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Mass number

The mass number, denoted by the symbol A and also known as the nucleon number, is defined as the total number of protons and neutrons (collectively called nucleons) present in the nucleus of an atom. It represents an integer value that approximates the atomic mass of the nuclide in unified atomic mass units (u), since both protons and neutrons have masses very close to 1 u each.^[1] The mass number is calculated as A = Z + N, where Z is the atomic number (the number of protons) and N is the number of neutrons.^[2] In standard chemical notation, the mass number appears as a left superscript to the chemical symbol of the element, often alongside the atomic number as a left subscript, in the form ^{A}_{Z}\mathrm{X}, where X is the element symbol; for example, carbon-12 is written as ^{12}_{6}\mathrm{C}.^[1] This notation uniquely identifies a nuclide, as the combination of atomic number and mass number specifies both the element and its isotopic form.^[3] The mass number plays a crucial role in nuclear physics and chemistry, particularly in distinguishing isotopes—variants of the same element that share the same atomic number but differ in mass number due to varying neutron counts.^[2] Unlike the atomic mass (or standard atomic weight), which is a weighted average of the atomic masses of an element's naturally occurring isotopes based on their abundances, the mass number applies specifically to individual isotopes and is always an integer.^[2] For instance, the two stable isotopes of carbon—^{12}_{6}\mathrm{C} (mass number 12) and ^{13}_{6}\mathrm{C} (mass number 13)—have the same atomic number of 6 but different mass numbers, leading to distinct nuclear properties. The radioactive isotope ^{14}_{6}\mathrm{C} (mass number 14) is used in radiometric dating.^[2] Mass numbers greater than 208 are typically unstable and radioactive, influencing stability trends across the periodic table.^[1]

Fundamentals

Definition

The mass number, denoted by the symbol A, is defined as the total number of protons and neutrons—collectively referred to as nucleons—in the nucleus of an atom.^[4]^[5] This integer value provides a fundamental measure of the nuclear composition, distinguishing it from the atomic number Z, which counts only the protons in the nucleus and determines the chemical identity of the element.^[5] The number of neutrons N is calculated as N = A - Z.^[5] As an approximation, the mass number represents the mass of the nucleus in atomic mass units (u), where each nucleon contributes roughly 1 u to the total, though actual isotopic masses deviate slightly due to binding effects. For instance, the carbon-12 isotope (^{12}\text{C}) has a mass number of 12, comprising 6 protons and 6 neutrons, which approximates its nuclear mass at 12 u.^[5] Isotopes of an element share the same atomic number but differ in mass number due to varying neutron counts.^[6]

Notation

The mass number A is conventionally represented as a left superscript preceding the chemical symbol of the element in nuclide notation, as in ^{A}X, where X denotes the element symbol. This form is used when the atomic number Z is either unnecessary or implied by context. For complete specification, the atomic number is included as a left subscript, yielding the standard nuclide symbol _{Z}^{A}X, which uniquely identifies a particular nuclide by indicating both the number of protons (Z) and the total number of nucleons (A).^[7] Alternative notations simplify representation in certain contexts. For instance, A X omits the atomic number subscript when Z is clear, while X-A places the mass number after the symbol separated by a hyphen, such as C-12 for carbon-12. In verbal or informal chemical nomenclature, a hyphenated form is common for isotopes, exemplified by uranium-235 or U-235, where the element name or symbol precedes the mass number. These variants facilitate readability in publications and discussions without altering the underlying meaning.^[7]^[8] Isobars, defined as nuclides sharing the same mass number A but differing in atomic number Z, are often denoted using the simplified ^{A}X form to emphasize the common A value across different elements, such as ^{14}\mathrm{C} and ^{14}\mathrm{N} for carbon-14 and nitrogen-14. This notation highlights structural similarities in nuclear physics contexts.^[8] The International Union of Pure and Applied Chemistry (IUPAC) provides guidelines for these notations in scientific literature, recommending the left-superscript placement for mass number to ensure precision and consistency, particularly in isotopically specified compounds and nuclear data compilations. Arabic numerals are used exclusively for A and Z, with the symbols italicized only if representing variables rather than fixed labels.^[7]

