Isothermal Process
An isothermal process is a change of state in which the temperature of the system remains constant, maintained by heat exchange with a reservoir. For an ideal gas, internal energy depends only on temperature, so ΔU = 0 and the heat absorbed equals the work done, Q = W = nRT ln(V2/V1); pressure and volume follow Boyle's law, pV = constant.
Curious
If you squeeze a gas slowly while it sits in good thermal contact with its surroundings, it stays at the surroundings' temperature: heat flows out as fast as compression would warm it. Under these conditions, halving the volume doubles the pressure, as Robert Boyle found in 1662 (see Ideal Gas Law).
Isothermal steps form half of the Carnot cycle. Phase changes such as boiling at fixed pressure are also isothermal (see Latent Heat).
Exploring
| Process | Held constant | Ideal-gas results |
|---|---|---|
| Isothermal | T | Q = W = nRT ln(V2/V1), ΔU = 0 |
| Adiabatic | Q = 0 | pVγ = const, W = −ΔU |
| Isochoric | V | W = 0, Q = ΔU = nCvΔT |
| Isobaric | p | W = pΔV, Q = nCpΔT |
Deep Dive
Work integral. W = ∫p dV = nRT ∫dV/V = nRT ln(V2/V1). On a p–V diagram an isotherm is a hyperbola, and it is less steep than an adiabat through the same point by the factor γ. For real gases, an isotherm below the critical temperature has a flat segment where liquid and vapour coexist, and the internal energy is no longer constant during isothermal compression because intermolecular forces matter (see Internal Energy).
Reversibility and speed. A truly isothermal process requires heat flow across an infinitesimal temperature difference, so it must be infinitely slow, an idealisation like the Carnot engine. In practice, processes are near-isothermal when they are slow compared with the thermal relaxation time of the system, as in slow gas compression with cooling, isothermal compressors with intercooling, and many processes in living cells. The isothermal compressibility κT = −(1/V)(∂V/∂p)T characterises a material's response.
Free energy. For any system at constant temperature, the maximum work obtainable equals the decrease in Helmholtz free energy F = U − TS, which is why F is the natural potential for isothermal processes.
Concept Relationship Map
Requires: first law, ideal gas law
Enables: Carnot cycle, Helmholtz free energy
Commonly confused with: adiabatic process
Sources
- Boyle, R. (1662). A Defence of the Doctrine Touching the Spring and Weight of the Air. Oxford.
- Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.
- Çengel, Y.A., Boles, M.A. (2015). Thermodynamics: An Engineering Approach (8th ed.). McGraw-Hill.