Comprehensive Notes on Atomic Theory, Nuclear Structure, and Radiochemistry

Historical Evolution of Atomic Theory

• Epistemological Foundations of Atomic Modeling

  • Scientific inquiry over 2,000 years ago lacked direct observational instrumentation to visualize atomic-scale phenomena.
  • Natural philosophers and early scientists relied upon indirect evidence to construct conceptual models.
  • Indirect Evidence: Evidence acquired without physically observing or touching the object of study.
  • Scientific Model: A conceptual framework that utilizes familiar ideas to elucidate unfamiliar phenomena observed in nature.
  • Philosophical perspective on scientific paradoxes, articulated by Niels Bohr: "The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth."

• The Ancient Greek Model (Democritus vs. Aristotle)

  • Proposed over 2,000 years ago by the Greek philosopher Democritus.
  • Posited that matter is discontinuous and cannot be subdivided into smaller pieces indefinitely; division ultimately terminates at an indivisible fundamental particle.
  • Designated the indivisible unit as the atom, derived from the Greek word atomos (meaning "not to be cut" or "indivisible").
  • Followers and proponents of Democritus's theoretical framework were known as Atomists.
  • Core Postulates of the Greek Atomic Hypothesis:
  • Atoms are small, hard particles.
  • Atoms are composed of the identical primordial material.
  • Atoms exist in an infinite variety of shapes and sizes.
  • Atoms are infinite in quantity.
  • Atoms remain in perpetual, continuous motion.
  • Atoms possess the capacity to collide and join together to constitute macroscopic matter.
  • Contemporaneous Opposition:
  • The atomic hypothesis garnered limited acceptance in antiquity.
  • Aristotle rejected atomic discontinuity, asserting instead that all matter is continuous and infinitely divisible without fundamental particulate boundaries.

• Dalton's Atomic Theory (1803)

  • Formulated in the early 1800s by English chemist John Dalton.
  • Grounded in experimental observations indicating that gases combine chemically as if composed of individual, discrete particles.
  • Formally proposed in 1803 as a comprehensive, five-postulate atomic model:
  • All elements are composed of extremely small, indivisible particles called atoms.
  • Atoms of a given element are identical in size, mass, and other physical/chemical properties. Atoms of differing elements differ systematically in size, mass, and properties.
  • Atoms cannot be subdivided, created, or destroyed during chemical processes.
  • Atoms of different elements combine in simple whole-number ratios to form chemical compounds.
  • In chemical reactions, atoms are combined, separated, or rearranged into new configurations.
  • Integration with Empirical Chemical Laws:
  • Law of Conservation of Mass: Supported by Postulates 3 and 5, demonstrating that chemical changes merely rearrange atomic groupings without net gain or loss of particulate matter.
  • Laws of Definite and Multiple Proportions: Supported by Postulate 4, confirming that specific chemical compounds consistently unite definite whole-number ratios of elemental atoms.

• Thomson's Plum Pudding Model and the Cathode Ray (1897)

  • Disproved the postulate of absolute atomic indivisibility through experimental physics conducted by English scientist J. J. Thomson.
  • Cathode Ray Tube Experimentation:
  • An electric discharge beam inside a vacuum tube deflected consistently toward a positively charged metal plate and away from a negatively charged plate.
  • Deflection behavior occurred identically regardless of the specific gas introduced into the cathode tube.
  • Demonstrated that cathode rays consist of universally occurring, negatively charged subatomic particles.
  • Thomson originally designated these negative entities corpuscles, subsequently renamed electrons (derived from the Greek term for amber).
  • Charge-to-mass quantification:
  • Robert Millikan experimentally determined the precise fundamental charge of the electron via the oil-drop experiment.
  • Calculations yielded an electron mass of $\frac{1}{1837}\,amu$ ($9.11 \times 10^{-28}\,g$).
  • The Structural Model:
  • Because bulk matter is electrically neutral, Thomson hypothesized the presence of balancing positive charge.
  • Proposed the Plum Pudding Model: the atom consists of a diffuse, uniform sphere of positive charge ("pudding") throughout which negatively charged electrons ("plums") are embedded.

