Nuclear Chemistry

LECTURE 1

Significant Figures and Units

• Measurement: Consists of a number and a unit.

• Significant Figures (sf): Digits that are meaningful to the precision of a measurement.

• Rules for Significant Figures:

  1. All non-zero digits are significant (e.g., $424$ has $3\,\text{sf}$).

  2. Zeros between non-zero digits are significant (e.g., $40024$ has $5\,\text{sf}$, $4.0024$ has $5\,\text{sf}$).

  3. Leading zeros are not significant (e.g., $024$ has $2\,\text{sf}$, $0.024$ has $2\,\text{sf}$).

  4. Trailing zeros in a number with a decimal point are significant (e.g., $42.400$ has $5\,\text{sf}$, $424.00$ has $5\,\text{sf}$).

  • Scientific Notation Check: If zeros disappear when written in scientific notation (e.g., $42400 = 4.24 \times 10^4$), they are not significant ($3\,\text{sf}$).

The International System of Units (SI)

• Fundamental Units:

  • Mass: kilogram ($kg$)

  • Length: metre ($m$)

  • Luminous intensity: candela ($cd$)

  • Time: second ($s$)

  • Electric current: ampere ($A$)

  • Temperature: Kelvin ($K$)

  • Amount of substance: mole ($mol$)

• Derived Units: All other units are generated from these (e.g., Force in Newtons ($N$): $F = m \times a = kg \times m\,s^{-2}$).

• Unit Analysis Examples:

  • Pressure ($P$): $\text{Force} \div \text{area} = (kg\,m\,s^{-2}) \times m^{-2} = kg\,m^{-1}\,s^{-2}$

  • Work ($W$): $\text{Force} \times \text{distance} = (kg\,m\,s^{-2}) \times m = kg\,m^2\,s^{-2}$

  • Energy Density: $\text{Work} \div \text{volume} = (kg\,m^2\,s^{-2}) \times m^{-3} = kg\,m^{-1}\,s^{-2}$

Calculations with Measured Quantities

• Precision Rule: Answers must be reported using the same number of significant figures as the datum with the fewest significant figures.

• Exact Numbers: Conversion factors (e.g., $1\,km = 1000\,m$, $1\,h = 60\,min$) are exact and do not limit precision.

• Rounding: Round only at the end of a calculation to avoid compounding errors.

• Worked Example (Express Train):

  • Constant speed: $62.3\,km/h$ ($3\,\text{sf}$).

  • Time: $38\,minutes$ ($2\,\text{sf}$).

  • Speed in $m\,s^{-1}$: $\frac{62.3 \times 1000}{3600} = 17.30556 \rightarrow 17.3\,m\,s^{-1}$.

  • Time in hours: $38 \div 60 = 0.633333 \rightarrow 0.63\,h$.

  • Distance between signals in $km$: $62.3 \times 0.633333 = 39.4567 \rightarrow 39\,km$ ($2\,\text{sf}$ due to time).

  • Distance if speed continues for 7 more minutes:

    • Total time: $38 + 7 = 45\,minutes$.

    • Total distance: $62.3 \times (45 \div 60) = 46.725 \rightarrow 47\,km$.

Scale and Orders of Magnitude

• Orders of Magnitude: Powers of 10 representing the size of a value. Useful for ‘sanity checks’ and approximate comparisons.

• Prefixes Table:

  • Tera ($T$): $10^{12}$

  • Giga ($G$): $10^{9}$

  • Mega ($M$): $10^{6}$

  • Kilo ($k$): $10^{3}$

  • Deca ($da$): $10^{1}$

  • Deci ($d$): $10^{-1}$

  • Centi ($c$): $10^{-2}$

  • Milli ($m$): $10^{-3}$

  • Micro ($\mu$): $10^{-6}$

  • Nano ($n$): $10^{-9}$

  • Pico ($p$): $10^{-12}$

• Magnitude Concepts:

  • A ‘googol’ refers to the order of magnitude of $10^{100}$.

  • Hydrogen atom diameter: Approximately $10^{-10}\,m$.

  • White blood cell diameter: Approximately $10^{-5}\,m$.

LECTURE 2

Sub-Atomic Particles and Scale

• An atom is an electrically neutral particle consisting of a positively charged nucleus and a cloud of negatively charged electrons.

• The nucleus consists of protons ($p^+$) and neutrons ($n^0$), which are collectively referred to as nucleons.

• Atomic Mass Unit ($a.m.u.$): Defined by setting the mass of carbon-12 to exactly $12 \, a.m.u.$. $1 \, a.m.u. \approx 1.66 \times 10^{-27} \, kg$.

• Comparison of Particles:

  • Proton ($p$): Charge $+1$, Mass $1.00726 \, a.m.u.$.

  • Neutron ($n$): Charge $0$, Mass $1.00865 \, a.m.u.$.

  • Electron ($e^-$): Charge $-1$, Mass $0.000549 \, a.m.u.$.

  • Positron ($e^+$): Charge $+1$, Mass $0.000549 \, a.m.u.$.

• Spatial Scale: If the diameter of a nucleus were $1 \, cm$, the atom's total size would be approximately $100 \, m$. The overall diameter of an atom is approximately $1 \, \text{Å}$ (Ångström), though almost all the mass is concentrated in the nucleus.