Isotopes and Identification

Isotopes

Isotopes are nuclides of the same chemical element that have the same atomic number Z but different mass numbers A, arising from variations in the number of neutrons in the nucleus.^[9] This difference in neutron count results in atoms that are chemically nearly identical but differ in mass and nuclear stability.^[9] Isotopes are classified as stable or unstable (radioactive), depending on whether they undergo spontaneous radioactive decay.^[10] Stable isotopes persist indefinitely without decaying, while unstable ones decay over time into other nuclides. In nature, elements typically occur as mixtures of isotopes with varying abundances; for instance, carbon consists primarily of the stable isotope carbon-12 (approximately 99% abundance) and a smaller fraction of carbon-13 (approximately 1% abundance).^[11] These natural abundances reflect the relative proportions in which the isotopes are found in Earth's materials. The relative atomic mass of an element, as listed in periodic tables, is calculated as the amount-weighted average of the atomic masses of its isotopes, using their fractional abundances as weights.^[12] This averaging accounts for the isotopic composition of the element in standard terrestrial samples, providing a practical value for chemical calculations rather than the mass of any single isotope. Isotopes play key roles in chemistry and biology due to subtle differences in their physical properties. For example, deuterium (hydrogen-2, with one proton and one neutron) exhibits a kinetic isotope effect, where reactions involving C–H bonds proceed faster than those with C–D bonds because of the heavier mass of deuterium, which strengthens the bond and raises the activation energy.^[13] In biology, stable isotopes like deuterium oxide are employed as tracers to study metabolic processes, such as water turnover in the body, without introducing radioactivity.^[14] The discovery of isotopes is credited to early 20th-century work using mass spectrometry, notably by Francis Aston, who in 1919 identified isotopes of neon (masses 20 and 22) with his improved mass spectrograph at the Cavendish Laboratory, confirming J.J. Thomson's earlier observations and enabling widespread isotopic analysis.^[15] Isotopes are conventionally denoted in isotopic notation, such as ^{A}_{Z}\mathrm{X}.^[9]

Isotopic Notation

Isotopic notation specifies the mass number A to distinguish isotopes of the same element, building on general nuclide symbolism where the mass number is placed as a left superscript to the element symbol, such as ^{12}\mathrm{C} for carbon-12 or ^{2}\mathrm{H} for deuterium.^[6]^[16] This convention, recommended by the International Union of Pure and Applied Chemistry (IUPAC), ensures precise identification by highlighting the total number of protons and neutrons, with the atomic number implied by the element symbol.^[16] An alternative verbal or abbreviated form uses a hyphen followed by the mass number, as in carbon-12 or C-12, which is commonly employed in textual descriptions and databases.^[6] For the hydrogen isotopes, standard notation designates protium as ^{1}\mathrm{H} or H-1, deuterium as ^{2}\mathrm{H} or H-2, and tritium as ^{3}\mathrm{H} or H-3, reflecting their increasing mass numbers due to additional neutrons.^[6] These examples illustrate how the superscript mass number directly conveys the isotopic variant without ambiguity.^[16] IUPAC guidelines in the Nomenclature of Organic Chemistry (Blue Book, Chapter P-8) recommend using roman type for the element symbol with the mass number as an italicized superscript in equations and tables, reserving italicization for locants in compound names.^[16] For instance, in chemical equations, ^{12}\mathrm{CH_4} denotes methane with a specific carbon isotope, and nuclides should be ordered alphabetically by symbol or by increasing mass number when multiple are present.^[16] Notation for isotopic mixtures or enriched samples extends these conventions to indicate composition variations. Natural mixtures are often denoted simply by the element symbol (e.g., C for natural carbon, comprising ~98.93% ^{12}\mathrm{C} and ~1.07% ^{13}\mathrm{C})^[17], while enriched samples use prefixes like "enr" followed by the nuclide symbol, such as enr-^{13}\mathrm{C} for carbon enriched in the 13 isotope.^[16] Isotopically deficient mixtures employ "def" (e.g., def-^{13}\mathrm{C}), adhering to IUPAC standards for labeling in analytical contexts.^[16] In spectroscopy and mass spectrometry, the mass number facilitates peak identification by correlating spectral peaks with specific isotopes, as each isotopologue produces distinct m/z values based on its mass.^[18] For example, in the mass spectrum of ethanol (C₂H₅OH), the molecular ion at m/z 46 corresponds to the most abundant ^{12}\mathrm{C}_2 form, while adjacent M+1 (m/z 47) and M+2 (m/z 48) peaks arise from ^{13}\mathrm{C} and ^{18}\mathrm{O} substitutions, respectively, allowing deduction of elemental composition from mass number patterns and natural abundances.^[18] This reliance on mass number enables quantitative analysis of isotopic distributions in complex samples.^[18]

