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A bond-dissociation enthalpy is the molar enthalpy change for the specified homolysis at temperature $T$, with reactant and products in specified standard states. Its usual units are $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The superscript $\circ$ specifies standard states; it does not fix $T$ to room temperature.^[ref]

Write $\delta h_i(T)=H_{m,i}^{\circ }(T)-H_{m,i}^{\circ }(0)$ for each species' molar thermal enthalpy increment. With $D_0$ an energy per parent molecule,Here $N_{\mathrm{A}}$ is the Avogadro constant. The thermal increments include translational kinetic energy, molecular rotation, thermally excited molecular vibration and electronic states when populated, and the $pV$ contribution to enthalpy. They are additional to the zero-point energies already included in $D_0$. If values of standard enthalpy of formation at the same $T$ are available, Hess's law gives the same quantity as ${\Delta }_f{H}^{\circ }(A)+{\Delta }_f{H}^{\circ }(B)-{\Delta }_f{H}^{\circ }(AB)$.

Units alone do not identify the convention. Multiplying either $D_e$ or $D_0$ by the Avogadro constant gives molar energy, but does not add thermal enthalpy. One electronvolt per dissociation corresponds to $96.485332\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. A spectroscopic wavenumber $\tilde{\nu }$ represents energy $hc\tilde{\nu }$, using the Planck constant $h$ and speed of light $c$; $1\mathrm{c}{\mathrm{m}}^{-1}$ corresponds to $0.011962657\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$.^[ref]

Real chemical bonds can stretch as molecular rotation increases. The resulting larger moment of inertia lowers the energy eigenvalues relative to a rigid rotor. Within one vibrational quantum state, a leading effective correction isBoth $B_v$ and $D_v$ are in frequency units here; $v$ labels the vibrational quantum state. For positive $D_v$, the adjacent-line spacing becomesso the departures grow with $J$. Molecular vibration also changes the effective rotational constant: even the lowest vibrational quantum state does not have motionless atomic nuclei.^[ref]

The quartic correction is a limited expansion, not a model to extrapolate to arbitrary $J$. It is small relative to the rigid contribution only while $(D_v/B_v)J(J+1)\ll 1$; eventually higher terms, changed internal motion, or chemical bond breaking matter. In the synthetic plot, the largest upper-state label is $J=5$, giving a stipulated correction ratio of $0.03$. No universal maximum valid $J$ follows without a particular molecule and precision target.

The quantum numbers also have limits of meaning. Here $J$ describes spin-free rotation. In a molecule with electronic spin, rotation alone is commonly labelled $N$, while $J$ includes electronic angular momentum; including nuclear spin introduces additional total-angular momentum labels and resolved splittings. An applied electric field can mix different field-free $J$ states through the Stark effect. For identical atomic nuclei, the Pauli exclusion principle can restrict combinations of rotational and nuclear-spin states. The simple $2J+1$ degeneracy and every-integer-$J$ ladder therefore describe the stated idealization, not all observed sublevels.^[ref]

An actual infrared spectroscopy route is to compare gaseous hydrogen chloride containing ordinary hydrogen with its deuterium counterpart. A documented teaching experiment uses a stopcock-equipped gas cell with potassium bromide windows and a Nicolet Avatar instrument for Fourier transform spectroscopy. A nitrogen-purged flask and drying tube supply the mixture; the sample also contains sulfur dioxide. An empty sample-chamber background precedes the filled-cell scan. The absorbance trace and peak-picked spectroscopic wavenumbers are the readout. The analysis assigns and indexes ${}^1{\mathrm{H}}^{35}\mathrm{Cl}$ and ${}^2{\mathrm{H}}^{35}\mathrm{Cl}$ transitions separately and propagates transition uncertainties. The instructions distinguish resolvable chlorine isotope peaks from adequate precision for the intended parameter analysis. ^[ref]

A separate published teaching dataset reports a Nicolet Nexus 670 instrument, a $10\mathrm{cm}$ gas cell at $20\mathrm{Torr}$, and chlorine isotope peaks resolved to baseline. These apparatus details describe that dataset, not the preceding experiment. ^[ref]

The inference connects several models. Fit correctly assigned transitions to separate molecular rotation from molecular vibration, then compare the inferred vibrational scales using isotope scaling of diatomic vibration. Agreement supports the approximate separation between a nearly unchanged chemical bond stiffness and a changed nuclear mass. It does not make a real molecule an exact classical spring: anharmonicity shifts the fundamental transition away from the local harmonic oscillator scale, and coupling to molecular rotation shifts individual lines.