Nuclear Notation and Isotopes

• Notation Format: ${}^{A}_{Z}X$

  • $A$ (Mass Number): The sum of protons and neutrons.

  • $Z$ (Atomic Number): The number of protons in the nucleus or the charge of the particles.

  • $N$ (Number of Neutrons): Calculated as $N = A - Z$.

• Nuclide: An atom with a specific mass number and atomic number.

• Isotopes: Nuclides that share the same atomic number ($Z$) but have different mass numbers ($A$).

• Isotopes of Carbon:

  • ${}^{12}_{6}C$: Stable; accounts for $98.89\%$ of natural carbon.

  • ${}^{13}_{6}C$: Stable; accounts for $1.11\%$ of natural carbon.

  • ${}^{14}_{6}C$: Unstable nucleus; found in trace amounts in living matter.

  • ${}^{9}_{6}C$ and ${}^{11}_{6}C$: Unstable; ${}^{11}_{6}C$ can be prepared by nuclear reaction in a cyclotron.

Mass Spectrometry and Spectroscopy

• Spectroscopy: A collection of techniques used to identify atoms and molecules.

  • Electronic Spectroscopy: Examines electrons.

  • Vibrational/Rotational Spectroscopy: Examines atomic and molecular motions.

• Mass Spectrometry: Used to measure atomic and molecular masses. The instrument operates under a vacuum and follows these steps:

  1. A neutral gas or vapor is introduced through a nozzle.

  2. An electron beam ionizes the sample.

  3. Ions are accelerated in an electric field.

  4. Ions are deflected by a magnet according to their mass-to-charge ratio ($m/z$). Lighter ions move faster and are deflected more.

  5. A position-sensitive detector (electron multipliers, photomultipliers, or photographic film) measures the number and location of ions.

• Historical Developments in Mass Spectrometry:

  • 1920: Aston measured isotopes of Neon ($20, 22$), Sulfur ($32, 33, 34$), Silicon ($28, 29, 30$), and Krypton ($78, 80, 82, 83, 84, 86$).

  • 1946: Pulsed gas injection and time-of-flight detectors.

  • 1956: Identification of complex organic molecules.

  • 1974: Viking Mars lander utilized mass spectrometry.

  • 1977: Accelerator mass spectrometry for trace analysis.

  • 1985: $C_{60}$ identified (Nobel Prize 1996).

  • 2002: Ionization developments for studying large biomolecules.

Atomic Mass Calculations

• The atomic mass of an element is the weighted average of the atomic masses and abundances of its naturally occurring isotopes.

• Relative Atomic Weight of Carbon: Calculated using isotope data:

  • ${}^{12}_{6}C$: $98.90\%$ abundance, $12.000000 \, a.m.u.$

  • ${}^{13}_{6}C$: $1.110\%$ abundance, $13.003354 \, a.m.u.$

  • ${}^{14}_{6}C$: <1 \times 10^{-10}\% abundance, $14.003421 \, a.m.u.$

  • Calculation: $(0.989 \times 12.000000) + (0.0111 \times 13.003354) + (1 \times 10^{-12} \times 14.003421) = 12.01 \, a.m.u.$

Nucleogenesis

• Elements are generated from the simplest nuclide, hydrogen (${}^{1}_{1}H$ or a proton), via nuclear reactions.

• Big Bang: Responsible for the formation of protons and neutrons, leading to isotopes of $H$, $He$, and $Li$.

• Stellar Nucleogenesis: Clouds of hydrogen are compressed by gravity; once temperatures are high enough, they ignite as stars. Hans Bethe studied stellar nucleogenesis (1967).

• Proton-Proton Chain (Helium Formation):

  1. Sequential reactions involving high-energy radiation ($\gamma$) which has no mass or charge.

  2. Overall Reaction: $4{}^{1}_{1}H \rightarrow {}^{4}_{2}He + 2{}^{0}_{1}e + 2\gamma + 2\nu$

  3. This reaction is exothermic, releasing energy as heat and radiation.

  4. Conservation must be maintained for both total mass number ($A$) and charge ($Z$).

Energy and Probability of Fusion

• Energy Calculation ($E = mc^2$):

  • Mass of $4$ protons: $6.69048632 \times 10^{-27} \, kg$

  • Mass of $1$ Helium nucleus: $6.64465675 \times 10^{-27} \, kg$

  • Mass of $2$ positrons: $1.82187638 \times 10^{-30} \, kg$ ($2 \times 9.10938188 \times 10^{-31} \, kg$)

  • Net mass of products: $6.64647862 \times 10^{-27} \, kg$

  • Change in mass ($\Delta m$): $0.0440 \times 10^{-27} \, kg$

  • Energy released: $E = (0.0440 \times 10^{-27} \, kg) \times (3.00 \times 10^{8} \, m \, s^{-1})^2 = 3.96 \times 10^{-12} \, J$ per reaction.

• Probabilistic Nature: Proton-proton fusion is the rate-limiting step in stars. Its cross-section (measured in 'barns') is so small ($10^{26}$ times smaller than deuterium-tritium fusion) that it is calculated theoretically rather than measured accurately.

• Energy Comparison:

  • Nuclear Hydrogen "burning": $2.5 \times 10^{12} \, J \, mol^{-1}$ of reaction or $6.2 \times 10^{11} \, J \, mol^{-1}$ of $H$ atoms.

