Nuclear Chemistry: Decay, Fission, Fusion, and Biological Effects Study Guide to Final Exams

Course Administration and Final Exam Structure

• Course Material Distribution: The course material is divided into five segments: Exam 1, Exam 2, Exam 3, Exam 4, and Post-Exam 4 material.

• Weighting: These five groups of material are equally weighted on the final exam.

• Final Exam Composition: Each of the five material groups corresponds to approximately eight questions each on the final exam.

• Exam 4 Grades: Grades for Exam 4 are planned to be posted the evening of the final lecture; physical exams will be brought to the final classroom meeting on Friday.

• Top Hat and Extra Credit Calculation:     - Top Hat scores are typically not directly integrated into Canvas because the platform prefers a percentage base (out of 100) and struggle to add points on top of existing totals without artificially deflating overall grades.     - Students can view their percentages on the Top Hat website.     - Scoring Breakdown: Students receive half of the potential points (5 points) just for answering questions. The remaining half (5 points) is based on the correctness of those answers.     - Example: Answering every question but getting them all wrong results in 5 points. Answering all questions correctly results in 10 points.

• Aleks Completion: The total Aleks completion has weight in the final grade according to the syllabus; this will be calculated alongside the final grade.

Radioactive Decay Processes and Quantitative Radio-Dating

• Rate Laws: All radioactive decay processes follow first-order rate laws.

• Rate Equation: The rate of decay is defined as $\text{Rate} = k[A]$, where $k$ is the decay constant and $[A]$ is the concentration or amount of the radioactive isotope.

• Integrated Rate Law: The first-order integrated rate law used to describe decay processes and determine the time elapsed (age) is:   $\ln\left(\frac{N_t}{N_0}\right) = -kt$   - $N_t$: Amount of substance remaining at time $t$.   - $N_0$: Initial amount of substance at time zero.   - $k$: Rate constant for decay.   - $t$: Time elapsed.

• Radioactive Decay Series (Uranium-238):   - The Uranium-238 series eventually decays into stable Lead-206 ($Pb^{206}$).   - The series involves multiple types of decay: alpha decay, beta decay, positron emission (also called beta emission), electron capture, and gamma decay.   - Gamma Decay: No particle is emitted, only energy. Neither the mass number nor the atomic number changes. It involves the transition of a higher energy nucleus (indicated by a star, e.g., $X^*$) to a lower energy state ($X$).   - Electron Capture: The nucleus "steals" an electron from the electron cloud and combines it with a proton to form a neutron.

• Half-Life Approximation in Series: If one half-life in a decay series is orders of magnitude larger than all other subsequent half-lives, the total half-life for the entire series can be approximated as the half-life of that first, slow step.   - Example: For Uranium-238, the first step has a half-life of $4.51 \times 10^9$ years. Even adding subsequent steps (at magnitudes of $10^4$ or $10^5$ years), the total remains approximately $4.51 \times 10^9$ years.

Case Study: Quantitative Dating of an Old Rock Formation

• Scenario: A rock sample contains $0.396\,g$ of Uranium-238 and $0.231\,g$ of Lead-206. Uranium-238 decays to Lead-206 with a half-life of $4.51 \times 10^9$ years.

• Step 1: Convert Lead mass to Uranium equivalent:   - Molar mass of $Pb^{206} = 205.97\,g/mol$.   - Molar mass of $U^{238} = 238.05\,g/mol$.   - Calculations: $\frac{0.231\,g\,Pb^{206}}{205.97\,g/mol} \times \left(\frac{1\,mol\,U^{238}}{1\,mol\,Pb^{206}}\right) \times 238.05\,g/mol = 0.267\,g\,U^{238}$.   - This represents the mass of Uranium-238 that has already decayed.

• Step 2: Determine initial mass ($N_0$):   - Initial mass = Current mass + Decayed mass = $0.396\,g + 0.267\,g = 0.663\,g\,U^{238}$.

• Step 3: Calculate the rate constant ($k$):   - $k = \frac{0.693}{t_{1/2}} = \frac{0.693}{4.51 \times 10^9\,years} = 1.54 \times 10^{-10}\,yr^{-1}$.

• Step 4: Solve for Time ($t$):   - $\ln\left(\frac{0.396}{0.663}\right) = -(1.54 \times 10^{-10}\,yr^{-1}) \times t$   - $-0.516 = -(1.54 \times 10^{-10}\,yr^{-1}) \times t$   - $t = 3.35 \times 10^9\,years$.

Nuclear Fission and Power Generation

• Definition: Nuclear fission is the process by which a heavy nucleus (mass number > 200) divides to form smaller nuclei of intermediate mass and one or more neutrons.

• Chain Reaction: The neutrons generated by the initial split act as reactants that split neighboring nuclei, leading to a self-sustaining chain reaction.

• Nuclear Reactor Components:   - Fuel: Often Uranium-235. The reaction is $U^{235} + n \rightarrow Xe^{143} + Sr^{90} + 3n$ (as one example of various potential products).   - Control Rods: Made of neutron-absorbing materials like Boron. They are inserted or withdrawn to control the rate of the reaction by regulating the number of available neutrons.   - Energy Production: The heat released from the reaction is used to heat water, turning it into steam. This steam spins a turbine to generate electricity.