For an independent test, calculate the reduced masses from the specified isotopes and compare the inferred values $k=\mu (2\pi c{\tilde{\nu }}_{\mathrm{harm}}{)}^2$. Agreement within uncertainty tests the common-curvature assumption; choosing separate $k$ values just to fit each frequency would not test it. The documented background addresses instrument and ambient contributions, but does not by itself subtract window absorption, identify contaminating species, or calibrate the spectroscopic wavenumber scale. Those require separate controls. Finite spectral resolution, blended peaks, incorrect isotope assignments, and departure from the harmonic approximation near a potential minimum limit the comparison. No numerical measured line positions are assumed here.

The gaseous hydrogen halide consisting of one hydrogen atom and one bromine atom. Its molecular vibration can be studied through its vibrational fundamental and weak vibrational overtones. The aqueous acid and the isolated diatomic molecule should be distinguished when specifying a dissociation process.^[ref]

Let $u_s=⟨ϵ{⟩}_s$ and $v_s={\mathrm{Var}}_s(ϵ)$ for the conditional distributions of parahydrogen and orthohydrogen, with $s=\mathrm{p},\mathrm{o}$. Differentiating the normalized Boltzmann factors gives $d{u}_s/dt=v_s/t^2$. Thus the rotational heat capacity at constant volume per molecule of a composition frozen at $x$ isAt conversion equilibrium, the law of total variance adds a nonnegative contribution from the changing species populations:The last term is the contribution from conversion. Compare the first two terms with a frozen sample at the same instantaneous composition; a frozen $3:1$ sample generally has different weights too. This formula follows directly from the variance of a mixture: subtract the overall mean inside each species, expand the square, and the cross terms vanish because each conditional deviation has zero mean. It also proves nonnegative rotational heat capacity in either model.

For an original worked example at $t=1/2$, summing the allowed ladders gives $q_{\mathrm{p}}\approx 1.00003072$, $q_{\mathrm{o}}\approx 0.05494692$, and $x_{\mathrm{o}}^{\mathrm{eq}}\approx 0.14150982$. The calculated rotational heat capacities are $c_{\mathrm{eq}}/k_{\mathrm{B}}\approx 1.94719633$ and $c_{\mathrm{fr}}/k_{\mathrm{B}}\approx 0.00110733$ for frozen normal hydrogen. The contrast is large because changing species accesses the $J=0$ to $J=1$ separation, whereas fixed species must use the larger $J=0$ to $2$ or $J=1$ to $3$ separations. These are model numbers at a chosen dimensionless temperature, not measured hydrogen data.

Both rotational heat capacities approach zero as $t\to 0$, although frozen normal hydrogen retains nonzero rotational internal energy. Both approach $k_{\mathrm{B}}$ as $t\to \infty$ within the rotor model. The high-temperature limit is a formal limit with all other excitations suppressed; actual molecular hydrogen eventually needs molecular vibration, nonrigidity, and dissociation. Multiplying the per-molecule result by the Avogadro constant gives rotational molar heat capacity in units of the molar gas constant $R$. A dilute classical ideal gas additionally contributes $3R/2$ from translation to its heat capacity at constant volume.

A Jablonski diagram arranges molecular energy eigenstates vertically and draws arrows for changes between them. In the usual closed-shell molecular example, $S_0$ is the electronic ground state, $S_1$ and $S_2$ are electronic excited states with singlet state spin, and $T_1$ has triplet state spin. Each electronic state also has levels of molecular vibration. The diagram records possible processes, not molecular paths in space.^[ref]

Photon absorption can populate an upper level of molecular vibration in $S_1$. Relaxation removes excess energy without photon emission; fluorescence commonly starts from relaxed $S_1$.^[ref] The drawing also includes relaxation from a higher electronic excited state and a competing spin-changing route to a triplet state.

Molecular vibration changes internal separations and bond angles, unlike the fixed internal molecular geometry of a rigid rotor. Its coupling to molecular rotation makes measured rotational constants depend on the vibrational quantum state.

A diatomic molecule has one internal stretching coordinate: its two atomic nuclei approach and separate about an equilibrium bond length. The diatomic bond stretch turns this motion into a particularly useful model. The local potential energy curvature supplies the restoring force, while the reduced mass accounts for both moving partners. Thus the same chemical bond can vibrate more slowly after an isotope substitution even when its electronic structure is nearly unchanged.