  • Chemical Hydrogen burning ($H_2 + \frac{1}{2}O_2 \rightarrow H_2O$): $1.4 \times 10^5 \, J \, mol^{-1}$ of $H$ atoms.

  • Nuclear reactions liberate approximately 4 million times more energy than chemical reactions.

Life Cycle of Stars and Element Distribution

• Hydrogen Burning: Occurs at temperatures ($T$) around $10^7 \, K$.

• Helium Burning: Fuses heavier nuclei to form larger atoms as hydrogen is exhausted; occurs at T < 2 \times 10^8 \, K.

• Heavier Nuclei Synthesis:

  • Red Giants: Produce isotopes like ${}^{13}C$, ${}^{13}N$, ${}^{14}N$, ${}^{15}N$, and ${}^{15}O$.

  • Supergiants: Form heavier nuclei (Carbon and Oxygen burning) up to $Ca$ and $Ni$ ($40Ca \dots 58Ni$) at T < 3 \times 10^9 \, K.

  • Supernovae: Responsible for true heavy elements (Z > 26).

• Summary of Origins:

  • Big Bang: $H$, $He$, $Li$.

  • Low Mass Stars: ${}^{4}He$, ${}^{12}C$, ${}^{14}N$.

  • High Mass Stars: All elements from $O$ up to $Fe$.

  • Supernovae: Everything heavier than $Fe$.

  • Human Manufactured: Elements with Z > 94 (e.g., ${}^{279}Nh$, ${}^{113}Nh$).

Questions & Discussion

• Question 1: Using NIST isotopic abundance data for germanium, calculate its relative atomic mass based on most common isotopes.

• Question 2: Nihonium ($113Nh$) synthesis: To attempt the synthesis of ${}^{279}Nh$, ${}^{209}Bi$ was bombarded with which nuclide?

• Question 3: In the fission of ${}^{235}U$ struck by a neutron, ${}^{141}Ba$ and ${}^{92}Kr$ are produced. Identify the number of neutrons also produced in this reaction.

LECTURE 3

Origins and Formation of the Chemical Elements

• Nucleogenesis and Stellar Evolution

  • The Big Bang: This event served as the initial source for the lightest elements.

  • Hydrogen Burning: Occurs within the cores of stars, converting hydrogen into heavier elements.

  • Helium Burning: As stars expand into red giants, helium burning begins in the core.

  • Carbon Burning: Takes place in the cores of red supergiants.

  • Supernova: Elements heavier than iron are produced during supernova explosions.

  • Human-Manufactured Elements: Elements with an atomic number Z > 94 are entirely human-manufactured and do not occur naturally in significant quantities.

Fundamental Concepts of Radioactivity

• Stability of Atomic Nuclei

  • Nuclei are classified as either stable or radioactive.

  • There are no stable nuclei for elements with an atomic number higher than Lead ($Pb$).

  • Elements lighter than Lead can have both stable and radioactive isotopes.

  • Until 2003, Bismuth-209 ($^{209}Bi$) was thought to be stable, but its radioactivity was discovered that year. Currently, no stable isotopes of Bismuth are known.

• The Decay Process

  • Radioactive nuclei undergo spontaneous decay through a decay series.

  • This series continues until a stable nucleus is reached.

  • Each step involves a parent radionuclide transforming into a daughter nuclide.

  • Each specific transformation is characterized by a unique mode of decay and a specific decay rate or half-life ($t_{1/2}$).

  • An example of a complex decay path is that of Uranium-238 ($^{238}U$).

Mechanisms of Radioactive Decay

• General Rules for Nuclear Reactions

  • The mass number ($A$) must be balanced on both sides of the equation.

  • The charge or atomic number ($Z$) must be balanced on both sides of the equation.

  • Equations do not typically include "normal" orbital electrons; they do not account for ion formation.

• Specific Decay Modes

  • Alpha ($\alpha$) Decay: The emission of an alpha particle, which is a helium nucleus with a mass of 4 and a charge of $2+$. It is represented as $^4_2He$, $^4_2\text{He}^{2+}$, or ^4_2\text{\alpha}.

  • Beta ($\beta^-$) Decay: An electron ($\beta$ or $\beta^-$) is ejected from the nucleus. During this reaction, a neutron is converted into a proton to maintain charge balance: $^1_{0}n \rightarrow ^1_{1}p + ^0_{-1}e$.

  • Positron ($\beta^+$) Decay: A positron is ejected from the nucleus. A proton is converted into a neutron: $^1_{1}p \rightarrow ^1_{0}n + ^0_{1}e$. The emitted positron typically annihilates upon colliding with an electron in the surrounding environment.

  • Electron Capture: An inner-shell electron is captured by the nucleus and combines with a proton to form a neutron. This process is followed by the emission of X-rays as outer-shell electrons drop to lower energy states to fill the vacancy.

  • Neutron Emission: The simple emission of a neutron ($^1_0n$). This changes the mass number ($A$) of the nuclide but does not change the atomic number ($Z$).

  • Gamma ($\gamma$) Emission: High-frequency electromagnetic radiation. It often accompanies other forms of decay and does not result in a change to $A$ or $Z$ by itself. A metastable or excited state is denoted by "m" (e.g., $^{99\text{m}}Tc$) or an asterisk (e.g., $Tc^*$).

