Radioactivity and Nuclear Decay: A Comprehensive Study Guide

Foundations of Nuclear Chemistry and Historical Context

The study of nuclear chemistry officially began in 1896 when Antoine Becquerel discovered radioactivity. His discovery occurred by accident when he left uranium ore on top of photographic plates; even without light exposure, the plates became fogged, indicating they were exposed to some form of radiation from the uranium. This work was extended by his two graduate students, Marie and Pierre Curie. Marie Curie and her husband Pierre discovered the elements Polonium and Radium and were responsible for coining the term "radioactivity."

Marie Curie's contributions extended into medicine and wartime service. During World War I, she created a mobile X-ray machine in a medical truck to examine the broken limbs of soldiers on the battlefield, a development that prevented many unnecessary amputations. Her scientific achievements earned her two Nobel Prizes: the Nobel Prize in Physics (shared with Pierre) and the Nobel Prize in Chemistry. However, her lifelong exposure to radioactive materials led to her death from aplastic anemia, a condition involving the failure of bone marrow to produce enough new blood cells.

Core Concepts and Terminology of Radioactivity

Radioactivity is defined as the process by which nuclei emit particles and rays. This occurs when a nucleus releases particles or energy to reach a more stable state. The particles and rays emitted from these radioactive sources are collectively referred to as nuclear radiation. Transmutation is a specific term describing the conversion of an atom of one element into an atom of another element. This process occurs naturally through radioactive decay, where unstable nuclei lose energy by spontaneously emitting radiation.

Radioisotopes, or radioactive isotopes, are atoms with unstable nuclei. The stability of a nucleus is primarily determined by its ratio of protons to neutrons. While most elements on the periodic table have stable isotopes, all elements with an atomic number ($Z$) greater than $83$ are naturally radioactive. Elements are generally most stable when their proton number is approximately equal to their neutron number ($Z=N$), particularly for lighter elements.

Subatomic Forces and Nuclear Stability

The nucleons (protons and neutrons) within an atomic nucleus are held together by the strong nuclear force. This interaction is powerful but operates only over very short distances. As a nucleus becomes larger and the number of protons increases, the repulsive electrostatic forces between those protons grow. Eventually, the strong nuclear force becomes insufficient to hold the large nucleus together, causing it to become unstable. This instability triggers radioactive decay. During this process, the starting radioactive isotope is referred to as the parent isotope, and the resulting atom produced after decay is called the daughter isotope.

Types of Radioactive Decay and Particles

Radioactive substances undergo several distinct types of decay, each characterized by the emission of different particles or energy. Alpha decay involves the emission of an alpha particle ($\text{α}$), which consists of $2$ protons and $2$ neutrons, identical to a helium nucleus. It is represented by the symbols $\text{α}$ or $^{4}_{2}\text{He}$. Alpha particles have a charge of $+2$ and a mass of $4$ amu. They have low penetrating power and can be stopped by a sheet of paper or skin cells, though they are dangerous if ingested. Common sources include Radium-226 ($^{226}\text{Ra}$).

Beta minus decay ($\text{β}^{-}$) involves the emission of a fast-moving electron formed by the decomposition of a neutron into a proton and an electron. The proton stays in the nucleus, and the electron is ejected. This is represented by $^{0}_{-1}\text{e}$ or $\text{β}$. These particles have a $-1$ charge and a negligible mass of approximately $\frac{1}{1837}$ amu. Beta particles have moderate penetrating power and can be stopped by aluminum foil or thin lead. Carbon-14 ($^{14}\text{C}$) is a common beta minus emitter.

Beta plus decay or positron emission involves a particle with the mass of an electron but a positive charge ($^{0}_{+1}\text{e}$ or $\text{β}^{+}$). This occurs when a proton changes into a neutron, typically because the neutron-to-proton ratio is too small. Electron capture is a related process where an inner orbital electron is captured by the nucleus, combining with a proton to form a neutron. Sodium-22 ($^{22}\text{Na}$) is a source for positron emission.

Gamma radiation ($\text{γ}$) is high-energy electromagnetic radiation (photons) emitted as a nucleus transitions from an excited state to a ground state. Gamma rays have no mass and no charge. They are extremely penetrating and dangerous, requiring several feet of concrete or several inches of lead to be stopped. Gamma rays are often emitted alongside alpha or beta particles and do not cause transmutation on their own. Cobalt-60 ($^{60}\text{Co}$) is a common gamma source.