• Rutherford's Gold Foil Experiment and the Nuclear Model (1911)
  • Conducted by British physicist Ernest Rutherford to probe the internal structure of Thomson's diffuse atomic model.
  • Experimental Protocol:
  • Fired a stream of heavy, positively charged alpha ($\alpha$) particles at an ultra-thin sheet of gold foil.
  • Theoretical Expectation: Under Thomson's plum pudding distribution, alpha particles would encounter negligible resistance and penetrate straight through the diffuse positive mass with minimal deflection.
  • Observed Results:
  • The vast majority of alpha particles passed directly through the gold foil undeflected.
  • A tiny fraction experienced extreme angular deflection, and occasionally an alpha particle rebounded backward toward the source.
  • Deductions and Conclusions:
  • The atom is composed predominantly of empty space, rather than a uniformly dense positive pudding.
  • The positive charge and virtually all atomic mass are concentrated within an extraordinarily dense, central core designated the nucleus.
  • Like charges repel; the alpha particle deflections were caused by intense electrostatic repulsion upon proximate encounters with the dense positive nucleus.
  • Scale analogy: The relative volume of the nucleus within an atom is proportional to that of a marble situated inside a baseball stadium.
  • Electrons occupy the vast volume surrounding the nucleus along the periphery of the atom.
  • Limitation: The Rutherford model failed to provide a stable mechanical description for the spatial configuration and non-radiating dynamics of orbiting electrons.

• The Bohr Planetary Model (1913)

  • Formulated by Danish scientist Niels Bohr (a student of Rutherford) to resolve electronic instability.
  • Postulates:
  • Electrons orbit the central nucleus in well-defined, discrete circular trajectories, analogous to planets revolving around the Sun.
  • Electronic orbits are constrained to specific radial distances from the nucleus.
  • Orbits possess discrete, quantized energy states termed energy levels.
  • Energy Level: The allowed discrete value of electron energy within an atom.
  • Limitation: While accurately predicting spectroscopic emissions for single-electron hydrogen, the planetary model failed to calculate or explain multi-electron atomic behavior.

• The Wave Mechanical and Modern Atomic Model

  • Advanced through principles of quantum wave mechanics.
  • Refuted the planetary trajectory hypothesis: electrons do not move in fixed, deterministic circular paths.
  • According to wave mechanics, the exact instantaneous position and momentum of an electron cannot be simultaneously determined.
  • Spatial distribution is represented as a probabilistic electron cloud, denoting orbital regions where the probability of finding an electron is maximized.
  • Synthesis of Modern Atomic Theory:
  • All matter is composed of atoms (Dalton).
  • Atoms of distinct elements differ in fundamental structure and mass (Dalton).
  • The atom contains a dense, central, positively charged nucleus containing nucleons (Rutherford).
  • The nucleus is surrounded by discrete energy levels occupied by electrons (Bohr).
  • Energy levels correspond to spatial electron clouds of varying orbital geometries (Wave Mechanics).
  • The total number of peripheral electrons balances nuclear protons, establishing overall atomic neutrality.

Fundamental Classical Laws of Chemical Combination

• Law of Conservation of Mass
  • Formal Definition: Mass is neither created nor destroyed during ordinary chemical reactions or physical transformations.
  • The total aggregate mass of reactants entering a chemical process is strictly identical to the total aggregate mass of the products formed.
  • Stoichiometric Demonstration:

$C + O_2 \rightarrow CO_2$

$12\,g\text{ (Carbon)} + 32\,g\text{ (Oxygen)} \rightarrow 44\,g\text{ (Carbon Dioxide)}$

• Demonstrates that the total initial mass ($12\,g + 32\,g = 44\,g$) perfectly balances the final product mass ($44\,g$).

  • Law of Multiple Proportions

• Formal Definition: When two or more distinct compounds are formed from combinations of the same two constituent elements, the ratio of the masses of the second element that combine with a fixed mass of the first element can always be reduced to a ratio of small whole numbers.

