Atom
An atom is the fundamental unit of matter that retains the chemical properties of an element, consisting of a central nucleus composed of positively charged protons and uncharged neutrons, surrounded by a cloud of negatively charged electrons.[1][2] The number of protons in the nucleus, known as the atomic number, uniquely identifies the element, while the total number of protons and neutrons defines the atom's mass number.[1] Electrons, which are much lighter than protons or neutrons, occupy probabilistic orbitals around the nucleus, enabling atoms to form bonds and molecules through electron interactions.[2] The concept of the atom originated in ancient Greek philosophy, with philosopher Democritus around 440 BCE proposing that matter consists of indivisible particles called atomos, meaning "uncuttable" in Greek.[2] Modern atomic theory was established in the early 19th century by John Dalton, who described atoms as indestructible spheres combining in fixed ratios to form compounds.[2] Key experimental discoveries followed: in 1897, J.J. Thomson identified the electron as a subatomic particle; Ernest Rutherford in 1911 revealed the nuclear structure through gold foil experiments; and James Chadwick discovered the neutron in 1932.[2] These advancements confirmed that atoms are not truly indivisible but are composed of smaller particles, with protons and neutrons themselves made of fundamental quarks.[2] Atoms form the basis of all ordinary matter in the universe, originating from the primordial nucleosynthesis following the Big Bang approximately 13.8 billion years ago, when light elements like hydrogen and helium were first produced.[2] Variations in neutron count create isotopes, some of which are stable and others radioactive, influencing nuclear processes like fission and fusion that power stars and atomic energy.[1] The diameter of an atom typically ranges from 0.1 to 0.5 nanometers, with the electron cloud extending vastly larger—up to 10,000 times the nucleus size—yet the nucleus holds over 99.9% of the atom's mass.[2] Understanding atomic structure underpins fields from chemistry and materials science to quantum mechanics and nuclear physics.[1]Historical Development
Philosophical Foundations
The concept of the atom originated in ancient Greek philosophy during the 5th century BCE, where Leucippus and his student Democritus proposed that all matter consists of indivisible, eternal particles called atomos, meaning "uncuttable" or "indivisible," moving through empty space to form the visible world through their combinations and interactions.[3] These atomists argued that atomos were solid, impenetrable, and varied in shape, size, and position, providing a mechanistic explanation for change and diversity in nature without invoking divine intervention or teleology.[3] Parallel ideas emerged independently in ancient Indian philosophy, particularly within the Nyāya-Vaiśeṣika school, which posited paramāṇu as the ultimate, indivisible units of matter that combine to form composite substances and explain the creation and dissolution of the physical world.[3] In texts like the Vaiśeṣika Sūtra (circa 2nd century BCE to 1st century CE), paramāṇu were described as eternal, partless particles possessing inherent qualities such as motion and combining under unseen forces to manifest grosser forms of matter.[3] This atomistic framework integrated with broader metaphysical categories, emphasizing qualities, actions, and universals as fundamental realities alongside these minimal material units.[3] Aristotle vehemently opposed this atomistic view in the 4th century BCE, advocating instead for matter as continuous and infinitely divisible, arguing that discrete particles would imply voids, which he deemed impossible since nature abhors a vacuum.[3] His hylomorphic theory, positing matter as a substrate informed by form and actualized through potentiality, dominated Western thought and suppressed atomism during the medieval period, where scholastic philosophers largely favored continuous substance over discrete particles.[3] The philosophical foundations of atomism experienced a revival in the 17th century through Pierre Gassendi, who reformulated Epicurean ideas into a Christian-compatible corpuscular theory, describing atoms as tiny, solid, indivisible bodies created by God and endowed with motion to account for natural phenomena.[4] Gassendi's Syntagma Philosophicum (published posthumously in 1658) emphasized sensory evidence and mechanistic principles, bridging ancient atomism with emerging scientific inquiry.[4] Isaac Newton further advanced these ideas in the 1704 edition of Opticks, where in a series of queries, he speculated on atoms as hard, impenetrable, movable particles whose attractions and repulsions underpin chemical affinities and the cohesion of bodies, laying conceptual groundwork for later empirical developments.[5]Classical Atomic Theory
