SCIENCE · INVESTIGATE AND EXPLAIN

Audit a complete electrical-to-thermal process across changing boundaries

A correct equation can still answer the wrong system question. An ideal constant-potential source charges an initially empty capacitor through resistance, then is disconnected. The charged capacitor later discharges into a thermal arrangement that transfers only a stated fraction of energy to a gas. The gas may expand or remain at fixed volume. Treat these as separate stages with separate boundaries before adding their ledgers.

Learning outcome: Retain both gas constraints at 8 V, then repeat at 12 V. Freeze each stage ledger first, check the V² scaling and compare temperature rise, pressure/volume ratios and work signs. Reconstruct the complete source account without counting an intermediate transfer twice.

Starting recap

Choose each system boundary and interval before applying an equation. Distinguish agreement from independent evidence and preserve earlier reports.

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Try

At V=8 V and C=0.020 F, source energy is 1.28 J, storage and charging heat are each 0.64 J. Of the storage, 0.48 J heats the gas and 0.16 J goes elsewhere. Constant-pressure gas receives 0.288 J internal-energy gain and performs 0.192 J work; fixed-volume gas instead gains the full 0.48 J internally.

Open the experiment

Predict

Does changing the gas constraint change the amount of energy the charging source supplied?

Your prediction and reason

Experiment

Retain both gas constraints at 8 V, then repeat at 12 V. Freeze each stage ledger first, check the V² scaling and compare temperature rise, pressure/volume ratios and work signs. Reconstruct the complete source account without counting an intermediate transfer twice.

Audit source, capacitor and gas energy stages

Worked teaching model · synthetic values · Initially uncharged C=0.020 F capacitor charges fully from an ideal constant-potential source through resistance, then is disconnected and fully discharged. 75% of stored energy heats 0.001 mol ideal monatomic gas initially at 300 K; 25% escapes elsewhere. Constant-volume or quasistatic constant-pressure gas. Aggregate endpoint energy only; no physical wiring, pressure vessel or RC transient calculation.

Ideal charging source potential: 8 V Gas constraint: 0 fixed volume, 1 constant pressure: 0 choice

Reset keeps your prediction, captures and practice history.

Cross-boundary evidence ledger

Each stage uses its own boundary; capacitor storage is not the entire charging-source energy.

No energy is counted twice: add charging heat, discharge leakage, gas internal-energy gain and gas work to recover source input. For the gas ΔU=Q+W_on=Q−W_by. Constant-pressure heating has a smaller temperature rise than fixed-volume heating for equal Q because expansion transfers energy as work. Ideal endpoint balances do not specify a time course or certify real equipment.

Every quantity retains its own unit
Quantity	Value	Unit
Charge delivered by source	0.16	C
Source energy delivered during charging	1.28	J
Stored capacitor energy before discharge	0.64	J
Charging resistance thermal conversion	0.64	J
Heat delivered to gas	0.48	J
Discharge energy to other surroundings	0.16	J
Gas internal-energy increase	0.48	J
Work done by gas	0	J
Work done on gas	0	J
Gas temperature increase	38.4872	K
Final gas temperature	338.487	K
Final to initial pressure ratio	1.12829	ratio
Final to initial volume ratio	1	ratio
Complete source-energy account	1.28	J

Follow each stage without adding its input twice

Charge from ideal source
1.28 J input = 0.64 J retained or passed to the named next subsystem + 0.64 J output.
Retained in capacitor; output is charging resistance thermal conversion.
Top outline: stage input. Lower solid: named retained share. Lower dashed: output share. Each stage has its own input-normalized scale.
Discharge isolated capacitor
0.64 J input = 0.48 J retained or passed to the named next subsystem + 0.16 J output.
Energy delivered as gas heat; output reaches other surroundings.
Top outline: stage input. Lower solid: named retained share. Lower dashed: output share. Each stage has its own input-normalized scale.
Gas thermal process
0.48 J input = 0.48 J retained or passed to the named next subsystem + 0 J output.
Retained gas internal energy; output is work by gas. Fixed volume gives zero work.
Top outline: stage input. Lower solid: named retained share. Lower dashed: output share. Each stage has its own input-normalized scale.

Each stage uses its own boundary; capacitor storage is not the entire charging-source energy.

Observation to attach to a capture

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

Understand

Select the evidence

Derive the relation

For capacitance C and source potential V, delivered charge is CV. Source energy is V times that charge, CV², whereas final capacitor storage is CV²/2. The other half becomes thermal energy in the charging resistance under the stated ideal full-charge process. Calling capacitor storage the entire source expenditure would lose a real energy term even though its formula is individually correct. These are endpoint balances; no intermediate-time RC mathematics is required.

Work the contrast

After isolation, the full stored energy is divided into gas heat Q and leakage to other surroundings. For an ideal monatomic gas, internal-energy change is 3nRΔT/2. At fixed volume, gas work is zero and all Q increases internal energy. For quasistatic constant-pressure heating, gas work by expansion is nRΔT, so Q=5nRΔT/2. In the course work-on convention, ΔU=Q+W_on and W_on is negative during expansion.

Challenge the inference

The final audit adds charging heat, discharge leakage, gas internal-energy gain and gas work. It must equal the source input. Do not add gas heat again on top of internal energy and work: that would double count an intermediate transfer. Equal first-law totals do not establish real component efficiency, a safe apparatus or second-law feasibility of an arbitrary reversed cycle. No physical capacitor, wiring or pressure vessel is requested.

Worked original case

Prediction and retained investigation

Does changing the gas constraint change the amount of energy the charging source supplied? Retain both gas constraints at 8 V, then repeat at 12 V. Freeze each stage ledger first, check the V² scaling and compare temperature rise, pressure/volume ratios and work signs. Reconstruct the complete source account without counting an intermediate transfer twice.

Changed-case practice

For C=0.010 F,V=10 V and gas heat fraction 60%, find source input, charging heat, gas heat and discharge leakage. For constant-pressure monatomic heating, split gas heat into internal energy and work.

Check after your attempt

Source 1.0 J; storage 0.50 J and charging heat 0.50 J. Gas heat 0.30 J, leakage 0.20 J. Constant-pressure gas gains 0.18 J internally and does 0.12 J work. Sum 0.50+0.20+0.18+0.12=1.0 J.

Retain course-wide evidence

Use the AP Physics 2 course evidence workspace for the seven-unit synthesis responses and cumulative project for the first report, critique, revision and frozen fresh assignment. These are retained records, not automated mastery judgments.

Which independent observations would be needed before replacing the stipulated 75% heat fraction with an empirical efficiency?

Sources and safety

Original supplied cases checked 30 October 2026 against current AP2 scope, OpenStax capacitor energy and first-law processes. Entirely virtual; no electrical, thermal, pressure, light or radiation apparatus is requested. No simulated result substitutes for supervised measurement technique or independent learner evidence.

Apply

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

Which independent observations would be needed before replacing the stipulated 75% heat fraction with an empirical efficiency?

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: Infer an ensemble half-life without predicting an individual decay Next: Challenge agreement between two energy measurements before defending a capstone

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