This separation makes infrared spectroscopy informative: transition frequencies constrain chemical bond stiffness, and isotope scaling of diatomic vibration tests the mass dependence. The University of Florida HCl and DCl FTIR experiment illustrates the measurement. Detectability also requires vibrational infrared activity; having vibrational energy eigenvalues alone does not guarantee an infrared absorption line.

A molecule, using the International Union of Pure and Applied Chemistry (IUPAC) convention, is an electrically neutral entity containing two or more chemical atoms. Its constituents form a bound unit through chemical bonds; merely drawing a boundary around neighboring particles does not make a molecule. The neutrality requirement concerns total electric charge, not the absence of internal charge separation: water is a polar molecule with zero total electric charge.^[ref]

One water molecule, ${\mathrm{H}}_2\mathrm{O}$, contains two hydrogen and one oxygen chemical atom. One molecular oxygen molecule, ${\mathrm{O}}_2$, contains two oxygen chemical atoms; a molecule need not contain different chemical elements. One carbon dioxide molecule, $\mathrm{C}{\mathrm{O}}_2$, contains one carbon and two oxygen chemical atoms. The molecular formula records these inventories, while molecular geometry describes their arrangement. Water and carbon dioxide are chemical compounds; molecular oxygen is a form of one chemical element.

Ordinary crystalline sodium chloride, $\mathrm{NaCl}$, contains an extended arrangement of sodium cations $\mathrm{N}{\mathrm{a}}^{+}$ and chlorine anions $\mathrm{C}{\mathrm{l}}^{-}$. Its formula unit specifies one of each for compositional accounting, without selecting a unique isolated pair. Bulk neutrality therefore does not imply a material consists of neutral molecules. This statement concerns the ionic crystal; the physical state and conditions matter when deciding what particles a chemical formula represents.^[ref]

A polyatomic ion is a charged bound group of chemical atoms, such as hydronium, ${\mathrm{H}}_3{\mathrm{O}}^{+}$. A molecular ion formed by removing an electron from intact water, ${\mathrm{H}}_2{\mathrm{O}}^{\bullet +}$, retains the same atom inventory but has net electric charge $+e$. Neither satisfies the neutral-molecule definition. Distinguishing these entities makes molecular count and amount of substance and mass spectrometry interpretable: say whether the count is of chemical atoms, neutral molecules, ions, or formula units.

The useful surprise is that a molecule can be treated as one particle for motion and counting while retaining internal structure. Its molar mass connects a particle inventory to bulk mass; molecular rotation and molecular vibration reveal internal organization. The continuum approximation uses bulk fields in place of individual molecular positions and motions.

A normal coordinate diagonalizes the linearized kinetic and potential forms so that its motion decouples as a harmonic oscillator.

A diatomic molecule has $3N-5=1$ vibrational normal mode: the bond stretch. For hydrogen chloride, opposite longitudinal displacements change the electric dipole moment, so the mode has vibrational infrared activity. The following original sketches show one phase of this motion and the three water modes; arrow lengths are schematic, not mass-weighted amplitudes.

Absorption depends on an allowed transition with a nonzero transition dipole moment and appreciable line strength at wavelengths carrying terrestrial thermal radiation. Molecular vibration and molecular rotation, level populations and pressure/temperature-dependent line shapes determine the spectrum. A band in an atmospheric window can be more effective incrementally than one already strongly absorbed by water or carbon dioxide. Symmetry can make a vibration infrared inactive even when the molecule has many normal modes. Atmospheric lifetime affects the persistence of a released gas, not the intrinsic absorption strength of one molecule.

Raman scattering is inelastic scattering of light in which the outgoing photon has a different frequency because matter gains or loses internal energy. Molecular vibration is a particularly useful recipient: a visible photon can reveal an internal energy spacing normally associated with infrared frequencies. Rayleigh scattering supplies the much stronger, approximately unchanged-frequency reference. The distinction concerns the transferred energy, rather than merely a change of direction or the colour of the illuminating beam.

The elementary bookkeeping is conservation of energy:Here $h$ is the Planck constant, ${\nu }_0$ and ${\nu }_s$ are the incident and scattered photon frequencies, and $E_i,E_f$ are initial and final internal molecular energies. This expression neglects translational recoil and Doppler broadening; those can matter in sufficiently precise gas measurements. A positive internal energy change gives Stokes Raman scattering, and a negative one gives Anti-Stokes Raman scattering. The energy difference determines a possible line position, but it does not by itself establish that a transition is optically allowed or predict its strength.

