SCIENCE · INVESTIGATE AND EXPLAIN

Derive the speed bias of a flux-selected sample

A bulk molecule selected uniformly from an equilibrium reservoir and a molecule selected because it crosses a small aperture are sampled differently. Faster molecules reach the opening more often. Assume a dilute gas, an aperture smaller than the mean free path, negligible depletion and speed-independent detection. This is a virtual kinetic-theory model, not an instruction to operate a vessel.

Learning outcome: Normalize the speed-weighted density, compare first and second moments, and freeze a threshold-tail prediction before inspecting the curve. State which detector calibration would test a speed-independent-response premise.

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 s=200 m/s, bulk mean speed is about 225.676 m/s while the emitted mean is about 265.868 m/s. Above 400 m/s, u0=2 and the emitted fraction is 5exp(−4)≈0.0915782. The reservoir scale is unchanged.

Open the experiment

Predict

Why is the emitted speed law different from the bulk law?

Your prediction and reason

Experiment

Normalize the speed-weighted density, compare first and second moments, and freeze a threshold-tail prediction before inspecting the curve. State which detector calibration would test a speed-independent-response premise.

Weight an equilibrium distribution by crossing flux

Worked teaching model · synthetic values · Equilibrium dilute Maxwell gas, collisionless effusive aperture much smaller than mean free path, negligible reservoir depletion and speed-independent detection. These are mathematical selection assumptions, not apparatus instructions. Emitted speed density is weighted by speed relative to the bulk.

Thermal speed scale: 200 m/s Speed threshold divided by scale: 2 ratio

Reset keeps your prediction, captures and practice history.

Selection changes the normalized speed law

Graph axes rescale independently; read units and values, not screen angles. 1: bulk density (solid).2: flux-selected density (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Selection changes the normalized speed law

bulk density

u = speed/scale	density per unit u
0	0
0.05	0.005627809
0.1	0.02234303
0.15	0.04964734
0.2	0.08673078
0.25	0.1325018
0.3	0.185627
0.35	0.2445795
0.4	0.3076932
0.45	0.3732205
0.5	0.4393913
0.55	0.5044712
0.6	0.5668153
0.65	0.624917
0.7	0.6774494
0.75	0.7232975
0.8	0.7615818
0.85	0.7916717
0.9	0.8131891
0.95	0.8260028
1	0.830215
1.05	0.8261406
1.1	0.8142806
1.15	0.7952914
1.2	0.7699517
1.25	0.7391285
1.3	0.703743
1.35	0.6647386
1.4	0.6230515
1.45	0.5795847
1.5	0.5351863
1.55	0.4906321
1.6	0.4466128
1.65	0.4037255
1.7	0.3624698
1.75	0.3232469
1.8	0.2863624
1.85	0.252032
1.9	0.2203886
1.95	0.1914915
2	0.1653359
2.05	0.1418634
2.1	0.120972
2.15	0.1025264
2.2	0.08636646
2.25	0.07231598
2.3	0.0601898
2.35	0.04980002
2.4	0.04096107
2.45	0.03349381
2.5	0.02722855
2.55	0.02200724
2.6	0.01768484
2.65	0.01413005
2.7	0.0112255
2.75	0.008867443
2.8	0.006965181
2.85	0.005440237
2.9	0.004225367
2.95	0.003263483
3	0.002506555
3.05	0.001914522
3.1	0.001454248
3.15	0.001098548
3.2	0.0008252956
3.25	0.0006166182
3.3	0.0004581904
3.35	0.0003386139
3.4	0.000248885
3.45	0.0001819421
3.5	0.000132286
3.55	0.00009566334
3.6	0.00006880715
3.65	0.00004922464
3.7	0.00003502652
3.75	0.00002479028
3.8	0.00001745179
3.85	0.00001222015
3.9	0.000008511286
3.95	0.000005896572
4	0.000004063435

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.