The classical atomic theory emerged in the early 19th century as chemists sought empirical explanations for the patterns observed in chemical reactions, shifting from philosophical speculation to quantitative evidence. John Dalton formulated the law of multiple proportions in 1803, observing that when two elements form more than one compound, the masses of one element that combine with a fixed mass of the other are in the ratios of small whole numbers, such as the 1:2 ratio of oxygen in carbon monoxide (CO) and carbon dioxide (CO₂).[6] This law provided key support for the existence of atoms as discrete units of matter.[7] In 1808, Dalton fully articulated his atomic theory in A New System of Chemical Philosophy, proposing that all matter consists of tiny, indivisible particles called atoms; atoms of the same element are identical in mass and properties, while atoms of different elements differ; atoms combine in simple whole-number ratios to form compounds; and atoms are indestructible and indivisible by chemical means.[8] These postulates explained the law of definite proportions and multiple proportions, establishing atoms as the fundamental building blocks of chemical substances.[9] Avogadro's hypothesis, proposed in 1811, advanced this framework by stating that equal volumes of all gases, at the same temperature and pressure, contain equal numbers of molecules, allowing chemists to differentiate between atomic and molecular forms of elements and to calculate relative atomic weights more accurately.[10] This idea resolved inconsistencies in Dalton's assignments of atomic weights and facilitated the quantitative study of gases.[11] Building on these foundations, early attempts to organize elements by atomic weights revealed periodic patterns. In 1829, Johann Wolfgang Döbereiner identified triads of elements with similar chemical properties, where the atomic weight of the middle element was approximately the average of the other two, such as chlorine (35.5), bromine (80), and iodine (127).[12] Later, in 1865, John Newlands arranged known elements in order of increasing atomic weight and noted that their properties repeated every eighth element, akin to the octaves in music, though his scheme was limited by incomplete atomic weight data.[12] These precursors laid the groundwork for the modern periodic table.Key Experimental Discoveries
In 1897, J.J. Thomson conducted experiments using cathode ray tubes, where streams of particles emitted from the cathode were deflected by electric and magnetic fields, revealing that these rays consisted of negatively charged particles much smaller than atoms. By measuring the deflection, Thomson calculated the charge-to-mass ratio e/m of these particles to be approximately $1.76 \times 10^{11} C/kg, establishing the existence of the electron as a fundamental component of matter. To determine the electron's absolute charge, Robert Millikan performed the oil drop experiment starting in 1909, ionizing oil droplets and balancing their gravitational fall against electrostatic forces in a controlled electric field. This yielded the elementary charge e = 1.602 \times 10^{-19} C, confirming that electric charge is quantized in discrete units. Ernest Rutherford's 1911 gold foil experiment further probed atomic structure by directing alpha particles at a thin sheet of gold foil and observing their scattering patterns on a fluorescent screen. While most particles passed through undeflected, a small fraction scattered at large angles, indicating that the atom's positive charge and mass were concentrated in a tiny, dense nucleus rather than diffusely distributed. Rutherford derived a formula for the differential scattering cross-section, \frac{d\sigma}{d\Omega} = \left( \frac{Z_1 Z_2 e^2}{8\pi \epsilon_0 E} \right)^2 \frac{1}{\sin^4(\theta/2)}, where Z_1 and Z_2 are the atomic numbers of the incident and target particles, e is the elementary charge, \epsilon_0 is the vacuum permittivity, and E is the kinetic energy of the alpha particle, which quantitatively matched the observed scattering distribution.[13] Between 1917 and 1919, Rutherford extended his scattering studies by bombarding light elements like nitrogen with alpha particles, detecting hydrogen nuclei ejected from the targets, which he identified as positively charged protons constituting the nucleus of the hydrogen atom. These experiments demonstrated that protons were fundamental building blocks present in all atomic nuclei. In 1932, James Chadwick irradiated beryllium with alpha particles, producing highly penetrating neutral radiation that knocked protons out of paraffin wax with energies inconsistent with gamma rays. Analyzing the momentum transfer, Chadwick concluded these were neutrons—uncharged particles with mass nearly equal to the proton—resolving discrepancies in atomic masses and explaining the existence of isotopes with the same atomic number but different masses.[14]Quantum Mechanical Model