• Distinguishing Modes

  • Positron decay and electron capture can be difficult to distinguish if only the parent and daughter nuclei are known, as both result in the same change in $Z$ and $A$.

Kinetics of Nuclear Decay

• Activity and Decay Rates

  • Activity ($A$) is the rate of emission or the negative rate of disappearance of the parent nuclide.

  • The number of nuclei decaying per unit time ($A$) is directly proportional to the number of nuclei present ($N$).

  • $A = - \frac{\Delta N}{\Delta t} = \lambda N$

  • As $\Delta t \rightarrow 0$, the differential form is $A = - \frac{dN}{dt} = \lambda N$.

  • The Decay Constant ($\lambda$): This is the proportionality constant, measured in units of inverse time ($s^{-1}$).

  • Units of Activity: The Becquerel ($Bq$) is defined as 1 disintegration per second ($1 Bq = 1 s^{-1}$).

• First-Order Kinetic Process

  • Radioactive decay follows first-order kinetics. The rate of reaction depends linearly on the amount of reactant ($N$) present.

  • Rearranging the differential equation: $\frac{dN}{N} = -\lambda dt$.

  • Integrating from $t = 0$ to time $t$ gives the integrated rate law: $\ln(N_t) - \ln(N_0) = -\lambda t$.

  • Expressed exponentially: $N_t = N_0 e^{-\lambda t}$.

• Half-Life ($t_{1/2}$)

  • Half-life is the time required for the number of nuclei to decrease from $N_0$ to $\frac{1}{2} N_0$.

  • Substituting into the integrated rate law: $\ln(\frac{\frac{1}{2} N_0}{N_0}) = -\lambda t_{1/2}$, which simplifies to $\ln(\frac{1}{2}) = -\lambda t_{1/2}$.

  • The relationship is defined as: $\lambda = \frac{\ln(2)}{t_{1/2}}$.

  • A large decay constant ($\lambda$) corresponds to a short half-life ($t_{1/2}$).

  • Both the number of nuclei and the total activity of the sample are halved after each half-life period.

Chemical Applications of Radioisotopes

• Isotopic Tracers and Labels

  • Isotopes are chemically almost identical. Highly active radioisotopes can therefore be used to label or trace chemical species through reactions.

  • Long-lived isotopes allow for the safe study of chemical structures and transformations when used in small quantities.

  • Example Case: The reduction of the solid structure of $SrUO_4$ can be studied using depleted Uranium ($^{238}U$, $t_{1/2} = 4.5 \times 10^9$ years) in the presence of $H_2(g)$ at $1000^\circ C$.

Radiometric Dating with Carbon-14

• The Carbon-14 Cycle

  • Cosmic rays in the upper atmosphere continuously generate unstable Carbon-14 ($^{14}C$).

  • Plants incorporate ${}^{14}CO_2$ during photosynthesis; animals then ingest these plants.

  • Living organisms maintain a constant ${}^{14}C:{}^{12}C$ ratio by constantly replenishing carbon from the environment.

  • Upon death, ingestion stops. $^{14}C$ begins to disappear via beta decay with a half-life of $5730$ years.

• Dating Methodology

  • The age of a sample is determined by comparing the remaining activity ($A_t$) or amount of $^{14}C$ ($N_t$) in the sample to a "fresh" or standard sample ($A_0$ or $N_0$).

  • The calculation uses the formula: $\ln(\frac{A_0}{A_t}) = \lambda t$.

• Measurement techniques for ${}^{14}C$

  • Scintillation Counter: Measures the intensity of light (scintillation) emitted when a sample is exposed to ionizing radiation. It is calibrated against a standard to read activity directly.

  • Accelerator Mass Spectrometry (AMS): A high-precision technique that measures the actual isotopic composition. It determines the ratios of $^{14}C/^{13}C$ and $^{13}C/^{12}C$. Because $^{13}C$ and $^{12}C$ are stable, these ratios provide an accurate age. It can measure very small samples and carries an effective limit of approximately $50,000$ years, whereas activity-based scintillation is limited to about $10,000$ years.

Normalized Activity Measures

• Specific and Molar Activity

  • Activity is dependent on the amount of material. Normalization allows for standardized comparison.

  • Specific Activity: The activity per unit mass of the radionuclide ($Bq/g$).

  • Molar Activity: The activity per mole of the radionuclide ($Bq/mol$).

  • Specific activity, molar activity, the decay constant ($\lambda$), and half-life ($t_{1/2}$) are intrinsic properties of each radionuclide and are interconvertible.

• Calculation Example: Uranium-234 ($^{234}U$)

  • Given: $117\,g$ of $^{234}U$ has an activity of $1.45 \times 10^6\,Bq$.

  • Specific Activity Calculation: $\frac{1.45 \times 10^6\,Bq}{117\,g} = 12,400\,Bq\,g^{-1}$.

  • Molar Activity Calculation: $\frac{1.45 \times 10^6\,Bq}{(117\,g / 234\,g\,mol^{-1})} = 2.90 \times 10^9\,Bq\,mol^{-1}$.

Quantitative Exercises and Scenarios

• Decay of Strontium-90 ($^{90}Sr$)

  • $^{90}Sr$ undergoes $\beta^-$ decay with a half-life of $28.8\,\text{years}$.

