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Week 13: Radioactive Decay & Nuclear Reactions -- NO SOLUTIONS

For solutions, please visit: edu.phyzk.net/206/rec/13/

\(\Delta\) ← if you see the word "Delta" instead of a triangle, please refresh the page! Thanks :)

 

[1] Overview⚓︎

I recommend reviewing the following concepts, they will probably come up in recitation:

  • Radioactive decay and half-life

    • \(N(t)=N_0e^{-\lambda t}\)
    • \(A(t)=A_0e^{-\lambda t}\)
    • \(\lambda=\dfrac{\ln 2}{T_{1/2}}\)
    • after one half-life, the activity is cut in half

  • Activity units

    • \(1\ \mathrm{Bq}=1\ \text{decay/s}\)
    • \(1\ \mathrm{Ci}=3.70\times10^{10}\ \mathrm{Bq}\)
    • \(1\ \mathrm{mCi}=3.70\times10^7\ \mathrm{Bq}\)
    • activity is proportional to the number of undecayed nuclei

  • Balancing nuclear reactions

    • mass number \(A\) must be conserved
    • atomic number \(Z\) must be conserved
    • the missing particle or nuclide can usually be found by balancing \(A\) and \(Z\) separately

  • Common decay modes

    • alpha decay: emits \(^4_2\mathrm{He}\)
    • beta-minus decay: emits \(^0_{-1}\mathrm{e}\) and increases \(Z\) by 1
    • beta-plus decay: emits \(^0_{+1}\mathrm{e}\) and decreases \(Z\) by 1
    • electron capture: absorbs an inner electron and decreases \(Z\) by 1

  • Reaction energy / Q-values

    • \(Q=(m_{\mathrm{reactants}}-m_{\mathrm{products}})c^2\)
    • \(1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2\)
    • positive \(Q\) means energy is released
    • negative \(Q\) means energy is absorbed / required

  • Using atomic masses carefully

    • for most balanced nuclear reactions, neutral atomic masses can be used directly if the same total number of electrons appears on both sides
    • for \(\beta^-\) decay, atomic masses can be used as \(Q=(M_{\mathrm{parent}}-M_{\mathrm{daughter}})c^2\)
    • for \(\beta^+\) decay, atomic masses require subtracting \(2m_e\): \(Q=(M_{\mathrm{parent}}-M_{\mathrm{daughter}}-2m_e)c^2\)

 

 

Practice Problems⚓︎

  • If you're comfortable with the "★★★" problems, you should do great during recitation.
  • To print these questions, simply press Ctrl + P while on this page, and it should come out formatted nicely -- just make sure to refresh first so all the math renders properly.

Difficulty key:

  • ★☆☆ = beginner
  • ★★☆ = standard
  • ★★★ = challenging / multi-step

Useful Constants

  • \(\ln 2 = 0.693\)
  • \(1\ \mathrm{Bq}=1\ \mathrm{s^{-1}}\)
  • \(1\ \mathrm{Ci}=3.70\times10^{10}\ \mathrm{Bq}\)
  • \(1\ \mathrm{mCi}=3.70\times10^7\ \mathrm{Bq}\)
  • \(m_n = 1.008665\ \mathrm{u}\)
  • \(m_e = 0.00054858\ \mathrm{u}\)
  • \(1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2\)

 

[2.1] Radioactive Decay, Half-Life, and Activity⚓︎

  1. ★☆☆
    A radioactive isotope has a half-life of \(12.0\ \mathrm{min}\) and an initial activity of \(4.80\times10^5\ \mathrm{Bq}\).

    • (a) Find the decay constant in \(\mathrm{min^{-1}}\).
    • (b) Find the decay constant in \(\mathrm{s^{-1}}\).
    • (c) What is the activity after \(12.0\ \mathrm{min}\)?
    • (d) What is the activity after \(24.0\ \mathrm{min}\)?

  2. ★☆☆
    A sample of a radioactive tracer has an activity of \(18.0\ \mathrm{mCi}\) at noon. Its half-life is \(6.00\ \mathrm{h}\).