Mass Relations

Isotopic Mass

The isotopic mass of a nuclide refers to the experimentally measured mass of its neutral atom, including the masses of its electrons, expressed in unified atomic mass units (u). This value is typically a non-integer, reflecting the mass defect caused by the conversion of a portion of the nucleons' rest mass into nuclear binding energy during the formation of the nucleus.^[19]^[20] The isotopic mass m approximates the mass number A, but a more precise conceptual relation is given by

m \approx Z m_\mathrm{H} + N m_\mathrm{n} - \frac{\Delta m c^2}{c^2},

where Z is the atomic number (number of protons), N = A - Z is the number of neutrons, m_\mathrm{H} is the mass of a neutral hydrogen atom, m_\mathrm{n} is the neutron mass, \Delta m is the mass defect, and c is the speed of light. The mass defect \Delta m is defined as the difference between the total mass of the separated protons, neutrons, and electrons (Z m_\mathrm{H} + N m_\mathrm{n}) and the actual measured atomic mass; this defect corresponds to the nuclear binding energy via Einstein's equation E_b = \Delta m c^2, which quantifies the energy required to disassemble the nucleus into its constituent particles.^[21]^[22] Isotopic masses are determined through high-precision mass spectrometry, a technique that ionizes atoms and separates them based on their mass-to-charge ratio in electric and magnetic fields to yield accurate mass values. A key reference is the isotope carbon-12 (^{12}\mathrm{C}), defined by the International Union of Pure and Applied Chemistry (IUPAC) in 1961 as having an isotopic mass of exactly 12 u for its neutral atom in the ground state, establishing the unified atomic mass unit as one-twelfth of this value.^[23]^[19] In distinction from the mass number A, which is the dimensionless integer approximating the total number of nucleons, the isotopic mass is an empirical quantity derived from measurement. For example, the isotopic mass of oxygen-16 (^{16}\mathrm{O}) is 15.994915 u, slightly less than 16 due to the mass defect.^[23]^[24]

Relative Atomic Mass

The relative atomic mass, denoted A_r(E), of an element E is defined as the ratio of the average mass per atom of the element in its standard isotope composition to one-twelfth the mass of an atom of the carbon-12 isotope.^[25] This value is dimensionless and represents a weighted average of the relative isotopic masses of the element's naturally occurring isotopes, scaled such that the relative mass of ^{12}\mathrm{C} is exactly 12.^[12] The calculation accounts for the fractional abundances of each isotope, ensuring the result reflects the typical composition found in normal terrestrial materials. The formula for relative atomic mass is given by

A_r(E) = \sum_i x_i \cdot A_i,

where x_i is the mole fraction (fractional abundance) of isotope i, and A_i is the relative isotopic mass of that isotope.^[12] The mass number A of each isotope, which approximates the integer sum of protons and neutrons, determines the range of isotopic masses around this average; for elements with multiple stable isotopes differing in A, the weighted average often falls between the nearest integers. For example, chlorine has two stable isotopes, ^{35}\mathrm{[Cl](/page/CL)} (A = 35, relative mass 34.96885 u, abundance approximately 75.8%) and ^{37}\mathrm{[Cl](/page/CL)} (A = 37, relative mass 36.96590 u, abundance approximately 24.2%), yielding a relative atomic mass of approximately 35.45.^[26] Standard atomic weights, as recommended by the Commission on Isotopic Abundances and Atomic Weights (CIAAW), incorporate uncertainties to account for natural variations in isotopic abundances across terrestrial sources, expressed as intervals like [35.446, 35.457] for chlorine.^[26] These variations arise from isotopic fractionation processes in nature, such as evaporation or biological uptake, differing from more uniform laboratory-synthesized samples where fixed abundances can yield precise values without brackets.^[12] This physics-based scale, with ^{12}\mathrm{C} = 12, was adopted internationally in 1961 by IUPAC and the International Union of Pure and Applied Physics, replacing the earlier chemical scale based on oxygen-16 set to 16 to unify measurements across disciplines and align with mass spectrometry standards.^[27]