  • Reaction: $^{90}_{38}Sr \rightarrow ^{90}_{39}Y + ^{0}_{-1}\beta$.

  • The daughter nuclide, Yttrium-90, is also radioactive, decaying via $\beta^-$ with a half-life of $60\,\text{hours}$ to a stable product ($^{90}Zr$).

  • Calculations involve determining the decay constant, specific activity, and the time required for activity to drop to $10\%$ ($0.10 A_0$).

• Sequential Decay Kinetics

  • Consider the decay of Potassium-37 ($^{37}K$) to Argon-37 ($^{37}Ar$) via positron emission ($t_{1/2} = 1.2\,\text{seconds}$), followed by Argon-37 decaying to Chlorine-37 ($^{37}Cl$) via electron capture ($t_{1/2} = 840\,\text{hours}$).

  • The change in the number of $^{37}Ar$ nuclei over time is represented by the differential equation: $\frac{dN_{Ar}}{dt} = \lambda_K N_K - \lambda_{Ar} N_{Ar}$.

  • This equation shows the rate of change is the difference between the rate of production (from Potassium) and the rate of decay (into Chlorine).

  • In a graph of $N_{Ar}$ versus time, the very short half-life of Potassium causes a rapid initial increase in Argon nuclei, while the much longer half-life of Argon governs the subsequent slow decay of the Argon population.

LECTURE 4

Atmospheric Carbon-14 Variations and Trends

• Carbon-14 ($^{14}C$) levels in the atmosphere are not constant and undergo fluctuations due to various anthropogenic and natural phenomena.

• The "bomb pulse" refers to a significant spike in tropospheric $^{14}C$ resulting from mid-20th-century nuclear testing. Research into carbon-14 in ivory helps track these levels, showing varying concentrations between the Northern and Southern Hemispheres.

• The relative abundance of $^{14}C$ compared to $^{12}C$ is currently decreasing.

• The "Suess effect," named after Hans Suess, describes the dilution of atmospheric $^{14}C$ caused by the burning of fossil fuels. Fossil fuels are derived from ancient organic matter ("old sources") and are therefore depleted in $^{14}C$.

• Solar activity also influences $^{14}C$ levels; data shows a correlation with the 11-year sunspot activity cycle.

• Large solar photon events, indicated by tropospheric measurements, can cause noticeable shifts in radiocarbon data, as seen in records spanning the last 3,000 years.

Fundamental Mechanisms of Radioactive Decay

• Radioactive nuclei undergo spontaneous decay to move toward a state of stability. This process often occurs via a decay series, where a radioactive nuclide produces daughter nuclides until a stable nucleus is reached.

• Naturally occurring Uranium-238 ($^{238}U$) is a primary example of a nuclide that initiates a decay series. The ratios of specific elements, such as the Uranium-Lead ($U-Pb$) ratio, are used to determine the age of minerals.

• In every nuclear reaction, the mass number ($A$) and the atomic number or charge ($Z$) must be balanced.

• Standard nuclear equations typically do not include neutrinos or antineutrinos, nor do they consider the formation of ions.

• The primary mechanisms of decay include:

  • Alpha ($\alpha$) decay: Involves the loss of two protons and two neutrons ($-2Z, -2N$).

  • Beta ($\beta^-$) decay: A neutron is converted into a proton, resulting in an increase of one in the atomic number and a decrease of one in the neutron number ($+1Z, -1N$).

  • Positron ($\beta^+$) decay: A proton is converted into a neutron.

  • Electron Capture: An inner-shell electron is captured by the nucleus, converting a proton into a neutron.

  • Neutron Emission: The ejection of a neutron from the nucleus.

  • Gamma ($\gamma$) Emission: The release of high-energy electromagnetic radiation without changing the number of protons or neutrons.

  • Spontaneous Fission: A process seen in very heavy synthetic isotopes (e.g., $^{266}Lr$) and some natural heavy isotopes where the nucleus splits into smaller nuclides.

  • Proton Emission: Though less common, it is a possible mode of decay.

Electromagnetic Radiation in Nuclear Chemistry

• Both X-rays and gamma ($\gamma$) radiation are forms of high-energy, high-frequency, short-wavelength electromagnetic radiation (light).

• X-rays possess wavelengths shorter than visible or ultraviolet light, typically ranging between $0.01\,nm$ and $10\,nm$.

• Gamma rays have even shorter wavelengths, categorized as less than $0.01\,nm$ (or less than $0.1\,\text{Å}$).

Quantitative Measures of Radioactivity

• Activity is proportional to the number of nuclei present. To standardize these measurements, normalized versions are utilized:

  • Specific Activity: The activity measured per gram of the radionuclide ($Bq\,g^{-1}$).

  • Molar Activity: The activity measured per mole of the radionuclide ($Bq\,mol^{-1}$).

• Specific activity, molar activity, the decay constant ($\lambda$), and the half-life ($t_{1/2}$) are intrinsic properties of each radionuclide and are interconvertible.