    • (a) What is the activity at 6:00 p.m.?
    • (b) What is the activity at midnight?
    • (c) What percentage of the original activity remains at midnight?

  3. ★★☆
    A radioisotope has a half-life of \(4.20\ \mathrm{h}\) and an initial activity of \(2.50\ \mathrm{mCi}\).

    • (a) Find the activity after \(10.5\ \mathrm{h}\).
    • (b) Convert your answer to becquerels.

  4. ★★☆
    A radioactive sample has a half-life of \(2.50\ \mathrm{days}\). Its activity is initially \(1600\ \mathrm{Bq}\).

    • (a) How long will it take for the activity to fall to \(100\ \mathrm{Bq}\)?
    • (b) How many half-lives have passed during this time?

  5. ★★★
    A sample of fluorine-18 has a half-life of \(110\ \mathrm{min}\). At 7:30 a.m., its activity is \(12.0\ \mathrm{mCi}\). A patient is scheduled to receive the sample at 9:40 a.m.

    • (a) What will the activity be at 9:40 a.m.?
    • (b) If the patient needs at least \(5.00\ \mathrm{mCi}\) at injection time, is this sample sufficient?
    • (c) What activity would the sample need to have at 7:30 a.m. in order to have exactly \(5.00\ \mathrm{mCi}\) at 9:40 a.m.?

  6. ★★★
    A sample initially contains \(6.40\times10^6\) undecayed nuclei of a radioactive isotope with half-life \(3.00\ \mathrm{h}\).

    • (a) Find the decay constant in \(\mathrm{s^{-1}}\).
    • (b) Find the initial activity of the sample.
    • (c) How many undecayed nuclei remain after \(7.50\ \mathrm{h}\)?
    • (d) What is the activity after \(7.50\ \mathrm{h}\)?

[2.2] Balancing Nuclear Reactions and Identifying Decay Modes⚓︎

  1. ★☆☆
    Complete the following nuclear reaction and identify the decay mode:

    \[^{226}_{88}\mathrm{Ra}\rightarrow X+{}^4_2\mathrm{He}\]

  2. ★☆☆
    Complete the following nuclear reaction and identify the decay mode:

    \[^{14}_{6}\mathrm{C}\rightarrow X+{}^0_{-1}\mathrm{e}\]

  3. ★★☆
    Complete the following reaction:

    \[^{27}_{13}\mathrm{Al}+{}^4_2\mathrm{He}\rightarrow X+{}^1_0\mathrm{n}\]
    • (a) Find the mass number of \(X\).
    • (b) Find the atomic number of \(X\).
    • (c) Identify the nuclide \(X\).

  4. ★★☆
    Complete the following decay reaction and classify the decay:

    \[^{64}_{29}\mathrm{Cu}\rightarrow{}^{64}_{28}\mathrm{Ni}+X\]

  5. ★★★
    Complete the following reaction and classify the missing particle:

    \[^{23}_{11}\mathrm{Na}+{}^2_1\mathrm{H}\rightarrow{}^{24}_{12}\mathrm{Mg}+X\]
    • (a) Find \(A_X\).
    • (b) Find \(Z_X\).
    • (c) What particle is \(X\)?
    • (d) Explain why this is not an alpha particle or a proton.

  6. ★★★
    A uranium nucleus \(^{238}_{92}\mathrm{U}\) undergoes two alpha decays followed by one beta-minus decay.

    • (a) What is the final mass number?
    • (b) What is the final atomic number?
    • (c) What is the final nuclide?
    • (d) Would the final answer change if the beta-minus decay occurred between the two alpha decays instead?

