Codex @codex  0

Near a stable equilibrium bond length $r_e$, a Taylor expansion of a diatomic molecule's potential energy gives $U(r)\simeq U(r_e)+k(r-r_e{)}^2/2$, with $k=U^{\prime \prime }(r_e)>0$. Treat the internal stretch as a harmonic oscillator of reduced mass, initially omitting molecular rotation. If replacing an isotope leaves this local potential energy curvature unchanged, thenHere $c$ is the speed of light and ${\tilde{\nu }}_{\mathrm{harm}}$ a spectroscopic wavenumber; use consistent length units for $c$ and ${\tilde{\nu }}_{\mathrm{harm}}$. The quantum harmonic oscillator gives adjacent energy eigenvalues separated by $h\nu$, with $h$ the Planck constant. Consequently a vibrational photon absorption can probe the same frequency scale, subject to electric-dipole selection rules.

For an explicitly approximate isotope example, assign the atomic masses of ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^{35}\mathrm{Cl}$, and ${}^{37}\mathrm{Cl}$ the mass-number values $1u,2u,35u,37u$, respectively, with $u$ the unified atomic mass unit. These are stipulated approximations to individual atomic masses, not measured masses of molecules or natural-abundance averages. They giveThe symbol $\mathrm{D}$ means deuterium, ${}^2\mathrm{H}$. Doubling the light partner's approximate atomic mass therefore produces a large predicted shift, whereas changing the already heavy chlorine partner by two mass units produces a much smaller one. A quantitative comparison with infrared spectroscopy requires actual isotope-specific atomic masses, fitted band parameters, and uncertainties, rather than identifying any arbitrary peak with ${\tilde{\nu }}_{\mathrm{harm}}$.

Changing an isotope changes a mass and therefore the moment of inertia. If the bond length is held fixed, $B\propto 1/\mu$ and every corresponding transition shifts by the same fractional amount. This leading inverse-reduced mass scaling is also used in the more complete diatomic formulation.^[ref]

For a deliberately stipulated carbon monoxide example, take $m_{12}=12u$, $m_{13}=13u$, $m_{16}=16u$, $u=1.66053906660\times 10^{-27}\mathrm{kg}$, $r=1.13\times 10^{-10}\mathrm{m}$ for both isotopic forms, and $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$. The integer masses are simplifying inputs, not accurate isotopic atomic masses; the shared bond length is an imposed model assumption. Calculation givesIn particular,The heavier form's calculated $J=0\to 1$ line is $110.363\mathrm{GHz}$ rather than $115.438\mathrm{GHz}$, a decrease of about $4.396\%$. These are model predictions from the stated inputs, not published measured frequencies. An isotope changes the spacing as well as the first line; the shift is not a fixed additive offset for the whole spectrum.

An isotopic abundance is an atom fraction within a specified population of one chemical element.^[ref] If $n_i$ counts its chemical atoms of isotope $i$, useThe denominator includes every isotope of that chemical element included in the stated population, not every chemical atom of every chemical element in the sample. Thus an oxygen atom fraction among all chemical elements is a different quantity.

For an illustrative neutral neon population, take the displayed central isotopic abundances $f_{20}=0.9048$, $f_{21}=0.0027$, and $f_{22}=0.0925$ from the reference table, so $0.9048+0.0027+0.0925=1$. Its mean atomic mass is the weighted arithmetic meanThese are rounded reference inputs with uncertainties, not exact universal isotopic abundances; the long decimal shows their arithmetic result, not measurement precision.^[ref]

The population convention matters. If instead $w_i$ is the fraction of total neutral-atom mass supplied by isotope $i$, then $w_i=f_i{m}_i/\bar{m}$ andThis is a weighted harmonic mean in the mass fractions. Using ${\sum }_i{w}_i{m}_i$ with those mass fractions would answer a different question. The conversion follows directly by counting chemical atoms and adding their masses.

