6.1 The Atomic Nucleus & Radioactive Decay

Every element on the periodic table is defined by the number of protons packed into the tiny, dense core of its atoms: the atomic nucleus. Content category 4E of the MCAT tests how well you understand what's inside that nucleus, how it can break apart, and how scientists measure atomic mass — all high-yield topics spanning General Chemistry and Physics passages.

Atomic Number, Mass Number, and Atomic Weight

The atomic number (Z) is the number of protons in an atom's nucleus. Because the number of protons determines which element an atom is, atomic number is the defining property found on the periodic table — every atom with Z = 6 is carbon, no exceptions. In a neutral atom, the number of electrons equals the atomic number as well.

The mass number (A) is the total count of protons plus neutrons in the nucleus. Mass number is always a whole number, written as a superscript before the element symbol (for example, ¹²C has A = 12). Do not confuse mass number with atomic weight, the weighted average mass (in atomic mass units, amu) of all naturally occurring isotopes of an element — this is the decimal value printed under each element's symbol on the periodic table (carbon's atomic weight is 12.011 amu, not exactly 12, because of the small natural abundance of ¹³C and ¹⁴C).

Isotopes

Isotopes are atoms of the same element (identical Z) with different numbers of neutrons, and therefore different mass numbers. Because chemical behavior is governed almost entirely by the number and arrangement of electrons — which is set by the number of protons — isotopes of an element share nearly identical chemical properties but can have very different nuclear stability. Hydrogen has three natural isotopes: protium (¹H, no neutrons), deuterium (²H, one neutron), and tritium (³H, two neutrons, radioactive). Isotope notation is typically written as ᴬZX, where X is the element symbol, A is the mass number, and Z is the atomic number.

Nuclear Forces and Binding Energy

Packing multiple positively charged protons into a nucleus only a few femtometers wide should cause them to repel each other violently via the Coulomb (electrostatic) force. The nucleus doesn't fly apart because of the strong nuclear force, an attractive force between nucleons (protons and neutrons) that is far stronger than electrostatic repulsion at very short range (roughly 1–3 femtometers) but drops off almost to zero beyond that range. Neutrons contribute to the strong force without adding any electrostatic repulsion, which is why stable nuclei need a growing neutron-to-proton ratio as atomic number increases — larger nuclei need extra neutrons to supply enough attractive "glue" to overcome the accumulating proton-proton repulsion.

If you compare the mass of an assembled nucleus to the sum of the masses of its individual protons and neutrons, the nucleus is always slightly lighter. This missing mass, called the mass defect (Δm), was converted into the energy that holds the nucleus together when it formed, according to Einstein's mass-energy equivalence, E = mc². This energy is the nucleus's binding energy — the energy that would be required to separate the nucleus completely into individual protons and neutrons. Binding energy per nucleon (binding energy divided by mass number) is a useful measure of nuclear stability: it rises sharply for the lightest elements, peaks around iron (mass number 56), and then slowly declines for heavier elements. This is exactly why both nuclear fission (splitting very heavy nuclei like uranium) and nuclear fusion (combining very light nuclei like hydrogen) release energy — both processes move nuclei toward the more stable, higher-binding-energy-per-nucleon region near iron.

Radioactive Decay: Alpha, Beta, Gamma, and Electron Capture

Nuclei with an unfavorable neutron-to-proton ratio, or that are simply too large to be stable, undergo radioactive decay, spontaneously transforming into a more stable configuration while releasing particles and/or energy. The MCAT expects you to recognize the major decay modes and their effects on mass number (A) and atomic number (Z):

In alpha decay, the nucleus ejects an alpha particle, identical to a helium-4 nucleus. Because the alpha particle carries away 2 protons and 2 neutrons, the daughter nucleus has a mass number 4 lower and an atomic number 2 lower than the parent — transmuting the atom into a different element two spots earlier on the periodic table. Alpha particles are relatively large, slow, and doubly charged, so they're stopped by something as thin as a sheet of paper or the outer layer of skin, but they are intensely ionizing and dangerous if an alpha-emitting isotope (such as radon) is inhaled or ingested.

