Know Which Law Each Process Obeys and Which Curve Is Steeper
An isothermal process holds temperature fixed and exchanges heat; an adiabatic process exchanges none and changes temperature. Each has its own equation, and the adiabatic curve is always steeper than the isotherm through the same point.
Isothermal and Adiabatic Processes is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
The two laws
A diatomic gas compressed to half its volume ends at P x 2¹.4, about 2.64 P.
Steepness and comparison of gases
The slope of an isotherm is -P/V and the slope of an adiabat is -γ P/V, so the adiabat is steeper by the factor γ.
When a figure shows a pair of adiabatic plots, identify the steeper one as the gas with larger γ, and use the degrees of freedom of each gas to rank them.
Isothermal against adiabatic expansion
If one sample is expanded isothermally to double its volume and an identical sample adiabatically to the same volume, the adiabatic sample cools while doing work, so it ends at a lower temperature and lower pressure than the isothermal one, and it does less work because the area under its curve is smaller. The isothermal sample therefore has both the higher final pressure and the larger work done. A curve along which a gas exchanges no heat with its surroundings is always the adiabatic one, never the isotherm.
OlympIQ exam lens
- ✓Isothermal: PV = constant, ΔU = 0 for an ideal gas; adiabatic: Q = 0 and PV^γ = constant
- ✓For adiabatic changes use TV^(γ - 1) = constant to find a final temperature
- ✓Adiabats are steeper than isotherms by a factor of γ; larger γ means a steeper adiabat
- ✓Same volume change: isothermal gives larger final pressure and more work than adiabatic expansion
- ✓No heat exchange with the surroundings identifies the adiabatic curve
Do not use PV = constant for an adiabatic process. Pressure and volume are linked by PV^γ = constant there, because the temperature changes as well; the isothermal PV = constant holds only when the temperature is fixed.
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