• Stoichiometric Demonstration:

  • Compare hydrogen peroxide ($H_2O_2$) and water ($H_2O$).
  • A constant fixed mass of hydrogen binds with $16\,g$ of oxygen in $H_2O_2$, whereas that same fixed mass of hydrogen binds with $8\,g$ of oxygen in $H_2O$.
  • Ratio of oxygen mass between the two compounds:

$\frac{16\,g}{8\,g} = \frac{2}{1} = 2:1$

• Law of Definite Proportions
  • Formal Definition: A specific chemical compound always contains its component elements in fixed, unvarying proportions by mass, regardless of sample size or source.
  • Worked Calculation for Magnesium Sulfide ($MgS$):
  • Given a sample of $100\,g$ of $MgS$, determine the constituent masses of magnesium ($Mg$) and sulfur ($S$).
  • Molar mass parameters: Molar mass of $Mg = 24.3\,g\,mol^{-1}$; Molar mass of $MgS = 56.4\,g\,mol^{-1}$.
  • Mass determination of Magnesium:

$\text{Mass of } Mg = 100\,g \times \left(\frac{\text{molar mass of } Mg}{\text{molar mass of } MgS}\right)$

$\text{Mass of } Mg = 100\,g \times \left(\frac{24.3\,g\,mol^{-1}}{56.4\,g\,mol^{-1}}\right) = 43.1\,g\text{ of } Mg$

* Mass determination of Sulfur:

$\text{Mass of } S = 100\,g - 43.1\,g\text{ of } Mg = 56.9\,g\text{ of } S$

* Elemental Mass Ratio in compound:

$\frac{43.1\,g}{56.9\,g} = 0.758 : 1$

* Conclusion: Regardless of initial sample size, magnesium and sulfur always combine in an exact mass ratio of $0.758:1$, corresponding to a fixed elemental mass percentage of $43.1\,\%$ $Mg$ and $56.9\,\%$ $S$.

Subatomic Structure and Isotopic Characterization

• Primary Subatomic Particles
  • Particles smaller than an individual atom, consisting of protons, neutrons, and electrons.
  • Proton ($p^+$ or $^1_1p$):
  • Discovered by Eugen Goldstein.
  • Located within the central nucleus.
  • Possesses an elementary relative electric charge of $+1$.
  • Relative mass is evaluated as $1\,amu$.
  • Absolute physical mass is $1.674 \times 10^{-24}\,g$.
  • Found in all atomic nuclei (every nucleus must contain at least one proton).
  • All protons are completely identical, irrespective of elemental origin.
  • Neutron ($n$ or $^1_0n$):
  • Discovered by James Chadwick in 1932.
  • Located within the central nucleus.
  • Possesses no electrical charge (electrically neutral, charge $0$).
  • Relative mass is evaluated as $1\,amu$.
  • Absolute physical mass is $1.675 \times 10^{-24}\,g$.
  • All neutrons are structurally identical across all atomic species.
  • Electron ($e^-$ or $^0_{-1}e$):
  • Discovered by J. J. Thomson.
  • Located in spatial energy levels / electron cloud external to the nucleus.
  • Possesses an elementary relative electric charge of $-1$.
  • Relative mass is evaluated as $\frac{1}{1837}\,amu$.
  • Absolute physical mass is $9.11 \times 10^{-28}\,g$.

• Subatomic Particle Properties Summary:

  • Electron: Symbol $e^-$; Relative charge $-1$; Mass $\frac{1}{1837}\,amu$; Actual mass $9.11 \times 10^{-28}\,g$; Location in energy levels.
  • Proton: Symbol $p^+$; Relative charge $+1$; Mass $1\,amu$; Actual mass $1.674 \times 10^{-24}\,g$; Location in nucleus.
  • Neutron: Symbol $n$; Relative charge $0$; Mass $1\,amu$; Actual mass $1.675 \times 10^{-24}\,g$; Location in nucleus.

• Metrological Definition of Atomic Mass Unit ($amu$)

  • Standard unit employed to quantify subatomic mass scales.
  • Defined precisely as exactly $\frac{1}{12}$ the mass of an unbound carbon-12 atom ($^{12}_6C$).