Nuclear Processes

Radioactive Decay

Radioactive decay involves the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, often altering the mass number A, which is the total number of protons and neutrons in the nucleus. In this process, the mass number of the daughter nucleus can change depending on the decay mode, while conservation of nucleon number ensures that the total A across all massive decay products remains constant, as neutrinos or antineutrinos emitted have negligible mass.^[28]^[29] Alpha decay occurs when a nucleus emits an alpha particle, which is a helium-4 nucleus (^{4}_{2}\alpha) consisting of two protons and two neutrons, thereby reducing the mass number of the parent nucleus by 4. This mode is common in heavy nuclides with A > [200](/page/200), as it helps achieve greater stability by lowering the proton-to-neutron ratio. For example, uranium-238 undergoes alpha decay to thorium-234: ^{238}_{92}\mathrm{U} \to ^{234}_{90}\mathrm{Th} + ^{4}_{2}\alpha.^[30]^[31] Beta minus (\beta^-) decay transforms a neutron into a proton, emitting an electron and an antineutrino, which leaves the mass number unchanged since the total number of nucleons remains the same, but increases the atomic number Z by 1. This process typically occurs in neutron-rich nuclei and shifts the nuclide toward stability. A representative example is the decay of carbon-14 to nitrogen-14: ^{14}_{6}\mathrm{C} \to ^{14}_{7}\mathrm{N} + e^- + \bar{\nu}_e.^[32]^[33] Beta plus (\beta^+) decay and electron capture both convert a proton into a neutron, resulting in no change to the mass number but a decrease in the atomic number by 1; in \beta^+ decay, a positron and neutrino are emitted, while electron capture involves an inner-shell electron combining with a proton to form a neutron and emitting a neutrino. These modes are prevalent in proton-rich nuclei lighter than bismuth.^[34]^[35] Gamma decay involves the emission of a high-energy photon from an excited nucleus, with no alteration to the mass number or atomic number, as it merely releases excess energy without changing the nuclear composition. This often follows other decay modes to de-excite the daughter nucleus. For instance, an excited cobalt-60 nucleus decays to its ground state: ^{60}_{27}\mathrm{Co}^* \to ^{60}_{27}\mathrm{Co} + \gamma.^[36] In decay chains, such as the uranium-238 series, alpha decays progressively reduce the mass number by 4 in each step, interspersed with beta decays that preserve A, ultimately leading to stable lead-206 after 14 decays (8 alpha and 6 beta). This stepwise reduction illustrates how mass number evolves toward stability in heavy-element series.^[37]

Nuclear Reactions

In nuclear reactions, the mass number A, defined as the total number of protons and neutrons in a nucleus, is strictly conserved, reflecting the preservation of baryon number under the strong nuclear force. This conservation principle ensures that the sum of the mass numbers of all reactants equals that of all products in any induced nuclear process, distinguishing it from mass-energy equivalence where slight differences in atomic masses contribute to the reaction's energy release via the Q-value, calculated as Q = (\sum m_{\text{reactants}} - \sum m_{\text{products}}) c^2, with the mass defect arising from binding energy differences while A remains unchanged.^[38] Nuclear fission exemplifies this conservation in the induced splitting of a heavy nucleus. For instance, in the thermal neutron-induced fission of uranium-235, the reaction ^{235}\mathrm{U} + n \rightarrow fission fragments (e.g., one with A \approx 95 and another with A \approx 139) + 2-3 neutrons results in products whose total mass number sums to 236, matching the reactants' combined A. This process, central to nuclear reactors and atomic bombs, releases energy from the mass defect but maintains A balance, with the distribution of mass numbers among fragments typically peaking around asymmetric splits due to nuclear shell effects.^[39]^[40]^[41] In contrast, nuclear fusion combines light nuclei to form heavier ones, again conserving total A. A key example is the deuterium-tritium reaction, ^2\mathrm{H} + ^3\mathrm{H} \rightarrow ^4\mathrm{He} + n, where the reactants' total A = 5 equals the products' A = 5, powering stars through sequential fusions that progressively build elements up to iron (mass number around 56) in stellar cores via processes like the proton-proton chain or CNO cycle. Neutron capture reactions further illustrate A alteration at the individual nucleus level: in ^{59}\mathrm{Co} + n \rightarrow ^{60}\mathrm{Co} + \gamma, the target's mass number increases by 1 as the neutron is incorporated, a process vital for producing isotopes like cobalt-60 used in medicine and contributing to the rapid buildup of heavier elements in stellar nucleosynthesis.^[42]^[43]^[40]

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