OPOpenPithMake curiosity visiblePhysics lab / Thermodynamics

About this lab

One cylinder, five experiments: from jostling molecules to the limit of engines.

Temperature measures the average kinetic energy of molecules; pressure is the average effect of their hits on the walls. So pressure, volume and temperature are tied together: pV = nRT. Heating a gas or doing work on it both change its internal energy, and the books read ΔU = Q + W. An engine turns heat into work, but only part of it, with a ceiling set by the two temperatures.

Temperatures here are always in kelvin and pressures always absolute; W always means work done on the gas. Every reading and every curve comes from one model.

Five results

Five results

  • Molecular speeds: heating shifts the distribution toward faster speeds and widens it, but not every molecule speeds up. In nitrogen at 800 K, 5.5 % of molecules are still slower than 300 m/s. At 400 K helium averages 1455 m/s and nitrogen 550 m/s, yet both have the same mean kinetic energy, 8.28 × 10⁻²¹ J.
  • Boyle’s law: squeezing slowly from 0.5 L to 0.2 L in a 300 K bath raises the pressure from 80 kPa to 200 kPa, with pV fixed at 40.00 J. A fast squeeze leaves no time to shed heat: the gas reaches 433 K and the pressure jumps to 289 kPa, settling back to about 200 kPa once it exchanges heat with the bath.
  • Charles’s law: with the piston locked, heating from 0 °C to 100 °C raises the pressure from 91.05 kPa to 124.38 kPa. Pressure is not proportional to the Celsius temperature: the line reaches zero only at −273.15 °C. With the piston free, the volume goes from 0.364 L to 0.498 L and extrapolates to the same temperature.
  • First law: isothermal expansion takes in 11.51 J of heat and does 11.51 J of work, temperature unchanged; adiabatic compression takes in no heat, 31.95 J of work is done on the gas, and it warms to 395.9 K. All four processes satisfy ΔU = Q + W.
  • Carnot cycle: with a 400 K source and a 300 K sink, one lap takes in 36.97 J, gives out 27.73 J and does 9.24 J of work: 25.0 % efficient, exactly 1 − 300/400. Even without friction it cannot reach 100 %.

Model and sources

Model and sources

  • Gas: ideal, air treated as diatomic, Cv = 5R/2, γ = 1.4; 0.0160 mol, filling 0.4 L at 300 K and 100 kPa; cylinder bore radius 3 cm.
  • Molecular speeds: Maxwell distribution; the 160 molecules shown take the distribution’s quantiles, their positions are only a sketch, and their motion is slowed by more than a million times.
  • Boyle: a slow squeeze is isothermal; a fast one is adiabatic, pV^γ = const, after which the temperature returns to the bath with a 0.8 s time constant (a teaching assumption).
  • Charles and Gay-Lussac: one point every 10 °C, a least-squares line against Celsius temperature, extrapolated.
  • Carnot cycle: two isothermal and two adiabatic legs, 2 s each; Qin = nRThot ln(V₂/V₁), efficiency η = 1 − Tcold/Thot.

Boltzmann constant k = 1.380649 × 10⁻²³ J/K and Avogadro constant 6.02214076 × 10²³ mol⁻¹ are exact SI values; R = kNA. The molecule picture, handwheel turning and bath colour are only illustrations and take no part in the calculation.

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