Work in Isothermal, Isobaric and Adiabatic Processes
Each idealised process fixes one variable and gives its own expression for the work, with the adiabatic case following a steeper curve than the isothermal one. Which curve applies decides how much a compressed gas heats up.
What a learner can do afterwards
- Derives the work done in isothermal and isobaric expansion by integration
- Uses the adiabatic relation between pressure and volume and says where the exponent comes from
- Explains why an adiabatic compression raises the temperature and an isothermal one does not
1 · Read
Work is the area under the pV curve, which is the integral of p dV. At fixed pressure p slides out front, so isobaric work is p times delta V. At fixed temperature the integral gives n R T times the log of V2 over V1, and the ideal gas keeps its internal energy, so every joule of heat becomes work.
A car engine squeezes its fuel-air mix from 240 to 40 cubic cm with no heat flow. Pressure jumps from 1.00e5 to 1.23e6 and temperature climbs from 293 K to 600 K, costing about 63 J of compression work. The same squeeze done isothermally would stay at 293 K, with the work leaving as heat instead.
Adiabatic means Q equals zero: p times V to the gamma stays constant, where gamma is Cp over Cv, about 1.4 for diatomic air. Because gamma tops one, the adiabat runs steeper than the isotherm on a pV plot. Compression work then has nowhere to go but inside, so the gas warms; expansion spends internal energy, so it cools.
Name the process by what stays fixed: T for isothermal, p for isobaric, Q equals zero for adiabatic. Expanding into a vacuum does zero work, since it pushes against zero pressure. Watch signs: work the gas does counts positive in the integral.
Isobaric work is p delta V, isothermal work is n R T log V2 over V1, and adiabatic work stays inside as temperature change.
2 · Watch
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