• Atomic Nomenclature and Quantities

  • Atomic Number ($Z$):
  • Defines the total number of protons housed in the nucleus.
  • Governs the elemental identity of the atom; atoms of different elements possess distinct atomic numbers.
  • Because all isolated atoms are electrically neutral:

$\text{Number of protons } (p^+) = \text{Number of electrons } (e^-)$

* Consequently, the atomic number also equals the total electron count in a neutral atom.

• Mass Number ($A$):
  • The integer sum of protons and neutrons in the nucleus:

$A = Z + N$

• Neutron Calculation:

$\text{Number of Neutrons } (N) = \text{Mass Number } (A) - \text{Atomic Number } (Z)$

• Atomic Mass: The weighted average mass of all naturally occurring stable and long-lived isotopes of an element.

• Nucleon: Generic term designating a constituent nuclear particle (protons and neutrons).

• Nuclide: An atomic species characterized by the specific constitution of its nucleus, defined by its nuclear composition ($A$ and $Z$).

  • Isotopes and Chemical Notations

• Isotopes: Atoms of the identical chemical element that possess identical atomic numbers ($Z$) but different mass numbers ($A$), caused by differences in neutron counts ($N$).

• Hyphen Notation: The formal name of the element joined by a hyphen to the mass number ($A$) (e.g., Uranium-235, Radium-288, Carbon-14).

• Nuclear Symbol Notation:

$^A_Z X$

• Where $A$ represents mass number, $Z$ represents atomic number, and $X$ represents elemental symbol (e.g., $^{235}_{92}U$, $^{288}_{88}Ra$, $^{14}_6C$).

  • Isotopic Variations of Hydrogen

• Protium ($Hydrogen-1$ or $^1_1H$):

  • Most prevalent natural form of hydrogen.
  • Nuclear metrics: Mass number $A = 1$, Atomic number $Z = 1$.
  • Constituent breakdown: $1\text{ proton}$, $0\text{ neutrons}$, $1\text{ electron}$.

• Deuterium ($Hydrogen-2$ or $^2_1H$):

  • Nuclear metrics: Mass number $A = 2$, Atomic number $Z = 1$.
  • Constituent breakdown: $1\text{ proton}$, $1\text{ neutron}$, $1\text{ electron}$.

• Tritium ($Hydrogen-3$ or $^3_1H$):

  • Radioactive isotope of hydrogen.
  • Nuclear metrics: Mass number $A = 3$, Atomic number $Z = 1$.
  • Constituent breakdown: $1\text{ proton}$, $2\text{ neutrons}$, $1\text{ electron}$.

Nuclear Forces, Mass Defect, and Binding Energy

• The Nuclear Force Mechanism

  • Protons carry identical positive charges and exert mutually repulsive electrostatic forces (Coulombic repulsion) across space.
  • Disassembly is prevented by the nuclear force (strong force).
  • Nuclear Force: An extremely powerful, short-range attractive force operating between proton-neutron, proton-proton, and neutron-neutron pairings that binds nucleons within the nucleus.
  • Functions analogously to nuclear "glue," overcoming long-range electrostatic repulsion provided nucleons are in immediate proximity.

• Mass Defect ($\Delta m$)

  • Direct experimental measurement reveals that the actual mass of an intact atom is invariably less than the theoretical sum of the separate individual masses of its constituent protons, neutrons, and electrons.
  • Mass Defect: The quantitative difference between the calculated sum of constituent nucleon/electron masses and the actual experimentally measured nuclear mass.
  • Origin: Attributable to the direct transformation of mass into energy occurring when free nucleons coalesce to form the bound nucleus.

• Nuclear Binding Energy

  • Quantified via Albert Einstein's mass-energy equivalence relation:

$E = mc^2$

• Binding Energy: The energy released when an atomic nucleus forms spontaneously from isolated nucleons, or conversely, the mechanical energy required to decompose an intact nucleus into free constituent nucleons.
• Binding Energy Per Nucleon:

$\text{Binding Energy per Nucleon} = \frac{\text{Total Nuclear Binding Energy}}{\text{Total Number of Nucleons } (A)}$

• Significance: Serves as the direct index of nuclear cohesion. A higher binding energy per nucleon indicates that nucleons are held together more tightly, signifying greater thermodynamic and nuclear stability.