• Example Calculation ($^{234}U$):

  • Mass ($m$) = $117\,g$

  • Activity ($A$) = $1.45 \times 10^6\,Bq$

  • Specific Activity = $\frac{1.45 \times 10^6\,Bq}{117\,g} = 12,400\,Bq\,g^{-1}$

  • Molar Activity ($\text{given molar mass} ≈ 234\,g\,mol^{-1}\text{ approx.}$) = $\frac{1.45 \times 10^6\,Bq}{117\,g / 234\,g\,mol^{-1}} = 2.90 \times 10^3\,Bq\,mol^{-1}$

Determinants of Nuclear Stability

• Nuclear Size: There are no stable nuclei heavier than Lead-208 ($^{208}_{82}Pb$). This represents the upper limit for nuclear stability in terms of proton count.

• Band of Stability: This is a graphical representation of stable nuclides plotted by neutron number ($N$) vs. proton number ($Z$).

  • For small atomic numbers, the stability ratio of neutrons to protons ($N:Z$) is approximately $1:1$.

  • As the nucleus grows larger, the band curves toward a higher $N:Z$ ratio to accommodate more neutrons for stability.

  • At the end of the band for stable nuclides ($^{208}_{82}Pb$), the $N/Z$ ratio reaches $1.54$.

• Elements lighter than Lead that have no stable isotopes include Technetium ($Tc$, $Z=43$) and Promethium ($Pm$, $Z=61$).

Predicting Modes of Decay

• Nuclides decay toward the band of stability by altering their $N/Z$ ratio while maintaining a constant mass number ($A$).

• High $N/Z$ Ratio (Neutron-rich): These nuclides typically undergo $\beta^-$ decay to convert a neutron into a proton.

  • Example: $^{14}_{6}C \rightarrow ^{14}_{7}N + e^- + \bar{\nu}$ (where $N/Z$ changes from $1.33$ to $1.0$).

• Low $N/Z$ Ratio (Proton-rich): These nuclides undergo $\beta^+$ (positron) decay or electron capture to convert a proton into a neutron.

  • Example Positron Decay: $^{11}_{6}C \rightarrow ^{11}_{5}B + e^+ + \nu$ (where $N/Z$ changes from $0.83$ to $1.2$).

  • Example Electron Capture: $^{37}_{18}Ar + e^- \rightarrow ^{37}_{17}Cl$ ($N/Z$ changes from $1.06$ to $1.18$).

Microscopic Models of Nuclear Forces

• Nuclear stability is determined by the competition between two fundamental forces:

  • Electrostatic Repulsive Force: Occurs between positively charged protons and tends to push the nucleus apart over long ranges.

  • Strong Nuclear Force (Binding Energy): An attractive force that acts between all nucleons (protons and neutrons). It is approximately 1,000 times stronger than the electrostatic force but acts only over a very short range (< 10^{-15}\,m).

• Mechanical Failure of Stability:

  1. If there are too few neutrons, the electrostatic repulsion between protons overcomes the strong nuclear attraction.

  2. As a nucleus increases in size, the cumulative long-range electrostatic repulsion eventually overwhelms the short-range strong nuclear attraction, regardless of the $N/Z$ ratio.

• Instability due to an excess of neutrons cannot be explained by this microscopic model and requires quantum mechanics to understand fully.

Nuclear Fission

• Spontaneous Fission: A low-probability decay process occurring in nuclides with a mass greater than $231$ (e.g., $^{232}Th$, $^{235}U$).

• Induced Fission: A nuclear reaction initiated by bombarding a nucleus with neutrons to create an excited, unstable isotope ($X^*$). This process produces two smaller daughter nuclides and additional neutrons, potentially triggering an exothermic chain reaction.

  • Example Equation: $^1_0n + ^{235}_{92}U \rightarrow ^{236}_{92}U^* \rightarrow ^{94}_{36}Kr + ^{139}_{56}Ba + 3^1_0n$

• OPAL (ANSTO Lucas Heights): A 20 MW multi-purpose research reactor used for the production of radioisotopes, neutron activation analysis, and providing neutron beams to study material structures.

Problems and Discussion

• The CNO Cycle: A catalytic cycle of nuclear reactions in massive stars (hotter than the sun) that converts hydrogen into helium using Carbon, Nitrogen, and Oxygen as catalysts.

• Sample Reactions in CNO Cycle:

  • Step 1: $^{12}_{6}C + ^1_1H \rightarrow ^{13}_{7}N$

  • Step 2: $^{13}_{7}N \rightarrow ^{13}_{6}C + \beta^+$

LECTURE 5

Radioactive Decay Series

• The decay series of all heavy elements must involve $\alpha$ decay, as it is the only form of decay capable of reducing the atomic mass number (by $4\,amu$).

• There are four distinct decay series for heavy elements:

  • Thorium series ($4n$): The mass number is divisible by 4.

  • Neptunium series ($4n+1$).

  • Radium/Uranium series ($4n+2$).

  • Actinium series ($4n+3$).

• These series result in four stable end-product isotopes:

  • $^{208}Pb$

  • $^{205}Tl$

  • $^{207}Pb$

  • $^{206}Pb$

• Decay chains also exist for unstable isotopes of light elements, such as $^{28}Mg$ and $^{39}Cl$.

Predicting Radioactive Modes of Decay

• Decay modes are determined by the Proton/Neutron ratio ($N/Z$) and the total mass of the nuclide:

  • If the $N/Z$ ratio is too high: The nuclide undergoes $\beta^{-}$ decay.

