Infer gas amount and challenge a straight-line calibration

Establish the model and evidence

An equation of state becomes an experimental inference only after its controlled quantities are specified. At fixed amount and volume, P=nRT/V predicts a pressure-versus-Kelvin-temperature line through the origin. A plot against degrees Celsius has a nonzero intercept because T = θ + 273.15. Absolute pressure, not gauge pressure, belongs in this equation.

The original packet uses four temperatures and small assigned pressure errors. The controls deliberately add a pressure-zero offset or reduce particle amount during warming. Neither intervention is a random uncertainty estimate. A zero offset can change the extrapolated zero-pressure temperature without appreciably changing slope. A changing amount can corrupt the inferred number of moles even if the points look almost straight.

The displayed least-squares line summarizes all four readings. The slope has units kPa/K, so multiplying by volume in litres gives J/K; divide by R to infer moles. A numerical fit is not a causal explanation. Residuals, leak checks, an independent pressure reference and knowledge of volume behavior are distinct evidence.

Extrapolating to zero pressure is especially delicate because it is far outside the observed temperature interval. Real gases can condense and cease to obey the simple model before such an extrapolated endpoint. Use the synthetic intervention to explain why a compelling straight line does not by itself establish an absolute-zero measurement.

Worked evidence

A fitted slope of 0.400 kPa/K in a 2.00 L vessel gives n=(0.400×2.00)/8.314≈0.0962 mol. A constant +6 kPa offset shifts the inferred zero but leaves that ideal slope unchanged.

Predict and investigate

Predict which fitted quantities change when a pressure offset is added and which change when particle loss increases.

Capture zero offset/no loss, then +6 kPa/no loss, then zero offset/20% final particle loss. Compare slopes, extrapolated zeros and residuals. Propose a separate measurement that distinguishes the two failure mechanisms.

Transfer to changed conditions

A correct fixed-amount slope is 0.250 kPa/K in 4.00 L. What amount follows, and would a gauge-pressure offset alone change that slope?

Compare the worked transfer

n=1.00/8.314≈0.1203 mol. A constant gauge offset does not change slope, though it changes intercept and makes direct substitution into PV=nRT invalid.

Explain the boundary

Explain why fitting more points cannot remove an unrecognized common sensor bias. State what must be known before interpreting slope as one fixed amount of gas.

State every ideal-gas assumption before scaling

The classical idealization uses randomly directed molecular velocities, molecular volumes negligible compared with the container volume, elastic collisions and no appreciable intermolecular forces except during collisions. It is not a claim that real molecules have exactly zero size. The two equivalent amount conventions are PV=nRT=NkBT, with N=nNA and R=NAkB.

Predict and check: at fixed absolute temperature and volume, doubling N doubles pressure; at fixed N and temperature, doubling volume halves pressure. At fixed N and volume, changing 300 K to 450 K multiplies pressure by 1.5, not by a ratio of Celsius readings. Sketch each changed constraint as a separate comparison. If attraction or occupied molecular volume becomes important, neither a clean sensor zero nor a straight segment alone proves the ideal assumptions. (Current clause 9.2.A.)

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. The numerical cases, text and computed representations are original. All browser evidence is synthetic or calculated, not a record of completed practical work. No hazardous apparatus is required.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.