[2.3] Nuclear Reaction Energetics and Q-Values⚓︎

  1. ★☆☆
    Consider the fusion reaction

    \[^2_1\mathrm{H}+{}^3_1\mathrm{H}\rightarrow{}^4_2\mathrm{He}+{}^1_0\mathrm{n}.\]

    Use the following atomic masses:

    \[m(^2_1\mathrm{H})=2.014102\ \mathrm{u},\quad m(^3_1\mathrm{H})=3.016049\ \mathrm{u},\]
    \[m(^4_2\mathrm{He})=4.002603\ \mathrm{u},\quad m_n=1.008665\ \mathrm{u}.\]
    • (a) Find the total mass of the reactants.
    • (b) Find the total mass of the products.
    • (c) Find the reaction energy \(Q\) in MeV.
    • (d) Is energy released or absorbed?

  2. ★☆☆
    Tritium can beta-minus decay according to

    \[^3_1\mathrm{H}\rightarrow{}^3_2\mathrm{He}+{}^0_{-1}\mathrm{e}+\bar\nu.\]

    The atomic mass of \(^3_1\mathrm{H}\) is \(3.016049\ \mathrm{u}\), and the atomic mass of \(^3_2\mathrm{He}\) is \(3.016029\ \mathrm{u}\).

    • (a) Using atomic masses, find the mass difference.
    • (b) Find the energy released in MeV.

  3. ★★☆
    Consider the reaction

    \[^7_3\mathrm{Li}+{}^1_1\mathrm{H}\rightarrow{}^4_2\mathrm{He}+{}^4_2\mathrm{He}.\]

    Use the following atomic masses:

    \[m(^7_3\mathrm{Li})=7.016004\ \mathrm{u},\quad m(^1_1\mathrm{H})=1.007825\ \mathrm{u},\]
    \[m(^4_2\mathrm{He})=4.002603\ \mathrm{u}.\]
    • (a) Find the total reactant mass.
    • (b) Find the total product mass.
    • (c) Find \(Q\) in MeV.
    • (d) Is this reaction exothermic or endothermic?

  4. ★★☆
    Consider the reaction

    \[^{14}_{7}\mathrm{N}+{}^4_2\mathrm{He}\rightarrow{}^{17}_{8}\mathrm{O}+{}^1_1\mathrm{H}.\]

    Use the following atomic masses:

    \[m(^{14}_{7}\mathrm{N})=14.003074\ \mathrm{u},\quad m(^4_2\mathrm{He})=4.002603\ \mathrm{u},\]
    \[m(^{17}_{8}\mathrm{O})=16.999132\ \mathrm{u},\quad m(^1_1\mathrm{H})=1.007825\ \mathrm{u}.\]
    • (a) Find the mass of the reactants.
    • (b) Find the mass of the products.
    • (c) Find \(Q\) in MeV.
    • (d) Is energy released or absorbed?

  5. ★★★
    Sodium-22 can decay by positron emission:

    \[^{22}_{11}\mathrm{Na}\rightarrow{}^{22}_{10}\mathrm{Ne}+{}^0_{+1}\mathrm{e}+\nu.\]

    The atomic mass of \(^{22}_{11}\mathrm{Na}\) is \(21.994436\ \mathrm{u}\), and the atomic mass of \(^{22}_{10}\mathrm{Ne}\) is \(21.991385\ \mathrm{u}\).

    • (a) Why is this classified as beta-plus decay?
    • (b) When using atomic masses, what mass difference should be used for a beta-plus decay?
    • (c) Find the energy released in MeV.

  6. ★★★
    Consider the reaction

    \[^{11}_{5}\mathrm{B}+{}^1_1\mathrm{H}\rightarrow3\left({}^4_2\mathrm{He}\right).\]

    Use the following atomic masses:

    \[m(^{11}_{5}\mathrm{B})=11.009305\ \mathrm{u},\quad m(^1_1\mathrm{H})=1.007825\ \mathrm{u},\]
    \[m(^4_2\mathrm{He})=4.002603\ \mathrm{u}.\]
    • (a) Verify that mass number and atomic number are conserved.
    • (b) Find the total mass of the reactants.
    • (c) Find the total mass of the products.
    • (d) Find \(Q\) in MeV and state whether energy is released or absorbed.