  • If the $N/Z$ ratio is too low: The nuclide undergoes $\beta^{+}$ decay or electron capture.

  • If the mass is too high (heavy elements): The nuclide typically undergoes $\alpha$ decay and $\beta^{-}$ decay.

• The stable nuclide $^{208}Pb$ ($Z=82$) serves as a common reference point for stability.

Interaction of Radiation with Matter

• Radioactive decay produces ionising radiation, creating reactive species through the process: $\text{Atom} \rightarrow \text{ion}^{+} + e^{-}$.

• Different forms of radiation possess varying penetrating powers and ionising abilities:

  • Alpha ($\alpha$): Massive and highly charged. It loses energy by causing many ionisation events (electron ejections) but has very low penetration depth.

  • Beta ($\beta^{-}, \beta^{+}$): Light but charged. These cause fewer ionising collisions than alpha particles but penetrate deeper into matter. Positrons ($\beta^{+}$) are eventually annihilated in an ionisation event.

  • Gamma ($\gamma$): Light and uncharged. These penetrate most deeply before causing an ionisation event, often through the photoelectric or Compton effect.

  • Neutrons ($n$): Heavy and uncharged. They penetrate deeply and interact primarily via neutron capture, which can create new radionuclides.

Ionising Ability and Biological Impact

• $\alpha$ and $\beta$ particles are very high energy. Ejecting an electron from a molecule typically requires $10\,eV$.

• A single $\alpha$ particle with $5\,MeV$ of energy can ionise approximately $5 \times 10^{5}$ molecules.

• $\gamma$ rays lose all energy in a single event, resulting in the ejection of a high-energy electron that behaves like a $\beta$ particle at a greater depth.

• Living matter is predominantly water. The ionisation of water is the dominant process when exposed to radiation:

  • $H_{2}O + \text{ionising radiation} \rightarrow H_{2}O^{+} + e^{-}$

  • $H_{2}O^{+} + H_{2}O \rightarrow H_{3}O^{+} + OH^{\bullet}$

  • $e^{-} + H_{2}O \rightarrow OH^{-} + H^{\bullet}$

• These reactions form highly-reactive free radicals ($H^{\bullet}$ and $OH^{\bullet}$), which cause:

  • Genetic damage or cancer by affecting DNA strands.

  • Cell disintegration by affecting cell membranes.

  • Loss of enzyme function by affecting proteins.

Radiation Damage Factors and Dosimetry

• The biological effect of radiation is measured in Sieverts ($Sv$), which accounts for the type of radiation, energy, and source activity.

• Factors Affecting Damage:

  1. Type of Radiation: Measured by Relative Biological Effectiveness ($W_{R}$).

  • $\alpha$ ($5\,MeV$): Air penetration $40\,mm$, tissue penetration $0.05\,mm$, $W_{R} = 2-20$ (depends on cell type).

  • $\beta$ ($0.05-1\,MeV$): Air penetration $6-300\,mm$, tissue penetration $0.06-4\,mm$, $W_{R} = 1-1.7$.

  • $\gamma$ ($1\,MeV$): Air penetration $400\,m$, tissue penetration $50\,cm$, $W_{R} = 1$.

  1. Length of Exposure:

  • Acute (short term): High doses cause radiation poisoning, cell damage, and death.

  • Chronic (long term): Interrupts DNA, potentially leading to cancer.

  1. Source of Exposure:

  • External: $\gamma$ radiation is the highest risk because $\alpha$ and $\beta$ cannot penetrate skin/air.

  • Internal (Ingestion/Inhalation): $\alpha$ and $\beta$ are most dangerous; $\gamma$ usually escapes the body.

Radiation Units and Health Effects

• Radiation Absorbed Dose (Gray, Gy): Defined as $1\,J$ of radiation absorbed per kg of body weight ($1\,Gy = 1\,J/kg$).

• Effective Dose (Sievert, Sv): Calculated as $\text{Effective Dose (Sv)} = W_{R} \times \text{absorbed dose (Gy)}$.

• Acute Biological Effects of Effective Dose:

  • $0.2\,Sv$: Drop in red blood cell count, bone marrow damage, fatigue.

  • $0.5\,Sv$: Immune system compromised, onset of illness.

  • $1.0\,Sv$: Nausea and fatigue for all; small percentage of deaths without medical care.

  • $3.0\,Sv$: Internal bleeding; $50\%$ mortality rate within 1 month.

  • $6.0\,Sv$: $100\%$ mortality rate within 2 weeks.

Sources of Everyday Exposure

• Natural background radiation is approximately $3\,mSv$ per year.

  • Radon: Part of the $^{238}U$ decay series ($t_{1/2} = 3.82\,d$); causes internal lung damage.

  • Potassium-40 ($^{40}K$): Natural abundance $0.01\%$, $t_{1/2} = 1.25 \times 10^{9}$ years. Found in bananas ($0.1\,\mu Sv$), kidney beans, and sunflower seeds.

  • $^{40}K$ decay paths: $\beta^{-}$ emission ($89.28\%$), electron capture ($10.72\%$), and positron emission ($0.001\%$).

  • Cosmic Rays: High kinetic energy radiation from the upper atmosphere; average ground exposure is $260\,\mu Sv/year$.

