SCIENCE · INVESTIGATE AND EXPLAIN

Separate classical mode counts from a supplied activation law

Classical equipartition assigns mean energy kT/2 to each independent quadratic energy term. Three translational terms give molar Cv=3R/2. A diatomic molecule with two active rotational terms adds R, giving 5R/2 before vibrational and other contributions. Quantum level spacing can suppress a mode at low temperature, so a classical count is a regime statement rather than a universal constant.

Learning outcome: Compare two temperatures at fixed activation scale and two scales at fixed temperature. Differentiate the proposed energy function and retain a regime where classical mode counting is an approximation.

Starting recap

Review definite integrals, probability densities, mean versus mean square, temperature units and the first-law work convention. State the domain and distinguish calibration data from unused validation.

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Try

At T=300 K and theta=200 K, Cv/R=1.5+90000/130000=57/26≈2.1923077. At T=theta it is exactly 2, halfway between the two limiting values. The energy change from 100 K follows the antiderivative, not Cv evaluated only at 300 K times the interval.

Open the experiment

Predict

At T=theta, what is the supplied Cv/R?

Your prediction and reason

Experiment

Compare two temperatures at fixed activation scale and two scales at fixed temperature. Differentiate the proposed energy function and retain a regime where classical mode counting is an approximation.

Audit a supplied mode-activation interpolation

Worked teaching model · synthetic values · Supplied molar Cv/R=3/2+T²/(T²+theta²), positive temperatures and fixed volume. This smooth interpolation illustrates one rotational contribution between classical limits; it is not an exact molecular quantum partition function or universal material heat capacity.

Temperature: 300 K Activation temperature scale: 200 K

Reset keeps your prediction, captures and practice history.

Supplied mode-activation response

Graph axes rescale independently; read units and values, not screen angles. 1: smooth illustrative constitutive law (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Supplied mode-activation response

smooth illustrative constitutive law

temperature (K)	molar Cv/R
100	1.7
105	1.716071
110	1.732246
115	1.748473
120	1.764706
125	1.780899
130	1.797012
135	1.81301
140	1.828859
145	1.844531
150	1.86
155	1.875244
160	1.890244
165	1.904983
170	1.919448
175	1.933628
180	1.947514
185	1.961098
190	1.974376
195	1.987344
200	2
205	2.012344
210	2.024376
215	2.036097
220	2.047511
225	2.058621
230	2.069429
235	2.079942
240	2.090164
245	2.1001
250	2.109756
255	2.119138
260	2.128253
265	2.137106
270	2.145704
275	2.154054
280	2.162162
285	2.170035
290	2.177679
295	2.185101
300	2.192308
305	2.199305
310	2.206098
315	2.212695
320	2.219101
325	2.225322
330	2.231363
335	2.237231
340	2.242931
345	2.248467
350	2.253846
355	2.259072
360	2.264151
365	2.269086
370	2.273884
375	2.278547
380	2.28308
385	2.287488
390	2.291775
395	2.295944
400	2.3
405	2.303946
410	2.307785
415	2.311521
420	2.315157
425	2.318697
430	2.322143
435	2.325499
440	2.328767
445	2.33195
450	2.335052
455	2.338073
460	2.341017
465	2.343887
470	2.346685
475	2.349412
480	2.352071
485	2.354664
490	2.357194
495	2.359661
500	2.362069

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Supplied molar Cv divided by R	2.192308	ratio
Energy difference from 100 K divided by R	396.1708	K
Low-temperature asymptote of Cv/R	1.5	ratio
High-temperature asymptote of Cv/R	2.5	ratio
Activated rotational contribution divided by R	0.6923077	ratio

Mode counting is a classical limiting model, not a promise that every mode is active at every temperature.

Classical quadratic degrees of freedom each contribute R/2 per mole to Cv. Three translations give 3R/2; two activated rotations add R. The supplied interpolation tends toward those two limits. Integrating it gives U/R=2.5T−theta atan(T/theta)+constant. The reference constant cancels in differences. This is not a derived quantum rotor spectrum and does not model vibrations or dissociation.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Mode counting is a classical limiting model, not a promise that every mode is active at every temperature.

Observation to attach to a capture

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

Understand

Declare the distribution or system

Derive the integral or response

Use the supplied smooth illustration Cv/R=3/2+T²/(T²+theta²). Its rotational contribution rises from zero toward one as temperature exceeds the scale theta. This is an original constitutive interpolation, not an exact quantum-rotor partition function, and no molecular identity is inferred from fitting theta.

Audit limits and evidence

Rewrite T²/(T²+theta²)=1−theta²/(T²+theta²). An antiderivative is U/R=2.5T−theta atan(T/theta)+constant. Differentiating recovers the stated capacity, which is the useful consistency check. The arbitrary reference energy cancels in any difference; it is not measured by a capacity experiment.

Predict a changed premise

At fixed temperature, increasing theta decreases the activated contribution. At fixed theta, raising temperature increases it. Predict both comparisons and distinguish the limiting classical values from a finite-temperature value. No claim about vibration, dissociation or arbitrary material phases follows from this one-mode interpolation.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At T=2theta, find Cv/R.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic cases checked 31 October 2026 against the primary teaching reference. No source exercises or graphics are reproduced. These bounded CPU calculations do not claim actual gas handling, pressure-vessel work, thermal exposure or practical measurement. Public feedback is practice, not independently verified reasoning; retain original work, critique, revision and a newly assigned case in the course evidence workspace.

Apply

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

Which distribution, constitutive law, path or calibration premise supports this inference, and what independent evidence would challenge it?

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: Derive the speed bias of a flux-selected sample Next: Integrate a temperature-dependent state-energy change

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