SCIENCE · INVESTIGATE AND EXPLAIN

Test a bond-stiffness claim against isotope evidence

Use actual spectroscopic reference coefficients to separate a mass effect from a local-curvature inference.

Learning outcome: Predict a stiffness ratio, reconstruct a target vibrational coefficient and compare naive versus reduced-mass-corrected inferences without treating frequency as dissociation energy.

Starting recap

A heavier oscillator can vibrate more slowly at the same local restoring stiffness. For a two-body oscillator, reduced mass is m1m2/(m1+m2); only ratios are needed here.

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Try

D35Cl has a much lower vibrational coefficient than H35Cl. Does squaring their ratio establish a bond roughly half as stiff?

Open the experiment

Predict

Predict k_target/k_reference before reading the corrected comparison. Explain which mass assumption makes the naive squared frequency ratio valid or invalid.

Your prediction and reason

Experiment

Start HCl→DCl with ratio 0.51, capture its frequency residual, then test ratio 1.00. Transfer to HBr→DBr without assuming exact equality. Use HCl→HBr as a changed-element comparison and explain why it is not an isotope-only control.

Separate isotope mass from inferred stiffness

Frozen reference values with a learner prediction and calculated comparison · Historical ground-state omega_e coefficients, not observed fundamental lines. Diatomic harmonic local-curvature ratio uses neutral-atom isotope masses; no dissociation energy, statistical significance or exact isotope invariance follows.

Spectroscopic reference comparison: H35Cl → D35Cl: isotope comparison pair Your proposed target/reference stiffness ratio: 0.51 ratio

Reset keeps your prediction, captures and practice history.

A mass change can shift vibration without halving stiffness

Reference H35Cl: ωₑ = 2990.9463 cm⁻¹. Target D35Cl: ωₑ = 2145.163 cm⁻¹. These are first-term spectroscopic constants in the ground electronic state, interpreted with a harmonic model; they are not observed fundamental transition frequencies or angular frequencies in radians per second.

Reduced mass μ = m₁m₂/(m₁ + m₂): reference 0.979592539 u, target 1.904413246 u. In the diatomic harmonic approximation ωₑ is proportional to √(k/μ), so k₂/k₁ = (μ₂/μ₁)(ω₂/ω₁)².

Frequency ratio = 0.717218828. Squaring it alone gives 0.514402847. Including reduced mass gives 1.000043954. Your proposed ratio 0.51 predicts target ωₑ = 1531.919 cm⁻¹, with residual -613.244 cm⁻¹ relative to the frozen target coefficient.

This comparison changes H to D while retaining the halogen isotope and electronic-state label. The corrected ratio is near one under this approximation; absent coefficient uncertainties, it does not establish exact equality or agreement within uncertainty.

Species / state	Selected ωₑ / cm⁻¹	Provenance / definition	Missing information
H35Cl X 1Sigma+	2990.9463	Rank, Eastman et al. (1962); Rank, Rao et al. (1965) Selected uncorrected tabulated omega_e. Note26 gives Dunham-corrected 2991.0904; not silently substituted.	Temperature, pressure and numeric coefficient uncertainty not supplied in the selected row.
D35Cl X 1Sigma+	2145.163	Pickworth and Thompson (1953); Van Horne and Hause (1956); Mould, Price et al. (1960) Selected ground-state omega_e, not the observed 1–0 fundamental.	Temperature, pressure and numeric coefficient uncertainty not supplied in the selected row.

Isotope	Mass / u	Original uncertainty notation and source
H1	1.00782503223	1.00782503223(9)
H2	2.01410177812	2.01410177812(12)
Cl35	34.968852682	34.968852682(37)

The parenthesized mass digits are the source uncertainty notation. The calculation uses neutral-atom isotope masses as an approximation to nuclear inertia; electron-mass and nonadiabatic corrections are omitted. No uncertainty is propagated into the displayed stiffness ratio because the required coefficient uncertainties are absent. Source precision is not a confidence interval.

k is local potential curvature, not a dissociation energy. The NIST table's Dₑ column denotes a centrifugal-distortion constant, despite the same symbol sometimes meaning well depth elsewhere. Read the source's definitions before making an energy claim. No bond energy, reaction rate or temperature is calculated here.

Frozen reference selection

Accessed 2026-10-28. Historical compiled reference values, not raw measurement records, current-best estimates or newly measured classroom data. Missing conditions/uncertainty remain null. Selected factual reference values from NIST CCCBDB SRD101 release22 (May2022), NIST Chemistry WebBook SRD69 Huber/Herzberg compilation, and NIST isotopic mass tables. Original derived comparisons; no source illustrations reproduced. NIST SRD compilations retain their stated copyright. Only selected factual values are used; all comparisons and diagrams are original. Record a capture of your numerical prediction and explain which variables or definitions remain unmatched.

The corrected local-stiffness ratio is 1.000044 under the stated model.

