4.6 Nuclear Chemistry: Stability, Decay, Transmutation, and Radioanalytical Methods

4.6 Nuclear Chemistry: Stability, Decay, Transmutation, and Radioanalytical Methods

Quick Answer: Nuclear chemistry is a 10-item area of the Inorganic Chemistry TOS. Analytical Chemistry adds radioanalytical methods. Expect to balance nuclear equations, predict decay modes from neutron-to-proton ratios, compute binding energies from mass defects, count steps in decay series, and handle first-order decay, isotope dilution, and activation analysis.

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The Nucleus and Binding Energy

A nuclide $^{A}_{Z}X$ has $Z$ protons and $N = A - Z$ neutrons. The mass of a nucleus is smaller than the sum of its free nucleons; this mass defect $\Delta m$ appears as binding energy via $E = \Delta m\,c^2$, with $1\text{ u} = 931.494\text{ MeV}$.

Worked example (helium-4, using atomic masses):

$\Delta m = 2(1.007825) + 2(1.008665) - 4.002603 = 0.030377\text{ u}$

$E_B = 0.030377 \times 931.494 = 28.30\text{ MeV} \quad (7.07\text{ MeV per nucleon})$

Binding energy per nucleon rises steeply for light nuclei and peaks near $^{56}Fe$ and $^{62}Ni$ (about 8.8 MeV). Fusion of light nuclei and fission of heavy nuclei both release energy by moving toward that peak.

The Band of Stability

• Stable light nuclei have $N/Z \approx 1$; heavier stable nuclei need extra neutrons ($N/Z \approx 1.5$ near lead).
• Nuclei with even numbers of both protons and neutrons are most often stable. Magic numbers (2, 8, 20, 28, 50, 82, 126) give extra stability, so $^{4}He$, $^{16}O$, $^{40}Ca$, and $^{208}Pb$ are especially stable.
• Every element beyond bismuth ($Z > 83$) has only radioactive isotopes.

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Decay Modes

Alpha emitters are relatively harmless outside the body but very damaging when inhaled or ingested, because their energy is deposited over a very short path.

Decay Series

Uranium-238 decays through a chain to stable lead-206. Because only alpha decay changes $A$:

• Number of alpha decays: $(238 - 206)/4 = 8$, lowering $Z$ by 16 to 76
• Number of beta-minus decays: $82 - 76 = 6$

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Nuclear Reactions: Transmutation, Fission, and Fusion

Balance nuclear equations by conserving mass number and charge.

• Rutherford's transmutation (1919): $^{14}_{7}N + ^{4}_{2}He \rightarrow ^{17}_{8}O + ^{1}_{1}H$
• Fission: $^{235}_{92}U + ^{1}_{0}n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3\,^{1}_{0}n$, releasing about 200 MeV per fission. The extra neutrons can sustain a chain reaction above the critical mass.
• Fusion: $^{2}_{1}H + ^{3}_{1}H \rightarrow ^{4}_{2}He + ^{1}_{0}n$, releasing 17.6 MeV.

The Q-value of a reaction is the mass difference between reactants and products, times 931.494 MeV per u. A positive Q means energy is released.

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Decay Kinetics and Units

Radioactive decay is first order and independent of temperature, pressure, and chemical form:

$N = N_0 e^{-\lambda t}, \qquad A = \lambda N, \qquad t_{1/2} = \frac{\ln 2}{\lambda}$

Activity is measured in becquerels ($1\text{ Bq} = 1$ decay per second) or curies ($1\text{ Ci} = 3.7 \times 10^{10}\text{ Bq}$). Absorbed dose is in grays ($1\text{ Gy} = 1\text{ J/kg}$), and equivalent dose, which weights radiation type, is in sieverts. Section 10.1 applies these equations to radiocarbon dating.

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Detection and Radioanalytical Methods

Isotope Dilution

Add a known mass $m_s$ of tracer with specific activity $S_s$ (activity per mass) to the sample, mix, isolate any pure portion of the analyte, and measure its specific activity $S_m$:

$m_x = m_s\left(\frac{S_s}{S_m} - 1\right)$

Recovery need not be complete, because only the ratio of activity to mass is measured.

Neutron Activation Analysis (NAA)

Irradiating a sample with neutrons converts stable nuclei into radioactive ones. Their characteristic gamma rays identify and quantify elements at trace levels. Induced activity is $A = N\sigma\phi(1 - e^{-\lambda t_{irr}})$, where $\sigma$ is the capture cross-section and $\phi$ the neutron flux. NAA is non-destructive and multi-element, and it is widely used for trace elements in geological, forensic, and environmental samples.

Radiotracers and Dating

Tracers such as $^{32}P$, $^{35}S$, and $^{14}C$ follow reaction pathways and metabolism. Radiometric dating uses known half-lives: $^{14}C$ (5730 y) for organic materials up to about 50,000 years, and the $^{40}K$/$^{40}Ar$ or $^{238}U$/$^{206}Pb$ systems for rocks.