• Man-made Exposure Doses:

  • Airport security X-ray scanner: $0.0001\,mSv$

  • 7-hour plane flight: $0.03\,mSv$

  • Chest X-ray: $0.1\,mSv$

  • Mammogram: $0.4\,mSv$

  • PET scan: $14\,mSv$

  • Cardiac stress-test: $41\,mSv$

  • Annual limit for radiation workers: $50\,mSv$

Medical Applications: Radiotherapy

• Focussed External Radiotherapy: Uses highly penetrating $\gamma$ radiation (e.g., $^{60}Co$) to target tumours through air and skin.

• Internal Radiopharmaceutical Therapy:

  • Radioactive drug targeted chemically to a tumour.

  • Uses short-range emitters ($\alpha$ or $\beta$) for localized effect.

  • Example: Iodine-131 ($\beta$ emission, $t_{1/2} = 8\,d$) is used for thyroid cancer because iodine naturally accumulates there.

• Contrast in pop culture: External radiation is likened to the Hulk (gamma rays), while internal is likened to Spider-Man (radioactive spider bite).

Medical Applications: Imaging

• Radioimaging maps the distribution of a radionuclide within the body. It requires highly penetrating, low-harm radiation, specifically $\gamma$ radiation.

• Ideal Imaging Isotope Properties:   i. $\gamma$ or $\beta^{+}$ emitter for tissue penetration.   ii. Low toxicity.   iii. Monoenergetic radiation.   iv. Optimal half-life (hours to days) for contrast and transport.   v. Chemical reactivity for incorporation into compounds.   vi. Good body clearance.   vii. Cheap and available.

• Technetium-99m ($^{99m}Tc$): The most common imaging isotope.

  • Metastable isomer decaying to $^{99}Tc$ via $\gamma$ emission ($t_{1/2} = 6\,h$).

  • Equation: $^{99m}_{43}Tc \rightarrow ^{99}_{43}Tc + \gamma$.

  • Produced from Molybdenum-99 ($^{99}Mo$, $t_{1/2} = 66\,h$) in a generator.

  • $^{99}Mo$ is adsorbed as molybdate ($MoO_{4}^{2-}$); decay forms pertechnetate which is extracted via elution.

  • Applications: Thyroid scans, bone cancer imaging (with methylene diphosphonic acid), kidney function tracing (with pentetic acid).

Positron Emission Tomography (PET)

• PET uses positron emitters ($\beta^{+}$). When a positron meets an electron inside the body, they annihilate to produce two high-energy $\gamma$ rays traveling in opposite directions.

• Standard Agent: Fludeoxyglucose (FDG), where an oxygen in glucose is replaced by Fluorine-18 ($^{18}F$).

• FDG targets high-glucose usage areas like the brain and tumours.

• Isotope Production: PET isotopes are formed in a cyclotron using proton or deuteron bombardment of gases.

  • $^{14}N + ^{1}H \rightarrow ^{11}C + \alpha$ ($t_{1/2} = 20.3\,min$).

  • $^{14}N + ^{2}H \rightarrow ^{15}O + n$ ($t_{1/2} = 2.07\,min$).

  • $^{18}O + ^{1}H \rightarrow ^{18}F + n$ ($t_{1/2} = 109.7\,min$).

• FDG synthesis takes approximately 35 minutes with a $50\%$ radiochemical yield.

Selected Medical Isotopes and Usage

• Bismuth-213 ($46\,min$): Targeted alpha therapy (TAT).

• Cobalt-60 ($5.27\,y$): External beam radiotherapy and sterilisation.

• Iodine-131 ($8\,d$): Thyroid cancer treatment and imaging.

• Lutetium-177 ($6.7\,d$): Combined imaging (low $\gamma$) and therapy ( $\beta$) for small tumours.

• Yttrium-90 ($64\,h$): Brachytherapy and arthritis pain relief; pure $\beta$ emitter.

• Xenon-133 ($5\,d$): Lung ventilation studies.

• Palladium-103 ($17\,d$): Permanent seeds for prostate cancer brachytherapy.

Questions & Discussion

• Question 1: Balancing Nuclear Reactions.

  • a. $^{18}_{8}O + ^{1}_{1}H \rightarrow ^{18}_{9}F + ^{1}_{0}n$

  • b. $^{16}_{8}O + ^{1}_{1}H \rightarrow ^{13}_{7}N + ^{4}_{2}He$

  • c. $^{10}_{5}B + ^{1}_{0}n \rightarrow ^{7}_{3}Li + ^{4}_{2}He$

  • d. $^{7}_{4}Be + ^{1}_{0}n \rightarrow ^{7}_{3}Li + ^{1}_{1}H$

  • e. $^{14}_{7}N + ^{4}_{2}He \rightarrow ^{17}_{8}O + ^{1}_{1}H$

• Question 2: Iodine isotopes in medicine.

  • $^{123}I$ ($t_{1/2} = 13.2\,h$): Predicted decay is electron capture or $\beta^{+}$ (low $N/Z$); used for imaging due to $\gamma$ emission.

  • $^{131}I$ ($t_{1/2} = 8.02\,d$): Predicted decay is $\beta^{-}$ (high $N/Z$); used for therapy due to $\beta$ production targeting the thyroid.

  • Considerations for $^{124}I$ and $^{125}I$ include half-life suitability and radiation type for specific clinical durations.