The quantum mechanical model of the atom represents a paradigm shift from classical descriptions, incorporating the wave-particle duality of electrons, where they exhibit both particle-like and wave-like properties, leading to probabilistic interpretations of their positions rather than definite orbits.[15] This framework emerged in the early 20th century to resolve inconsistencies in classical physics, such as the stability of atoms and the discrete nature of atomic spectra, by treating electrons as solutions to wave equations that yield probability distributions for their locations.[16] A foundational step was Niels Bohr's 1913 model, which introduced quantized angular momentum for the electron in hydrogen, given by L = n [\hbar](/page/H-bar), where n is a positive integer (the principal quantum number) and \hbar = h / 2\pi is the reduced Planck's constant. This quantization condition, combined with balancing centripetal and Coulomb forces, yielded discrete energy levels E_n = -\frac{13.6 \, \text{eV}}{n^2} for the hydrogen atom, accurately predicting its emission spectrum. Although semi-classical, Bohr's model laid the groundwork for full quantum treatments by imposing discreteness on continuous classical motion.[16] The complete quantum description arrived with Erwin Schrödinger's 1926 wave equation, a partial differential equation governing the electron's wave function \psi(r, \theta, \phi) in spherical coordinates for the hydrogen atom: -\frac{\hbar^2}{2m} \nabla^2 \psi - \frac{e^2}{4\pi \epsilon_0 r} \psi = E \psi, where m and e are the electron mass and charge, respectively, and \nabla^2 is the Laplacian operator.[17] Solutions to this time-independent equation separate into radial and angular parts, producing hydrogen-like atomic orbitals characterized by quantum numbers n, l (azimuthal), and m_l (magnetic), with |\psi|^2 giving the probability density for finding the electron./11%3A_Quantum_Mechanics_and_Atomic_Structure/11.10%3A_The_Schrodinger_Wave_Equation_for_the_Hydrogen_Atom) These orbitals replace Bohr's circular orbits with three-dimensional probability clouds, explaining atomic stability through standing wave interference. For multi-electron atoms, the Pauli exclusion principle, formulated by Wolfgang Pauli in 1925, states that no two electrons can occupy the same quantum state simultaneously, ensuring distinct orbitals for each electron.[18] This principle, initially proposed to explain periodic trends in atomic spectra, relies on electrons possessing an intrinsic angular momentum or spin. In 1925, George Uhlenbeck and Samuel Goudsmit introduced electron spin as a fourth quantum number s = \pm 1/2, attributing the anomalous Zeeman effect to this magnetic moment, with spin angular momentum \mathbf{S} = \frac{\hbar}{2} \boldsymbol{\sigma} (where \boldsymbol{\sigma} are Pauli matrices).[19] Combining spin with orbital quantum numbers, the exclusion principle limits each orbital to at most two electrons of opposite spin.[18] The Heisenberg uncertainty principle, articulated by Werner Heisenberg in 1927, further underscores the probabilistic nature of the quantum atom: \Delta x \Delta p \geq \frac{\hbar}{2}, where \Delta x and \Delta p are uncertainties in position and momentum.[15] This implies that precise knowledge of an electron's position precludes exact momentum determination, rendering classical trajectories impossible and confining electrons to delocalized orbitals with inherent spread, which stabilizes atoms against radiation.[20] To approximate solutions for multi-electron atoms, where the Schrödinger equation becomes intractable due to electron-electron interactions, the Hartree-Fock method employs a mean-field approach. Developed by Douglas Hartree in 1928 and refined by Vladimir Fock in 1930, it assumes each electron moves in an effective potential from the nucleus and the averaged charge distribution of other electrons, yielding self-consistent single-particle orbitals via iterative solution of coupled equations.[21] This approximation captures much of the electronic structure, though it neglects instantaneous correlations between electrons.[22]Atomic Structure
The atomic structure describes the organization of an atom's subatomic components, featuring a central, positively charged nucleus composed of protons and neutrons, which accounts for nearly all of the atom's mass, surrounded by a probabilistic cloud of negatively charged electrons. This configuration, established through early 20th-century experiments and quantum mechanics, determines the atom's chemical and physical properties.[23][24]Subatomic Particles
Atoms are composed of three primary types of subatomic particles: protons, neutrons, and electrons, which together determine the atom's fundamental properties such as charge and mass. These particles are the building blocks that form the nucleus and surrounding electron cloud, with protons and neutrons residing in the dense central nucleus and electrons occupying the outer regions.[25] Protons carry a positive electric charge of +e, where e is the elementary charge with a value of $1.602176634 \times 10^{-19} C, and have a rest mass of approximately $1.6726 \times 10^{-27} kg.[26][27] The number of protons in an atom's nucleus, denoted by the atomic number Z, uniquely identifies the chemical element and governs its chemical behavior.[28] Neutrons are electrically neutral particles with no net charge and a rest mass of approximately $1.6749 \times 10^{-27} kg, slightly greater than that of the proton.[29] They contribute to the atom's mass without affecting its charge; the total number of protons and neutrons, known as the mass number A, approximates the atomic mass in atomic mass units.