in gas lantern mantles and in metal alloys used in jets (${}^{\text{228}}\text{Ra}$ is also radioactive).

When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum. (a) Confirm that the annihilation equation $e^{+}+e^{-}\to \gamma +\gamma$ conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation. (b) Find the energy of each $\gamma$ ray, assuming the electron and positron are initially nearly at rest. (c) Explain why the two $\gamma$ rays travel in exactly opposite directions if the center of mass of the electron-positron system is initially at rest.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for ${\beta }^{-}$ decay given in the equation ${}_Z^A{\mathrm{X}}_N\to {}_{Z+1}^A{\text{Y}}_{N-1}+{\beta }^{-}+{\overline{\nu }}_e$. To do this, identify the values of each before and after the decay.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for electron capture given in the equation ${}_Z^A{\mathrm{X}}_N+e^{-}\to {}_{Z-1}^A{\text{Y}}_{N+1}+{\nu }_e$. To do this, identify the values of each before and after the capture.

(a) Write the complete $\alpha$ decay equation for ${}^{\text{226}}\text{Ra}$.

(b) Find the energy released in the decay.

(a) Write the complete ${\beta }^{-}$ decay equation for the neutron. (b) Find the energy released in the decay.

Calculate the energy released in the ${\beta }^{+}$ decay of ${}^{\text{22}}\text{Na}$, the equation for which is given in the text. The masses of ${}^{\text{22}}\text{Na}$ and ${}^{\text{22}}\text{Ne}$ are 21.994434 and 21.991383 u, respectively.

(a) Calculate the energy released in the $\alpha$ decay of ${}^{\text{238}}\text{U}$.

(b) What fraction of the mass of a single ${}^{\text{238}}\text{U}$ is destroyed in the decay? The mass of ${}^{\text{234}}\text{Th}$ is 234.043593 u.

(c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?

(a) Write the complete reaction equation for electron capture by ${}^{\text{15}}\text{O}$.

(b) Calculate the energy released.

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 $2^{\text{10}}=\text{1024}$.

(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.

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?

(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.)

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

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?

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?

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?

(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.

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.)

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.

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?

${}^2\text{H}$ is a loosely bound isotope of hydrogen. Called deuterium or heavy hydrogen, it is stable but relatively rare—it is 0.015% of natural hydrogen. Note that deuterium has $Z=N$, which should tend to make it more tightly bound, but both are odd numbers. Calculate $\mathrm{BE/}A$, the binding energy per nucleon, for ${}^2\text{H}$ and compare it with the approximate value obtained from the graph in Figure 31.24.

${}^{\text{209}}\text{Bi}$ is the heaviest stable nuclide, and its $\text{BE}/A$ is low compared with medium-mass nuclides. Calculate $\mathrm{BE/}A$, the binding energy per nucleon, for ${}^{\text{209}}\text{Bi}$ and compare it with the approximate value obtained from the graph in Figure 31.24.

(a) Calculate $\text{BE}/A$ for ${}^{\text{12}}\text{C}$. Stable and relatively tightly bound, this nuclide is most of natural carbon. (b) Calculate $\text{BE}/A$ for ${}^{\text{14}}\text{C}$. Is the difference in $\text{BE}/A$ between ${}^{\text{12}}\text{C}$ and ${}^{\text{14}}\text{C}$ significant? One is stable and common, and the other is unstable and rare.

The purpose of this problem is to show in three ways that the binding energy of the electron in a hydrogen atom is negligible compared with the masses of the proton and electron. (a) Calculate the mass equivalent in u of the 13.6-eV binding energy of an electron in a hydrogen atom, and compare this with the mass of the hydrogen atom obtained from Appendix A. (b) Subtract the mass of the proton given in Table 31.2 from the mass of the hydrogen atom given in Appendix A. You will find the difference is equal to the electron’s mass to three digits, implying the binding energy is small in comparison. (c) Take the ratio of the binding energy of the electron (13.6 eV) to the energy equivalent of the electron’s mass (0.511 MeV). (d) Discuss how your answers confirm the stated purpose of this problem.

Unreasonable Results

A particle physicist discovers a neutral particle with a mass of 2.02733 u that he assumes is two neutrons bound together. (a) Find the binding energy. (b) What is unreasonable about this result? (c) What assumptions are unreasonable or inconsistent?

Integrated Concepts

A 2.00-T magnetic field is applied perpendicular to the path of charged particles in a bubble chamber. What is the radius of curvature of the path of a 10 MeV proton in this field? Neglect any slowing along its path.

(a) Write the decay equation for the $\alpha$ decay of ${}^{235}\text{U}$. (b) What energy is released in this decay? The mass of the daughter nuclide is 231.036298 u. (c) Assuming the residual nucleus is formed in its ground state, how much energy goes to the $\alpha$ particle?

Unreasonable Results

A physicist scatters $\gamma$ rays from a substance and sees evidence of a nucleus $7.5\times {10}^{\mathrm{–13}}\text{m}$ in radius. (a) Find the atomic mass of such a nucleus. (b) What is unreasonable about this result? (c) What is unreasonable about the assumption?

Unreasonable Results

A frazzled theoretical physicist reckons that all conservation laws are obeyed in the decay of a proton into a neutron, positron, and neutrino (as in ${\beta }^{+}$ decay of a nucleus) and sends a paper to a journal to announce the reaction as a possible end of the universe due to the spontaneous decay of protons. (a) What energy is released in this decay? (b) What is unreasonable about this result? (c) What assumption is responsible?

Critical Thinking (a) What is the density of ${}^{235}\text{U}$ in ${\text{u/fm}}^3$? (b) If ${}^{235}\text{U}$ splits into ${}^{141}\text{Ba}$ and ${}^{92}\text{Kr}+3$ neutrons, how does the density of ${}^{235}\text{U}$ compare with the density of each element it splits into?