  • The Band of Stability and Nucleon Ratios

• Band of Stability: The narrow zone on a graph plotting neutron count ($N$) versus proton count ($Z$) where stable, non-radioactive nuclides cluster.

• Low Atomic Number Nuclides ($Z \le 20$):

  • Nuclei achieve maximum stability when maintaining an exact $1:1$ neutron-to-proton ratio ($\frac{n}{p} = 1.0$).
  • Short-range nuclear attractive forces between adjacent nucleons effectively counter electrostatic repulsion.

• High Atomic Number Nuclides ($Z > 20$):

  • Stability shifts toward an elevated neutron-to-proton ratio approaching $1.5:1$ ($\frac{n}{p} = 1.5$).
  • Physical Cause: Electrostatic repulsion operates over longer physical ranges than the strong force. As proton quantity ($Z$) escalates, cumulative Coulombic repulsion expands across the entire volume of the nucleus. Additional neutrons are required to supply extra short-range attractive nuclear forces without introducing additive destabilizing positive charge.

• Threshold of Stability:

  • Beyond atomic number $83$ ($Z > 83$, elements heavier than bismuth), electrostatic repulsion overwhelms nuclear attraction.
  • All known nuclides possessing $Z > 83$ are unstable and undergo spontaneous radioactive decay.

Radioactive Decay Modes and Transmutation Reactions

• Fundamental Principles of Nuclear Reactions

  • Radioactive Decay: The spontaneous nuclear disintegration of an unstable nucleus into a lighter, more stable daughter nucleus, accompanied by the emission of particulate matter, electromagnetic radiation, or both.
  • Nuclear Radiation: Energetic particles or photons emitted from a decaying nucleus during a nuclear event.
  • Radioactive Nuclide: An unstable atomic species prone to spontaneous radioactive breakdown (e.g., all uranium isotopes).
  • Transmutation: A nuclear transformation that changes the proton number ($Z$), thereby altering the chemical identity of the parent element into an entirely different element.
  • Nuclear Conservation Laws: All nuclear equations must satisfy two invariant balance conditions:
  • Conservation of Mass Number: The sum of mass numbers ($A$) on the reactant side must equal the sum of mass numbers on the product side.
  • Conservation of Nuclear Charge: The sum of atomic numbers/charges ($Z$) on the reactant side must equal the sum of atomic numbers/charges on the product side.

• Comprehensive Survey of Decay Modes

  • Alpha Particle ($\alpha$ or $^4_2He$) Emission:
  • Nature: A helium-4 nucleus composed of 2 protons and 2 neutrons bound together.
  • Physical Constants: Electric charge is $+2$; physical mass is $4.0015062\,amu$.
  • Trigger: Restricted predominantly to very heavy nuclei ($Z > 83$) burdened by excess mass and total nucleon overpopulation.
  • Nuclear Alteration: Mass number decreases by 4; atomic number decreases by 2.
  • Exemplary Reactions:

$^{238}_{92}U \rightarrow ^{234}_{90}Th + ^4_2He$

$^{220}_{87}Fr \rightarrow ^{216}_{85}At + ^4_2He$

$^{212}_{84}Po \rightarrow ^{208}_{82}Pb + ^4_2He$

$^{222}_{86}Rn \rightarrow ^{218}_{84}Po + ^4_2He$

• Beta Particle ($\beta^-$ or $^0_{-1}e$ or $^0_{-1}\beta$) Emission:
  • Nature: A high-speed electron ejected directly from the nucleus.
  • Physical Constants: Electric charge is $-1$; physical mass is $0.0005486\,amu$.
  • Trigger: Occurs in neutron-rich nuclides positioned above the band of stability where the $\frac{n}{p}$ ratio is too large.
  • Nuclear Alteration: A nuclear neutron spontaneously converts into a nuclear proton and an ejected beta electron:

