Problems

Silver has two stable isotopes. The nucleus, ${}_{47}^{107}\text{A}\text{g},$ has atomic mass 106.905095 g/mol with an abundance of $51.83\text{\%}$; whereas ${}_{47}^{109}\text{A}\text{g}$ has atomic mass 108.904754 g/mol with an abundance of $48.17\text{\%}$. Find the atomic mass of the element silver.

A particle has a mass equal to 10 u. If this mass is converted completely into energy, how much energy is released? Express your answer in mega-electron volts (MeV). (Recall that $1\text{eV}=1.6\times {10}^{\mathrm{-19}}\text{J}$.)

The detail that you can observe using a probe is limited by its wavelength. Calculate the energy of a particle that has a wavelength of $1\times {10}^{\mathrm{-16}}\text{m}$, small enough to detect details about one-tenth the size of a nucleon.

Find the mass defect and the binding energy for the helium-4 nucleus.

${}^{209}\text{Bi}$ is the heaviest stable nuclide, and its BEN is low compared with medium-mass nuclides. Calculate BEN for this nucleus and compare it with the approximate value obtained from the graph in Figure 10.7.

The fact that BEN peaks at roughly $A=60$ implies that the range of the strong nuclear force is about the diameter of this nucleus.

(a) Calculate the diameter of $A=60$ nucleus.

(b) Compare BEN for ${}^{58}\text{Ni and}{}^{90}\text{Sr}$. The first is one of the most tightly bound nuclides, whereas the second is larger and less tightly bound.

Show that: $\bar{T}=\frac{1}{\text{λ}}$.

A sample of pure carbon-14 $\left(T_{1\text{/}2}=5730\text{y}\right)$ has an activity of $1.0\mu \text{Ci}.$ What is the mass of the sample?

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

Natural uranium consists of ${}^{235}\text{U}$ $(\text{percent abundance}=0.7200\text{\%}$, $\lambda =3.12\times {10}^{\mathrm{-17}}\text{/}\text{s})$ and ${}^{238}\text{U}$ $(\text{percent abundance}=99.27\text{\%}$, $\lambda =4.92\times {10}^{\mathrm{-18}}\text{/}\text{s}).$ What were the values for percent abundance of ${}^{235}\text{U}$ and ${}^{238}\text{U}$ when Earth formed $4.5\times {10}^9$ years ago?

The ${}^{210}\text{P}\text{o}$ source used in a physics laboratory is labeled as having an activity of $1.0\text{μCi}$ 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?

${}^{249}\text{C}\text{f}$ undergoes alpha decay. (a) Write the reaction equation. (b) Find the energy released in the decay.

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?

Write a nuclear ${\beta }^{-}$ decay reaction that produces the ${}^{90}\text{Y}$ nucleus. (Hint: The parent nuclide is a major waste product of reactors and has chemistry similar to calcium, so that it is concentrated in bones if ingested.)

If a 1.50-cm-thick piece of lead can absorb $90.0\text{\%}$ of the rays from a radioactive source, how many centimeters of lead are needed to absorb all but $0.100\text{\%}$ of the rays?

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

(b) Calculate the energy released.

A large power reactor that has been in operation for some months is turned off, but residual activity in the core still produces 150 MW of power. If the average energy per decay of the fission products is 1.00 MeV, what is the core activity?

(a) Calculate the energy released in the neutron-induced fission reaction $n+{}^{235}\text{U}\to {}^{92}\text{K}\text{r}+{}^{142}\text{B}\text{a}+2n$, given $m\left({}^{92}\text{K}\text{r}\right)=91.926269\text{u}$ and $m\left({}^{142}\text{B}\text{a}\right)=141.916361\text{u}$. (b) Confirm that the total number of nucleons and total charge are conserved in this reaction.

Find the total energy released if 1.00 kg of ${}_{92}^{235}\text{U}$ were to undergo fission.

Calculate the energy output in each of the fusion reactions in the proton-proton chain, and verify the values determined in the preceding problem.

Two fusion reactions mentioned in the text are $n+{}^3\text{H}\text{e}\to {}^4\text{H}\text{e}+\gamma$ and $n+{}^1\text{H}\to {}^2\text{H}+\gamma$. Both reactions release energy, but the second also creates more fuel. Confirm that the energies produced in the reactions are 20.58 and 2.22 MeV, respectively. Comment on which product nuclide is most tightly bound, ${}^4\text{H}\text{e}$ or ${}^2\text{H}$.

Another set of reactions that fuses hydrogen into helium in the Sun and especially in hotter stars is called the CNO cycle:

${}^{12}\text{C}+{}^1\text{H}\to {}^{13}\text{N}+\gamma$

${}^{13}\text{N}\to {}^{13}\text{C}+e^{+}+v_{\text{e}}$

${}^{13}\text{C}+{}^1\text{H}\to {}^{14}\text{N}+\gamma$

${}^{14}\text{N}+{}^1\text{H}\to {}^{15}\text{O}+\gamma$

${}^{15}\text{O}\to {}^{15}\text{N}+e^{+}+v_{\text{e}}$

${}^{15}\text{N}+{}^1\text{H}\to {}^{12}\text{C}+{}^4\text{H}\text{e}$

This process is a “cycle” because ${}^{12}\text{C}$ appears at the beginning and end of these reactions. Write down the overall effect of this cycle (as done for the proton-proton chain in $2e^{-}+4{}^1\text{H}\to {}^4\text{H}\text{e}+2v_{\text{e}}+6\gamma$). Assume that the positrons annihilate electrons to form more $\gamma$ rays.

What is the dose in mSv for: (a) a 0.1-Gy X-ray? (b) 2.5 mGy of neutron exposure to the eye? (c) 1.5m Gy of $\alpha$ exposure?

Find the mass of ${}^{239}\text{P}\text{u}$ that has an activity of $1.00\text{μCi}$.

What is the dose in Sv in a cancer treatment that exposes the patient to 200 Gy of $\gamma$ rays?

How many Gy of exposure is needed to give a cancerous tumor a dose of 40 Sv if it is exposed to $\alpha$ activity?

Calculate the dose in rem/y for the lungs of a weapons plant employee who inhales and retains an activity of $1.00\mu \text{Ci}$ ${}^{239}\text{Pu}$ in an accident. The mass of affected lung tissue is 2.00 kg and the plutonium decays by emission of a 5.23-MeV $\alpha$ particle. Assume a RBE value of 20.