Predicting Decay and Balancing Nuclear Equations

To predict whether an isotope is stable or what type of decay it will undergo, scientists use a specific logical framework. If the atomic number ($Z$) is greater than $83$, the isotope will likely undergo alpha emission. If the mass of the isotope is significantly greater than the average atomic mass found on the periodic table, it will likely undergo beta emission. If the mass is significantly less than the periodic table mass, it will likely undergo positron emission or electron capture. Other factors include whether the number of protons and neutrons are even or odd, with even-even combinations being the most stable.

Nuclear equations must be balanced according to the law of conservation of mass, meaning the sum of the mass numbers (top) and the sum of the atomic numbers (bottom) must be equal on both sides of the equation. For example, the alpha decay of Uranium-238 is written as:

$^{238}_{92}\text{U} \rightarrow ^{4}_{2}\text{He} + ^{234}_{90}\text{Th}$

Similarly, the beta decay of Carbon-14 is:

$^{14}_{6}\text{C} \rightarrow ^{0}_{-1}\text{e} + ^{14}_{7}\text{N}$

The positron emission of Neon-18 is:

$^{18}_{10}\text{Ne} \rightarrow ^{0}_{+1}\text{e} + ^{18}_{9}\text{F}$

Mathematical Principles of Radioactive Half-Life

A half-life is the duration of time required for half of a radioactive sample to decay into daughter products. After one half-life, $50\%$ of the parent isotope remains. After two half-lives, $25\%$ remains ($\frac{1}{2}$ of $\frac{1}{2}$), and three-quarters of the sample has decayed. The length of a half-life varies wildly between isotopes, ranging from less than one second to over $10^{14}$ years. For example, Phosphorus-32 has a half-life of $14.3$ years, while Carbon-14 has a half-life of approximately $5700$ years.

The amount of a substance remaining after a certain number of half-lives can be calculated using the formula:

$\text{Amount remaining} = (\text{initial amount}) \times (\frac{1}{2})^{n}$

In this formula, $n$ represents the number of half-lives that have elapsed. For instance, if you start with $2.00$ g of Nitrogen-13 and it undergoes $4$ half-lives, the amount remaining would be:

$2.00 \times (\frac{1}{2})^{4} = 0.125$ g.

If Phosphorus-32 ($14.3$ year half-life) decays for $57.2$ years, we first find $n$:

$n = \frac{57.2}{14.3} = 4$ half-lives.

A $4.0$ g sample would then become $0.25$ g after those $4$ cycles.

Radioactive Decay Chains

Many unstable isotopes do not become stable after a single decay event. Instead, they undergo a radioactive decay chain, which is a series of successive decays. For example, in the Uranium-238 decay chain, Astatine-218 undergoes alpha decay, followed by beta minus decay, and another alpha decay before eventually reaching a stable state. In the Thorium-232 decay series, Radium-224 goes through three alpha decays and one beta minus decay to produce a specific daughter isotope. For instance, Radium-224 ($^{224}_{88}\text{Ra}$) first produces Radon-220 ($^{220}_{86}\text{Rn}$), which decays to Polonium-216 ($^{216}_{84}\text{Po}$), then to Lead-212 ($^{212}_{82}\text{Pb}$), and finally through beta decay to Bismuth-212 ($^{212}_{83}\text{Bi}$).

Applications in Radiometric Dating

Radiometric dating utilizes the known, constant half-lives of isotopes to estimate the age of ancient materials. Carbon-14 dating is a prominent method used for organic matter. Carbon-14 is produced in the upper atmosphere when high-energy neutrons collide with Nitrogen-14. It enters the food chain through photosynthesis and ingestion. While an organism is alive, the ratio of Carbon-14 to Carbon-12 remains constant as it is continuously replaced. Upon death, the intake stops, and Carbon-14 begins to decay with a half-life of roughly $5700$ years.

A famous case study is the discovery of "Iceman," a prehistoric man whose remains were found with roughly half the Carbon-14 concentration expected in a living organism. Since exactly one half-life had passed ($50\%$ remaining), scientists estimated he died approximately $5700$ years ago. For older applications, different isotopes are used. To date early hominid bones from $3$ million years ago, isotopes with much longer half-lives than Carbon-14 are required. In other archaeological examples, a fossil containing only $25\%$ of its original Plutonium-240 ($6,850$ year half-life) would be calculated as being two half-lives old, or $13,700$ years. Similarly, an Inuit glass vase with only $\frac{1}{8}$ of its initial Cesium-137 ($130$ year half-life) would be three half-lives old, or $390$ years.