In beta-minus decay, a neutron converts into a proton, an electron, and an antineutrino; the electron is ejected as the beta particle. Atomic number increases by 1 while mass number stays the same — the atom transmutes into the next element up. In beta-plus decay (positron emission), a proton converts into a neutron and a positron, so atomic number decreases by 1. Positron emission is the nuclear event that powers positron emission tomography (PET) imaging, discussed below.

Electron capture is a competing pathway for proton-rich nuclei: the nucleus captures an inner-shell electron (often from the K shell), converting a proton into a neutron and emitting a neutrino. Mass number is unchanged and atomic number falls by 1 — the same net nuclear change as beta-plus decay — but no positron is emitted. After capture, an outer electron often drops into the vacancy and emits a characteristic X-ray or an Auger electron, which passages sometimes use as a detection handle.

A common trap: gamma decay changes neither the mass number nor the atomic number — it does not transmute the element at all. Gamma decay simply releases excess energy as a photon when an already-transmuted nucleus (often the product of a prior alpha or beta decay) relaxes from an excited nuclear state to its ground state, much like an electron emitting a photon when it drops to a lower energy level. Metastable nuclear isomers that emit pure gamma radiation (notably technetium-99m) are ideal clinical tracers because they report location without permanently changing elemental identity.

Half-Life, Exponential Decay, and Semi-Log Plots

Radioactive decay is a random, first-order process: at any instant, each unstable nucleus in a sample has the same fixed probability of decaying, regardless of how long it has already existed and regardless of how many other nuclei are present. This produces exponential decay, described by N(t) = N₀(1/2)^(t/t₁/₂), where N₀ is the initial quantity, N(t) is the amount remaining at time t, and t₁/₂ (half-life) is the time required for exactly half of the sample to decay. Equivalently, N(t) = N₀e^(−λt), where the decay constant λ = ln(2)/t₁/₂ ≈ 0.693/t₁/₂. A critical exam trap: half-life is a constant property of the isotope — it does not depend on the starting amount, temperature, or chemical environment. You cannot "speed up" a radiotracer's decay by heating the patient or diluting the dose.

Because decay is exponential, a plot of N versus t curves downward and flattens, but a semi-log plot (plotting ln N, or log₁₀ N, on the y-axis against t on a linear x-axis) turns that curve into a straight line with slope −λ (or −λ/2.303 for log₁₀). MCAT passages frequently show a semi-log decay graph and ask you to extract the half-life from the slope or to read off how many half-lives have elapsed between two data points. Activity (decays per unit time) follows the same exponential law as number of nuclei, because activity A = λN.

Worked example: Iodine-131, used diagnostically and therapeutically for thyroid conditions, has a half-life of about 8 days. Starting with a 160 mg dose, how much remains after 32 days? First find the number of half-lives elapsed: 32 days ÷ 8 days = 4 half-lives. Then repeatedly halve the starting amount: 160 mg → 80 mg (day 8) → 40 mg (day 16) → 20 mg (day 24) → 10 mg (day 32). So only 10 mg remains — matching the shortcut N = N₀(1/2)⁴ = 160 mg × (1/16) = 10 mg. If a question instead asks about a time that isn't a clean multiple of the half-life (say, 5 days), you need the exponential form: N/N₀ = (1/2)^(5/8) ≈ 0.65, so about 65% of the sample remains. Clinically, the same arithmetic schedules imaging times, residual activity, and when a patient can safely re-enter public spaces after radioiodine therapy.

Medical Imaging: PET, SPECT, and Clinical Radiotracers

Nuclear medicine converts the decay modes above into maps of physiology rather than anatomy alone. Two high-yield modalities on the MCAT are positron emission tomography (PET) and single-photon emission computed tomography (SPECT) (or planar gamma-camera imaging).

