Test which inference survives isolated vacuum expansion
An insulated gas expands into an initially evacuated part of a rigid enclosure. The external boundary does no work and no heat crosses it, so delta U=0. The actual expansion is not a sequence of equilibrium piston states. A pressure-volume curve drawn through intermediate equilibrium states must not be mistaken for its actual work path.
Learning outcome: Derive the model-consistency identity, retain ideal and nonideal endpoint predictions and explain why integrating equilibrium p dV along the displayed endpoint family would assign the wrong actual work.
Starting recap
Review logarithmic integrals, equilibrium states, absolute temperature and the first-law work convention. State the domain and distinguish calibration data from unused validation.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For A=200 J L and r=3, T2=3500/9≈388.888889 K. The entropy change is 12 ln(35/36)+8 ln 3≈8.450848 J/K. The pressure is 8T2/3−200/9≈1014.814815 kPa; the endpoint energy equals its initial value.
Open the experimentPredict
Which statement follows for both stipulated models in isolated vacuum expansion?
Experiment
Derive the model-consistency identity, retain ideal and nonideal endpoint predictions and explain why integrating equilibrium p dV along the displayed endpoint family would assign the wrong actual work.
Test an ideal inference against a consistent nonideal model
Worked teaching model · synthetic values · Insulated expansion into a vacuum from V1=1 litre and T1=400 K. Supplied consistent model U=12T−A/V joule and p=8T/V−A/V² kPa for V in litres. No boundary work or heat. Endpoints lie in a positive-pressure, locally stable supported region; this is not a full phase diagram.
Reset keeps your prediction, captures and practice history.
Energy-conserving endpoint family, not intermediate equilibrium states
Graph axes rescale independently; read units and values, not screen angles. 1: supplied nonideal model (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Energy-conserving endpoint family, not intermediate equilibrium states
supplied nonideal model
| final volume ratio | final temperature (K) |
|---|---|
| 1 | 400 |
| 1.0375 | 399.3976 |
| 1.075 | 398.8372 |
| 1.1125 | 398.3146 |
| 1.15 | 397.8261 |
| 1.1875 | 397.3684 |
| 1.225 | 396.9388 |
| 1.2625 | 396.5347 |
| 1.3 | 396.1538 |
| 1.3375 | 395.7944 |
| 1.375 | 395.4545 |
| 1.4125 | 395.1327 |
| 1.45 | 394.8276 |
| 1.4875 | 394.5378 |
| 1.525 | 394.2623 |
| 1.5625 | 394 |
| 1.6 | 393.75 |
| 1.6375 | 393.5115 |
| 1.675 | 393.2836 |
| 1.7125 | 393.0657 |
| 1.75 | 392.8571 |
| 1.7875 | 392.6573 |
| 1.825 | 392.4658 |
| 1.8625 | 392.2819 |
| 1.9 | 392.1053 |
| 1.9375 | 391.9355 |
| 1.975 | 391.7722 |
| 2.0125 | 391.6149 |
| 2.05 | 391.4634 |
| 2.0875 | 391.3174 |
| 2.125 | 391.1765 |
| 2.1625 | 391.0405 |
| 2.2 | 390.9091 |
| 2.2375 | 390.7821 |
| 2.275 | 390.6593 |
| 2.3125 | 390.5405 |
| 2.35 | 390.4255 |
| 2.3875 | 390.3141 |
| 2.425 | 390.2062 |
| 2.4625 | 390.1015 |
| 2.5 | 390 |
| 2.5375 | 389.9015 |
| 2.575 | 389.8058 |
| 2.6125 | 389.7129 |
| 2.65 | 389.6226 |
| 2.6875 | 389.5349 |
| 2.725 | 389.4495 |
| 2.7625 | 389.3665 |
| 2.8 | 389.2857 |
| 2.8375 | 389.207 |
| 2.875 | 389.1304 |
| 2.9125 | 389.0558 |
| 2.95 | 388.9831 |
| 2.9875 | 388.9121 |
| 3.025 | 388.843 |
| 3.0625 | 388.7755 |
| 3.1 | 388.7097 |
| 3.1375 | 388.6454 |
| 3.175 | 388.5827 |
| 3.2125 | 388.5214 |
| 3.25 | 388.4615 |
| 3.2875 | 388.403 |
| 3.325 | 388.3459 |
| 3.3625 | 388.29 |
| 3.4 | 388.2353 |
| 3.4375 | 388.1818 |
| 3.475 | 388.1295 |
| 3.5125 | 388.0783 |
| 3.55 | 388.0282 |
| 3.5875 | 387.9791 |
| 3.625 | 387.931 |
| 3.6625 | 387.884 |
| 3.7 | 387.8378 |
| 3.7375 | 387.7926 |
| 3.775 | 387.7483 |
| 3.8125 | 387.7049 |
| 3.85 | 387.6623 |
| 3.8875 | 387.6206 |
| 3.925 | 387.5796 |
| 3.9625 | 387.5394 |
| 4 | 387.5 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Nonideal final temperature | 388.8889 | K |
| Ideal-model final temperature | 400 | K |
| State entropy change | 8.450848 | J/K |
| Ideal-model entropy change | 8.788898 | J/K |
| Energy residual | 3.126388e-13 | J |
| Final pressure | 1014.815 | kPa |
Zero heat and zero work imply constant internal energy, not universally constant temperature.
For U=12T−A/V, conservation gives T2=400−(A/12)(1−1/r). The pressure law satisfies (∂U/∂V)T=T(∂p/∂T)V−p=A/V², so dS=12 dT/T+8 dV/V is consistent. The ideal A=0 limit is isothermal; all supported endpoint entropy changes are positive. No quasistatic pressure path is assigned to the actual vacuum expansion.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Zero heat and zero work imply constant internal energy, not universally constant temperature.
For U=12T−A/V, conservation gives T2=400−(A/12)(1−1/r). The pressure law satisfies (∂U/∂V)T=T(∂p/∂T)V−p=A/V², so dS=12 dT/T+8 dV/V is consistent. The ideal A=0 limit is isothermal; all supported endpoint entropy changes are positive. No quasistatic pressure path is assigned to the actual vacuum expansion.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which system boundary, reference path, constitutive law or independently calibrated record supports this entropy inference?
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.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.