The spontaneous molecular process developed here has a connection to the Raman transition used in atomic measurement methods: two photon interactions transfer an internal energy difference. Coherently driving that process with two applied light fields is a different experimental regime from collecting weak spontaneous Raman scattering beside an illuminating beam.

The signed Raman shift subtracts scattered from incident spectroscopic wavenumber:Here $c$ is the speed of light, and both wavelengths are in vacuum. Positive shifts denote Stokes Raman scattering; negative shifts denote Anti-Stokes Raman scattering. Some plots reverse the horizontal direction or report separate unsigned magnitudes, so check the convention. Spectroscopic wavenumber has no $2\pi$ factor, unlike the common angular wavenumber $k=2\pi /\lambda$. A shift of $1000\mathrm{c}{\mathrm{m}}^{-1}$ means $100000{\mathrm{m}}^{-1}$, not a frequency of $1000\mathrm{Hz}$.

For a one-quantum change of molecular vibration with ordinary frequency ${\nu }_v$, the magnitude is ${\tilde{\nu }}_v={\nu }_v/c$. This makes Raman spectroscopy attractive: the same molecular energy spacing can be read using different visible excitation wavelengths. Equal and opposite Raman shifts are not equal and opposite changes measured in nanometres, because wavelength is the reciprocal of spectroscopic wavenumber.

Raman spectroscopy measures shifted scattering of light to infer internal energy spacings and the molecular response that makes them visible. In a modern illustrative arrangement, narrowband laser illumination replaces the historical arc, an optical filter rejects the strong excitation line, and a spectrograph resolves the weak shifted spectral lines. The illumination, rejection and spectral dispersion are separate jobs. Increasing rejection can conceal small Raman shifts close to the excitation line.

A proposed measurement should record the excitation wavelength, a dark signal, a blank container and the liquid, then compare calibrated line positions and integrated signals under the same collection conditions. Measure optical filter transmission and detector response before using the Thermal Raman line ratio for temperature. Repeat at lower incident power to look for heating or altered sample response. These are suggested checks, not an assertion that the 1928 apparatus implemented modern calibration procedures.

Its appeal is the conversion of internal molecular energy differences into visible spectral lines: changing the excitation can move the whole pattern while preserving characteristic Raman shifts. Combined with vibrational infrared activity, Raman vibrational activity supplies complementary information about molecular vibration. A line pattern can constrain a material's identity or environment, but overlapping bands, fluorescence backgrounds and the finite spectral resolution element prevent an unqualified identification from one peak alone.

The simplest off-resonant explanation is a vibrating polarizability. Let $Q(t)=Q_0\cos (2\pi {\nu }_{v}t)$ be a small normal mode coordinate of molecular vibration, and use one scalar component of polarizability, $\alpha (Q)\simeq {\alpha }_0+{\alpha }_0^{\prime }Q$. For incident electric field $E(t)=E_0\cos (2\pi {\nu }_{0}t)$, the induced electric dipole moment isThe product-to-sum formula produces the two displaced frequencies. A nonzero derivative ${\alpha }_0^{\prime }$ gives vibrational Raman scattering in this linear model. Real polarizability is generally a tensor, and the selected components depend on incident and observed wave polarization. This condition differs from vibrational infrared activity, which requires a changing molecular electric dipole moment without an applied optical electric field. A vibration can therefore appear in one measurement and be absent in the other.

This classical modulation argument explains line locations but supplies equal sideband wave amplitudes; it does not explain their thermal asymmetry or the nonzero Stokes Raman scattering from a vibrational ground state. Those require the quantum harmonic oscillator population and transition strengths. The far-off-resonance drawing uses dashed virtual intermediate levels: they represent terms in a scattering probability amplitude, not a long-lived occupied electronic energy eigenstate. Near an electronic resonance, a more detailed treatment of the intermediate response is needed.

The reduced mass is the effective mass for the two-particle relative coordinate of two bodies with constant positive masses $m_1,m_2$:Both bodies can respond to their mutual force, so their relative acceleration is greater than if one were fixed. Consequently $0<\mu <\min (m_1,m_2)$. This explains why using either individual mass, or their sum, generally gives the wrong internal oscillation rate. The same reduced mass describes a two-body problem, a free two-mass spring, and the approximate molecular vibration of a diatomic molecule.^[ref]

For equal masses $m_1=m_2=m$, $\mu =m/2$. Two $1\mathrm{kg}$ bodies therefore have $\mu =0.5\mathrm{kg}$: in their stationary centre of mass frame a relative speed $2m/s$ means individual speeds $1m/s$ and total kinetic energy $1\mathrm{J}$. With $m_1=1\mathrm{kg}$ and $m_2=100\mathrm{kg}$, instead $\mu =100/101\mathrm{kg}\approx 0.990099\mathrm{kg}$. At the same relative speed, the individual speeds are $200/101$ and $2/101m/s$, and the total kinetic energy is $200/101\mathrm{J}$. These are calculations for stipulated masses and velocities.