flux-selected density

u = speed/scale	density per unit u
0	0
0.05	0.0002493758
0.1	0.0019801
0.15	0.006599821
0.2	0.01537263
0.25	0.02935666
0.3	0.04935228
0.35	0.07586353
0.4	0.1090744
0.45	0.1488411
0.5	0.1947002
0.55	0.2458918
0.6	0.3013962
0.65	0.3599819
0.7	0.4202617
0.75	0.4807543
0.8	0.5399474
0.85	0.5963607
0.9	0.6486031
0.95	0.6954246
1	0.7357589
1.05	0.7687555
1.1	0.7938012
1.15	0.8105299
1.2	0.8188223
1.25	0.8187945
1.3	0.8107788
1.35	0.7952975
1.4	0.773031
1.45	0.7447832
1.5	0.7114448
1.55	0.6739576
1.6	0.6332804
1.65	0.5903575
1.7	0.5460919
1.75	0.5013226
1.8	0.4568077
1.85	0.4132114
1.9	0.3710972
1.95	0.3309246
2	0.2930502
2.05	0.2577324
2.1	0.2251382
2.15	0.1953526
2.2	0.1683886
2.25	0.1441988
2.3	0.1226862
2.35	0.1037152
2.4	0.08712193
2.45	0.07272364
2.5	0.06032669
2.55	0.04973369
2.6	0.04074922
2.65	0.03318445
2.7	0.02686053
2.75	0.02161106
2.8	0.01728365
2.85	0.01374066
2.9	0.01085944
2.95	0.00853195
3	0.006664129
3.05	0.005174937
3.1	0.003995261
3.15	0.003066724
3.2	0.002340477
3.25	0.001776007
3.3	0.00134
3.35	0.001005297
3.4	0.0007499331
3.45	0.0005562848
3.5	0.0004103238
3.55	0.000300967
3.6	0.0002195235
3.65	0.0001592283
3.7	0.0001148534
3.75	0.0000823868
3.8	0.00005877173
3.85	0.00004169483
3.9	0.00002941743
3.95	0.00002064152
4	0.0000144045

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
Bulk mean speed	225.6758	m/s
Effusive mean speed	265.8681	m/s
Bulk mean squared speed	60000	m²/s²
Effusive mean squared speed	80000	m²/s²
Effusive probability above threshold	0.09157819	ratio
Effusive most probable speed	244.949	m/s
Threshold speed	400	m/s

Faster molecules cross an aperture more often; an emitted sample is not a random bulk sample.

For an isotropic reservoir, the angular crossing factor integrates separately, leaving speed weighting v fbulk(v). Dividing by bulk mean speed normalizes the emitted law to 2u³e^(−u²). Its second moment is 2, compared with bulk 3/2; an emitted molecule carries greater mean translational energy. Integrating above threshold gives (1+u0²)e^(−u0²). A detector response varying with speed introduces yet another selection law.

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

Faster molecules cross an aperture more often; an emitted sample is not a random bulk sample.

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

Crossing flux contains the normal velocity v cos(theta). For an isotropic bulk distribution, integrate the outward hemisphere angular factor separately. The normalized speed density among emitted molecules is v fbulk(v)/mean(v). Using u=v/s and the Maxwell density produces feff(u)=2u³ exp(−u²), whose integral is one. The extra power of u is selection, not a higher reservoir temperature.

Audit limits and evidence

Integrating moments gives mean effusive speed (3sqrt(pi)/4)s and mean squared speed 2s². Bulk values are (2/sqrt(pi))s and 3s²/2. Thus mean translational energy per emitted molecule is larger by a factor of 4/3. Integrating from threshold u0 to infinity with w=u² gives probability (1+u0²)exp(−u0²). This exact tail belongs to the flux-selected distribution, not the bulk.

Predict a changed premise

A detector whose response favors slow or fast particles adds another weighting. A directional acceptance changes the angular factor, and collisions near a large opening invalidate the ideal effusion premise. A hotter-looking emitted histogram alone therefore does not establish a different reservoir temperature.

Worked original case

At s=200 m/s, bulk mean speed is about 225.676 m/s while the emitted mean is about 265.868 m/s. Above 400 m/s, u0=2 and the emitted fraction is 5exp(−4)≈0.0915782. The reservoir scale is unchanged.

Investigation

Changed-case practice

Try the changed case before feedback

For threshold speed equal to s, find the emitted fraction above threshold.

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.

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Previous: Normalize speed evidence without hiding the omitted tail Next: Separate classical mode counts from a supplied activation law

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