[30] Electrons possess a negative charge of -e and have a much smaller rest mass of approximately $9.109 \times 10^{-31} kg, which is about $1/[1836](/page/1836) that of a proton, rendering their contribution to the atom's total mass negligible.[31][26] Despite their light weight, electrons play a crucial role in atomic interactions and chemical bonding due to their position outside the nucleus. At a deeper level, protons and neutrons are not elementary but composite particles made of quarks bound by the strong nuclear force mediated by gluons. A proton consists of two up quarks and one down quark (uud), while a neutron comprises one up quark and two down quarks (udd); electrons, however, are fundamental leptons with no known substructure.[25] In the context of atoms, protons and neutrons function as effective units, as quarks are confined within hadrons and not observed in isolation. Antimatter counterparts of these particles exist in exotic contexts, such as particle accelerators or cosmic rays, including the positron (antielectron) with charge +e and mass identical to the electron, the antiproton with charge -e and quark content \bar{u}\bar{u}\bar{d}, and the antineutron with neutral charge and \bar{u}\bar{d}\bar{d} composition. These antiparticles annihilate upon contact with their matter counterparts, releasing energy, but play no role in ordinary atomic matter.Nuclear Composition
The atomic nucleus constitutes the central core of an atom, with a diameter on the order of $10^{-15} meters (1 femtometer), containing Z protons and N = A - Z neutrons, where A is the mass number representing the total number of nucleons.[32][33] This compact structure, vastly smaller than the surrounding electron cloud, holds over 99.9% of the atom's mass due to the close packing of these subatomic particles.[34] The protons and neutrons, collectively known as nucleons, are bound together by the strong nuclear force, a fundamental interaction that acts over extremely short distances of approximately 1 femtometer.[35] This force originates from the underlying color force mediated by gluons between quarks within the nucleons, resulting in a residual attraction that overcomes the electromagnetic Coulomb repulsion between the positively charged protons.[35] Without this overpowering attraction, the electrostatic repulsion would cause the nucleus to disintegrate, highlighting the strong force's essential role in maintaining nuclear stability.[35] The stability of the nucleus arises from its binding energy, defined as the energy required to disassemble it into its individual protons and neutrons. This energy is calculated using the mass-energy equivalence principle: E_b = \left[ Z m_p + N m_n - M \right] c^2 where m_p and m_n are the masses of the proton and neutron, respectively, M is the measured mass of the nucleus, and c is the speed of light.[36] The difference Z m_p + N m_n - M, known as the mass defect, reflects the conversion of mass into binding energy during nucleus formation. To approximate this binding energy across different nuclei, the semi-empirical mass formula, developed by Hans Bethe and Carl Friedrich von Weizsäcker, incorporates terms for volume, surface, Coulomb repulsion, asymmetry, and pairing effects, providing a macroscopic description of nuclear masses. Isotopes are variants of an element with the same atomic number Z (and thus the same number of protons) but different numbers of neutrons N, leading to distinct mass numbers A. For instance, carbon-12 (^{12}\mathrm{C}, with 6 protons and 6 neutrons) and carbon-14 (^{14}\mathrm{C}, with 6 protons and 8 neutrons) are isotopes of carbon, differing in nuclear stability and applications such as dating in archaeology.[33] Two primary theoretical models describe the nucleus's composition and behavior: the liquid drop model and the shell model. The liquid drop model, analogous to a charged incompressible fluid, treats the nucleus as a droplet where nucleons interact collectively, accounting for bulk properties like binding energy through surface tension and electrostatic terms; it was notably applied by Niels Bohr to explain nuclear fission. In contrast, the shell model views nucleons as occupying discrete energy levels or "shells" governed by quantum mechanics, similar to electrons in atoms, with strong spin-orbit coupling explaining periodic variations in nuclear properties such as magic numbers (e.g., 2, 8, 20, 28); this model was independently developed by Maria Goeppert Mayer and J. Hans D. Jensen. These models complement each other, with the liquid drop providing a global view and the shell model capturing microscopic details.Electron Arrangement