• Energy Comparison:   - Fission releases approximately $2 \times 10^{13}\,J/mol$.   - Burning coal releases approximately $5 \times 10^7\,J/mol$.   - Fission is nearly twice as energetic per mole (specifically orders of magnitude higher) than fossil fuels.

• Environmental Impact:   - Positive: No $CO_2$ generation; the visible emission from stacks is water vapor (steam).   - Negative: Production of nuclear waste with extremely long half-lives ($10^9$ years) that requires safe, permanent storage.

Nuclear Fusion: Principles and Comparison

• Definition: The combination of two light nuclei (atomic number < 56) to form a heavier nucleus.

• Example: Fusion of Hydrogen in the sun to produce Helium ($H + H \rightarrow He$).

• Challenges:   - Repulsion: Two positively charged nuclei naturally repel each other like magnets.   - High Temperatures: To overcome this repulsion, temperatures must reach approximately $100,000,000^\circ Celsius$ (100 million degrees).   - Containment: At these temperatures, matter exists as plasma, which can destroy physical containers.   - Net Energy Gain: Currently, terrestrial research focuses on achieving a "net energy gain" where the energy output exceeds the energy required to reach fusion temperatures.

• Fission vs. Fusion Comparison:   - Fission: One major nucleus produces two major nuclei (increasing the number of nuclei); uses heavy elements like Uranium (can be toxic).   - Fusion: Two major nuclei produce one heavier major nucleus (decreasing the number of nuclei); uses light elements like Hydrogen (helium waste is inert).

Applications of Isotopes in Medical Imaging

• Imaging Signals: Often utilize gamma rays for detection.

• Sodium-24 ($Na^{24}$): Used to track obstructions in the circulatory system.

• Technetium ($Tc$): Used for general body imaging.

• Iodine-131 ($I^{131}$): Used to monitor thyroid gland activity.

• Iodine-123 ($I^{123}$): Used for imaging brain gray matter via the following path:   - \text{Step 1: } I^{123} \text{ (Electron Capture)} \rightarrow Te^{123}^   - \text{Step 2: } Te^{123}^ \text{ (Gamma Decay)} \rightarrow Te^{123} + \gamma   - The resulting gamma ray serves as the imaging signal.

• Targeting: Radioactive atoms are attached to larger molecules (like iodoamphetamine) designed to bind to specific receptors/parts of the body to ensure targeted imaging rather than general systemic emission.

Biological Effects of Radiation and Safety Standards

• Units of Measure:   - REM (Roentgen Equivalent Man): Describes human exposure to ionizing radiation.   - Sievert (Sv): $1\,rem = 0.01\,Sv = 0.01\,J/kg$ of energy absorbed per kilogram of organic matter.

• Safety Limits: The US maximum safe radiation dose is specified as $5,000\,mrem/year$. Above this level, risks of radiation poisoning and DNA damage increase dramatically.

• Common Exposure Levels:   - X-rays: $50$ to $75\,mrem$ per procedure.   - Body/Nature: Natural radioactive elements exist in the body, ground (radon), and air.   - Air Travel: Increases exposure due to cosmic rays from the sun at high altitudes.   - Nuclear Waste: Provides a negligible amount of radiation to the general public under proper storage.

• Molecular Damage Mechanism: Radiation creates free radicals (highly reactive species).   - Radiolysis of Water: Water absorbs a gamma ray, releasing an electron to form $H_2O^+$. This reacts to create $H_3O^+$ and Hydroxyl radicals ($OH^\bullet$). Electrons can react with oxygen to create Superoxide radicals ($O_2^{-\bullet}$).   - These radicals damage cell membranes, enzymes, and DNA, potentially leading to cancer.

Shielding and Protection from Radiation Types

• Alpha particles ($\alpha$): Heavy; requires minimal shielding (paper or clothing).

• Beta decay ($\beta$): High energy electrons/positrons; requires moderate shielding (plastic or drywall).

• Gamma rays ($\gamma$) and Neutrons ($n$): High penetration; requires extreme shielding (thick lead or several feet of concrete).

• Historical Anecdote: Marie Curie and fellow researchers often kept radium on their nightstands because it emitted a faint blue glow. This lack of shielding led to significant health issues (cancer).

Questions & Discussion

• Student Question: "Are you going to put the Alec's completion in the Top Hat extra credit in when you put in the final exam so we can see, like, whatever it will be going into the final exam?"

• Professor Response: The professor explained that Top Hat scores are hard to incorporate directly into Canvas because Canvas prefers a percentage out of 100 rather than adding extra points. However, the completion of Aleks (the PIE completion) is weighted in the final grade according to the syllabus and will be reflected accordingly. Top Hat specifically grants 5 points for participation and 5 for correctness.

• Question regarding Fission/Fusion Equation Identification: The professor provided a practice question to distinguish fission.   - Identification Tip: Fission occurs when there is one major nucleus on the reactant side and two or more on the product side (mass decrease/splitting). Fusion occurs when two light nuclei combine into one heavier nucleus. E.g., Identify $_2^4He + <em>{13}^{27}Al \rightarrow </em>{15}^{30}P + <em>0^1n$ as a transformation/capture, whereas $</em>{92}^{235}U + <em>0^1n \rightarrow </em>{54}^{143}Xe + _{38}^{90}Sr + 3_0^1n$ is fission.