$^1_0n \rightarrow ^1_1p + ^0_{-1}e$

* *Net Result:* Mass number ($A$) remains constant; atomic number ($Z$) increases by 1.
* *Exemplary Reactions:*

$^{14}_6C \rightarrow ^{14}_7N + ^0_{-1}e$

$^{16}_7N \rightarrow ^{16}_8O + ^0_{-1}e$

$^{27}_{12}Mg \rightarrow ^{27}_{13}Al + ^0_{-1}e$

$^{42}_{19}K \rightarrow ^{42}_{20}Ca + ^0_{-1}e$

• Positron ($\beta^+$ or $^0_{+1}e$ or $^0_{+1}\beta$) Emission:
  • Nature: An emitted anti-electron (antimatter) bearing an identical mass to an electron with an opposite positive charge.
  • Physical Constants: Electric charge is $+1$; physical mass is $0.0005486\,amu$.
  • Trigger: Occurs in proton-rich nuclides situated below the band of stability where the $\frac{n}{p}$ ratio is too small.
  • Nuclear Alteration: A nuclear proton converts into a neutron while ejecting a positron:

$^1_1p \rightarrow ^1_0n + ^0_{+1}e$

* *Net Result:* Mass number ($A$) remains unchanged; atomic number ($Z$) decreases by 1.
* *Exemplary Reactions:*

$^{11}_6C \rightarrow ^{11}_5B + ^0_{+1}e$

$^{38}_{19}K \rightarrow ^{38}_{18}Ar + ^0_{+1}e$

$^{14}_8O \rightarrow ^{14}_7N + ^0_{+1}e$

• Electron Capture (EC):
  • Nature: The capture of an inner atomic orbital electron by the parent nucleus.
  • Trigger: Alternative mechanism for proton-rich nuclides below the band of stability ($\frac{n}{p}$ ratio too small).
  • Nuclear Alteration: The captured inner electron combines with a nuclear proton to synthesize a neutron:

$^1_1p + ^0_{-1}e \rightarrow ^1_0n$

* *Net Result:* Mass number ($A$) remains unchanged; atomic number ($Z$) decreases by 1.
* *Exemplary Reactions:*

$^{106}_{47}Ag + ^0_{-1}e \rightarrow ^{106}_{46}Pd$

$^{142}_{61}Pm + ^0_{-1}e \rightarrow ^{142}_{60}Nd$

• Gamma Ray ($\gamma$ or $^0_0\gamma$) Emission:

  • Nature: High-energy electromagnetic waves emitted directly from the nucleus.

  • Physical Constants: Electric charge is $0$; mass is $0$.

  • Mechanism: Emitted when a nucleus shifts from a metastable, highly excited nuclear state to a stable ground state.

  • Theoretical Link: Directly supports nuclear shell theory; mirrors photon emission when atomic electrons cascade from higher energy orbitals to lower energy levels.

  • Occurrence: Typically follows immediately after preceding alpha or beta decay processes that leave the daughter nucleus in an excited energetic configuration.

  • Net Result: Induces zero change in either mass number ($A$) or atomic number ($Z$).

  • Radiation Penetration and Attenuation

• Alpha Particles: Due to large mass ($\sim 4\,amu$) and high charge ($+2$), alpha radiation rapidly ionizes surrounding matter; travel distance is restricted to a few centimeters in air; fully shielded and blocked by a single sheet of paper.

• Beta Particles: Ejected at velocities approaching the speed of light; propagate several meters in air; exhibit penetrating capabilities approximately 100 times greater than alpha particles; effectively blocked by lead plates or thick glass.

• Gamma Rays: Pure high-energy photons possessing superior penetrating capability; travel vast distances unimpeded; require dense, high-Z shielding such as thick lead walls and/or substantial concrete barriers to achieve significant attenuation.