Electrons in an atom occupy specific regions of space around the nucleus known as atomic orbitals, which are solutions to the Schrödinger equation describing the probability distribution of electron positions. These orbitals are organized into shells and subshells based on quantum mechanical principles, providing the framework for the atom's electronic structure. The arrangement of electrons is specified by four quantum numbers. The principal quantum number n determines the energy level and average distance from the nucleus, taking positive integer values (n = 1, 2, [3](/page/3), \ldots). The azimuthal quantum number l, also called the angular momentum quantum number, defines the subshell and orbital shape, ranging from 0 to n-1 (with l = 0 for s, 1 for p, 2 for d, and 3 for f orbitals). The magnetic quantum number m_l specifies the orbital's orientation in space, ranging from -l to +l in integer steps. The spin quantum number m_s describes the electron's intrinsic spin, with values of +\frac{1}{2} or -\frac{1}{2}. These quantum numbers ensure that no two electrons in an atom share the same set, as per the Pauli exclusion principle. Atomic orbitals have characteristic shapes determined by the azimuthal quantum number l. S orbitals (l = 0) are spherical, symmetric around the nucleus. P orbitals (l = 1) have a dumbbell shape with two lobes along one axis (x, y, or z). D orbitals (l = 2) exhibit more complex cloverleaf or double-dumbbell shapes, while f orbitals (l = 3) are even more intricate with multiple lobes. The electron's location is probabilistic, given by the square of the wave function |\psi|^2, which represents the probability density of finding the electron at a particular point in space. Electrons fill orbitals according to the Aufbau principle, which states that orbitals are occupied in order of increasing energy, starting with the lowest available. The typical filling sequence is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on, accommodating up to 2 electrons per orbital. Hund's rule governs the arrangement within degenerate orbitals (those of equal energy), requiring electrons to occupy separate orbitals with parallel spins before pairing up, thereby maximizing the total spin angular momentum. This filling pattern results in ground-state electron configurations that correlate directly with the periodic table's block structure. The s-block elements (groups 1 and 2) have valence electrons in ns orbitals, typically ns¹ or ns² configurations. The p-block elements (groups 13–18) fill np orbitals, leading to ns²np¹ through ns²np⁶ valence setups. In heavy atoms, relativistic effects become significant due to high nuclear charge, causing inner electrons to move at speeds approaching the speed of light. This leads to relativistic contraction of s and p orbitals, stabilizing them relative to d and f orbitals and altering electron configurations. For example, in gold (atomic number 79), these effects narrow the energy gap between 5d and 6s orbitals, shifting absorption of blue light and resulting in the metal's characteristic yellow color.Physical Properties
Mass and Charge
The mass of an atom is predominantly contributed by its nucleus, where protons and neutrons account for nearly all of the total mass, as the mass of electrons is negligible—approximately 1/1836 that of a proton.[37] Atomic masses are conventionally expressed in unified atomic mass units (u), defined such that 1 u equals one-twelfth the mass of a carbon-12 atom at rest in its ground state, corresponding to 1.660539 × 10^{-27} kg.[38] For instance, the protium isotope (hydrogen-1) has an atomic mass of approximately 1.0078 u, reflecting the slight excess over the proton's mass due to the binding energy and electron contribution.[39] While isotopic masses refer to the specific mass of a single isotope, the standard atomic weight for an element is a weighted average based on its natural isotopic abundances. This distinction is evident in elements like chlorine, where the atomic weight of 35.45 u arises from the prevalence of chlorine-35 (about 75.8%) and chlorine-37 (about 24.2%), rather than an exact integer value.[40] Such weighted averages are crucial for chemical calculations, as they represent the effective mass encountered in typical samples. A neutral atom possesses zero net charge due to the exact balance between the positive charges of its Z protons in the nucleus and the negative charges of its Z electrons orbiting the nucleus.[41] Ionization disrupts this neutrality: loss of one or more electrons produces a cation with a net positive charge, while gain of electrons yields an anion with a net negative charge. Precise measurement of atomic and isotopic masses relies on mass spectrometry, a technique developed by Francis Aston in 1919 using his mass spectrograph to ionize atoms and separate them by mass-to-charge ratio.[42] Aston's work revealed the "whole number rule," showing that isotopic masses are very close to integer multiples of the hydrogen mass but deviate slightly due to nuclear binding effects, enabling the discovery of non-radiogenic isotopes and refining atomic mass scales.[43]Size and Shape