In PET, a positron-emitting isotope (β⁺ decay) is chemically incorporated into a biologically active molecule — most famously ¹⁸F-fluorodeoxyglucose (¹⁸F-FDG), a glucose analog. After intravenous injection, tissues with high glucose uptake (brain, myocardium, many tumors) concentrate the tracer. When the ¹⁸F nucleus emits a positron, that positron almost immediately annihilates with a nearby electron, producing two 511 keV gamma photons traveling in opposite directions. Coincident detection of those photon pairs reconstructs a three-dimensional map of metabolic activity. Fluorine-18's half-life of about 110 minutes is long enough to synthesize FDG, transport it within a city, and image a patient, yet short enough that residual activity falls quickly after the scan. Carbon-11 (t₁/₂ ≈ 20 min) and oxygen-15 (t₁/₂ ≈ 2 min) appear in research PET but require an on-site cyclotron.

SPECT and conventional nuclear scans use gamma emitters rather than positrons. Technetium-99m (⁹⁹mTc) is the workhorse: it is a metastable nuclear isomer that decays by gamma emission (no change in Z or A) with a half-life of about 6 hours, can be chemically chelated into many organ-specific complexes (bone, cardiac perfusion, renal function, thyroid), and produces a clean monoenergetic gamma well matched to clinical cameras. Iodine-123 (gamma, t₁/₂ ≈ 13 hours) images the thyroid when a pure diagnostic scan is desired; iodine-131 (β⁻ plus gamma, t₁/₂ ≈ 8 days) doubles as a thyroid therapy agent because its beta particles deposit ionizing energy in thyroid tissue that concentrates iodide via the sodium–iodide symporter.

Ideal clinical tracers share design rules the MCAT likes to reverse-engineer from a passage: (1) a half-life matched to the procedure window — hours, not years; (2) emission type that escapes the body for external detection (gamma or annihilation photons) rather than purely local alpha damage; (3) chemistry that tracks a real physiologic pathway (glucose uptake, iodide trapping, bone mineral turnover); and (4) chemistry and radiation dose safe enough for outpatient use. Alpha emitters are generally avoided for routine diagnostic imaging precisely because their short range dumps energy into surrounding tissue without providing useful external photons.

Common MCAT traps for nuclear medicine: confusing PET (positron → annihilation gammas) with SPECT/gamma imaging (nuclear gamma emission alone); thinking half-life depends on dose size; mixing up electron capture with beta-minus (electron capture lowers Z, beta-minus raises Z); and assuming any radioactive iodine is therapeutic — diagnostic ¹²³I and therapeutic ¹³¹I are different isotopes chosen for different decay properties.

The Mass Spectrometer

A mass spectrometer measures the mass-to-charge ratio (m/z) of ions, which is how chemists experimentally determine isotopic masses and abundances — the very data used to calculate an element's atomic weight. Operation happens in three stages. First, ionization: the sample is vaporized and bombarded with an electron beam (or another ionization method), knocking electrons off molecules or atoms to create positively charged ions. Second, acceleration: those ions are accelerated through an electric potential difference, so every ion of a given charge gains the same kinetic energy — but because kinetic energy is (1/2)mv², lighter ions end up moving faster than heavier ions. Third, magnetic deflection: the ion beam passes through a magnetic field oriented perpendicular to its velocity. The magnetic force curves each ion's path into a circular arc with radius r = mv/(qB), so heavier ions (larger m) curve less sharply (larger radius) than lighter ions of the same charge. A detector records where each ion lands, producing a spectrum of peaks at each mass-to-charge ratio, with peak height proportional to the relative abundance of that isotope.

Mass spectrometry has major biomedical applications relevant to MCAT passages: it is central to proteomics, where digesting a protein into peptide fragments and measuring their exact masses (peptide mass fingerprinting) identifies the protein; it underlies newborn metabolic screening for inborn errors of metabolism; and isotope-ratio methods track nutrient or drug metabolism with stable (non-radioactive) heavy isotopes such as ¹³C and ¹⁵N. Unlike PET or SPECT, mass spectrometry does not require radioactivity — it separates ions by physics alone — but the same careful distinction between isotopes of identical chemistry and different mass that underpins radiotracer design also underpins every mass spectrum you will interpret on exam day.