For fixed $m_1$ and $q=m_2/m_1>0$, $\mu /m_1=q/(1+q)$. The heavy-partner limit $q\to \infty$ gives $\mu \to m_1$ from below; the relative error incurred by replacing $\mu$ with $m_1$ is $(m_1-\mu )/\mu =1/q$. At $q=100$ this is $1\%$. Treating the heavy body as fixed is thus a controlled approximation to internal motion, rather than an exact consequence of its being heavier.

In this article the rotational constant is a frequency,Its dimensions are ${\mathrm{s}}^{-1}$, since the Planck constant has dimensions $\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and a moment of inertia has dimensions $\mathrm{kg}{\mathrm{m}}^2$. Some authors instead call $hB={\hslash }^2/(2I)$ the energy-valued rotational constant; this is the convention in the earlier synthetic molecular beam electric resonance diagram. A third convention uses the spectroscopic wavenumber $\tilde{B}=B/c$, with $c$ the speed of light. Always check units before using $\nu =2B(J+1)$.

If $B$ is known and the two masses are specified, the rigid two-atom model gives the bond length $r=\sqrt{h/(8{\pi }^2\mu B)}$. This is why rotational spectroscopy can constrain molecular geometry. A real fitted constant can depend on molecular vibration, so this inference alone does not establish an equilibrium bond length.

Take an invented diatomic molecule with a Morse potential, setting the separated-atom energy to zero:Here $a$ is an inverse length, $r_e$ the equilibrium bond length, and ${\omega }_e=a\sqrt{2D_e/\mu }$, with reduced mass $\mu$. These stipulated parameters describe a simple model, not a measured chemical substance. For the one-dimensional Morse oscillator, energy eigenvalues above the minimum areConsequently,The harmonic estimate $A/2=0.1000\mathrm{eV}$ is slightly larger than this model's exact zero-point energy. The physical bound-level restriction matters: the quadratic expression is not an infinite ladder of vibrational quantum states.

For a separate thermal illustration using this same stipulated $D_0$, assume an ideal gas, fully active molecular rotation of the parent, negligible thermal molecular vibration, and negligible electronic excitation for all species. Each monatomic product contributes $\delta h=5RT/2$; the diatomic molecule contributes $7RT/2$. Thus the net increment is $3RT/2$, with molar gas constant $R$. At $T=300\mathrm{K}$,This last result follows from the stated thermal model, not a universal correction for every chemical bond. Frozen rotations, accessible excited states, different fragment composition, or a condensed phase change the thermal increments.

In Stokes Raman scattering, matter gains internal energy and the photon loses the same amount. For a quantum harmonic oscillator representing one molecular vibration, $E_v=h{\nu }_v(v+1/2)$ with $v=0,1,\dots$. The transition $v\to v+1$ gives ${\nu }_s={\nu }_0-{\nu }_v$, requiring ${\nu }_0>{\nu }_v$ for an outgoing photon. Even a molecule initially in its vibrational ground state can undergo this process. The shared name does not turn it into the absorption/emission-band Stokes shift used for fluorescence.

The Stokes shift compares the lowest absorption-band maximum with the fluorescence emission spectrum maximum. Relaxation often makes $\Delta \lambda >0$. This comparison differs from using an arbitrary excitation wavelength. A spectral-density maximum can move when changing from per unit wavelength to per unit spectroscopic wavenumber; specify the spectral convention.^[ref]

An original illustrative energy budget makes the distinction concrete. Excitation at $450\mathrm{nm}$ and emission at $520\mathrm{nm}$ correspond to about $2.76$ and $2.38\mathrm{eV}$ per photon, using $E=hc/\lambda$ with Planck constant $h$ and speed of light $c$ in vacuum. The difference, about $0.37\mathrm{eV}$, must go elsewhere in the complete energy balance. These chosen wavelengths are not measured band maxima for a particular fluorophore.

Anti-Stokes emission has shorter wavelength than excitation. Additional energy, for example from a thermally populated level of molecular vibration or multiple excitation photons, makes it possible. The one-photon, initially relaxed drawing does not cover these cases.^[ref]