The size of an atom is typically on the order of $10^{-10} meters, or 1 angstrom (Å), representing the scale at which atomic dimensions are measured. This scale encompasses different measures of atomic radius depending on the context: the covalent radius is half the distance between nuclei of two identical atoms sharing a bond, the van der Waals radius is half the distance between non-bonded atoms in close contact, and the ionic radius applies to atoms in ionic compounds where size varies with charge.[44] These variants highlight how atomic size adapts to bonding environments, with covalent radii generally smaller than van der Waals radii due to orbital overlap. In the quantum mechanical description, the spatial extent of an atom is defined by the electron cloud rather than a fixed boundary. For the hydrogen atom in its ground state, the Bohr radius a_0 = 5.29 \times 10^{-11} m serves as a fundamental unit, representing the most probable distance of the electron from the nucleus. For hydrogen-like atoms, the electron cloud's extent scales with the square of the principal quantum number n, such that approximately 99% of the electron probability density is contained within a radius on the order of n^2 a_0. This probabilistic distribution arises from the wave nature of electrons, where electron orbitals define regions of high probability rather than precise paths. Atoms lack a definite shape due to the probabilistic nature of the electron cloud, but the distribution can exhibit non-spherical characteristics influenced by quantum numbers and molecular contexts. For instance, in molecules, hybridization of atomic orbitals—such as sp³ hybridization in carbon—results in a tetrahedral arrangement of electron density to minimize repulsion and optimize bonding.[45] However, for isolated atoms, the overall electron cloud is often effectively spherical for closed shells, with deviations arising from unfilled orbitals. Direct visualization of atomic surfaces has been enabled by scanning tunneling microscopy (STM), invented by Gerd Binnig and Heinrich Rohrer in 1981, which maps electron density variations at the atomic scale by measuring tunneling currents between a sharp tip and the sample surface. This technique provides empirical evidence of the irregular, probabilistic contours of atomic electron clouds, confirming sizes and subtle shape features in real materials.Stability and Decay
Nuclear stability arises from the balance of attractive strong nuclear forces and repulsive Coulomb forces within the nucleus, with stable isotopes exhibiting specific patterns in proton (Z) and neutron (N) numbers.[46] Nuclides with even values of both Z and N, known as even-even nuclei, are particularly stable due to nucleon pairing effects that lower their energy.[46] Additionally, certain "magic numbers" of protons or neutrons—2, 8, 20, 28, 50, 82, and 126—correspond to filled nuclear shells, enhancing stability analogous to electron shells in atoms; nuclei with these configurations, such as helium-4 (N=2, Z=2) or lead-208 (N=126, Z=82), are exceptionally long-lived.[47] The neutron-to-proton ratio (N/Z) also governs stability: for light nuclei (Z < 20), stable isotopes have N/Z ≈ 1, while heavier nuclei require N/Z > 1 (up to about 1.5) to counterbalance increasing Coulomb repulsion.[46] Unstable nuclei undergo radioactive decay to achieve a more stable configuration, primarily through three modes: alpha, beta, and gamma decay.[48] Alpha decay involves the emission of an alpha particle, a helium-4 nucleus (two protons and two neutrons), typically from heavy nuclei (Z > 82) to reduce Coulomb repulsion; for example, uranium-238 decays via alpha emission to thorium-234.[48] Beta decay occurs in two variants: beta-minus (β⁻) emission of an electron and antineutrino, converting a neutron to a proton (increasing Z by 1), or beta-plus (β⁺) emission of a positron and neutrino, converting a proton to a neutron (decreasing Z by 1); this mode adjusts the N/Z ratio toward stability.[48] Gamma decay follows alpha or beta decay, releasing a high-energy photon from an excited nucleus to reach its ground state, without altering Z or N.[48] The energetics of decay are quantified by the Q-value, the energy released or absorbed, calculated asQ = (M_\text{parent} - M_\text{daughters}) c^2,
where M denotes atomic masses and c is the speed of light; a positive Q indicates an energetically favorable, spontaneous process.[49] For decay to occur, Q must exceed zero, with the mass difference reflecting the binding energy differences between parent and daughter nuclei.[49] Radioactive decay follows an exponential law governed by the decay constant λ, where the number of undecayed nuclei N at time t is N = N_0 e^{-\lambda t}, with N_0 the initial number.[50] The half-life t_{1/2}, the time for half the nuclei to decay, relates to λ by t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}.[50] Half-lives span vast ranges, from fractions of a second to billions of years; for instance, uranium-238 has a half-life of 4.468 billion years, making it key for geochronology.[51] Nuclear stability also influences fission and fusion processes, determined by the binding energy per nucleon curve, which peaks at nickel-62 (approximately 8.80 MeV per nucleon), near iron-56, rendering it the most stable nucleus.[52] Lighter elements (A < 56) can release energy via fusion by forming heavier nuclei toward the peak, while heavier elements (A > 56) undergo fission to lighter, more stable fragments, also releasing energy; these thresholds explain stellar nucleosynthesis and nuclear power.[52]