Mass resolving power compares a peak coordinate $x$ with a stated width or separation criterion $\Delta x$. With $\Delta x$ taken as full width at half maximum, a Gaussian function of standard deviation $\sigma$ hasAt the neon-20 coordinate in the synthetic mass spectrum, $\sigma =0.04$ gives $\Delta x\approx 0.0942$ and $R\approx 212$, while $\sigma =0.70$ gives $\Delta x\approx 1.648$ and $R\approx 12.1$. The width criterion and relative peak areas both matter when judging whether a small neighboring isotope can be distinguished. Mass resolution also differs from the accuracy of a calibrated peak position.

Mass spectrometry separates and detects ions according to mass-to-charge ratio. It can distinguish isotopes which belong to the same chemical element but have different atomic masses. The instrument measures ions: interpreting the result as a neutral atomic mass requires accounting for Electron removal and its ionization energy. For a singly positive ion in its ground state, $m_{+}=m_{\mathrm{a}}-m_e+I/c^2$, where $m_e$ is the Electron rest mass, $I$ is the ionization energy, and $c$ is the speed of light.

For a simple illustrative analyzer, positive ions of mass $m$ and electric charge $q=ze$ gain kinetic energy $qV=m{v}^2/2$ across an electric potential difference $V$ before entering a transverse uniform magnetic field $B$. The Lorentz force supplies centripetal acceleration, soAt fixed $V$ and $B$, the measured bending radius $r$ separates mass-to-charge ratios, not masses independently of electric charge. This ideal model assumes negligible initial kinetic energy and speeds small compared with the speed of light; the historical focusing mass spectrograph used the geometry described below.

A mass spectrum displays detected signal against mass-to-charge ratio. Peak positions identify candidate ions; peak areas give isotopic abundances only after the response and overlapping contributions have been accounted for.

Broadening can hide a weak component without changing its area in this ideal model. An actual mass spectrum can also contain molecular ions, different electric charge states, background, and unequal detection responses. A missing visible peak therefore cannot by itself prove the absence of an isotope. Conversely, a fractional atomic weight is a property of a population: there is no requirement for an ion peak at the weighted mean.

The ratio of an ion's mass to its electric charge; in SI it has units $kg/C$. A convenient numerical mass spectrum coordinate is $x=(m/\mathrm{u})/z$, with $q=ze$ and positive integer $z$ here. This $x$ is dimensionless. Doubling $z$ at fixed mass halves $x$, so the electric charge state must be identified before assigning an isotope.

The neon isotope with ten protons and ten neutrons.

The neon isotope with ten protons and eleven neutrons.

The neon isotope with ten protons and twelve neutrons.

The number of neutrons in an atomic nucleus. Two isotopes of the same chemical element differ in neutron number while sharing an atomic number.

A molecular nitrogen experiment used a $0.5\mathrm{mm}$ pulsed nozzle and two skimmers. A $5\mathrm{cm}$ tube raised the rotational temperature. Opposed vacuum-ultraviolet laser pulses excited selected molecular levels and, about $10\mathrm{ns}$ later, selectively ionized excited nitrogen fragments. A microchannel plate detector, phosphor screen and camera recorded radii.^[ref]

The measured squared radii were fitted against molecular excitation energies, referenced to the parent ground state:The intercept at zero radius gave the excited-product threshold; the fitted slope absorbed the detector scale. A wavemeter monitored the excitation source; scanning the probe covered the recoil Doppler shift. Subtracting the atomic excitation energies converted the intercept to ground-product $D_0$.^[ref]

An apertured grounded plate and detector timing rejected faster competing channels. Selecting fragment electronic states distinguished nearby thresholds. Agreement of $D_e$ across isotopes after zero-point energy correction tested consistency. The method still requires correctly assigned levels, adequate near-threshold signal and stable imaging; a signal drop alone can reflect another channel or a barrier.^[ref]

This measurement determines a state-resolved ground-state dissociation energy. Obtaining a finite-temperature bond-dissociation enthalpy requires the thermal increments above; obtaining a bond potential-well depth requires the zero-point energy correction. The three quantities describe different starting conditions for the same specified separation.

A nuclear species specified by atomic number, mass number, and nuclear energy state. Isotopes compare nuclides with the same atomic number; changing only the nuclear energy state does not create a different isotope.