Vibrational frequency depends on mass as well as local curvature. Neither this ratio nor the source centrifugal-distortion column is a dissociation energy.

Observation to attach to a capture

Captures retain the selected frozen reference pair and your numerical prediction. Replay recalculates the same comparison; it creates no new physical measurement, fitted uncertainty or independent causal evidence.

Understand

Slower vibration does not identify its cause

The selected NIST ground-state coefficients are 2990.9463 cm⁻¹ for H³⁵Cl and 2145.163 cm⁻¹ for D³⁵Cl. D is deuterium, hydrogen with a heavier nucleus. Both species have the same number of electrons and the same chlorine isotope. The lower deuterated coefficient could tempt us to say the bond is roughly half as stiff: (2145.163/2990.9463)² is about 0.5144. That reasoning silently holds inertia fixed when it actually changes.

Try a proposed stiffness ratio of 0.51, capture the predicted coefficient and residual, then try 1.00. Explain which proposal better reproduces the selected target under this model. The graph combines your numerical hypothesis with the fixed reference data; it does not change or refit the source values to make your answer agree.

A two-body oscillator needs reduced mass

Both nuclei move. Their relative separation behaves, in a simple two-body model, as if one effective mass were moving. This reduced mass, μ, is m₁m₂/(m₁ + m₂). The masses must use the same unit. Here we use isotope masses in unified atomic mass units, u, from the linked NIST tables. These are isotope masses, not the abundance-weighted molecular weights printed near the top of WebBook pages.

For H³⁵Cl, μ is about 0.979592539 u. For D³⁵Cl it is 1.904413246 u. The heavier molecule’s relative motion has almost twice the effective inertia. A familiar spring illustrates the direction of the effect: a larger mass oscillates more slowly under the same spring stiffness. The molecule is not literally a macroscopic spring, but the small-displacement harmonic approximation uses the same mathematical relationship.

Here k means local potential curvature: how rapidly the restoring force changes near the minimum. The first-term coefficient ωₑ is proportional to √(k/μ). Taking a ratio cancels the shared constants. Rearranging gives:

k₂/k₁ = (μ₂/μ₁)(ω₂/ω₁)².

Read this as: multiply the target-to-reference reduced-mass ratio by the square of the coefficient ratio. For DCl/HCl, the answer is approximately 1.000044, not 0.5144. Conversely, your proposed k ratio predicts the target coefficient by multiplying the reference coefficient by √[(k₂/k₁)(μ₁/μ₂)]. That is the prediction shown beside the target record.

What the reference coefficients are

NIST HCl and DCl list historical spectroscopic constants compiled by Huber and Herzberg. ωₑ here is in inverse centimetres, a spectroscopic wavenumber convention. It is not an angular frequency in radians per second, and it is not the observed 1–0 fundamental transition. Higher terms describe departures from the harmonic approximation. The selected HCl table coefficient is uncorrected; note 26 gives a different Dunham-corrected value. We do not silently mix the two conventions.

The selected rows do not supply a numeric coefficient uncertainty or a common temperature/pressure pair. Their state and isotopologue labels are explicit; complete measurement conditions are not. Thus “near one under this model” is justified, while “exactly equal” or “equal within experimental uncertainty” is not established here. Neutral-atom isotope masses approximate nuclear inertia; electron-mass and nonadiabatic corrections are omitted. Parenthesized mass uncertainty digits remain visible, but missing spectroscopic uncertainty prevents the displayed ratio from being a complete uncertainty analysis.

Transfer and a deliberately unmatched pair

Transfer to H⁸¹Br and D⁸¹Br: the selected coefficients are 2648.975 and 1884.75 cm⁻¹. Predict before inspecting the correction. The naive squared ratio is about 0.506235, while mass correction gives 0.999416. The small departure from one must not be overinterpreted without uncertainties and a more complete model.

Then compare H³⁵Cl with H⁸¹Br. The corrected ratio is about 0.797081. This time the halogen element changes, so both inertia and electronic interactions can change. Calling it an isotope-only experiment would hide that confound.

Stiffness is not dissociation energy

Local curvature describes behavior near a minimum. Separating the atoms to infinity asks about an entire potential and its energy baseline. A curvature ratio alone gives neither well depth nor dissociation energy. In these NIST tables Dₑ labels a centrifugal-distortion constant; the same symbol can mean something else in another source. Read the table legend before assigning an energy interpretation.

Use the changed-case practice to calculate a new ratio, then explain what evidence remains absent. No new measurement, reaction rate, material classification, full Unit 2 assessment or independent mastery is claimed by a matching number.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Separate reference coefficients, neutral-isotope mass approximation, harmonic model, local stiffness and dissociation energy. Explain why near-one is not equality within an unreported uncertainty.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Compare bond distances without hiding unmatched evidence Next: Test alloy-property claims against matched reference data

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