SCIENCE · INVESTIGATE AND EXPLAIN

Compare dispersion contacts without a mass-only shortcut

Separate polarizability, distance and contact multiplicity in a declared attractive term.

Learning outcome: Compare one and four contacts at reference separation and twice it. Derive ratios before capture, then identify the missing terms needed for a stable condensed structure.

Starting recap

Review the electron and formal-charge budget, bond multiplicity, signed interaction energies and the distinction between a model and a measured property.

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Try

Four identical contacts at twice the reference separation contribute−4/64=−1/16 energy unit, weaker in magnitude than one reference contact.

Open the experiment

Predict

Can four contacts always outweigh one if their separations differ?

Your prediction and reason

Experiment

Compare one and four contacts at reference separation and twice it. Derive ratios before capture, then identify the missing terms needed for a stable condensed structure.

Compare contact count with separation in a long-range model

Worked teaching model · synthetic values · Supplied pair attraction U=−C6/r⁶, normalized so one reference contact is −1 energy unit. Independent identical contacts add. Long-range attractive term only; short-range repulsion, many-body effects, molecular geometry and entropy are omitted. Not a boiling-point or equilibrium-distance predictor.

Separation relative to reference: 1 ratio Independent pair contacts: 1 count

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Axes rescale. Connected samples represent the supplied continuous parameter relationship; stems denote discrete pairs or steps, not a time trajectory.

Numerical plot coordinates

separation / reference	attractive energy term
1	-1
1.05	-0.7462154
1.1	-0.5644739
1.15	-0.4323276
1.2	-0.334898
1.25	-0.262144
1.3	-0.2071762
1.35	-0.1651952
1.4	-0.1328103
1.45	-0.107595
1.5	-0.0877915
1.55	-0.07211238
1.6	-0.05960464
1.65	-0.04955601
1.7	-0.04142919
1.75	-0.03481543
1.8	-0.02940119
1.85	-0.0249442
1.9	-0.02125585
1.95	-0.01818831
2	-0.015625

Calculated evidence under the declared representation
Quantity	Value	Unit
Total attractive term	-1	reference energy units
One pair attractive term	-1	reference energy units
Contact count	1	pairs

More contact pairs can compete with a steep distance dependence.

At twice the separation, one pair contribution is only 1/64 as large in magnitude. Polarizability, molecular shape and accessible contact area affect real dispersion; molecular mass alone is not a force law. Dispersion exists even for polar molecules. The missing repulsive term prevents this attraction-only model from predicting equilibrium spacing.

More contact pairs can compete with a steep distance dependence.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Dispersion is not restricted to nonpolar molecules

Fluctuating electronic distributions and their correlations give dispersion contributions between particles. Polar molecules have dispersion as well as other interactions. Polarizability depends on how an electron distribution responds, while molecular shape changes which contacts are accessible. Molecular mass often correlates with some trends within a related series, but mass itself is not a complete interaction law.

Isolate one mathematical dependence

The supplied long-range attractive term is U=−C6/r⁶. Normalize one contact at a reference separation to−1 energy unit and hold its coefficient fixed. Doubling separation reduces the magnitude by64. This is an asymptotic pair contribution, not a complete short-range intermolecular potential. Extending it alone to zero distance would predict an unphysical collapse.

Add declared contacts

If four identical independent contacts all remain at the reference distance, the total is−4. If all four move to twice that separation, the total is−4/64=−1/16. One closer contact can therefore outweigh several more distant ones in this stipulated comparison. The independence and identical-coefficient assumptions are part of the model, not discoveries about a real molecule’s packing.

Compare structural explanations

A more extended molecule may present different accessible contacts than a compact isomer, but contact count is not simply its number of atoms. Rotational freedom, orientation, packing and other interactions affect bulk properties. A useful explanation should identify which geometric or electronic factor changed rather than use ‘larger molecule’ as a substitute for a mechanism.

Transfer and limit

At separation1.5 times the reference, two contacts contribute−2/(1.5⁶)≈−0.17558 units. This does not determine boiling temperature: phase equilibrium compares complete phases and includes entropy and temperature dependence. It also does not supply an equilibrium distance without repulsion. Retain the numerical ratio alongside these omitted terms so that a correct small model does not become an unjustified whole-material claim.

Sources and evidence boundary

Conceptual sources checked 30 October 2026: Primary dispersion-coefficient study, Intermolecular forces. All prose, cases, code and diagrams are original. The supplied models are not measured structures, a performed laboratory investigation or certification of independent mastery. No chemical handling, heating, pressure vessel or cryogenic experiment is requested. Retain predictions, contrasting evidence and a changed-case explanation.

Apply

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

Why cannot one attractive pair term predict boiling point or determine an equilibrium spacing?

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: Use orbital occupancy to compare bond order and magnetism Next: Separate donor identity, acceptance and contact geometry

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