• Nuclear Fission
  • Definition: A reaction in which a very heavy, unstable nucleus divides into two or more intermediate-mass nuclei of higher relative stability.
  • Accompanied by the liberation of massive quantities of kinetic and radiative energy alongside excess free neutrons.
  • Can proceed spontaneously in select super-heavy nuclides or be artificially induced via bombardment with external thermal neutrons.

Decay Kinetics, Radiometric Dating, and Quantitative Applications

• Half-Life ($t_{1/2}$) Dynamics
  • Formal Definition: The duration of time required for exactly one-half ($50\,\%$) of the radioactive atomic nuclei in a given sample to undergo spontaneous decay.
  • Stability Correlation:
  • Highly stable nuclides decay slowly, exhibiting long half-lives.
  • Unstable, highly active nuclides disintegrate rapidly, exhibiting short half-lives.
  • Mathematical progression of mass decay:

$m(t) = m_0 \times \left(\frac{1}{2}\right)^n$

• Where $n = \frac{t}{t_{1/2}}$ represents the total elapsed half-life cycles.

• Empirical Demonstration with Radium-226 ($t_{1/2} = 1599\text{ years}$):

  • At elapsed time $t = 0\text{ years}$: Sample contains $10\,g$.

  • At elapsed time $t = 1599\text{ years}$ (1 half-life): Sample contains $5\,g$.

  • At elapsed time $t = 3198\text{ years}$ (2 half-lives): Sample contains $2.5\,g$.

  • Principles of Radiometric Dating

• Quantitative technique used to deduce the geological or historical age of materials based on fixed radioactive decay rates.

• Relies upon evaluating the precise ratio between remaining parent nuclides and accumulated stable daughter decay products.

• Carbon-14 Dating:

  • Carbon-14 is a beta-emitting radioactive nuclide with a half-life of $t_{1/2} = 5715\text{ years}$.

  • Used to establish the chronological age of historical organic remains up to approximately $50,000\text{ years}$.

  • Systematic Quantitative Half-Life Problem Portfolio

• Problem 1: Remaining Mass of Gold-198

  • Scenario: Initial mass of $100.0\,g$ of $^{198}Au$; half-life $t_{1/2} = 2.70\text{ days}$; elapsed duration $8.10\text{ days}$.
  • Cycle Calculation: $n = \frac{8.10\text{ days}}{2.70\text{ days}} = 3\text{ half-lives}$.
  • Residual Mass: $100.0\,g \times (0.5)^3 = 100.0\,g \times 0.125 = 12.5\,g$.

• Problem 2: Determination of Nitrogen-16 Half-Life

  • Scenario: A $50.0\,g$ sample of $^{16}N$ decays to $12.5\,g$ in $14.4\text{ seconds}$.
  • Cycle Determination: $\frac{12.5\,g}{50.0\,g} = 0.25 = \left(\frac{1}{2}\right)^2 \rightarrow n = 2\text{ cycles}$.
  • Half-Life Metric: $t_{1/2} = \frac{14.4\text{ seconds}}{2} = 7.20\text{ seconds}$.

• Problem 3: Residual Quantity of Potassium-42

  • Scenario: Initial mass of $750.\,g$ of $^{42}K$; half-life $t_{1/2} = 12.4\text{ hours}$; elapsed time $62.0\text{ hours}$.
  • Cycle Calculation: $n = \frac{62.0\text{ hours}}{12.4\text{ hours}} = 5\text{ half-lives}$.
  • Residual Mass: $750.\,g \times (0.5)^5 = 750.\,g \times \frac{1}{32} = 23.4375\,g \approx 23.4\,g$.

• Problem 4: Technetium-99 Half-Life Evaluation

  • Scenario: $500.\,g$ of $^{99}Tc$ decays to $62.5\,g$ across an epoch of $639,000\text{ years}$.
  • Cycle Determination: $\frac{62.5\,g}{500.\,g} = 0.125 = \left(\frac{1}{2}\right)^3 \rightarrow n = 3\text{ cycles}$.
  • Half-Life Metric: $t_{1/2} = \frac{639,000\text{ years}}{3} = 213,000\text{ years}$.

• Problem 5: Thorium-232 Original Sample Mass

  • Scenario: Half-life $t_{1/2} = 1.4 \times 10^{10}\text{ years}$; remaining mass is $25.0\,g$ after an interval of $2.8 \times 10^{10}\text{ years}$.
  • Cycle Calculation: $n = \frac{2.8 \times 10^{10}\text{ years}}{1.4 \times 10^{10}\text{ years}} = 2\text{ half-lives}$.
  • Initial Mass: $m_0 = 25.0\,g \times 2^2 = 25.0\,g \times 4 = 100.0\,g$.

• Problem 6: Iodine-131 Original Source Mass

  • Scenario: $5.0\,g$ of $^{131}I$ remains after $40.35\text{ days}$; half-life $t_{1/2} = 8.07\text{ days}$.
  • Cycle Calculation: $n = \frac{40.35\text{ days}}{8.07\text{ days}} = 5\text{ half-lives}$.
  • Initial Mass: $m_0 = 5.0\,g \times 2^5 = 5.0\,g \times 32 = 160\,g$.

• Problem 7: Zinc-71 Elapsed Decay Time

  • Scenario: Half-life of $^{71}_{30}Zn$ is $2.4\text{ minutes}$; determines time to reach $\frac{1}{16}$ initial mass.
  • Cycle Determination: $\frac{1}{16} = \left(\frac{1}{2}\right)^4 \rightarrow n = 4\text{ cycles}$.
  • Elapsed Time: $t = 4 \times 2.4\text{ minutes} = 9.6\text{ minutes}$.

• Problem 8: Sodium-24 Circulatory Tracer Atom Count

  • Scenario: Initial count $N_0 = 5.85 \times 10^{23}\text{ atoms}$ of $^{24}_{11}Na$; half-life $t_{1/2} = 4.0\text{ days}$; elapsed duration $12.0\text{ days}$.
  • Cycle Calculation: $n = \frac{12.0\text{ days}}{4.0\text{ days}} = 3\text{ half-lives}$.
  • Surviving Atoms: $N = 5.85 \times 10^{23} \times (0.5)^3 = 5.85 \times 10^{23} \times 0.125 = 7.3125 \times 10^{22} \approx 7.31 \times 10^{22}\text{ atoms}$.

• Problem 9: Iodine-131 Therapeutic Dose Depletion

  • Scenario: Initial activity of $7.73 \times 10^{12}\text{ atoms}$ of $^{131}_{53}I$ reduces to $7.55 \times 10^9\text{ atoms}$; half-life $t_{1/2} = 8.07\text{ days}$.

  • Fractional Survival: $\frac{7.55 \times 10^9}{7.73 \times 10^{12}} \approx 0.0009767 \approx \frac{1}{1024} = \left(\frac{1}{2}\right)^{10} \rightarrow n = 10\text{ half-lives}$.

  • Elapsed Time: $t = 10 \times 8.07\text{ days} = 80.7\text{ days}$.

  • Selected Balanced Transmutation Reference Equations

• Transmutation of Beryllium by alpha bombardment:

$^9_4Be + ^4_2He \rightarrow ^{12}_6C + ^1_0n$

• Synthesis of Mendelevium via Einsteinium bombardment:

$^{253}_{99}Es + ^4_2He \rightarrow ^{256}_{101}Md + ^1_0n$

• Deuterium-Lithium fusion:

$^6_3Li + ^2_1H \rightarrow ^4_2He + ^4_2He$

• Deuterium-Tritium thermonuclear fusion:

$^2_1H + ^3_1H \rightarrow ^4_2He + ^1_0n$

• Alpha-induced transmutation of Nitrogen-14:

$^{14}_7N + ^4_2He \rightarrow ^{17}_8O + ^1_1p$

• Alpha-induced transmutation of Aluminum-27:

$^{27}_{13}Al + ^4_2He \rightarrow ^{30}_{15}P + ^1_0n$

• Thermal neutron-induced fission of Uranium-235:

$^{235}_{92}U + ^1_0n \rightarrow ^{142}_{56}Ba + ^{91}_{36}Kr + 3\,^1_0n$